Saxon math course 1 pdf - Hacker Middle School

17 years of classroom experience as a teacher in grades 5 through 12 and as a ... Stephen has been writing math curriculum since 1975 and for Saxon since ... California Mathematics Council. ... University and his MA from Chapman College.
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Course 1

Student Edition

Stephen Hake

ACKNOWLEDGEMENTS

This book was made possible by the significant contributions of many individuals and the dedicated efforts of a talented team at Harcourt Achieve.

Special thanks to: • Melody Simmons and Chris Braun for suggestions and explanations for problem solving in Courses 1–3, • Elizabeth Rivas and Bryon Hake for their extensive contributions to lessons and practice in Course 3, • Sue Ellen Fealko for suggested application problems in Course 3. The long hours and technical assistance of John and James Hake on Courses 1–3, Robert Hake on Course 3, Tom Curtis on Course 3, and Roger Phan on Course 3 were invaluable in meeting publishing deadlines. The saintly patience and unwavering support of Mary is most appreciated. – Stephen Hake

Staff Credits Editorial: Jean Armstrong, Shelley Farrar-Coleman, Marc Connolly, Hirva Raj, Brooke Butner, Robin Adams, Roxanne Picou, Cecilia Colome, Michael Ota Design: Alison Klassen, Joan Cunningham, Deborah Diver, Alan Klemp, Andy Hendrix, Rhonda Holcomb Production: Mychael Ferris-Pacheco, Heather Jernt, Greg Gaspard, Donna Brawley, John-Paxton Gremillion Manufacturing: Cathy Voltaggio Marketing: Marilyn Trow, Kimberly Sadler E-Learning: Layne Hedrick, Karen Stitt

ISBN 1-5914-1783-X © 2007 Harcourt Achieve Inc. and Stephen Hake All rights reserved. No part of the material protected by this copyright may be reproduced or utilized in any form or by any means, in whole or in part, without permission in writing from the copyright owner. Requests for permission should be mailed to: Paralegal Department, 6277 Sea Harbor Drive, Orlando, FL 32887. Saxon is a trademark of Harcourt Achieve Inc. 2 3 4 5 6 7 8 9 048 12 11 10 09 08 07 06

ABOUT THE AUTHOR Stephen Hake has authored five books in the Saxon Math series. He writes from 17 years of classroom experience as a teacher in grades 5 through 12 and as a math specialist in El Monte, California. As a math coach, his students won honors and recognition in local, regional, and statewide competitions. Stephen has been writing math curriculum since 1975 and for Saxon since 1985. He has also authored several math contests including Los Angeles County’s first Math Field Day contest. Stephen contributed to the 1999 National Academy of Science publication on the Nature and Teaching of Algebra in the Middle Grades. Stephen is a member of the National Council of Teachers of Mathematics and the California Mathematics Council. He earned his BA from United States International University and his MA from Chapman College.

EDUCATIONAL CONSULTANTS Nicole Hamilton

Heidi Graviette

Melody Simmons

Consultant Manager Richardson, TX

Stockton, CA

Nogales, AZ

Brenda Halulka

Benjamin Swagerty

Atlanta, GA

Moore, OK

Marilyn Lance

Kristyn Warren

East Greenbush, NY

Macedonia, OH

Ann Norris

Mary Warrington

Wichita Falls, TX

East Wenatchee, WA

Joquita McKibben Consultant Manager Pensacola, FL

John Anderson Lowell, IN

Beckie Fulcher Gulf Breeze, FL

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CONTENTS OVERVIEW

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Letter from the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii How to Use Your Textbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Introduction to Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Section 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Lessons 1–10, Investigation 1 Section 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Lessons 11–20, Investigation 2 Section 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Lessons 21–30, Investigation 3 Section 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Lessons 31–40, Investigation 4 Section 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Lessons 41–50, Investigation 5 Section 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 Lessons 51–60, Investigation 6 Section 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 Lessons 61–70, Investigation 7 Section 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 Lessons 71–80, Investigation 8 Section 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Lessons 81–90, Investigation 9 Section 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 Lessons 91–100, Investigation 10 Section 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Lessons 101–110, Investigation 11 Section 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 Lessons 111–120, Investigation 12 Glossary with Spanish Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666

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TA B LE O F CO N T E N T S Integrated and Distributed Units of Instruction

Section 1

Lessons 1–10, Investigation 1

Math Focus: Number & Operations • Algebra Distributed Strands: Number & Operations • Algebra • Geometry • Measurement • Data Analysis & Probability Lesson 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 • Adding Whole Numbers and Money • Subtracting Whole Numbers and Money • Fact Families, Part 1

Maintaining & Extending

Lesson 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 • Multiplying Whole Numbers and Money • Dividing Whole Numbers and Money • Fact Families, Part 2

Mental Math Strategies pp. 7, 12, 18, 23, 28, 32, 36, 42, 46, 50

Lesson 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 • Unknown Numbers in Addition • Unknown Numbers in Subtraction Lesson 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 • Unknown Numbers in Multiplication • Unknown Numbers in Division

Power Up Facts pp. 7, 12, 18, 23, 28, 32, 36, 42, 46, 50

Problem Solving Strategies pp. 7, 12, 18, 23, 28, 32, 36, 42, 46, 50

Enrichment Early Finishers pp. 17, 22, 31, 41 Extensions p. 57

Lesson 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 • Order of Operations, Part 1 Lesson 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 • Fractional Parts Lesson 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 • Lines, Segments, and Rays • Linear Measure Activity Inch Ruler Lesson 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 • Perimeter Activity Perimeter Lesson 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 • The Number Line: Ordering and Comparing Lesson 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 • Sequences • Scales Investigation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 • Frequency Tables • Histograms • Surveys

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Section 2

Lessons 11–20, Investigation 2

Math Focus: Number & Operations • Problem Solving Distributed Strands: Number & Operations • Algebra • Measurement • Data Analysis & Probability • Problem Solving Lesson 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 • Problems About Comparing • Problems About Separating Lesson 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 • Place Value Through Trillions • Multistep Problems Lesson 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 • Problems About Comparing • Elapsed-Time Problems Lesson 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 • The Number Line: Negative Numbers Lesson 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 • Problems About Equal Groups Lesson 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 • Rounding Whole Numbers • Estimating Lesson 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 • The Number Line: Fractions and Mixed Numbers Activity Inch Ruler to Sixteenths Lesson 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 • Average • Line Graphs Lesson 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 • Factors • Prime Numbers Activity Prime Numbers Lesson 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 • Greatest Common Factor (GCF) Investigation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 • Investigating Fractions with Manipulatives Activity Using Fraction Manipulatives

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Maintaining & Extending

Power Up Facts pp. 58, 63, 68, 73, 78, 82, 87, 93, 99, 105 Mental Math Strategies pp. 58, 63, 68, 73, 78, 82, 87, 93, 99, 105 Problem Solving Strategies pp. 58, 63, 68, 73, 78, 82, 87, 93, 99, 105

Enrichment Early Finishers pp. 77, 81, 92, 98, 104, 108 Extensions p. 111

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Section 3

Lessons 21–30, Investigation 3

Math Focus: Number & Operations • Geometry Distributed Strands: Number & Operations • Geometry • Measurement • Problem Solving Lesson 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 • Divisibility

Maintaining & Extending

Lesson 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 • “Equal Groups” Problems with Fractions

Facts pp. 112, 117, 122, 127, 132, 136, 141, 145, 150, 156

Power Up

Lesson 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 • Ratio • Rate

Mental Math Strategies pp. 112, 117, 122, 127, 132, 136, 141, 145, 150, 156

Lesson 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 • Adding and Subtracting Fractions That Have Common Denominators

Problem Solving Strategies pp. 112, 117, 122, 127, 132, 136, 141, 145, 150, 156

Lesson 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 • Writing Division Answers as Mixed Numbers • Multiples Lesson 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 • Using Manipulatives to Reduce Fractions • Adding and Subtracting Mixed Numbers

Enrichment Early Finishers pp. 116, 126, 131, 144, 155, 160 Extensions p. 163

Lesson 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 • Measures of a Circle Activity Using a Compass Lesson 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 • Angles Lesson 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 • Multiplying Fractions • Reducing Fractions by Dividing by Common Factors Lesson 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 • Least Common Multiple (LCM) • Reciprocals Investigation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 • Measuring and Drawing Angles with a Protractor Activity Measuring Angles

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Section 4

Lessons 31–40, Investigation 4

Math Focus: Number & Operations • Measurement Distributed Strands: Number & Operations • Geometry • Measurement • Data Analysis & Probability • Problem Solving Lesson 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 • Areas of Rectangles

Maintaining & Extending

Lesson 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 • Expanded Notation • More on Elapsed Time

Facts pp. 164, 169, 174, 178, 182, 187, 191, 195, 200, 205

Lesson 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 • Writing Percents as Fractions, Part 1 Lesson 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 • Decimal Place Value Lesson 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 • Writing Decimal Numbers as Fractions, Part 1 • Reading and Writing Decimal Numbers Lesson 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 • Subtracting Fractions and Mixed Numbers from Whole Numbers Lesson 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 • Adding and Subtracting Decimal Numbers Lesson 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 • Adding and Subtracting Decimal Numbers and Whole Numbers • Squares and Square Roots Lesson 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 • Multiplying Decimal Numbers Lesson 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 • Using Zero as a Placeholder • Circle Graphs Investigation 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 • Collecting, Organizing, Displaying, and Interpreting Data

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Power Up

Mental Math Strategies pp. 164, 169, 174, 178, 182, 187, 191, 195, 200, 205 Problem Solving Strategies pp. 164, 169, 174, 178, 182, 187, 191, 195, 200, 205

Enrichment Early Finishers pp. 177, 194, 204 Extensions p. 214

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Section 5

Lessons 41–50, Investigation 5

Math Focus: Number & Operations Distributed Strands: Number & Operations • Algebra • Geometry • Measurement • Data Analysis & Probability • Problem Solving Lesson 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 • Finding a Percent of a Number

Maintaining & Extending

Lesson 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 • Renaming Fractions by Multiplying by 1

Facts pp. 216, 221, 225, 231, 235, 239, 244, 250, 254, 259

Power Up

Lesson 43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 • Equivalent Division Problems • Finding Unknowns in Fraction and Decimal Problems

Mental Math Strategies pp. 216, 221, 225, 231, 235, 239, 244, 250, 254, 259

Lesson 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 • Simplifying Decimal Numbers • Comparing Decimal Numbers

Problem Solving Strategies pp. 216, 221, 225, 231, 235, 239, 244, 250, 254, 259

Lesson 45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 • Dividing a Decimal Number by a Whole Number Lesson 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 • Writing Decimal Numbers in Expanded Notation • Mentally Multiplying Decimal Numbers by 10 and by 100

Enrichment Early Finishers pp. 238, 249, 253, 258, 263 Extensions p. 267

Lesson 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 • Circumference • Pi (π) Activity Circumference Lesson 48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 • Subtracting Mixed Numbers with Regrouping, Part 1 Lesson 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 • Dividing by a Decimal Number Lesson 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 • Decimal Number Line (Tenths) • Dividing by a Fraction Investigation 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 • Displaying Data

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Section 6

Lessons 51–60, Investigation 6

Math Focus: Number & Operations • Geometry Distributed Strands: Number & Operations • Geometry • Measurement • Data Analysis & Probability • Problem Solving Lesson 51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 • Rounding Decimal Numbers

Maintaining & Extending

Lesson 52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 • Mentally Dividing Decimal Numbers by 10 and by 100

Facts pp. 268, 272, 276, 280, 285, 289, 295, 299, 306, 310

Lesson 53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 • Decimals Chart • Simplifying Fractions Lesson 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 • Reducing by Grouping Factors Equal to 1 • Dividing Fractions Lesson 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 • Common Denominators, Part 1 Lesson 56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 • Common Denominators, Part 2 Lesson 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 • Adding and Subtracting Fractions: Three Steps Lesson 58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 • Probability and Chance Lesson 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 • Adding Mixed Numbers Lesson 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 • Polygons Investigation 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 • Attributes of Geometric Solids Activity Comparing Geometric Solids

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Power Up

Mental Math Strategies pp. 268, 272, 276, 280, 285, 289, 295, 299, 306, 310 Problem Solving Strategies pp. 268, 272, 276, 280, 285, 289, 295, 299, 306, 310

Enrichment Early Finishers pp. 279, 284, 294, 298, 305, 309 Extensions p. 318

TA B LE O F CO NTE NT S

Section 7

Lessons 61–70, Investigation 7

Math Focus: Number & Operations • Geometry Distributed Strands: Number & Operations • Algebra • Geometry • Measurement • Problem Solving Lesson 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 • Adding Three or More Fractions

Maintaining & Extending

Lesson 62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 • Writing Mixed Numbers as Improper Fractions

Facts pp. 320, 324, 329, 333, 337, 342, 346, 349, 353, 358

Lesson 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 • Subtracting Mixed Numbers with Regrouping, Part 2 Lesson 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 • Classifying Quadrilaterals Lesson 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 • Prime Factorization • Division by Primes • Factor Trees Lesson 66 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 • Multiplying Mixed Numbers

Power Up

Mental Math Strategies pp. 320, 324, 329, 333, 337, 342, 346, 349, 353, 358 Problem Solving Strategies pp. 320, 324, 329, 333, 337, 342, 346, 349, 353, 358

Enrichment Early Finishers pp. 323, 332, 352, 357, 362 Extensions p. 367

Lesson 67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 • Using Prime Factorization to Reduce Fractions Lesson 68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 • Dividing Mixed Numbers Lesson 69 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 • Lengths of Segments • Complementary and Supplementary Angles Lesson 70 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 • Reducing Fractions Before Multiplying Investigation 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 • The Coordinate Plane Activity Drawing on the Coordinate Plane

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Section 8

Lessons 71–80, Investigation 8

Math Focus: Number & Operations • Geometry Distributed Strands: Number & Operations • Geometry • Measurement • Problem Solving Lesson 71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 • Parallelograms Activity Area of a Parallelogram Lesson 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 • Fractions Chart • Multiplying Three Fractions Lesson 73 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 • Exponents • Writing Decimal Numbers as Fractions, Part 2 Lesson 74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 • Writing Fractions as Decimal Numbers • Writing Ratios as Decimal Numbers Lesson 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 • Writing Fractions and Decimals as Percents, Part 1 Lesson 76 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 • Comparing Fractions by Converting to Decimal Form Lesson 77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 • Finding Unstated Information in Fraction Problems Lesson 78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 • Capacity Lesson 79 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 • Area of a Triangle Activity Area of a Triangle Lesson 80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 • Using a Constant Factor to Solve Ratio Problems Investigation 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 • Geometric Construction of Bisectors Activity 1 Perpendicular Bisectors Activity 2 Angle Bisectors Activity 3 Constructing Bisectors

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Maintaining & Extending

Power Up Facts pp. 368, 375, 380, 385, 390, 395, 399, 404, 408, 413 Mental Math Strategies pp. 368, 375, 380, 385, 390, 395, 399, 404, 408, 413 Problem Solving Strategies pp. 368, 375, 380, 385, 390, 395, 399, 404, 408, 413

Enrichment Early Finishers pp. 379, 384, 394, 403, 412 Extensions p. 420

TA B LE O F CO NTE NT S

Section 9

Lessons 81–90, Investigation 9

Math Focus: Algebra • Measurement Distributed Strands: Number & Operations • Algebra • Geometry • Measurement • Data Analysis & Probability • Problem Solving Lesson 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 • Arithmetic with Units of Measure

Maintaining & Extending

Lesson 82 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 • Volume of a Rectangular Prism

Facts pp. 421, 426, 431, 436, 441, 447, 452, 456, 460, 465

Lesson 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 • Proportions Lesson 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 • Order of Operations, Part 2 Lesson 85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 • Using Cross Products to Solve Proportions Lesson 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 • Area of a Circle Lesson 87 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 • Finding Unknown Factors

Power Up

Mental Math Strategies pp. 421, 426, 431, 436, 441, 447, 452, 456, 460, 465 Problem Solving Strategies pp. 421, 426, 431, 436, 441, 447, 452, 456, 460, 465

Enrichment Early Finishers pp. 435, 440, 446, 451, 455, 469 Extensions p. 472

Lesson 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 • Using Proportions to Solve Ratio Word Problems Lesson 89 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 • Estimating Square Roots Lesson 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 • Measuring Turns Investigation 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 • Experimental Probability Activity Probability Experiment

Table of Contents

xiii

TA B LE O F CO N T E N T S

Section 10

Lessons 91–100, Investigation 10

Math Focus: Numbers & Operations • Geometry Distributed Strands: Number & Operations • Algebra • Geometry • Measurement • Data Analysis & Probability Lesson 91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 • Geometric Formulas

Maintaining & Extending

Lesson 92 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 • Expanded Notation with Exponents • Order of Operations with Exponents

Facts pp. 474, 479, 484, 488, 493, 497, 503, 508, 513, 517

Lesson 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 • Classifying Triangles Lesson 94 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 • Writing Fractions and Decimals as Percents, Part 2 Lesson 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 • Reducing Rates Before Multiplying Lesson 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 • Functions • Graphing Functions Lesson 97 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 • Transversals Lesson 98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 • Sum of the Angle Measures of Triangles and Quadrilaterals Lesson 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 • Fraction-Decimal-Percent Equivalents Lesson 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 • Algebraic Addition of Integers Investigation 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 • Compound Experiments

xiv

Saxon Math Course 1

Power Up

Mental Math Strategies pp. 474, 479, 484, 488, 493, 497, 503, 508, 513, 517 Problem Solving Strategies pp. 474, 479, 484, 488, 493, 497, 503, 508, 513, 517

Enrichment Early Finishers pp. 478, 483, 487, 496, 516, 523 Extensions p. 527

TA B LE O F CO NTE NT S

Section 11

Lessons 101–110, Investigation 11

Math Focus: Algebra • Geometry Distributed Strands: Number & Operations • Algebra • Geometry • Measurement • Problem Solving Lesson 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 • Ratio Problems Involving Totals

Maintaining & Extending

Lesson 102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 • Mass and Weight

Facts pp. 528, 533, 538, 548, 553, 557, 561, 566, 573

Lesson 103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 • Perimeter of Complex Shapes Lesson 104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 • Algebraic Addition Activity Activity Sign Game Lesson 105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 • Using Proportions to Solve Percent Problems Lesson 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 • Two-Step Equations

Power Up

Mental Math Strategies pp. 528, 533, 538, 548, 553, 557, 561, 566, 573 Problem Solving Strategies pp. 528, 533, 538, 543, 548, 553, 557, 561, 566, 573

Enrichment Early Finishers pp. 532, 572 Extensions p. 581

Lesson 107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 • Area of Complex Shapes Lesson 108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 • Transformations Activity Transformations Lesson 109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 • Corresponding Parts • Similar Figures Lesson 110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 • Symmetry Investigation 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 • Scale Factor: Scale Drawings and Models

Table of Contents

xv

TA B LE O F CO N T E N T S

Section 12

Lessons 111–120, Investigation 12

Math Focus: Measurement • Problem Solving Distributed Strands: Number & Operations • Algebra • Geometry • Measurement • Problem Solving Lesson 111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582 • Applications Using Division

Maintaining & Extending

Lesson 112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 • Multiplying and Dividing Integers

Facts pp. 582, 587, 592, 597, 602, 606, 612, 617, 621, 626

Lesson 113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 • Adding and Subtracting Mixed Measures • Multiplying by Powers of Ten

Mental Math Strategies pp. 582, 587, 592, 597, 602, 606, 612, 617, 621, 626

Lesson 114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 • Unit Multipliers

Problem Solving Strategies pp. 582, 587, 592, 597, 602, 606, 612, 617, 621, 626

Lesson 115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 • Writing Percents as Fractions, Part 2 Lesson 116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 • Compound Interest Lesson 117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 • Finding a Whole When a Fraction is Known Lesson 118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617 • Estimating Area Lesson 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 • Finding a Whole When a Percent is Known Lesson 120 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 • Volume of a Cylinder Investigation 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630 • Volume of Prisms, Pyramids, Cylinders and Cones • Surface Area of Prisms and Cylinders

xvi

Power Up

Saxon Math Course 1

Enrichment Early Finishers pp. 591, 601, 605, 611, 616, 629 Extensions p. 635

LETTER FROM AUTHOR STEPHEN HAKE

Dear Student, We study mathematics because of its importance to our lives. Our school schedule, our trip to the store, the preparation of our meals, and many of the games we play involve mathematics. You will find that the word problems in this book are often drawn from everyday experiences. As you grow into adulthood, mathematics will become even more important. In fact, your future in the adult world may depend on the mathematics you have learned. This book was written to help you learn mathematics and to learn it well. For this to happen, you must use the book properly. As you work through the pages, you will see that similar problems are presented over and over again. Solving each problem day after day is the secret to success. Your book is made up of daily lessons and investigations. Each lesson has three parts. 1. The first part is a Power Up that includes practice of basic facts and mental math. These exercises improve your speed, accuracy, and ability to do math “in your head.” The Power Up also includes a problem-solving exercise to familiarize you with strategies for solving complicated problems. 2. The second part of the lesson is the New Concept. This section introduces a new mathematical concept and presents examples that use the concept. The Practice Set provides a chance to solve problems involving the new concept. The problems are lettered a, b, c, and so on. 3. The final part of the lesson is the Written Practice. This problem set reviews previously taught concepts and prepares you for concepts that will be taught in later lessons. Solving these problems helps you remember skills and concepts for a long time. Investigations are variations of the daily lesson. The investigations in this book often involve activities that fill an entire class period. Investigations contain their own set of questions instead of a problem set. Remember, solve every problem in every practice set, written practice set, and investigation. Do not skip problems. With honest effort, you will experience success and true learning that will stay with you and serve you well in the future.

Temple City, California

Letter from the Author

xvii

HOW TO USE YOUR TEXTBOOK Saxon Math Course 1 is unlike any math book you have used! It doesn’t have colorful photos to distract you from learning. The Saxon approach lets you see the beauty and structure within math itself. You will understand more mathematics, become more confident in doing math, and will be well prepared when you take high school math classes.

Power Yourself Up! Start off each lesson by practicing your basic skills and concepts, mental math, and problem solving. Make your math brain stronger by exercising it every day. Soon you’ll know these facts by memory!

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PROBLEM-SOLVING OVERVIEW

Focus on Problem Solving As we study mathematics we learn how to use tools that help us solve problems. We face mathematical problems in our daily lives, in our work, and in our efforts to advance our technological society. We can become powerful problem solvers by improving our ability to use the tools we store in our minds. In this book we will practice solving problems every day. This lesson has three parts: Problem-Solving Process The four steps we follow when solving problems. Problem-Solving Strategies Some strategies that can help us solve problems. Writing and Problem Solving Describing how we solved a problem or formulating a problem.

four-step problemsolving process

Solving a problem is like arriving at a destination, so the process of solving a problem is similar to the process of taking a trip. Suppose we are on the mainland and want to reach a nearby island.

Island Mainland

Problem-Solving Process

Taking a Trip

Step 1: Know where you are and where you want to go. Understand

We are on the mainland and want to go to the island.

Step 2: Plan

Plan your route.

We might use the bridge, the boat, or swim.

Step 3: Solve

Follow the plan.

Take the journey to the island.

Step 4: Check Check that you have reached the right place.

Verify that you have reached your desired destination.

Problem-Solving Overview

1

When we solve a problem, it helps to ask ourselves some questions along the way. Ask Yourself Questions

Follow the Process Step 1: Understand

What information am I given? What am I asked to find or do?

Step 2: Plan

How can I use the given information to solve the problem? What strategy can I use to solve the problem?

Step 3: Solve

Am I following the plan? Is my math correct?

Step 4: Check (Look Back)

Does my solution answer the question that was asked? Is my answer reasonable?

Below we show how we follow these steps to solve a word problem.

Example 1 Carla wants to buy a CD player that costs $48.70 including tax. She has saved $10.50. Carla earns $10 each weekend babysitting. How many weekends does she need to babysit to earn enough money to buy the CD player?

Solution Step 1: Understand the problem. The problem gives the following information: • The CD player costs $48.70. • Carla has saved $10.50. • Carla earns $10.00 every weekend. We are asked to find out how many weekends Carla needs to babysit to have enough money to buy the CD player. Step 2: Make a plan. We see that we cannot get to the answer in one step. We plan how to use the given information in a manner that will lead us toward the solution. One way to solve the problem is: • Find out how much more money Carla needs. • Then find out how many weekends it will take to earn the needed amount. Step 3: Solve the problem. (Follow the plan.) First we subtract $10.50 from $48.70 to find out how much more money Carla needs. Carla needs $38.20 more than she has.

2

Saxon Math Course 1

$48.70 cost − $10.50 Carla has $38.20 Carla needs

Thinking Skill Verify

What strategy did we use to find the number of weekends?

Now we find the number of weekends Carla needs to work. One way is to divide $38.20 by $10.00. Another way is to find the multiple of $10.00 that gives Carla enough money. We can make a table to do this. Weekend 1

Weekend 2

Weekend 3

Weekend 4

$10

$20

$30

$40

After one weekend Carla earns $10, after two weekends $20, three weekends $30, and four weekends $40. Carla needs an additional $38.20. She will need to work four weekends to have enough money to buy the CD player. Step 4: Check your answer. (Look back.) We read the problem again to see if our solution answers the question. We decide if our answer is reasonable. The problem asks how many weekends Carla will need to work to earn the rest of the money for the CD player. Our solution, 4 weekends, answers the question. Our solution is reasonable because $40 is just a little over the $38.20 that Carla needs. After four weekends Carla has $10.50 + $40 = $50.50. This is enough money to buy the CD player.

Example 2 Howard is planning to tile the top of an end table. The tabletop is 1 ft by 2 ft. He wants to show his initial “H” using the tiles. The 4-inch tiles come in black and white. How many tiles of each color does Howard need to cover the table with his initial?

Solution Step 1: Understand the problem. The problem gives the following information: • The table is 1 ft × 2 ft. • The tiles are 4-inches on a side. • The tiles are black and white. • We need to model the letter “H”. We are asked to find how many tiles of each color we need to show the letter “H” on the tabletop. Step 2: Make a plan. We see that we cannot get to the answer in one step. We plan how to use the given information in a manner that will lead us toward the solution. • Change the table dimensions to inches and determine how many tiles will cover the table. • Use tiles to model the tabletop and decide how many tiles are needed to show the letter “H”.

Problem-Solving Overview

3

Step 3: Solve the problem. (Follow the plan.) There are 12 inches in 1 foot, so the table is 12 in. by 24 in. 12 in. ÷ 4 in. = 3

24 in. ÷ 4 in. = 6

3 × 6 = 18

Howard needs 18 tiles to cover the table. Thinking Skill Verify

How can we use 18 tiles to model the letter “H”? Here are two possibilities.

What strategy did we use to design the tabletop? 10 white tiles, 8 black tiles

14 white tiles, 4 black tiles

Since the problem does not specify the number of each color, we can decide based on the design we choose. Howard needs 10 white tiles and 8 black tiles. Step 4: Check your answer. (Look back.) We read the problem again to see if our solution answers the question. We decide if our answer is reasonable. The problem asks for the number of tiles of each color Howard needs to show his initial on the tabletop. Our solution shows the letter “H” using 10 white tiles and 8 black tiles. 1. List in order the four steps in the problem-solving process. 2. What two questions do we answer to understand a problem? Refer to the following problem to answer questions 3–8. Mrs. Rojas is planning to take her daughter Lena and her friend Natalie to see a movie. The movie starts at 4:30 p.m. She wants to arrive at the theater 20 minutes before the movie starts. It will take 15 minutes to drive to Natalie’s house. It is 10 minutes from Natalie’s house to the theater. At what time should Mrs. Rojas leave her house? 3. Connect What information are we given? 4. Verify What are you asked to find? 5. Which step of the four-step problem-solving process did you complete when you answered questions 3 and 4? 6. Describe your plan for solving the problem. 7. Explain Solve the problem by following your plan. Show your work. Write your solution to the problem in a way someone else will understand. 8. Check your work and your answer. Look back to the problem. Be sure you use the information correctly. Be sure you found what you were asked to find. Is your answer reasonable? 4

Saxon Math Course 1

problemsolving strategies

As we consider how to solve a problem we choose one or more strategies that seem to be helpful. Referring to the picture at the beginning of this lesson, we might choose to swim, to take the boat, or to cross the bridge to travel from the mainland to the island. Other strategies might not be as effective for the illustrated problem. For example, choosing to walk or bike across the water are strategies that are not reasonable for this situation. When solving mathematical problems we also select strategies that are appropriate for the problem. Problem-solving strategies are types of plans we can use to solve problems. Listed below are ten strategies we will practice in this book. You may refer to these descriptions as you solve problems throughout the year. Act it out or make a model. Moving objects or people can help us visualize the problem and lead us to the solution. Use logical reasoning. All problems require reasoning, but for some problems we use given information to eliminate choices so that we can close in on the solution. Usually a chart, diagram, or picture can be used to organize the given information and to make the solution more apparent. Draw a picture or diagram. Sketching a picture or a diagram can help us understand and solve problems, especially problems about graphs or maps or shapes. Write a number sentence or equation. We can solve many word problems by fitting the given numbers into equations or number sentences and then finding the unknown numbers. Make it simpler. We can make some complicated problems easier by using smaller numbers or fewer items. Solving the simpler problem might help us see a pattern or method that can help us solve the complex problem. Find a pattern. Identifying a pattern that helps you to predict what will come next as the pattern continues might lead to the solution. Make an organized list. Making a list can help us organize our thinking about a problem. Guess and check. Guessing the answer and trying the guess in the problem might start a process that leads to the answer. If the guess is not correct, use the information from the guess to make a better guess. Continue to improve your guesses until you find the answer. Make or use a table, chart, or graph. Arranging information in a table, chart, or graph can help us organize and keep track of data. This might reveal patterns or relationships that can help us solve the problem. Work backwards. Finding a route through a maze is often easier by beginning at the end and tracing a path back to the start. Likewise, some problems are easier to solve by working back from information that is given toward the end of the problem to information that is unknown near the beginning of the problem. 9. Name some strategies used in this lesson.

Problem-Solving Overview

5

The chart below shows where each strategy is first introduced in this textbook.

writing and problem solving

Strategy

Lesson

Act It Out or Make a Model

Problem Solving Overview Example 2

Use Logical Reasoning

Lesson 3

Draw a Picture or Diagram

Lesson 17

Write a Number Sentence or Equation

Lesson 17

Make It Simpler

Lesson 4

Find a Pattern

Lesson 1

Make an Organized List

Lesson 8

Guess and Check

Lesson 5

Make or Use a Table, Chart, or Graph

Problem Solving Overview Example 1

Work Backwards

Lesson 84

Sometimes, a problem will ask us to explain our thinking. This helps us measure our understanding of math. • Explain how you solved the problem. • Explain how you know your answer is correct. • Explain why your answer is reasonable. For these situations, we can describe the way we followed our plan. This is a description of the way we solved example 1. Subtract $10.50 from $48.70 to find out how much more money Carla needs. $48.70 – $10.50 = $38.20. Make a table and count by 10s to determine that Carla needs to work 4 weekends so she can earn enough money for the CD player. 10. Write a description of how we solved the problem in example 2. Other times, we will be asked to write a problem for a given equation. Be sure to include the correct numbers and operations to represent the equation. 11. Write a word problem for the equation 32 + 32 = 64.

6

Saxon Math Course 1

LESSON

1

Adding Whole Numbers and Money Subtracting Whole Numbers and Money Fact Families, Part 1

Power Up 1 facts mental math

Building Power Power Up A a. Number Sense: 30 + 30 b. Number Sense: 300 + 300 c. Number Sense: 80 + 40 d. Number Sense: 800 + 400 e. Number Sense: 20 + 30 + 40 f. Number Sense: 200 + 300 + 400 g. Measurement: How many inches are in a foot? h. Measurement: How many millimeters are in a centimeter?

problem solving

Sharon made three square patterns using 4 coins, 9 coins, and 16 coins. If she continues forming larger square patterns, how many coins will she need for each of the next three square patterns? We are given 4, 9, and 16 as the first three square numbers. We are asked to extend the sequence an additional three terms. Understand

We will find the pattern in the first three terms of the sequence, then use the pattern to extend the sequence an additional three terms. Plan

We see that the number of coins in each square can be found by multiplying the number of coins in each row by the number of rows: 2 × 2 = 4, 3 × 3 = 9, and 4 × 4 = 16. We use this rule to find the next three terms: 5 × 5 = 25, 6 × 6 = 36, and 7 × 7 = 49. Solve

We found that Sharon needs 25, 36, and 49 coins to build each of the next three squares in the pattern. We can verify our answers by drawing pictures of each of the next three terms in the pattern and counting the coins. Check

1

For instructions on how to use the Power Up, please consult the preface.

Lesson 1

7

New Concepts adding whole numbers and money

Increasing Knowledge

To combine two or more numbers, we add. The numbers that are added together are called addends. The answer is called the sum. Changing the order of the addends does not change the sum. For example, 3+5=5+3 This property of addition is called the Commutative Property of Addition. When adding numbers, we add digits that have the same place value.

Example 1 Add: 345 + 67

Solution When we add whole numbers on paper, we write the numbers so that the place values are aligned. Then we add the digits by column. Changing the order of the addends does not change the sum. One way to check an addition answer is to change the order of the addends and add again.

11

345 addend + 67 addend 412 sum 11

67 + 345 412 check

Example 2 Thinking Skill Connect

$5 means five dollars and no cents. Why does writing $5 as $5.00 help when adding money amounts?

Add: $1.25 + $12.50 + $5

Solution When we add money, we write the numbers so that the decimal points are aligned. We write $5 as $5.00 and add the digits in each column.

$1.25 $12.50 + $5.00 $18.75

If one of two addends is zero, the sum of the addends is identical to the nonzero addend. This property of addition is called the Identity Property of Addition. 5+0=5

subtracting whole numbers and money

We subtract one number from another number to find the difference between the two numbers. In a subtraction problem, the subtrahend is taken from the minuend. 5−3=2 In the problem above, 5 is the minuend and 3 is the subtrahend. The difference between 5 and 3 is 2. Does the Commutative Property apply to subtraction? Give an example to support your answer. Verify

8

Saxon Math Course 1

Example 3 Subtract: 345 ∙ 67

Solution When we subtract whole numbers, we align the digits by place value. We subtract the bottom number from the top number and regroup when necessary.

2 13

1

3 4 5 − 6 7 2 7 8 difference

Example 4 Jim spent $1.25 for a hamburger. He paid for it with a five-dollar bill. Find how much change he should get back by subtracting $1.25 from $5.

Solution Thinking Skill When is it necessary to line up decimals?

Order matters when we subtract. The starting amount is put on top. We write $5 as $5.00. We line up the decimal points to align the place values. Then we subtract. Jim should get back $3.75.

4 9

1

$5.0 0 − $1.2 5 $3.7 5

We can check the answer to a subtraction problem by adding. If we add the answer (difference) to the amount subtracted, the total should equal the starting amount. We do not need to rewrite the problem. We just add the two bottom numbers to see whether their sum equals the top number. Subtract Down To find the difference

fact families, part 1

$5.00 − $1.25 $3.75

Add Up To check the answer

Addition and subtraction are called inverse operations. We can “undo” an addition by subtracting one addend from the sum. The three numbers that form an addition fact also form a subtraction fact. For example, 4+5=9

9−5=4

The numbers 4, 5, and 9 are a fact family. They can be arranged to form the two addition facts and two subtraction facts shown below. 4 +5 9

5 +4 9

9 −5 4

9 −4 5

Example 5 Rearrange the numbers in this addition fact to form another addition fact and two subtraction facts. 11 + 14 = 25

Lesson 1

9

Solution We form another addition fact by reversing the addends. 14 + 11 = 25 We form two subtraction facts by making the sum, 25, the first number of each subtraction fact. Then each remaining number is subtracted from 25. 25 ∙ 11 = 14 25 ∙ 14 = 11

Example 6 Rearrange the numbers in this subtraction fact to form another subtraction fact and two addition facts. 11 ∙ 6 5

Solution The Commutative Property does not apply to subtraction, so we may not reverse the first two numbers of a subtraction problem. However, we may reverse the last two numbers.

11 ∙ 5 6

11 − 6 5 For the two addition facts, 11 is the sum. 5 +6 11

Practice Set

6 +5 11

Simplify: a. 3675 + 426 + 1357

b. $6.25 + $8.23 + $12

c. 5374 − 168

d. $5 − $1.35

e. f.

Represent Arrange the numbers 6, 8, and 14 to form two addition facts and two subtraction facts.

Rearrange the numbers in this subtraction fact to form another subtraction fact and two addition facts. Connect

25 − 10 = 15

Written Practice

Strengthening Concepts

1. What is the sum of 25 and 40? 2. At a planetarium show, Johnny counted 137 students and 89 adults. He also counted 9 preschoolers. How many people did Johnny count in all? 10

Saxon Math Course 1

3.

Generalize

What is the difference when 93 is subtracted

from 387? 4. Keisha paid $5 for a movie ticket that cost $3.75. Find how much change Keisha should get back by subtracting $3.75 from $5. 5.

Tatiana had $5.22 and earned $4.15 more by taking care of her neighbor’s cat. How much money did she have then? Explain how you found the answer. Explain

6. The soup cost $1.25, the fruit cost $0.70, and the drink cost $0.60. To find the total price of the lunch, add $1.25, $0.70, and $0.60. 7.

63 47 + 50

8.

9.

632 57 + 198

78 9 + 987

10.

432 579 + 3604

11. 345 − 67

12. 678 − 416

13. 3764 − 96

14. 875 + 1086 + 980

15. 10 + 156 + 8 + 27 16.

$3.47 − $0.92

17.

$24.15 − $1.45

18.

$0.75 + $0.75

19.

$0.12 $0.46 + $0.50

20. What is the name for the answer when we add? 21. What is the name for the answer when we subtract? * 22.

The numbers 5, 6, and 11 are a fact family. Form two addition facts and two subtraction facts with these three numbers.

* 23.

Connect Rearrange the numbers in this addition fact to form another addition fact and two subtraction facts.

Represent

27 + 16 = 43 * 24.

Connect Rearrange the numbers in this subtraction fact to form another subtraction fact and two addition facts.

50 − 21 = 29 25. Describe a way to check the correctness of a subtraction answer.

* We encourage students to work first on the exercises on which they might want help, saving the easier exercises for last. Beginning in this lesson, we star the exercises that cover challenging or recently presented content. We suggest that these exercises be worked first.

Lesson 1

11

LESSON

2

Power Up facts mental math

Multiplying Whole Numbers and Money Dividing Whole Numbers and Money Fact Families, Part 2 Building Power Power Up A a. Number Sense: 500 + 40 b. Number Sense: 60 + 200 c. Number Sense: 30 + 200 + 40 d. Number Sense: 70 + 300 + 400 e. Number Sense: 400 + 50 + 30 f. Number Sense: 60 + 20 + 400 g. Measurement: How many inches are in 2 feet? h. Measurement: How many millimeters are in 2 centimeters?

problem solving

Sam thought of a number between ten and twenty. Then he gave a clue: You say the number when you count by twos and when you count by threes, but not when you count by fours. Of what number was Sam thinking?

New Concepts multiplying whole numbers and money

Increasing Knowledge

Courtney wants to enclose a square garden to grow vegetables. How many feet of fencing does she need? When we add the same number several times, we get a sum. We can get the same result by multiplying.

15 ft 15 ft

15 ft 15 ft

15 + 15 + 15 + 15 = 60 Four 15s equal 60.

4 × 15 = 60 Numbers that are multiplied together are called factors. The answer is called the product. To indicate multiplication, we can use a times sign, a dot, or write the factors side by side without a sign. Each of these expressions means that l and w are multiplied: l × w l∙w lw

12

Saxon Math Course 1

Notice that in the form l ∙ w the multiplication dot is elevated and is not in the position of a decimal point. The form lw can be used to show the multiplication of two or more letters or of a number and letters, as we show below. lwh

4s

4st

The form lw can also be used to show the multiplication of two or more numbers. To prevent confusion, however, we use parentheses to separate the numbers in the multiplication. Each of the following is a correct use of parentheses to indicate “3 times 5,” although the first form is most commonly used. Without the parentheses, we would read each of these simply as the number 35. 3(5) Thinking Skill Discuss

Why do we multiply 28 by 4, by 10, and then add to find the product?

(3)(5)

(3)5

When we multiply by a two-digit number on paper, we multiply twice. To multiply 28 by 14, we first multiply 28 by 4. Then we multiply 28 by 10. For each multiplication we write a partial product. We add the partial products to find the final product. 28 × 14 112 280 392

factor factor partial product (28 × 4) partial product (28 × 10) product (14 × 28)

When multiplying dollars and cents by a whole number, the answer will have a dollar sign and a decimal point with two places after the decimal point. $1.35 × 6 $8.10

Example 1 Find the cost of two dozen pencils at 35¢ each.

Solution Two dozen is two 12s, which is 24. To find the cost of 24 pencils, we multiply 35¢ by 24. 35¢ × 24 140 700 840¢ The cost of two dozen pencils is 840¢, which is $8.40. The Commutative Property applies to multiplication as well as addition, so changing the order of the factors does not change the product. For example, 4×2=2×4

Lesson 2

13

One way to check multiplication is to reverse the order of factors and multiply. 23 × 14 92 230 322

14 × 23 42 280 322

check

The Identity Property of Multiplication states that if one of two factors is 1, the product equals the other factor. The Zero Property of Multiplication states that if zero is a factor of a multiplication, the product is zero. Represent

Give an example for each property.

Example 2 Multiply:

Thinking Skill Discuss

Why does writing trailing zeros not change the product?

400 ∙ 874

Solution

dividing whole 2  24 and numbers money

21

874 × 400 349,600

S S

To simplify the multiplication, we reverse the order of the factors and write trailing zeros so that they “hang out” to the right.

When we separate a number into a certain number of equal parts, we divide. We can indicate division 24 with a division symbol (÷), a division box aQ  Rb, or a 2 expressions below means “24 divided by 2”: division bar (−). Each of the 24 24 24 ÷ 2 2 24 2 24   2 2  24 2

quotient The answer to a division problem is the quotient. The number that is divided 24 divisor  dividendis the dividend. The number by which the dividend is divided is the divisor. 2  24 2 quotient quotient quotient divisor  dividend divisor  dividend divisor  dividenddividend ÷ divisor = quotient dividend  quotient divisor

quotient divisor  dividend dividend dividend  dividend quotient quotient  quotient divisor  divisor divisor

When the dividend is zero, thedividend quotient is zero. The divisor may not be zero.  (and quotient When the dividend and divisordivisor are equal not zero), the quotient is 1.

Example 3 Divide: 3456 ∙ 7

Solution On the next page, we show both the long-division and short-division methods.

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Saxon Math Course 1

24 2

a

b

0 0 Long Division 493 R 5 7  3456 28 65 63 26 21 5

Thinking Skill Discuss

Why must the remainder always be less than the divisor?

fact families, part 2

Short Division 493R5 7  346526

493 R 5 7  3456 28 65 63 26 21 5

$.90 $1.60 4  $3.60 $4.80 3  36 3 Using the short-division method, we perform the multiplication and 00 1 8 0 of each subtraction. subtraction steps mentally, recording only the result 1 8 0 $1.60 the quotient by the divisor. Then we To check our work, we multiply 00add the 3  $4.80 remainder to this answer. The result should be the dividend. For this0 example 3 0 we multiply 493 by 7. Then we add 5. 18 62 18 493 00 × 7 0 493R5 0 3451 7  346526 + 5 3456 493R5 When dividing dollars7and $.90 526 cents will be  346cents, 4  $3.60 included in the answer. Notice that the decimal 36 point in the quotient is directly above the decimal 00 point in the division box, separating the dollars 0 from the cents. $.90 0 4  $3.60 Multiplication and division 3 6are inverse operations, so there are multiplication and division fact families just 00 as there are addition and subtraction fact families. The numbers 5, 6, 0 and 30 are a fact family. We can form two multiplication facts and two0division facts with these numbers. 5 × 6 = 30

30 ÷ 5 = 6

6 × 5 = 30

30 ÷ 6 = 5

Example 4 Rearrange the numbers in this multiplication fact to form another multiplication fact and two division facts. 5 ∙ 12 = 60

Solution By reversing the factors, we form another multiplication fact. 12 ∙ 5 = 60 By making 60 the dividend, we can form two division facts. 60 ∙ 5 = 12 60 ∙ 12 = 5

Lesson 2

15

Practice Set

a. 20 × 37¢

b. 37 ∙ 0

d. 5  $8.40 5  $8.40

e. 200 ÷ 12

c. 407(37) 234 234 3 f. 3

g. Which numbers are the divisors in problems d, e, and f? h.

Represent Use the numbers 8, 9, and 72 to form two multiplication facts and two division facts.

Written Practice 1

Strengthening Concepts

1. If the factors are 7 and 11, what is the product?

(2)

2. (1)

Generalize

What is the difference between 97 and 79?

3. If the addends are 170 and 130, what is the sum?

(1)

4. If 36 is the dividend and 4 is the divisor, what is the quotient?

(2)

5. Find the sum of 386, 98, and 1734.

(1)

6. Fatima spent $2.25 for a book. She paid for it with a five-dollar bill. Find how much change she should get back by subtracting $2.25 from $5.

(1)

7. Luke wants to buy a $70.00 radio for his car. He has $47.50. Find how much more money he needs by subtracting $47.50 from $70.00.

(1)

8.

Explain Each energy bar costs 75¢. Find the cost of one dozen energy bars. Explain how you found your answer.

9.

312 − 86

10.

4106 + 1398

4000 − 1357

12.

$10.00 − $2.83

(2)

(1)

11. (1)

14. 25 ∙ 25

288 15. 288 (2) 6 6 17. $1.25 × 8

16. 225 225 (2) 15 15 18. 400 × 50

19. 1000 ÷ 8

20. $45.00 ÷ 20

(2)

(2)

9 9 4  364  36

16

(1)

13. 405(8) (2)

1

(1)

(2)

(2)

(2)

The italicized numbers within parentheses underneath each problem number are called lesson reference numbers. These numbers refer to the lesson(s) in which the major concept of that particular problem is introduced. If additional assistance is needed, refer to the discussion, examples, or practice problems of that lesson.

Saxon Math Course 1

* 21.

Use the numbers 6,288 8, and 48 to form two multiplication facts and two division facts. 6

* 22.

Rearrange the numbers in this division fact to form another division fact and two multiplication facts.

(2)

(2)

Represent

225 15

Connect

9 225 4  36 15 * 23. Connect Rearrange the numbers in this addition fact to form another (1) addition fact and two subtraction facts.

288 6

12 + 24 = 36

9 4  36

24. a. Find the sum of 9 and 6. (1)

b. Find the difference between 9 and 6. 25. The divisor, dividend, and quotient are in these positions when we use a (2) division sign: dividend ÷ divisor = quotient On your paper, draw a division box and show the positions of the divisor, dividend, and quotient. 26. Multiply to find the answer to this addition problem: (2)

39¢ + 39¢ + 39¢ + 39¢ + 39¢ + 39¢

27. 365 × 0 (2)

* 30. (2)

Early Finishers

Real-World Application

28. 0 ÷ 50 (2)

29. 365 ÷ 365 (2)

How can you check the correctness of a division answer that has no remainder? Explain

A customer at a bank deposits 2 one hundred-dollar bills, 8 twenty-dollar bills, 5 five-dollar bills, 20 one-dollar bills, 2 rolls of quarters, 25 dimes and 95 pennies. How much money will be deposited in all? Note: One roll of quarters = 40 quarters.

Lesson 2

17

LESSON

3 Power Up facts mental math

Unknown Numbers in Addition Unknown Numbers in Subtraction Building Power Power Up B a. Number Sense: 3000 + 4000 b. Number Sense: 600 + 2000 c. Number Sense: 20 + 3000 d. Number Sense: 600 + 300 + 20 e. Number Sense: 4000 + 300 + 200 f. Number Sense: 70 + 300 + 4000 g. Measurement: How many inches are in 3 feet? h. Measurement: How many millimeters are in 3 centimeters?

problem solving

Tad picked up a number cube. His thumb and forefinger covered opposite faces. He counted the dots on the other four faces. How many dots did he count? We must first establish a base of knowledge about standard number cubes. The faces of a standard number cube are numbered with 1, 2, 3, 4, 5, or 6 dots. The number of dots on opposite faces of a number cube always total 7 (1 dot is opposite 6 dots, 2 dots are opposite 5 dots, and 3 dots are opposite 4 dots). Tad’s thumb and forefinger covered opposite faces. We are asked to find how many dots were on the remaining four faces altogether. Understand

We will use logical reasoning about a number cube and write an equation to determine the number of dots Tad counted. Plan

Logical reasoning tells us that the four uncovered faces form two pairs of opposite faces. Each pair of opposite faces has 7 dots, so two pairs of opposite faces have 2 × 7, or 14 dots. Solve

We determined that Tad counted 14 dots. We can check our answer by subtracting the number of dots Tad’s fingers covered from the total number of dots on the number cube: 21 − 7 = 14 dots. Check

New Concepts unknown numbers in addition

Increasing Knowledge

Below is an addition fact with three numbers. If one of the addends were missing, we could use the other addend and the sum to find the missing number. 4 +3 7

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Saxon Math Course 1

addend addend sum

Cover the 4 with your finger. How can you use the 7 and the 3 to find that the number under your finger is 4? Now cover the 3 instead of the 4. How can you use the other two numbers to find that the number under your finger is 3? Notice that we can find a missing addend by subtracting the known addend from the sum. We will use a letter to stand for a missing number.

Example 1 Find the value of m:

12 +m 31

Solution One of the addends is missing. The known addend is 12. The sum is 31. If we subtract 12 from 31, we find that the missing addend is 19. We check our answer by using 19 in place of m in the original problem. 21

31 – 12 19

Use 19 in place of m.

1

12 + 19 31

check

Example 2 Find the value of n: 36 + 17 + 5 + n = 64

Solution First we add all the known addends. 36 +17 + 5 + n = 64 + n = 64

58

Then we find n by subtracting 58 from 64. 64 − 58 = 6

So n is 6.

We check our work by using 6 in place of n in the original problem. 36 + 17 + 5 + 6 = 64

unknown numbers in subtraction

The answer checks.

Cover the 8 with your finger, and describe how to use the other two numbers to find that the number under your finger is 8. Discuss

8 −3 5

minuend subtrahend difference

Now cover the 3 instead of the 8. Describe how to use the other two numbers to find that the covered number is 3. As we will show below, we can find a missing minuend by adding the other two numbers. We can find a missing subtrahend by subtracting the difference from the minuend.

Lesson 3

19

Example 3 Find the value of w:

w ∙ 16 24

Solution We can find the first number of a subtraction problem by adding the other two numbers. We add 16 and 24 to get 40. We check our answer by using 40 in place of w. 1

16 + 24 40

Use 40 in place of w.

31

40 – 16 24

check

Example 4 Find the value of y: 236 ∙ y = 152

Solution One way to determine how to find a missing number is to think of a simpler problem that is similar. Here is a simpler subtraction fact: 5−3=2 In the problem, y is in the same position as the 3 in the simpler subtraction fact. Just as we can find 3 by subtracting 2 from 5, we can find y by subtracting 152 from 236. 11

236 – 152 84

Thinking Skill Discuss

What is another way we can check the subtraction?

We find that y is 84. Now we check our answer by using 84 in place of y in the original problem. 11

236 – 84 152

Use 84 in place of y. The answer checks.

Statements such as 12 + m = 31 are equations. An equation is a mathematical sentence that uses the symbol = to show that two quantities are equal. In algebra we refer to a missing number in an equation as an unknown. When asked to find the unknown in the exercises that follow, look for the number represented by the letter that makes the equation true.

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Saxon Math Course 1

Practice Set Math Language We can use a lowercase or an uppercase letter as an unknown: a+3=5 A+3=5 The equations have the same meaning.

Analyze Find the unknown number in each problem. Check your work by using your answer in place of the letter in the original problem.

a.

b.

32 + B 60

c.

C − 15 24

e. e + 24 = 52

f. 29 + f = 70

g. g − 67 = 43

h. 80 − h = 36

d.

38 − D 29

i. 36 + 14 + n + 8 = 75

Written Practice Math Language Remember that factors are multiplied together to get a product.

A + 12 45

Strengthening Concepts

1. If the two factors are 25 and 12, what is the product?

(2)

2. If the addends are 25 and 12, what is the sum? (1)

3. What is the difference of 25 and 12?

(1)

4. Each of the 31 students brought 75 aluminum cans to class for a recycling drive. Find how many cans the class collected by multiplying 31 by 75.

(2)

5. Find the total price of one dozen pizzas at $7.85 each by multiplying $7.85 by 12.

(2)

6.

Explain The basketball team scored 63 of its 102 points in the first half of the game. Find how many points the team scored in the second half. Explain how you found your answer.

7.

$3.68 × 9

9.

28¢ × 14

(1)

(2)

(2)

11. 100 ∙ 100 (2)

8.

407 × 80

10.

370 × 140

(2)

(2)

12. 144 ÷ 12 (2)

13. (12)(5) (2)

14.

3627 598 + 4881

16.

$10.00 − $0.26

(1)

(1)

15. (1)

5010 − 1376

Find the unknown number in each problem. 17. (3)

A + 16 48

18. (3)

23 + B 52

Lesson 3

21

19. (3)

20.

C − 17 31

(3)

42 − D 25

21. x + 38 = 75

22. x − 38 = 75

23. 75 − y = 38

24. 6 + 8 + w + 5 = 32

(3)

(3)

* 25. (1)

(2)

(3)

Connect Rearrange the numbers in this addition fact to form another addition fact and two subtraction facts.

24 + 48 = 72 * 26. (2)

Connect Rearrange the numbers in this multiplication fact to form another multiplication fact and two division facts.

6 × 15 = 90 Math Language Remember the divisor is divided into the dividend. The resulting answer is the quotient.

27. Find the quotient when the divisor is 20 and the dividend is 200. (2)

28. (2)

15 + 15 + 15 + 15 + 15 + 15 + 15 + 15

(2)

(3)

Real-World Application

Multiply to find the answer to this addition problem:

29. 144 ÷ 144 * 30.

Early Finishers

Connect

Explain

How can you find a missing addend in an addition problem?

Petrov’s family has a compact car that gets an average of 30 miles per gallon of gasoline. The family drives an average of 15,000 miles a year. a. Approximately how many gallons of gas do they purchase every year? b. If the average price of gas is $2.89 a gallon, how much should the family expect to spend on gas in a year?

22

Saxon Math Course 1

LESSON

4 Power Up facts mental math

Unknown Numbers in Multiplication Unknown Numbers in Division Building Power Power Up A a. Number Sense: 600 + 2000 + 300 + 20 b. Number Sense: 3000 + 20 + 400 + 5000 c. Number Sense: 7000 + 200 + 40 + 500 d. Number Sense: 700 + 2000 + 50 + 100 e. Number Sense: 60 + 400 + 30 + 1000 f. Number Sense: 900 + 8000 + 100 + 50 g. Measurement: How many feet are in a yard? h. Measurement: How many centimeters are in a meter?

problem solving

The diagram below is called a Jordan curve. It is a simple closed curve (think of a clasped necklace that has been casually dropped on a table). Which letters are on the inside of the curve, and which letters are on the outside of the curve?

C

B

E F

A

D

Understand We must determine if A, B, C, D, E and F are inside or outside the closed curve. Plan We will make the problem simpler and use the simpler problems to find a pattern.

Lesson 4

23

We will draw less complicated closed curves, and place an X inside and a Y outside each of the closed curves. Solve

2

1

X

Y

0

1

X

Y 3

X

3

Y

4

On our simpler curves, we notice that lines drawn from the outside of the curve to the X cross 1 or 3 lines. Lines drawn to the Y cross 0, 2, or 4 lines. We see this pattern: lines drawn to letters on the inside of the curve cross an odd number of lines, and lines drawn to letters on the outside of the curve cross an even number of lines. We look at the Jordan curve again. • A line drawn to A crosses 1 line, so it is inside the closed curve. • A line drawn to B crosses 4 lines, so it is outside. • A line drawn to C crosses 3 lines, so it is inside. • A line drawn to D crosses 5 lines, so it is inside. • A line drawn to E crosses 9 lines, so it is inside. • A line drawn to F crosses 10 lines, so it is outside. We determined that A, C, D, and E are inside the closed curve and that B and F are outside of the closed curve. We found a pattern that can help us quickly determine whether points are on the inside or outside of a closed curve. Check

New Concepts unknown numbers in multiplication

Increasing Knowledge

This multiplication fact has three numbers. If one of the factors were unknown, we could use the other factor and the product to figure out the unknown factor. 4 × 3 12

Thinking Skill Discuss

With your finger, cover the factors in this multiplication fact one at a time. Describe how you can use the two uncovered numbers to find the covered number. Notice that we can find an unknown factor by dividing the product by the known factor. Explain

Why can we use division to find a missing factor? Example 1

Find the value of A:

24

Saxon Math Course 1

A ∙ 6 72

Solution The unknown number is a factor. The product is 72. The factor that we know is 6. Dividing 72 by 6, we find that the unknown factor is 12. We check our work by using 12 in the original problem. 1

12 × 6 72 check

12 6  712

Example 2 Find the value of w: 6w = 84

Solution Reading Math In this problem 6w means “6 times w.”

unknown numbers in division

14 2 6 8  We divide 84 by 6 and find4 that the unknown factor is 14. We check our work by multiplying. 14 46  824 6  24

2

14 × 6 14 84 check 6  824

This division fact has three numbers. If one of the numbers were unknown, 4 we could figure out the third number. 1 65 24 4 6  930 14 6  24 6  824 Cover each of the numbers with your finger, and describe how to use the 1 5covered number. Notice that we can find the other two numbers to find the 3 15 6  9the 0 division box) by multiplying the other two dividend (the number inside 15 6 k numbers. We can find either the divisor 3or quotient (the numbers outside of 6  9 04 6  24 the box) by dividing.

Example 3

15 k 6 k  15 Find the value of k: = 6

15 5 6  k114 6  9230 Solution 6 8 4 The letter k is in the position k of the dividend. If we rewrite this problem with a  15 division box, it looks like this: 6 k 15  15 66  k4 6  24 We find an unknown dividend by multiplying the divisor and quotient. We multiply 15 by 6 and find that the unknown number is 90. Then we check our work. k 3 1515 15 66 930 check × 6 90 15 6 k

Lesson 4

25

Example 4 Find the value of m: 126 ∙ m = 7

Solution The letter m is in the position of the divisor. If we were to rewrite the problem with a division box, it would look like this:7 m  1267 m  126

Practice Set

We can find m by dividing 126 by 7. 7 7 18 m  126 m  126 5 7  1218 6 7  1256 We find that m is 18. We can check our division by multiplying as7follows: m  126 5 18 18 18 15 7  1256 7  1256 7 ×15 C7 7 126 C 18 7 In the original equation we can replace the 7  1256and test the m letter  126 with our answer 15 truth of 15 the resulting equation. 8 7 C 7 C D 144 87 126 ÷ 18 = D  144 7 = 7 18 15 7 C 7  1256 8 8 Analyze Find each unknown number. Check your work by using your answer D  144 D  144 in place of the letter in the original problem. a.

A × 7 91

e. 7w = 84 g.

360  30 x =

i.

Formulate

Written Practice

b.

20 × B 440

15 c. 7  C

8 d. D  144

360 n  60 x  30 5n 360  60  30 x 5 8 8m f. 112 = D  144 n 360  60x  30 h. n =  60 5 5

Write a word problem using the equation in exercise h. 360 x  30 Strengthening Concepts

n  60 5

1. Five dozen carrot sticks are to be divided evenly among 15 children. Find how many carrot sticks each child 360 should receive n by dividing  60 x  30 60 by 15. 5

(2)

2. Matt separated 100 pennies into 4 equal piles. Find how many pennies were in each pile. Explain how you found your answer.

(2)

3. Sandra put 100 pennies into stacks of 5 pennies each. Find how many stacks she formed by dividing 100 by 5.

(2)

4. For the upcoming season, 294 players signed up for soccer. Find the number of 14-player soccer teams that can be formed by dividing 294 by 14.

(2)

26

Saxon Math Course 1

5. Angela is reading a 280-page book. She has just finished page 156. Find how many pages she still has to read by subtracting 156 from 280.

(1)

6. Each month Bill earns $0.75 per customer for delivering newspapers. Find how much money he would earn in a month in which he had 42 customers by multiplying $0.75 by 42.

(2)

*

Analyze

7.

J ×5 60

9.

L + 36 37

(4)

(3)

Find each unknown number. Check your work. 8.

(3)

10. (3)

27 + K 72 64 − M 46 6  $12.36

11. n − 48 = 84

12. 7p = 91

13. q ÷ 7 = 0

14. 144 ÷ r = 6

15. 6  $12.36

16.

17. 526 ÷ 18

18. 563 + 563 + 563 + 563

19. $3.75 ⋅ 16

20. $3 + $2.86 + $0.98

(3)

(4)

(4)

(4)

(2)

(2)

(2)

5760 8

(1)

(2)

5760 21. $10 8 − $6.43 (1)

(1)

22. If the divisor is 3 and the quotient is 12, what is the dividend? (4)

23. If the product is 100 and one factor is 5, what is the other factor? (4)

* 24. (1)

Rearrange the numbers in this subtraction fact to form another subtraction fact and two addition facts. Connect

17 − 9 = 8 * 25. (2)

Rearrange the numbers in this division fact to form another division fact and two multiplication facts. Connect

72 ÷ 8 = 9 26. w + 6 + 8 + 10 = 40 (3)

27. Find the answer to this addition problem by multiplying: (2)

23¢ + 23¢ + 23¢ + 23¢ + 23¢ + 23¢ + 23¢ 28. 25m = 25 (4)

* 30. (4)

29. 15n = 0 (4)

Explain How can you find an unknown factor in a multiplication problem?

Lesson 4

27

LESSON

5 Power Up facts mental math

Order of Operations, Part 1 Building Power Power Up B a. Number Sense: 560 + 200 b. Number Sense: 840 + 30 c. Number Sense: 5200 + 2000 d. Number Sense: 650 + 140 e. Number Sense: 3800 + 2000 f. Number Sense: 440 + 200 g. Measurement: How many days are in a week? h. Measurement: How many hours are in a day?

problem solving

Use the digits 5, 6, 7, and 8 to complete this addition problem. There are two possible arrangements.

__ + 9 __

We are shown an addition problem with several digits missing. We are asked to complete the problem using the digits 5, 6, 7, and 8. Because the bottom addend is 9, we know that the ones digit of the sum will be one less than the ones digit of the top addend. Understand

We will intelligently guess and check for the ones place in the top addend by trying the numbers in an orderly way. We will then use logical reasoning to fill in the remaining digits of the problem. Plan

We quickly eliminate 5 as a possibility for the ones digit of the top addend because we do not have a 4 to place in the sum. We try 6 for the ones digit of the top addend. Six plus 9 is 15, so we write a 5 as the ones digit of the sum. If we write 7 as the tens digit of the top addend, we get 76 + 9. We add the two numbers and get 85. Placing an 8 in the sum, we see that we have used all the digits 5, 6, 7, and 8. We have found the first of two possible arrangements. Solve

Next, we try 7 as the ones digit of the top addend. Seven plus 9 is 16, so we place a 6 in the sum. Now we must use the digits 5 and 8 in the tens column. We try 57 + 9 = 66. That does not work, because it does not use the 8. We try 87 + 9 = 96. That also does not work, because it omits the 5. Finally, we try 8 in the top addend and 7 in the sum. This leaves 5 and 6 for the tens column. We try 58 + 9 = 67, and find the second solution to the problem. The digits 5, 6, 7, and 8 can be used to form two solutions for our missing digit problem: Check

28

Saxon Math Course 1

76 + 9 85

58 + 9 67

Increasing Knowledge

New Concept Thinking Skill Analyze

Why is it important to have rules for the order of operations?

When there is more than one addition or subtraction step within a problem, we take the steps in order from left to right. In this problem we first subtract 4 from 9. Then we add 3. 9−4+3=8 If a different order of steps is desired, parentheses are used to show which step is taken first. In the problem below, we first add 4 and 3 to get 7. Then we subtract 7 from 9. 9 − (4 + 3) = 2 These two rules are part of the rules for the Order of Operations in mathematics.

Example 1 a. 18 ∙ 6 ∙ 3

b. 18 ∙ (6 ∙ 3)

Solution a. We subtract in order from left to right. �� � � � � First subtract 6 from 18. ��

� � Then subtract 3 from 12. 9 The answer is 9.

b. We subtract within the parentheses first. 18 − (6 – 3) First subtract 3 from 6. 18 −

3

Then subtract 3 from 18. 15 The answer is 15.

When there is more than one multiplication or division step within a problem, we take the steps in order from left to right. In this problem we divide 24 by 6 and then multiply by 2. 24 ÷ 6 × 2 = 8 If there are parentheses, then we first do the work within the parentheses. In the problem below, we first multiply 6 by 2 and get 12. Then we divide 24 by 12. 24 ÷ (6 × 2) = 2

Example 2 a. 18 ∙ 6 ∙ 3

b. 18 ∙ (6 ∙ 3)

Solution a. We take the steps in order from left to right. 18 ÷ 6 ÷ 3 First divide 18 by 6. 3

÷ 3 1

Then divide 3 by 3. The answer is 1. Lesson 5

29

b. We divide within the parentheses first. 18 ÷ (6 ÷ 3)

First divide 6 by 3.

18 ÷

Then divide 18 by 2.

2 9

The answer is 9.

Only two numbers are involved in each step of a calculation. If three numbers are added (or multiplied), changing the two numbers selected for the first addition (or first multiplication) does not change the final sum (or product). (2 + 3) + 4 = 2 + (3 + 4)

(2 × 3) × 4 = 2 × (3 × 4)

This property applies to addition and multiplication and is called the Associative Property. As shown by examples 1 and 2, the Associative Property does not apply to subtraction or to division.

Example 3 57 12

5  7 12  4 12 3

Solution Before dividing we perform the operations above the bar and below the bar. Then we divide 12 by 3. 57 12

Practice Set

5  7 12  4 12 3 a. 16 − 3 + 4

b. 16 − (3 + 4)

c. 24 ÷ (4 × 3)

d. 24 ÷ 4 × 3

e. 24 ÷ 6 ÷ 2  96  9 6 g. 3 3 i.

Connect

Written Practice

f. 24 ÷ (6 ÷ 2) 8 8 h. 12 12 12 12 8 8

Rewrite exercise g using parentheses instead of a bar.

Strengthening Concepts

1. Jack paid $5 for a sandwich that cost $1.25 and milk that cost $0.60. How much change should he get back?

(1)

2. In one day the elephant ate 82 kilograms of hay, 8 kilograms of apples, (1) and 12 kilograms of leaves and raw vegetables. How many kilograms of food did it eat in all? 3. What is the difference of 110 and 25?

(1)

4. What is the total price of one dozen apples that cost 25¢ each?

(2)

5. What number must be added to 149 to total 516?

(3)

30

Saxon Math Course 1

* 6. (2)

Judy plans to read a 235-page book in 5 days. How can you find the average number of pages she needs to read each day. Explain

7. 5 + (3 × 4)

(5)

9. 800 − (450 − 125)

(5)

8. (5 + 3) × 4

(5)

10. 600 ÷ (20 ÷ 5) (5)

11. 800 − 450 − 125

12. 600 ÷ 20 ÷ 5

13. 144 ÷ (8 × 6)

14. 144 ÷ 8 × 6

(5)

(5)

(5)

(5)

15. $5 − ($1.25 + $0.60) (5)

* 16. (2)

Use the numbers 63, 7, and 9 to form two multiplication facts and two division facts. Represent

17. If the quotient is 12 and the dividend is 288, what is the divisor? (4)

Reading Math Read expressions such as (4)(6) as “four times six.” The parentheses indicate multiplication.

18. 25  $10.00

19. (378)(64)

20.

21.

(2)

(2)

*

506 × 370

Analyze

(2)

(1)

$10.10 − $9.89

Find each unknown number. Check your work.

22. n − 63 = 36

23. 63 − p = 36

24. 56 + m = 432

25. 8w = 480

(3)

(3)

(3)

(4)

26. 5 + 12 + 27 + y = 50 (3)

27. 36 ÷ a = 4 (4)

* 29. (1)

28. x ÷ 4 = 8 (4)

Use the numbers 7, 11, and 18 to form two addition facts and two subtraction facts. Represent

30. 3 ⋅ 4 ⋅ 5 (5)

Early Finishers

Real-World Application

A painter is painting three exam rooms at a veterinarian’s office. If each exam room requires 2 gallons of paint and the total cost of the paint is $270, how much does each gallon of paint cost?

Lesson 5

31

LESSON

6

Fractional Parts Building Power

Power Up facts

Power Up C

mental math

a. Number Sense: 2500 + 400 b. Number Sense: 6000 + 2400 c. Number Sense: 370 + 400 d. Number Sense: 9500 + 240 e. Number Sense: 360 + 1200 f. Number Sense: 480 + 2500 g. Measurement: How many seconds are in a minute? h. Measurement: How many minutes are in an hour?

problem solving

Carrisa’s school library received a gift of 500 new reference books. The books were arranged on a bookcase as shown in the diagram at right. How many books are on each shelf?

New Concept

270 books 230 books 180 books 130 books

Increasing Knowledge

As young children we learned to count objects using whole numbers. As we grew older, we discovered that there are parts of wholes—like sections of an orange—that cannot be named with whole numbers. We can name these parts with fractions. A common fraction is written with two numbers and a fraction bar. The “bottom” number is the denominator. The denominator shows the number of equal parts in the whole. The “top” number, the numerator, shows the number of the parts that are being represented.

Example 1 What fraction of this circle is shaded?

Solution The circle has been divided into 6 equal parts. We use 6 for the bottom of the fraction. One of the parts is shaded, so we use 1 for the top of the fraction. 1 1 The fraction of the circle that is 4 shaded is one sixth, which we write as 6.

32

Saxon Math Course 1

1 2 $0.90 5 � $4.50 We can also use a fraction to name a part of a group. There are 6 members in this group. We can divide this group in half by dividing it into two equal groups with 3 in each half. We write1 that 21 of 6 is 3. 2

1 3

1 2

Thinking Skill Explain 1 2

How would we find1 16 of 6?

of 6 is13. 2

1 13 2 1 3

1 2

Example 2 1 2 1 3

1 2 225

2 � 450 1 13 5

1 3

of 6 is 2.

1 2 150

a. What number is of 450? 1 2

1 3

1 3

We can divide this group into thirds by dividing the 6 members into three 1 equal groups. We write that 31 of 6 is 2. 5

1 2

2

1 12 3

1 3

1 5

1 3 1 2

1 2

1 3

b. What number is

225

1 2 450 3 �

3 � 450

of 450?

1 1 2 2

1 16 3 1 5

1 1 2 2

1 3 1 5

c. How much money is 15 of $4.50? 225 $0.90 225 2 450 � 5 � $4.50 2251 2 � 450 a. To find 21 of 450, we divide 450 into 2 � 225 4503 2 � 450 two equal parts and find the amount in one of the parts. We find that 21 of 450 is 225. 1 150 150 3 � 450 1 b. To find 13 of 450,1 we divide 450 into 2 150115 3 � 450 150 2 2 3 � 4503 three equal parts. Since each part is 3 � 450 1 150, we find that 3 of 450 is 150.1

1 3 1 5 1 3

Solution 225 2 � 450 150 3 � 450 1 2

1 3 1 2

1 3

150 $0.90 3 � 450 1 $4.50 5 � 2 1 2

1 3

$0.90 1 5 � 1 $4.50 2 2 1 2

225 2 � 450

1 3

1 2

1 12 225 3 2 � 450

225 2 450 � 150 1 3 � 450 2

1 2

150 3 � 450

150 3 450 � $0.90 5 � $4.50 12

c. To find of1$4.50, we divide $4.50 $0.90 1 1 $0.90 3 1 $4.50 5 by 5. We find that of $4.50 is � 1 1 $0.90 1 55� $4.50 3 2 5 3 225 $0.90. $0.90 2 5 � $4.50 1 2 � 450 5 � $4.50 1 3 3 225 1 Example 3 2 � 450 1 1 5 1 1 3 1 1 22 1 5 and shade 3 of it: 1 2 Copy the figure at right, 2 225 150 1 1 2 3 � 450 2 � 450 2 3 1 150 5 3 450 � 1 Solution 1 2 225 1 1 1 2 $0.90 150 The rectangle has six parts of equal size. Since 1 2 � 450 2 3 2 1 3 � 450 � $4.50any of 6 is 2, we 5shade two of 2 3 the parts. $0.90 5 � $4.50 1 1 2

150 3 � 450

$0.90 5 � $4.50 $0.90 5 � $4.50 1 3

1 2

1

2

1 5

1 2

1 2

1 1 2 1 23

$0.90 1 3 5 � $4.50

1 3

1 2251 3 2 � 4502

1 2

1 5 1

of 450 is 225. 1 3

1 1 3 2

1 5

1 5

1 3

1 3

225 1 1 2 � 450 5 5 1 5

1 3

of 450 is 150.

of $4.50 is $0.90.

1 3 1 150 5 3 � 450

1 5

1 5

1 3

1 3

$0.90 5 � $4.50

2

1 2

1 2 $0.90 5 � $4.50

1 2

1 3

1 3

1 2

1Lesson 6 1 3 2 1

1 3 33

1 3 1 Practice Set

1 3

1 3 3

1

3 Use both words and numbers to write the fraction that is shaded in problems a–c. 1 3

a.

1 3

b.

c.

1

1 3

d. What number is 2 of 72? 1 2

1 1 2 2 1 3

1

f. What number1is 1 2

1 3

e. What number is 2 of 1000?

g. h.

2

Explain

1 1 1 3 3 3 of

180?

1 3

1

1 3 1 1 6 6

1 6

How much money is 3 of $3.60?

Represent

1 6

Copy this figure and shade one

half of it.

Written Practice 1 2

Strengthening Concepts

(6)

1. What number is 21 of 540?

1 3

2. What number is 31 of 540?

1 6

(6)

3. In four days of sight-seeing the Richmonds drove 346 miles, 417 miles, 289 miles, and 360 miles. How many miles did they drive in all?

(1)

4. Tanisha paid $20 for a book that cost $12.08. How much money should she get back?

(1)

5. How many days are in 52 weeks?

(2)

* 6. (2)

Analyze

How many $20 bills would it take to make $1000?

7. Use words and numbers to write the fraction of this circle that is shaded.

(6)

8.

3604 5186 + 7145

10.

376 × 87

(1)

(2)

34

Saxon Math Course 1

9.

(1)

11. (2)

$30.01 − $15.76

470 × 203

12. $20 − $11.98

13. 596 − (400 − 129)

14. 32 ÷ (8 × 4)

15. 8  4016

15 

8 4016 8  6009 4016 15 16.  15 6009

15  6009 6009 36 9000 17. 3615  9000 4016 8  (2)

36 15

(1)

(5)

8  4016 4016

(2)

(5)

(2)

Find each unknown number. Check your work. 18. 8w = 480 (4)

49 N

49 M49 20.M = 7 N N (4) 7 7 22. 365 + P = 653 (3)

19.49x − 64 = 46 (3) N M M 21. = 15 7 7 (4) 49 N 23. 36¢ + 25¢ + m = 99¢

The square at right was divided in half. Then each half was divided in half. What fraction of the square is shaded?

* 25.

Copy this figure on your paper, and shade one fourth of it.

(6)

M 7

(3)

* 24. (6)

M 7

Conclude

Represent

26. $6.35 ∙ 12 (2)

27. Use the numbers 2, 4, and 6 to form two addition facts and two (1) subtraction facts. 28. Write two multiplication facts and two division facts using the numbers (2) 2, 4, and 8. * 29. (2)

* 30. (6)

Connect

Write a multiplication equation to solve this addition problem.

38 + 38 + 38 + 38 + 38 + 38 + 38 + 38 + 38 + 38 Make up a fractional-part question about money, as in Example 2 part c. Then find the answer. Formulate

Lesson 6

35

LESSON

7 Power Up facts mental math

Lines, Segments, and Rays Linear Measure Building Power Power Up C a. Number Sense: 800 − 300 b. Number Sense: 3000 − 2000 c. Number Sense: 450 − 100 d. Number Sense: 2500 − 300 e. Number Sense: 480 − 80 f. Number Sense: 750 − 250 g. Measurement: How many weeks are in a year? h. Measurement: How many days are in a year?

problem solving

A pulley is in equilibrium when the total weight suspended from the left side is equal to the total weight suspended from the right side. The two pulleys on the left are both in equilibrium. Is the pulley on the right in equilibrium, or is one side heavier than the other? We are shown three pulleys on which three kinds of weights are suspended. The first two pulleys are in equilibrium. We are asked to determine if the third pulley is in equilibrium or if one side is heavier than the other. Understand

We will use logical reasoning to determine whether the third pulley is in equilibrium. Plan

From the first pulley we see that four cylinders are equal in weight to five cubes. This means that cylinders are heavier than cubes. The second pulley shows that two cubes weigh the same as two cones. This means that cubes and cones weigh the same. Solve

On the third pulley, the bottom cubes on either side have the same weight. We are left with two cones and two cubes on one side and four cylinders on the other. We know that cylinders are heavier than cubes and cones, so the pulley is not in equilibrium. The right side is heavier, so the pulley will pull to the right. We can confirm our conclusion by looking at the third pulley as five cubes on the left (because the two cones are equal in weight to two cubes). From the first pulley, we know that five cubes are equal in weight to four cylinders. Another cube on the right side makes the right side heavier. Check

36

Saxon Math Course 1

Increasing Knowledge

New Concepts lines, segments, and rays Thinking Skill Conclude

If two oppositefacing rays are joined at their endpoints, what is the result? What do those endpoints become?

linear measure

In everyday language the following figure is often referred to as a line: However, using mathematical terminology, we say that the figure represents a segment, or line segment. A segment is part of a line and has two endpoints. A mathematical line has no endpoints. To represent a line, we use arrowheads to indicate a line’s unending quality. A ray has one endpoint. We represent a ray with one arrowhead. A ray is roughly represented by a beam of sunlight. The beam begins at the sun (which represents the endpoint of the ray) and continues across billions of light years of space. Line segments have length. In the United States we have two systems of units that we use to measure length. One system is the U.S. Customary System. Some of the units in this system are inches (in.), feet (ft), yards (yd), and miles (mi). The other system is the metric system (International System). Some of the units in the metric system are millimeters (mm), centimeters (cm), meters (m), and kilometers (km). Some Units of Length and Benchmarks U.S. Customary System

Metric System

inch (in.)

width of thumb

millimeter (mm)

thickness of a dime

foot (ft)

length of ruler, 12 inches

centimeter (cm)

thickness of little finger tip, 10 millimeters

yard (yd)

a long step, 3 ft or 36 inches

meter (m)

a little over a yard, 100 centimeters

miles (mi)

distance walked in 20 minutes, 5280 feet

kilometer (km)

distance walked in 12 minutes, 1000 meters

In this lesson we will practice measuring line segments with an inch ruler and with a centimeter ruler, and we will select appropriate units for measuring lengths.

Activity

Inch Ruler Materials needed: • inch ruler • narrow strip of tagboard about 6 inches long and 1 inch wide • pencil

Lesson 7

37

Use your pencil and ruler to draw inch marks on the strip of tagboard. Number the inch marks. When you are finished, the tagboard strip should look like this: Model

2

1

3

4

Now set aside your ruler. We will use estimation to make the rest of the marks on the tagboard strip. Estimate the halfway point between inch marks, and make the half-inch marks slightly shorter than the inch marks, as shown below. Estimate

1

2

3

4

Now show every quarter inch on your tagboard ruler. To do this, estimate the halfway point between each mark on the ruler, and make the quarter-inch marks slightly shorter than the half-inch marks, as shown below. 1

2

3

4

Save your tagboard ruler. We will be making more marks on it in a few days. A metric ruler is divided into centimeters. There are 100 centimeters in a meter. Each centimeter is divided into 10 millimeters. So 1 centimeter equals 10 millimeters, and 2 centimeters equals 20 millimeters. Connect

By comparing an inch ruler with a centimeter ruler, we see that an inch is 1 1 72 about 2 2 centimeters. 1

inch

cm

1

2

2

3

4

1

5

3

7

6

8

1

22 A cinnamon stick that is 3 inches long is about 7 2 cm long. A foot-long ruler is about 30 cm long.

Example 1 How long is the line segment? inch

1

2

Solution Reading Math 1 7The 2 abbreviation for inches (in.) ends with a period so it is not confused with the word in. 38

The line is one whole inch plus a fraction. The fraction is one fourth. So the 1 length of the line is 1 4 in.

Saxon Math Course 1

1

14

1

14

Example 2 How long is the line segment? cm

1

2

3

Solution We simply read the scale to see that the line is 2 cm long. The segment is also 20 mm long.

Example 3 Select the appropriate unit for measuring the length of a soccer field. A centimeters

B meters

C kilometers

Solution An appropriate unit can give us a good sense of the measure of an object. Describing a soccer field as thousands of centimeters or a small fraction of 1 2 a kilometer can be accurate without being 2appropriate. The best choice is B meters for measuring the length of a soccer field.

Practice Set

1

72

How long is each line segment? a. inch

1

2

b. mm 10

c.

20

30

40

Measure the following segment twice, once with an inch ruler and once with a centimeter ruler. Connect

Use the words line, segment, or ray to describe each of these figures: d. e. f. g. Which of these units is most appropriate for measuring the length of a pencil? A inches

B yards

C miles

h. Select the appropriate unit for measuring the distance between two towns. A centimeters

B meters

C kilometers

Lesson 7

39

Written Practice

Strengthening Concepts

1. To earn money for gifts, Debbie sold decorated pinecones. If she sold 100 pinecones at $0.25 each, how much money did she earn?

(2)

2. There are 365 days in a common year. April 1 is the 91st day. How many (1) days are left in the year after April 1? 3. The Cardaso family is planning to complete a 1890-mile trip in 3 days. If they drive 596 miles the first day and 612 miles the second day, how far must they travel the third day? (Hint: This is a two-step problem. First find how far they traveled the first two days.)

(5)

1 2

4. What number is 12 of 234?

(6)

1 2

1 3

5. How much money is 13 of $2.34?

(6)

6. Use words and digits to write the fraction of this circle that is shaded.

(6)

1 2

7.

(1)

6  5000 9.

(2)

1 3

8.

3654 2893 + 5614

1 3

(1)

60  3174 10.

28¢ × 74

(2)

906 × 47

11. 6  5000

6016 12. 800 ÷  3174

13. 60  3174

14. 3 + 6 + 5 + w + 4 = 30

15. 300 − 30 + 3

16. 300 − (30 + 3)

(2)

0

$41.01 6  5000 – $15.76

(2)

(2)

(3)

(5)

17. $4.32 ∙ 20 (2)

(5)

18. 24(48¢) (2)

19. $8.75 ÷ 25 (2)

Find each unknown number. Check your work. 20. W ÷ 6 = 7 (4)

* 23. (1)

21. 6n = 96 (4)

22. 58 + r = 213 (4)

Rearrange the numbers in this subtraction fact to form another subtraction fact and two addition facts. Connect

60 − 24 = 36 24. How long is the line segment below? (7)

inch

40

Saxon Math Course 1

1

2

60  317

25. Find the length, in centimeters and in millimeters, of the line segment (7) below. cm

* 26. (2)

* 27. (4)

1

2

3

4

Use the numbers 9, 10, and 90 to form two multiplication facts and two division facts. Connect

Explain

How can you find a missing dividend in a division problem?

28. w − 12 = 8 (3)

29. 12 − x = 8 (3)

30. a. A meterstick is 100 centimeters long. One hundred centimeters is (7) how many millimeters? b. The length of which of the following would most likely be measured in meters? A a pencil

Early Finishers

Real-World Application

B a hallway

C a highway

The district championship game will be played on an artificial surface. One-fifth of the team needs new shoes for the game. There are 40 players on the team, and each pair of shoes sells for $45. a. How many players need new shoes? b. How much money must the booster club raise to cover the entire cost of the shoes?

Lesson 7

41

LESSON

8

Perimeter Building Power

Power Up facts

Power Up A

mental math

a. Number Sense: 400 + 2400 b. Number Sense: 750 + 36 c. Number Sense: 8400 + 520 d. Number Sense: 980 − 60 e. Number Sense: 4400 − 2000 f. Number Sense: 480 − 120 g. Measurement: How many feet are in 2 yards? h. Measurement: How many centimeters are in 2 meters?

problem solving

The digits 2, 4, and 6 can be arranged to form six different three-digit numbers. Each ordering is called a permutation of the three digits. The smallest permutation of 2, 4, and 6 is 246. What are the other five permutations? List the six numbers in order from least to greatest. We have been asked to find five of the six permutations that exist for three digits, and then list the permutations from least to greatest. Understand

To make sure we find all permutations possible, we will make an organized list. Plan

We first write the permutations of 2, 4, and 6 that begin with 2, then those that begin with 4, then those that begin with 6: 246, 264, 426, 462, 624, 642. Solve

We found all six permutations of the digits 2, 4, and 6. Writing them in an organized way helped us ensure we did not overlook any permutations. Because we wrote the numbers from least to greatest as we went along, we did not have to re-order our list to solve the problem. Check

New Concept

Increasing Knowledge

The distance around a shape is its perimeter. The perimeter of a square is the distance around it. The perimeter of a room is the distance around the room.

Activity

Perimeter Walk the perimeter of your classroom. Start at a point along a wall of the classroom, and, staying close to the walls, walk around the room until you return to your starting point. Count your steps as you travel around the room. How many of your steps is the perimeter of the room? Model

42

Saxon Math Course 1

Discuss

a. Did everyone count the same number of steps? b. Does the perimeter depend upon who is measuring it? c. Which of these is the best real-world example of perimeter? 1. The tile or carpet that covers the floor. 2. The molding along the base of the wall. Here we show a rectangle that is 3 cm long and 2 cm wide. 3 cm

2 cm

If we were to start at one corner and trace the perimeter of the rectangle, our pencil would travel 3 cm, then 2 cm, then 3 cm, and then 2 cm to get all the way around. We add these lengths to find the perimeter of the rectangle. 3 cm + 2 cm + 3 cm + 2 cm = 10 cm

Example 1 What is the perimeter of this triangle? 30 mm A

20 mm

30 mm

Solution The perimeter of a shape is the distance around the shape. If we trace around the triangle from point A, the point of the pencil would travel 30 mm, then 20 mm, and then 30 mm. Adding these distances, we find that the perimeter is 80 mm.

Example 2 The perimeter of a square is 20 cm. What is the length of each side?

Solution The four sides of a square are equal in length. So we divide the perimeter by 4 to find the length of each side. We find that the length of each side is 5 cm. 5 cm

5 cm

5 cm

5 cm

Lesson 8

43

Practice Set Thinking Skill

What is the perimeter of each shape? a. square

b. rectangle

c. trapezoid

Verify

15 mm

Why can we find the perimeter of a regular polygon when we know the length of one side?

15 mm 12 mm

10 mm

10 mm 20 mm

20 mm

Figures d and e below are regular polygons because all of their sides are the same length and all of their angles are the same size. Find the perimeter of each shape. d. equilateral triangle

e. pentagon

1 cm

2 cm

f. g.

The perimeter of a square is 60 cm. How long is each side of the square? Conclude

Represent Draw two different figures that have perimeters that are the same length.

h. Select the appropriate unit for measuring the perimeter of a classroom. A inches

Written Practice

B feet

C miles

Strengthening Concepts

1. In an auditorium there are 25 rows of chairs with 18 chairs in each row. How many chairs are in the auditorium?

(2)

* 2. (1)

Explain The sixth-graders collected 765 cans of food for the food pantry last year. This year they collected 1750 cans. How many fewer cans did they collect last year than this year? Explain how you found the answer.

3. A basketball team is made up of 5 players. Suppose there are 140 players signed up for a tournament. How many teams will there be of 5 players per team?

(2)

4. What is the perimeter of this triangle?

(8)

20 mm

15 mm 25 mm

1 4 1 2

5. How much money is 21 of $6.54?

(6)

6. What number is 13 of 654?

(6)

4 � $9.00 44

1 3

Saxon Math Course 1 10 � 373

10 � 373 12 � 1500

12 � 1500

$9.00

� 800

1 4

1 2

* 7. (6)

4 � $9.00

1 4

1 3

1 2

Represent

What fraction of this rectangle is

10 � 373 11. 39 � 800 12 � 1500 10. 12 � 1500

9. 10 � 373

(2)

(2)

(2)

12. 400 ÷ 20 ÷ 4

39 � 800

(2)

13. 400 ÷ (20 ÷ 4)

(5)

(2)

1 4 � $9.00 3

shaded?

108.� 373 4 � $9.00

* 14.

1 2

(5)

Connect Use the numbers 240, 20, and 12 to form two multiplication facts and two division facts. 39 � 800

15. Rearrange the numbers in this addition fact to form another addition (1) fact and two subtraction facts. 60 + 80 = 140 16. The ceiling tiles used in many classrooms have sides that are 12 inches (8) long. What is the perimeter of a square tile with sides 12 inches long? 17. a. Find the sum of 6 and 4.

(1, 2)

b. Find the product of 6 and 4. 18. $5 − M = $1.48 (3)

19. 10 × 20 × 30 (5)

20. 825 ÷ 8 (2)

Find each unknown number. Check your work. 21. w − 63 = 36 (3)

22. 150 + 165 + a = 397 (3)

23. 12w = 120 (4)

24. If the divisor is 8 and the quotient is 24, what is the dividend? (4)

* 25. (7)

a. Measure the length of the line segment below to the nearest centimeter. Estimate

b. Measure the length of the segment in millimeters. * 26. (7)

Model

3

Use a ruler to draw a line segment that is 2 4 in. long.

27. w − 27 = 18 (3)

28. 27 − x = 18 (3)

29. Multiply to find the answer to this addition problem: (2)

* 30. (8)

35 + 35 + 35 + 35 Explain

How can you calculate the perimeter of a rectangle?

Lesson 8

45

LESSON

9

The Number Line: Ordering and Comparing Building Power

Power Up facts

Power Up C

mental math

a. Number Sense: 48 + 120 b. Number Sense: 76 + 10 + 3 c. Number Sense: 7400 + 320 d. Number Sense: 860 − 50 e. Number Sense: 960 − 600 f. Number Sense: 365 − 200 g. Geometry: A square has a length of 5 inches. What is the perimeter of the square? h. Measurement: How many days are in a leap year?

problem solving

As you sit at your desk facing forward, you can describe the locations of people and objects in your classroom compared to your position. Perhaps a friend is two seats in front and one row to the left. Perhaps the door is directly to your right about 6 feet. Describe the location of your teacher’s desk, the pencil sharpener, and a person or object of your choice.

New Concept

Increasing Knowledge

A number line is a way to show numbers in order.

Thinking Skill Connect

Name some reallife situations in which we would use negative numbers.

–2

–1

0

1

3

2

4

5

6

7

The arrowheads show that the line continues without end and that the numbers continue without end. The small marks crossing the horizontal line are called tick marks. Number lines may be labeled with various types of numbers. The numbers we say when we count (1, 2, 3, 4, and so on) are called counting numbers. All the counting numbers along with the number zero make up the whole numbers. To the left of zero on this number line are negative numbers, which will be described in later lessons. As we move to the right on this number line, the numbers are greater in value. As we move to the left, the numbers are lesser in value.

Example 1 Arrange these numbers in order from least to greatest: 121

46

Saxon Math Course 1

112

211

Solution On a number line, these three numbers appear in order from least (on the left) to greatest (on the right). 112 121

211

For our answer, we write 112 Thinking Skill Discuss

When using place value to compare two numbers, what do you need to do first?

121

211

When we compare two numbers, we decide whether the numbers are equal; if they are not equal, we determine which number is greater and which is lesser. We show a comparison with symbols. If the numbers are equal, the comparison symbol we use is the equal sign (=). 1+1=2 If the numbers are not equal, we use one of the greater than/less than symbols (> or <). When properly placed between two numbers, the small end of the symbol points to the lesser number.

Example 2 Compare: 5012

5102

Solution In place of the circle we should write =, >, or < to make the statement true. Since 5012 is less than 5102, we point the small end to the 5012. 5012  5102

Example 3 Compare: 16 ∙ 8 ∙ 2

16 ∙ (8 ∙ 2)

Solution Reading Math Remember to follow the order of operations. Do the work within the parentheses first. Then divide in order from left to right.

Before we compare the two expressions, we find the value of each expression. 16 ÷ 8 ÷ 2

16 ÷ (8 ÷ 2)

1

4

Since 1 is less than 4, the comparison symbol points to the left. 16 ∙ 8 ∙ 2  16 ∙ (8 ∙ 2)

Example 4 Use digits and symbols to write this comparison: One fourth is less than one half.

Lesson 9

47

Solution We write the numbers in the order stated. 11 11  6 6 44 22

Practice Set

11 44

6 6

11 22

a. Arrange these amounts of money in order from least to greatest. 12¢ b. Compare: 16 − 8 − 2 c. Compare: 8 ÷ 4 × 2 d. 2 × 3 f.

$12

16 − (8 − 2) 8 ÷ (4 × 2)

2+3

Represent

$1.20

e. 1 × 1 × 1

1+1+1

Use digits and symbols to write this comparison: One half is greater than one fourth.

g. Compare the lengths: 10 inches

Written Practice

1 foot 1 2

Strengthening Concepts

1. Tamara arranged 144 books into 8 equal stacks. How many books were in each stack?

(2)

2. (1)

Find how many years there were from 1492 to 1603 by subtracting 1492 from 1603. Generalize

* 3.

Martin is carrying groceries in from the car. If he can carry 2 bags at a time, how many trips will it take him to carry in 9 bags?

* 4.

Use a centimeter ruler to measure the length and width of the rectangle below. Then calculate the perimeter of the rectangle.

(2)

(7, 8)

Conclude

Conclude

5. How much 1money is 21 of $5.80?

(6)

1 2

1 4

1 4

2

6. How many cents is 1 of a dollar? 4

(6)

* 7. 6)

Use words and digits to name the fraction of this triangle that is shaded. Represent

8. Compare:

(7, 9)

a. 5012

5120

b. 1 mm

1 cm

9. Arrange these numbers in order from least to greatest:

(9)

1

1 4

1, 0, 2 10. Compare: 100 − 50 − 25 (9)

48

Saxon Math Course 1

100 − (50 − 25) 1 2

11. (1)

1 2

1 2 (2)

9 � 7227

12.

$4.20 × 60

14.

(1)

2

13.

9 � 7227

478 3692 + 45 1

(2)

15. 9 � 722725 � 7600

(2)

18. 7136 ÷ 100 (2)

1 2

78 × 36 25 �17. 7600 20 � 8014

16. 25 � 7600

(2)

$50.00 – $31.76

20 � 8014

(2)

9 � 7227 19. 736 ÷ 736

25 �

(2)

1 2

Find each unknown number. Check your work. d 15

d 15

20. 165 d + a = 300 (3) 15 22. 9c = 144 (4)

* 24. (7)

21. b − 68 = 86 (3)

23. d = 7 (4) 15

9 � 7227 Use an inch ruler to draw a line segment two inches long. Then use a centimeter ruler to find the length of the segment to the nearest centimeter. Estimate

25. Which of the figures below represents a ray? (7)

d 15

A B C

* 26. (9)

Represent

Use digits and symbols to write this comparison: One half is greater than one third.

* 27. (2)

Connect Arrange the numbers 9, 11, and 99 to form two multiplication facts and two division facts.

28. Compare: 25 + 0 (9)

25 × 0

29. 100 = 20 + 30 + 40 + x (3)

* 30. (9)

How did you choose the positions of the small and large ends of the greater than/less than symbols that you used in problem 8? Explain

Lesson 9

49

LESSON

10 Power Up facts mental math

Sequences Scales Building Power Power Up C a. Number Sense: 43 + 20 + 5 b. Number Sense: 670 + 200 c. Number Sense: 254 + 20 + 5 d. Number Sense: 100 − 50 e. Number Sense: 300 − 50 f. Number Sense: 3600 − 400 g. Measurement: How many feet are in 3 yards? h. Measurement: How many centimeters are in 3 meters?

problem solving

A parallax is an error that can result from an observer changing their position and not reading a measurement tool while directly in front of it. Hold a ruler near the width of your textbook on your desk. Then read the ruler from several angles as the diagram shows. How short can you make the width of the book appear? How tall can you make it appear? What is your best measure of the width of the book?

New Concepts sequences

Increasing Knowledge

A sequence is an ordered list of numbers, called terms, that follows a certain rule. Here are two different sequences: a. 5, 10, 15, 20, 25, . . . b. 5, 10, 20, 40, 80, . . . Sequence a is an addition sequence because the same number is added to each term of the sequence to get the next term. In this case, we add 5 to the value of a term to find the next term. Sequence b is a multiplication sequence because each term of the sequence is multiplied by the same number to get the next term. In b we find the value of a term by multiplying the preceding term by 2. When we are asked to find unknown numbers in a sequence, we inspect the numbers to discover the rule for the sequence. Then we use the rule to find other numbers in the sequence.

50

Saxon Math Course 1

Example 1 Generalize Describe the following sequence as an addition sequence or a multiplication sequence. State the rule of the sequence, and find the next term.

1, 3, 9, 27, _____ , . . .

Solution The sequence is a multiplication sequence because each term in the sequence can be multiplied by 3 to find the next term. Multiplying 27 by 3, we find that the term that follows 27 in the sequence is 81. Thinking Skill Generalize

The numbers . . ., 0, 2, 4, 6, 8, . . . form a special sequence called even numbers. We say even numbers when we “count by twos.” Notice that zero is an even number. Any whole number with a ones digit of 0, 2, 4, 6, or 8 is an even number. Whole numbers that are not even numbers are odd numbers. The odd numbers form the sequence . . ., 1, 3, 5, 7, 9, . . .. An even number of objects can be divided into two equal groups. An odd number of objects cannot be divided into two equal groups.

Is the sum of an even number and an odd number even or odd? Give some examples to support Example 2 your answer. Think of a whole number. Double that number. Is the answer even

or odd?

Solution The answer is even. Doubling any whole number—odd or even—results in an even number.

scales

Numerical information is often presented to us in the form of a scale. A scale is a display of numbers with an indicator to show the value of a certain measure. To read a scale, we first need to discover the value of the tick marks on the scale. Marks on a scale may show whole units or divisions of a unit, such as one fourth (as with the inch ruler from Lesson 7). We study the scale to find the value of the units before we try to read the indicated number. Two commonly used scales on thermometers are the Fahrenheit scale and the Celsius scale. A cool room may be 68°F (20°C). The temperature at which water freezes under standard conditions is 32 degrees Fahrenheit (abbreviated 32°F) and zero degrees Celsius (0°C). The boiling temperature of water is 212°F, which is 100°C. Normal body temperature is 98.6°F (37°C). The thermometer on the right shows three temperatures that are helpful to know: Boiling temperature of water

212°F

100°C

Normal body temperature

98.6°F

37°C

Freezing temperature of water

32°F

0°C

212°F

100°C

98.6°F

37°C

32°F

0°C

Lesson 10

51

Example 3 What temperature is shown on this thermometer?

20°F

Solution

10°F

As we study the scale on this Fahrenheit thermometer, we see that the tick marks divide the distance from 0°F to 10°F into five equal sections. So the number of degrees from one tick mark to the next must be 2°F. Since the fluid in the thermometer is two marks above 0°F, the temperature shown is 4∙F.

Practice Set

0°F

For the sequence in problems a and b, determine whether the sequence is an addition sequence or a multiplication sequence, state the rule of the sequence, and find the next three terms. Generalize

a. . . ., 18, 27, 36, 45,

,

,

b. 1, 2, 4, 8,

,

, ...

,

, ...

c. Think of a whole number. Double that number. Then add 1 to the answer. Is the final number even or odd? Thinking Skill

d.

Explain

Why is the thermometer shown with a broken scale?

This thermometer indicates a comfortable room temperature. Find the temperature indicated on this thermometer to the nearest degree Fahrenheit and to the nearest degree Celsius. Estimate

90°F

30°C

80°F 70°F

20°C

60°F 50°F

10°C

40°F

Written Practice * 1. (10)

Strengthening Concepts Generalize State the rule of the following sequence. Then find the next three terms. 16, 24, 32, , , , ...

2. Find how many years there were from the year the Pilgrims landed (1) in 1620 to the year the colonies declared their independence in 1776. * 3. (10)

52

Explain

Saxon Math Course 1

Is the number 1492 even or odd? How can you tell?

4. What weight is indicated on this scale?

(10)

16

pounds

1 4 1 4

* 5. (8)

0

0 15

If the perimeter of a square is 40 mm, how long is each side of the square? Conclude

6. How much money is 12 of1 $6.50?

(6)

2

7. Compare: 4 × 3 + 2

(9)

* 8. (6)

4 × (3 + 2)

Represent Use words and digits to write the fraction of this circle that is not shaded.

9. What is the

(1, 2)

a. product of 100 and 100? b. sum of 100 and 100?

10. (2)

4260 10

4260 10

14. (2)

11.

365 × 100 4260 10

(2)

4260 20

146 × 240 15. (2)

4260 10

907 × 36 4260 15

19. $10 − $0.75 20. $0.56 × 60 21. $6.20 ÷ 4 (1) (2) (2) d 4260 12 12 4  12 your work.4  124260 4  12 number. Find each unknown 4260 Check 4260 8 10 4 4 10 10 20 22. 56 + 28 + 37 + n = 200 4260 23. a − 67 = 49 20 (3)

4260 20

4260 24. 67 − b = 49 15 (3)

26. d = 24 d d (4) 8 8 8 27. Here are three ways to write “12 divided by 4.” (4)

d 8

(2)

4260 4260 16. 20 (2) 15

25. 8c = 120 d 8

13.

(3)

(3)

4260 10

78¢ × 48

18. $8 + w = $11.49

(1)

d 8

(2)

4260 20

17. 28,347 − 9,637 d 8

12.

426 20

4260 15

4 1

4  12

(2)

4  12

4  12

12 ÷ 4

12 4

12 4

Show three ways to write “20 divided by 5.” 28. What number is one third of 36? (6)

* 29. (1)

Arrange the numbers 346, 463, and 809 to form two addition equations and two subtraction equations. Connect

30. At what temperature on the Fahrenheit scale does water freeze? (10)

Lesson 10

53

INVESTIGATION 1

Focus on Frequency Tables Histograms Surveys frequency tables

Mr. Lawson made a frequency table to record student scores on a math test. He listed the intervals of scores (bins) he wanted to tally. Then, as he graded each test, he made a tally mark for each test in the corresponding row. Frequency Table Number Correct

Tally

Frequency

19–20

9

17–18

7

15–16

4

13–14

2

When Mr. Lawson finished grading the tests, he counted the number of tally marks in each row and then recorded the count in the frequency column. For example, the table shows that nine students scored either 19 or 20 on the test. A frequency table is a way of pairing selected data, in this case specified test scores, with the number of times the selected data occur. 1.

Can you tell from this frequency table how many students had 20 correct answers on the test? Why or why not?

2.

Conclude Mr. Lawson tallied the number of scores in each interval. How wide is each interval? Suggest a reason why Mr. Lawson arranged the scores in such intervals.

3. 4.

histograms

54

Justify

Represent

Show how to make a tally for 12.

Represent As a class activity, make a frequency table of the birth months of the students in the class. Make four bins by grouping the months: Jan.–Mar., Apr.–Jun., Jul.–Sep., Oct.–Dec.

Using the information in the frequency table, Mr. Lawson created a histogram to display the results of the test.

Saxon Math Course 1

Frequency

10 9 8 7 6 5 4 3 2 1

Scores on Math Test

13–14 15 –16 17–18 19–20 Number of Correct Answers

Bar graphs display numerical information with shaded rectangles (bars) of various lengths. Bar graphs are often used to show comparisons. A histogram is a special type of bar graph. This histogram displays the data (test scores) in equal-size intervals (ranges of scores). There are no spaces between the bars. The break in the horizontal scale ( ) indicates that the portion of the scale between 0 and 13 has been omitted. The height of each bar indicates the number of test scores in each interval. Refer to the histogram to answer problems 5–7. 5. Which interval had the lowest frequency of scores? 6. Which interval had the highest frequency of scores? 7. Which interval had exactly twice as many scores as the 13–14 interval? 8.

Make a frequency table and a histogram for the following set of test scores. (Use 50–59, 60–69, 70–79, 80–89, and 90–99 for the intervals.) Represent

63, 75, 58, 89, 92, 84, 95, 63, 78, 88, 96, 67, 59, 70, 83, 89, 76, 85, 94, 80

surveys

A survey is a way of collecting data about a population. Rather than collecting data from every member of a population, a survey might focus on only a small part of the population called a sample. From the sample, conclusions are formed about the entire population. Mrs. Patterson’s class conducted a survey of 100 students to determine what sport middle school students most enjoyed playing. Survey participants were given six different sports from which to choose. The surveyors displayed the results on the frequency table shown on the following page.

Investigation 1

55

Frequency Table Tally

Sport

Frequency

Basketball

16

Bowling

12

Football

15

Softball

26

Table Tennis

12

Volleyball

19

From the frequency table, Mrs. Patterson’s students constructed a bar graph to display the results. Percent of Students Surveyed

Favorite Participant Sports 30% 25% 20% 15% 10% 5% 0

Basketball Bowling

Football

Softball

Table Tennis

Volleyball

Since 16 out of 100 students selected basketball as their favorite sport to play, basketball was the choice of 16% (which means “16 out of 100”) of the students surveyed. Refer to the frequency table and bar graph for this survey to answer problems 9–12. 9. Which sport was the favorite sport of about 14 of the students surveyed? 10. Which sport was the favorite sport of the girls who were surveyed?

11.

A softball

B volleyball

C basketball

D cannot be determined from given information

Evaluate How might changing the sample group change the results of the survey?

12. How might changing the survey question—the choice of sports— change the results of the survey? This survey was a closed-option survey because the responses were limited to the six choices offered. An open-option survey does not limit the choices. An example of an open-option survey question is “What is your favorite sport?”

56

Saxon Math Course 1

a.

Make a histogram based on the frequency table created in problem 4. What intervals did you use? What questions can be answered by referring to the histogram?

b.

Conduct a survey of favorite foods of class members. If you choose to conduct a closed-option survey, determine which food choices will be offered. What will be the size of the sample? How will the data gathered by the survey be displayed?

c.

The table below uses negative integers to express the estimated greatest depth of each of the Great Lakes.

Represent

Formulate

Analyze

U.S. Great Lakes Est. Greatest Depth (in meters) Lake

Depth

Erie Huron

−65 m −230 m

Michigan

−280 m

Ontario Superior

−245 m −400 m

• Write the depths of the lakes in order from the deepest to shallowest. • Is a bar graph an appropriate way to represent the data in the table? • Are the data in the table displayed correctly on the bar graph below? Explain why or why not.

Estimated Depth (in m)

extensions

d.

50 0 −50 −100 −150 −200 −250 −300 −350 −400 −450 −500

U.S. Great Lakes

Erie

Huron

Michigan Lake

Ontario

Superior

Analyze Choose between mental math, paper and pencil, or estimation to answer each of the following questions. Explain your choice. Use the table in extension c as a reference.

• Lake Superior is how much deeper than Lake Erie? • Is the depth of Lake Michigan closer to the depth of Lake Ontario or Lake Huron? • How much deeper is Lake Michigan than Lake Huron?

Investigation 1

57

LESSON

11 Power Up facts mental math

Problems About Combining Problems About Separating Building Power Power Up D a. Number Sense: 3 × 40 b. Number Sense: 3 × 400 c. Number Sense: $4.50 + $1.25 d. Number Sense: 451 + 240 e. Number Sense: 4500 − 400 f. Number Sense: $5.00 − $1.50 g. Geometry: A rectangle has a length of 3 cm and a width of 2 cm. What is the perimeter of the rectangle? h. Calculation: Start with 10. Add 2; divide by 2; add 2; divide by 2; then subtract 2. What is the answer?

problem solving

Sitha began building stair-step structures with blocks. She used one block for a onestep structure, three blocks for a two-step structure, and six blocks for a three-step structure. She wrote the information in a table. Copy the table and complete it through a ten-step structure.

Number of Steps

Total Blocks Number Added to of Blocks Previous Structure

1

1

N/A

2

3

2

3

6

3

Increasing Knowledge

New Concepts

Like stories in your reading books, many of the stories we analyze in mathematics have plots. We can use the plot of a word problem to write an equation for the problem. Problems with the same plot are represented by the same equation. That is why we say there are patterns for certain plots.

problems about combining

58

Many word problems are about combining. Here is an example: Before he went to work, Pham had $24.50. He earned $12.50 more putting up a fence. Then Pham had $37.00. (Plot: Pham had some money and then he earned some more money.)

Saxon Math Course 1

Problems about combining have an addition pattern. Some + some more = total s+m=t There are three numbers in the pattern. In a word problem one of the numbers is unknown, as in the story below. Katya had 734 stamps in her collection. Then her uncle gave her some more stamps. Now Katya has 813 stamps. How many stamps did Katya’s uncle give her? This problem has a plot similar to the previous one. (Plot: Katya had some stamps and was given some more stamps.) In this problem, however, one of the numbers is unknown. Katya’s uncle gave her some stamps, but the problem does not say how many. We use the four-step process to solve the problem. Step 1: Understand addition pattern. Step 2:

Since this problem is about combining, it has an

Plan We use the pattern to set up an equation.

Pattern: Some + some more = total Equation: 734 stamps + m = 813 stamps Step 3: Solve The answer to the question is the unknown number in the equation. Since the unknown number is an addend, we subtract 734 from 813 to find the number. Find answer: 813 stamps − 734 stamps 79 stamps

Check answer: 813 stamps − 79 stamps 734 stamps

Step 4: Check We review the question and write the answer. Katya’s uncle gave her 79 stamps.

Example 1 Thinking Skill Connect

How is an odometer the same as a ruler? How is it different?

Jenny rode her bike on a trip with her bicycling club. After the first day Jenny’s trip odometer showed that she had traveled 86 miles. After the second day the trip odometer showed that she had traveled a total of 163 miles. How far did Jenny ride the second day?

Solution Step 1: Jenny rode some miles and then rode some more miles. The distances from the two days combine to give a total. Since this is a problem about combining, it has an addition pattern. Step 2: The trip odometer showed how far she traveled the first day and the total of the first two days. We record the information in the pattern. Some + Some more Total

86 miles + m miles 163 miles

Lesson 11

59

Step 3: We solve the equation by finding the unknown number. From Lesson 3 we know that we can find the missing addend by subtracting 86 miles from 163 miles. We check the answer. Find answer: 163 miles − 86 miles 77 miles

Check answer: 86 miles + 77 miles 163 miles

Step 4: We review the question and write the answer. Jenny rode 77 miles on the second day of the trip.

problems about separating

Another common plot in word problems is separating. There is a beginning amount, then some goes away, and some remains. Problems about separating have a subtraction pattern. Beginning amount − some went away = what remains b−a=r Here is an example: Waverly took $37.00 to the music store. She bought headphones for $26.17. Then Waverly had $10.83. (Plot: Waverly had some money, but some of her money went away when she spent it.) This is a problem about separating. Thus it has a subtraction pattern. Pattern: Beginning amount − some went away = what remains Equation: $37.00 − $26.17 = $10.83

Example 2 On Saturday 47 people volunteered to clean up the park. Some people chose to remove trash from the lake. The remaining 29 people left to clean up the hiking trails. How many people chose to remove trash from the lake?

Solution Step 1: There were 47 people. Then some went away. This problem has a subtraction pattern. Step 2: We use the pattern to write an equation for the given information. We show the equation written vertically. Beginning amount − Some went away What remains

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Saxon Math Course 1

47 people − p people 29 people

Step 3: We find the unknown number by subtracting 29 from 47. We check the answer. Check answer: 47 people − 18 people 29 people

Find answer: 47 people − 29 people 18 people

Step 4: We review the question and write the answer. There were 18 people who chose to remove trash from the lake.

Practice Set

Follow the four-step method to solve each problem. Along with each answer, include the equation you use to solve the problem. Formulate

a. When Tim finished page 129 of a 314-page book, how many pages did he still have to read? b. The football team scored 19 points in the first half of the game and 42 points by the end of the game. How many points did the team score in the second half of the game? c.

Write a word problem about combining. Solve the problem and write the answer. Formulate

Written Practice * 1. (11)

Strengthening Concepts Juan ran 8 laps and rested. Then he ran some more laps. If Juan ran 21 laps in all, how many laps did he run after he rested? Write an equation and solve the problem. Formulate

2. a. Find the product of 8 and 4.

(1, 2)

b. Find the sum of 8 and 4. 3. The expression below means “the product of 6 and 4 divided by the difference of 8 and 5.” What is the quotient?

(5)

(6 × 4) ÷ (8 − 5) * 4. Marcia went to the store with $20.00 and returned home with $7.75. (11) How much money did Marcia spend at the store? Write an equation and solve the problem. * 5. When Franklin got his Labrador Retriever puppy, it weighed 8 pounds. (11) A year later, it weighed 74 pounds. How much weight did Franklin’s dog gain in that year? Write the equation and solve the problem. 6. $0.65 + $0.40

(1)

Analyze

Find each unknown number. Check your work.

7. 87 + w = 155

(3)

9. y − 1000 = 386

(3)

* 11. Compare: 1000 − (100 − 10) (9)

8. 1000 − x = 386

(3)

10. 42 + 596 + m = 700 (3)

1000 − 100 − 10 Lesson 11

61

8  1000 12. 88 1000  1000 (2)

8  1000

13. 10  987 (2)

15. 600 × 300

10  987

(4)

16. 365w = 365

(2)

35 12  W

35 10 987 14. 12   10  987 W (4)

35next three numbers in the following sequence? * 17. Predict What are the 1 1 (10) 12  W 2, 4 35 2, 6, 10, , , 35 ⋅⋅⋅ W 12 12  W 18. 2 × 3 × 4 × 5 (5)

19. What number is (6)

11

1 2

1 2

of 360?

1 4

1

1 20.4 2What number is4 14 of 360? 2 (6)

21. What is the product of eight and one hundred twenty-five? (2)

* 22. (7)

Connect

How long is the line segment below?

inch

1

2

3

23. What fraction of the circle at right is not (6) shaded?

24 4

4  24

24. What is the24 perimeter of the square 4(6)  24shown? 4 4  24

9 mm

24 4

24 17 4  24 4 sum of the first five odd numbers greater * 25. What is the than zero? 30 2424 (10) 4 424  24 44 26. Here are three ways to write “24 divided by 4”: (2)

17 30

17 30

24 4  24 4  2424 ÷ 4 4

24 4

Show three ways to write “30 divided by 6.”

27. Seventeen of the 30 students in the class are girls. So 17 of the students 30 (6) in the class are girls. What fraction of the students in the class are boys? 1717

3030 * 28. At what temperature on the Celsius scale does water freeze? (10)

29. (2)

* 30. (11)

62

Use the numbers 17 17 24, 6, and 4 to write two multiplication 30 facts and two division facts.30 Represent

Formulate In the second paragraph of this lesson there is a problem with an addition pattern. Rewrite the problem by removing one of the numbers from the problem and asking a question instead.

Saxon Math Course 1

LESSON

12 Power Up facts mental math

Place Value Through Trillions Multistep Problems Building Power Power Up D a. Number Sense: 6 × 40 b. Number Sense: 6 × 400 c. Number Sense: $12.50 + $5.00 d. Number Sense: 451 + 24 e. Number Sense: 7500 − 5000 f. Number Sense: $10.00 − $2.50 g. Measurement: How many inches are in a yard? h. Calculation: Start with 12. Divide by 2; subtract 2; divide by 2; then subtract 2. What is the answer?

problem solving

When he was a boy, German mathematician Karl Friedrich Gauss (1777–1855) developed a method for quickly adding a sequence of numbers. Like Gauss, we can sometimes solve difficult problems by making the problem simpler. What is the sum of the first ten natural numbers? Understand

We are asked to find the sum of the first ten natural numbers.

We will begin by making the problem simpler. If the assignment had been to add the first four natural numbers, we could simply add 1 + 2 + 3 + 4. However, adding columns of numbers can be time consuming. We will try to find a pattern that will help add the natural numbers 1–10 more quickly. Plan

We can find pairs of addends in the sequence that have the same sum and multiply by the number of pairs. We try this pairing technique on the sequence given in the problem: Solve

11 11 11

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 11 × 5 = 55 11 11 We found the sum of the first ten natural numbers by pairing the addends and multiplying. We can verify our solution by adding the numbers one-by-one with pencil and paper or a calculator. Check

Lesson 12

63

New Concepts

In our number system the value of a digit depends upon its position. The value of each position is called its place value.

,

,

,

hundreds tens ones

hundred millions ten millions millions

hundred trillions ten trillions trillions

Math Language Our number system is based on a pattern of tens. In a place value chart, each place has a value ten times greater than the place to its right.

hundred thousands ten thousands thousands

Whole-Number Place Values

hundred billions ten billions billions

place value through trillions

Increasing Knowledge

,

Example 1 In the number 123,456,789,000, which digit is in the ten-millions place?

Solution Either by counting places from the right or looking at the chart, we find that the digit in the ten-millions place is 5.

Example 2 In the number 5,764,283, what is the place value of the digit 4?

Solution By counting places from the right or looking at the chart, we can see that the place value of 4 is thousands. Thinking Skill Connect

Make a list of real-life situations in which large number are used.

Large numbers are easy to read and write if we use commas to group the digits. To place commas, we begin at the right and move to the left, writing a comma after every three digits. Putting commas in 1234567890, we get 1,234,567,890. Commas help us read large numbers by marking the end of the trillions, billions, millions, and thousands. We read the three-digit number in front of each comma and then say “trillion,” “billion,” “million,” or “thousand” when we reach the comma.

,

,

“trillion”

“billion”

,

“million” “thousand”

Example 3 Use words to write the number 1024305.

64

Saxon Math Course 1

,

Solution First we insert commas. 1,024,305 We write one million, twenty-four thousand, three hundred five. Note: We write commas after the words trillion, billion, million, and thousand. We hyphenate compound numbers from 21 through 99. We do not say or write “and” when naming whole numbers.

Example 4 Use digits to write the number one trillion, two hundred fifty billion.

Solution When writing large numbers, it may help to sketch the pattern before writing the digits.

,

,

“trillion”

“billion”

,

,

“million” “thousand”

We write a 1 to the left of the trillions comma and 250 in the three places to the left of the billions comma. The remaining places are filled with zeros. 1,250,000,000,000

multistep problems

The operations of arithmetic are addition, subtraction, multiplication, and division. In this table we list the terms for the answers we get when we perform these operations: Sum Difference Product Quotient

the answer when we add the answer when we subtract the answer when we multiply the answer when we divide

We will use these terms in problems that have several steps.

Example 5 What is the difference between the product of 6 and 4 and the sum of 6 and 4?

Solution Math Language Sometimes it is helpful to rewrite a question and underline the phrases that indicate operations of arithmetic.

We see the words difference, product, and sum in this question. We first look for phrases such as “the product of 6 and 4.” We will rewrite the question, emphasizing these phrases. What is the difference between the product of 6 and 4 and the sum of 6 and 4?

Lesson 12

65

For each phrase we find one number. “The product of 6 and 4” is 24, and “the sum of 6 and 4” is 10. So we can replace the two phrases with the numbers 24 and 10 to get this question: What is the difference between 24 and 10? We find this answer by subtracting 10 from 24. The difference between 24 and 10 is 14.

Practice Set

a. Which digit is in the millions place in 123,456,789? b. What is the place value of the 1 in 12,453,000,000? c. Use words to write 21,350,608. d. Use digits to write four billion, five hundred twenty million. e.

When the product of 6 and 4 is divided by the difference of 6 and 4, what is the quotient? Analyze

Written Practice

Strengthening Concepts

* 1. What is the difference between the product of 1, 2, and 3 and the sum (12) of 1, 2, and 3? * 2. Earth is about ninety-three million miles from the Sun. Use digits to write (12) that distance. * 3.

Gilbert and Kadeeja cooked 342 pancakes for the pancake breakfast. If Gilbert cooked 167 pancakes, how many pancakes did Kadeeja cook? Write an equation and solve the problem.

* 4.

The two teams scored a total of 102 points in the basketball game. If the winning team scored 59 points, how many points did the losing team score? Write an equation and solve the problem.

(11)

(11)

Formulate

Formulate

* 5. What is the perimeter of the rectangle (8) at right?

10 SXN_M6_L012_61.qxd mm 18 mm SXN_M6_L012_61.qxd

6. 6m = 60

(4)

7. a. What number is 1 of 100?

1 4

2 1 4

(6)

1 b. What number is of 100? 2

8. Compare: 300 × 1

(9)

300 ÷ 1

9. (3 × 3) − (3 + 3)

(5)

* 10. (10)

Predict

What are the next three numbers in the following sequence? 1, 2, 4, 8,

11. 1 + m + 456 = 480 (3)

66

Saxon Math Course 1

,

,

,…

12. 1010 − n = 101 (3)

13. 1234 ÷ 10

14. 1234 ÷ 12

(2)

(2)

* 15. What is the sum of the first five even numbers greater than zero? (10)

16. (7)

Connect

How many millimeters long is the line segment

below? mm 10

20

30

40

* 17. In the number 123,456,789,000, which digit is in the ten-billions (12) place? * 18. In the number 5,764,283,000, what is the place value of the digit 4? (12)

* 19. Which digit is in the hundred-thousands place in the number (12) 987,654,321? 20. 1 × 10 × 100 × 1000 (5)

21. $3.75 × 3 (2)

22. 22y = 0

25 m  625

(4)

23. 100 + 200 + 300 + 400 + w = 2000 (3)

24. 24 × 26 (2)

25 25. m  625 (4)

3  27

27 3

26. If the divisor is 4 and the quotient is 8, what is the dividend? (4)

27. Show three ways to write “27 divided by 3.” 27 (2) 3  27 3 7 28. Explain Seven of the ten marbles in a bag are red. So 10 of the marbles (6) are red. What fraction of the marbles are not red? Explain why your answer is correct. * 29. Use digits to write four trillion. (12)

* 30. (12)

7 10

3 10

Formulate Using different numbers, make up a question similar to example 5 in this lesson. Then find the answer. 3 10

Lesson 12

67

LESSON

13

Problems About Comparing Elapsed-Time Problems

Power Up facts mental math

Building Power Power Up A a. Number Sense: 5 × 300 b. Number Sense: 5 × 3000 c. Number Sense: $7.50 + $1.75 d. Number Sense: 3600 + 230 e. Number Sense: 4500 − 500 f. Number Sense: $20.00 − $5.00 g. Measurement: How many millimeters are in a meter? h. Measurement: How many years are in a decade?

problem solving

A pair of number cubes is tossed. The total number of dots on the two top faces is 6. What is the total of the dots on the bottom faces of the pair of number cubes?

New Concepts

Increasing Knowledge

We practiced using patterns to solve word problems in Lesson 11. For problems about combining, we used an addition pattern. For problems about separating, we used a subtraction pattern. In this lesson we will look at two other kinds of math word problems.

problems about comparing

Some word problems are about comparing the size of two groups. They usually ask questions such as “How many more are in the first group” and “How many fewer are in the second group?” Comparison problems such as these have a subtraction pattern. We write the numbers in the equation in this order: Greater − lesser = difference In place of the words, we can use letters. We use the first letter of each word. g−l=d

Example 1 There were 324 girls and 289 boys in the school. How many fewer boys than girls were there in the school?

Solution Again we use the four-step process to solve the problem.

68

Saxon Math Course 1

Step 1: We are asked to compare the number of boys to the number of girls. The question asks “how many fewer?” This problem has a subtraction pattern. Step 2: We use the pattern to write an equation. There are more girls than boys, so the number of girls replaces “greater” and the number of boys replaces “lesser.” Pattern: Greater − lesser = difference Equation: 324 − 289 = d Step 3: We find the missing number by subtracting. 324 girls − 289 boys 35 fewer boys Step 4: We review the question and write the answer. There were 35 fewer boys than girls in the school. We can also state that there were 35 more girls than boys in the school. Explain

elapsed-time problems

How can we check the answer?

Elapsed time is the length of time between two events. We illustrate this on the ray below. elapsed time Time earlier date

later date

The time that has elapsed since the moment you were born until now is your age. Subtracting your birth date from today’s date gives your age.

Thinking Skill Verify

When you subtract your birth year from the current year, why don’t you always get your exact age?

Today’s date (later) − Your birth date (earlier) Your age (difference) Elapsed-time problems are like comparison problems. They have a subtraction pattern. Later − earlier = difference We use the first letter of each word to represent the word. l−e=d

Example 2 How many years were there from the year Columbus landed in America in 1492 to the year the Pilgrims landed in 1620?

Solution Step 1: This is an elapsed-time problem. It has a subtraction pattern. We use l, e, and d to stand for “later,” “earlier,” and “difference.” l−e=d

Lesson 13

69

Step 2: The later year is 1620. The earlier year is 1492. 1620 − 1492 = d Step 3: We find the missing number by subtracting. We can check the answer by adding. 1620 − 1492 128

1492 + 128 1620

Step 4: We review the question and write the answer. There were 128 years from 1492 to 1620.

Example 3 Abraham Lincoln was born in 1809 and died in 1865. How many years did he live?

Solution Step 1: This is an elapsed-time problem. It has a subtraction pattern. We use l for the later time, e for the earlier time, and d for the difference of the times. l−e=d Step 2: We write an equation using 1809 for the earlier year and 1865 for the later year. 1865 − 1809 = d Step 3: We find the missing number by subtracting. We may add his age to the year of his birth to check the answer. 1865 − 1809 56

1809 + 56 1865

We also note that 56 is a reasonable age, so our computation makes sense. Step 4: We review the question and write the answer. Abraham Lincoln lived 56 years.

Practice Set

Formulate Follow the four-step method to solve each problem. Along with each answer, include the equation you use to solve the problem.

a. The population of Castor is 26,290. The population of Weston is 18,962. How many more people live in Castor than live in Weston? b. Two important dates in British history are the Norman Conquest in 1066 and the signing of the Magna Carta in 1215, which limited the power of the king. How many years were there from 1066 to 1215?

70

Saxon Math Course 1

Written Practice

Strengthening Concepts

* 1. When the sum of 8 and 5 is subtracted from the product of 8 and 5, (12) what is the difference? * 2. The Moon is about two hundred fifty thousand miles from the Earth. Use (12) digits to write that distance. * 3. Use words to write 521,000,000,000. (12)

* 4. Use digits to write five million, two hundred thousand. (12)

5.

(2)

Robin entered a tennis tournament when she was three-score years old. How old was Robin when she entered the tournament? How do you know your answer is correct? (Recall that one score equals 20 years.) Explain

* 6. The auditorium at the Community Cultural Center has seats for (11) 1000 people. For a symphony concert at the center, 487 tickets have already been sold. How many more tickets are still available? Write an equation and solve the problem. * 7. (13)

It is 405 miles from Minneapolis, Minnesota to Chicago, Illinois. It is 692 miles from Minneapolis to Cincinnati, Ohio. Cincinnati is how many miles farther from Minneapolis than Chicago? Write an equation and solve the problem. Formulate

Use mental math to solve exercises 8 and 9. Describe the mental math strategy you used for each exercise. Justify

8. 99 + 100 + 101

9. 9 × 10 × 11

(1)

(5)

* 10. Which digit is in the thousands place in 54,321? (12)

* 11. What is the place value of the 1 in 1,234,567,890? (12)

12. The three sides of an equilateral triangle are (8) equal in length. What is the perimeter of the equilateral triangle shown? 18 mm

60,000 30

13. 5432 ÷ 100

14.

15. 1000 ÷ 7

16. $4.56 ÷ 3

(2)

(2)

(2)

(2)

17. Compare: 3 + 2 + 1 + 0 (9)

* 18. (10)

1 2

3×2×1×0

The rule for the sequence below is different from the rules for addition sequences or multiplication sequences. What is the next number in the sequence? Predict

1, 4, 3, 6, 5, 8,

, ⋅⋅⋅ b  12 4

12 c 4

Lesson 13

71

19.

60,000 What Generalize

30 20. 365 ÷ w = 365 (6)

1

is 2 of 5280?

60,000 30

1 2

b  12 4

12 c 4

(4)

21. (5 + 6 + 7) ÷ 3 (5)

22. Use a ruler to find the length in inches of the rectangle below. (7)

width

b  12 4 23. (8)

60,000 30

12 c 4

length

Write two ways to find the perimeter one way by 1 60,000 of a square: 2 adding and the other way by multiplying. 30 Explain

1 60,000 1 1 2 24. Multiply to find the answer to60,000 this addition problem: 2 2 (2) 3030 125 + 125 + 125 + 125 + 125 + 125

25. At what temperature on the Fahrenheit scale does water boil? (10)

26. Show three ways to write “21 divided by 7.” (2)

Find each unknown number. Check your work. b 12  12 27. 8a = 816 28. c 4 (4) (4) 4 12 b b30. 29. c  4 1212 d − 16 = 61  4 1212 (4) c c 4 4 4 (3) Analyze

b  12 4

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Saxon Math Course 1

LESSON

14

The Number Line: Negative Numbers Building Power

Power Up facts

Power Up D

mental math

a. Number Sense: 8 × 400 b. Number Sense: 6 × 3000 c. Number Sense: $7.50 + $7.50 d. Number Sense: 360 + 230 e. Number Sense: 1250 − 1000 f. Number Sense: $10.00 − $7.50 g. Measurement: How many years are in a century? h. Calculation: Start with 10. Add 2; divide by 3; multiply by 4; then subtract 5. What is the answer?

problem solving

It takes the local hardware store 8 seconds to cut through a piece of round galvanized steel pipe. How long will it take to cut a piece of pipe in half? Into quarters? Into six pieces? (Each cut must be perpendicular to the length of the pipe.)

New Concept

Increasing Knowledge

We have seen that a number line can be used to arrange numbers in order. –5

Reading Math Negative numbers are represented by writing a minus sign before a number: −5.

–4

–3

–2

–1

0

1

2

3

4

5

On the number line above, the points to the right of zero represent positive numbers. The points to the left of zero represent negative numbers. Zero is neither positive nor negative. Negative numbers are used in various ways. A temperature of five degrees below zero Fahrenheit may be written as −5°F. An elevation of 100 feet below sea level may be indicated as “elev. −100 ft.” The change in a stock’s price from $23.00 to $21.50 may be shown in a newspaper as −1.50.

Example 1 Arrange these numbers in order from least to greatest: 0, 1, ∙2

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73

Solution All negative numbers are less than zero. All positive numbers are greater than zero. ∙2, 0, 1

Example 2 Compare: ∙3 Visit www. SaxonPublishers. com/ActivitiesC1 for a graphing calculator activity.

∙4

Solution Negative three is three less than zero, and negative four is four less than zero. So ∙3  ∙4

Math Language Zero is neither positive nor negative. Zero has no opposite.

The number −5 is read “negative five.” Notice that the points on the number line marked 5 and −5 are the same distance from zero but are on opposite sides of zero. We say that 5 and −5 are opposites. Other opposite pairs include −2 and 2, −3 and 3, and −4 and 4. The tick marks show the location of numbers called integers. Integers include all of the counting numbers and their opposites, as well as the number zero. If you subtract a larger number from a smaller number (for example, 2 − 3), the answer will be a negative number. One way to find the answer to such questions is to use the number line. We start at 2 and count back (to the left) three integers. Maybe you can figure out a faster way to find the answer.

–5

–4

–3

–2

–1

0

1

2

3

4

5

2 − 3 = −1

Example 3 Subtract 5 from 2.

Solution Order matters in subtraction. Start at 2 and count to the left 5 integers. You should end up at ∙3. Try this problem with a calculator by entering . What number is displayed after the is pressed? We see that the calculator displays –3 as the solution.

Example 4 Arrange these four numbers in order from least to greatest: 1, ∙2, 0, ∙1

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Saxon Math Course 1

Solution A number line shows numbers in order. By arranging these numbers in the order they appear on a number line, we arrange them in order from least to greatest. ∙2, ∙1, 0, 1

Example 5 What number is 7 less than 3?

Solution The phrase “7 less than 3” means to start with 3 and subtract 7. 3−7 We count to the left 7 integers from 3. The answer is ∙4.

Practice Set

a. Compare: −8

−6

b. Use words to write this number: −8. c. What number is the opposite of 3? d. Arrange these numbers in order from least to greatest: 0, −1, 2, −3 e. What number is 5 less than 0? f. What number is 10 less than 5? g. 5 − 8 i.

Verify

h. 1 − 5 All five of the numbers below are integers. True or false? −3, 0, 2, −10, 50

j. The temperature was twelve degrees below zero Fahrenheit. Use a negative number to write the temperature. k. The desert floor was 186 feet below sea level. Use a negative number to indicate that elevation. l. The stock’s price dropped from $18.50 to $16.25. Use a negative number to express the change in the stock’s value.

Written Practice * 1. (12)

Strengthening Concepts Connect What is the quotient when the sum of 15 and 12 is divided by the difference of 15 and 12?

* 2. What is the place value of the 7 in 987,654,321,000? (12)

Lesson 14

75

* 3. Light travels at a speed of about one hundred eighty-six thousand miles (12) per second. Use digits to write that speed. * 4. (14)

What number is three integers to the left of 2 on the number

Connect

line? –5

* 5. (14)

–4

–3

–2

–1

0

1

2

3

4

5

Arrange these numbers in order from least to greatest:

Connect

5, −3, 1, 0, −2

* 6. What number is halfway between −4 and 0 on the number line? (14)

* 7. (11)

Formulate There are 140 sixth-grade students in the school. Seventytwo play on school sports teams. How many are not on school sports teams? Write an equation and solve the problem.

8. Compare: 1 + 2 + 3 + 4

(9)

1×2×3×4

9. What is the perimeter of this right triangle?

(8)

25 mm

15 mm

20 mm

10. (10)

Predict

What are the next two numbers in the following sequence? . . ., 16, 8, 4,

* 11. (13)

,

, ...

36,180 There are 365 days in a common year. How much less than 12 500 is 365? Write an equation and solve the problem. Formulate

* 12. What number is 8 less than 6? (14)

13. 1020 ÷ 100 (2)

14. (2)

36,180 12

15. 18  564 (2)

16. 1234 + 567 + 89

17. n − 310 = 186

18. 10 ∙ 11 ∙ 12

19. $3.05 − m = $2.98

(1)

(5)

(3)

(3)

18  564 20. Estimate About how long is this nail in centimeters? Use a centimeter (7) ruler to find its length to the nearest centimeter and to the nearest millimeter.

21. (100)(100)(100) (5)

22. What digit in 123,456,789 is in the ten-thousands place? (12)

23.

Verify If you know the length of an object in centimeters, how can you figure out the length of the object in millimeters without remeasuring?

24.

Use the numbers 19, 21, and 399 to write two multiplication facts and two division facts.

(7)

(2)

76

Connect

Saxon Math Course 1

2 2

25. Compare: 12÷ 6 × 2 (9)

12 ÷ (6 × 2)

26. Show three ways to write “60 divided by 6.” (2)

* 2 7. In January, 2005, the world’s population was about six billion, (12) four hundred million people. Use digits to write this number of people. 1

28. One third of the 12 eggs in the carton were cracked. How many2eggs (6) were cracked? * 29. What number is the opposite of 10? (14)

* 30. Arrange these numbers in order from least to greatest: (14)

Early Finishers

Real-World Application

1, 0, −1,

1 2

At dawn the temperature was 42ºF. By 5:00 p.m. the temperature had risen 33ºF to its highest value for the day. Between 5:00 p.m. and dusk the temperature fell 12ºF. a. What was the temperature at dusk? b. A cold front passes through during the night causing the temperature to drop 32ºF just before dawn. What is the temperature at that time?

Lesson 14

77

LESSON

15

Problems About Equal Groups Building Power

Power Up facts

Power Up C

mental math

a. Number Sense: 7 × 4000 b. Number Sense: 8 × 300 c. Number Sense: $12.50 + $12.50 d. Number Sense: 80 + 12 e. Number Sense: 6250 − 150 f. Number Sense: $20.00 − $2.50 g. Measurement: How many decades are in a century? h. Calculation: Start with a dozen. Subtract 3; divide by 3; subtract 3; then multiply by 3. What is the answer?

problem solving

Copy this subtraction problem and fill in the missing digits:

New Concept

4_7 − _9_ 21

Increasing Knowledge

We have studied several types of mathematical word problems. Problems about combining have an addition pattern. Problems about separating, comparing, and elapsed-time have subtraction patterns. Another type of mathematical problem is the equal groups problem. Here is an example: In the auditorium there were 15 rows of chairs with 20 chairs in each row. Altogether, there were 300 chairs in the auditorium. The chairs were arranged in 15 groups (rows) with 20 chairs in each group. Here is how we write the pattern: 15 rows × 20 chairs in each row = 300 chairs Number of groups × number in group = total n×g=t In a problem about equal groups, any one of the numbers might be unknown. We multiply to find the unknown total. We divide to find an unknown factor.

78

Saxon Math Course 1

Example At Russell Middle School there were 232 seventh-grade students in 8 classrooms. If there were the same number of students in each classroom, how many students would be in each seventh-grade classroom at Russell Middle School?

Solution Step 1: A number of students is divided into equal groups (classrooms). This is a problem about equal groups. The words in each often appear in “equal groups” problems. Step 2: We draw the pattern and record the numbers, writing a letter in place of the unknown number. Pattern

Equation

Number in each group × Number of groups Number in all groups

n in each classroom × 8 classrooms 232 in all classrooms

Step 3: We find the unknown factor by dividing. Then we check our work. 29 8  232

29 × 8 232

Step 4: We review the question and write the answer. If there were the same number of students in each classroom, there would be 29 students in each seventh-grade classroom at Russell Middle School.

Practice Set Reading Math Sometimes it is helpful to write dollars and cents in cents-only form.

Follow the four-step method to solve each problem. Along with each answer, include the equation you use to solve the problem. Formulate

a. Marcie collected $4.50 selling lemonade at 25¢ for each cup. How many cups of lemonade did Marcie sell? (Hint: Record $4.50 as 450¢.) b. In the store parking lot there were 18 parking spaces in each row, and there were 12 rows of parking spaces. Altogether, how many parking spaces were in the parking lot?

Written Practice

Strengthening Concepts

* 1.

The second paragraph of this lesson contains an “equal groups” situation. Write a word problem by removing one of the numbers in the problem and writing an “equal groups” question.

* 2.

On the Fahrenheit scale, water freezes at 32°F and boils at 212°F. How many degrees difference is there between the freezing and boiling points of water? Write an equation and solve the problem. Explain why your answer is reasonable.

(15)

(13)

Formulate

Explain

Lesson 15

79

* 3. (15)

There are about three hundred twenty little O’s of cereal in an ounce. About how many little O’s are there in a one-pound box? Write an equation and solve the problem (1 pound = 16 ounces). Formulate

* 4. There are 31 days in August. How many days are left in August after (11) August 3? Write an equation and solve the problem. * 5. Compare: 3 − 1 (14)

* 6. (14)

1−3

Subtract 5 from 2. Use words to write the answer.

Represent

* 7. The stock’s value dropped from $28.00 to $25.50. Use a negative (14) number to show the change in the stock’s value. * 8. (10)

Predict

What are the next three numbers in the following sequence? . . ., 6, 4, 2, 0,

,

,

, ...

* 9. What is the temperature reading on this thermometer? Write the answer twice, once with digits and an abbreviation and once with words.

(10, 14)

0

�5

�10

°F

10. $10 − 10¢ (1)

11. How much money is 12 of $3.50? (6)

12. (9)

Connect

To which hundred is 587 closest? 587 400

13. 9 + 87 + 654 + 3210 (1) 4320 4320 15. 9 9 (2) Analyze

21. 3 + n + 12 + 27 = 50

80

Saxon Math Course 1

600

700

14. 574 × 76 (2)

16. 36  493 36  493 (2)

Find each unknown number. Check your work.

17. 1200 ÷ w = 300 (4) 76 76  1 19. m  1 m (4) (3)

500

18. 63w = 63 (4)

20. w + $65 = $1000 (3)

22. There are 10 millimeters in 1 centimeter. How many millimeters long is (7) this paper clip?

cm

1

2

3

4

5

23. (8 + 9 + 16) ÷ 3 (5)

24. What is the place value of the 5 in 12,345,678? (12)

4320 25. Which digit occupies the ten-billions place in 123,456,789,000? 36  493 (12) 9 26. (1)

Use the numbers 19, 21, and 40 to write two addition facts and two subtraction facts. Represent

* 27. Arrange these numbers 76 in order from least to greatest: (14) m 1 0, −1, 2, −3 28. Of the seventeen students in Angela’s class, eight play in the school (6) band. What fraction of the total number of students in Angela’s class are in the band? * 29. (15)

Reggie sold buttons with a picture of his school’s mascot for 75¢ each. If Reggie sold seven buttons, how much money did he receive? Analyze

* 30. What number is neither positive nor negative? (14)

Early Finishers

Real-World Application

Franklin D. Roosevelt, the thirty-second president of the United States, was born on January 30, 1882 in Hyde Park, New York. His presidency began March 4, 1933 and ended April 12, 1945. a. How old was Franklin D. Roosevelt when he took office? b. How many years did he serve as president?

Lesson 15

81

LESSON

16

Rounding Whole Numbers Estimating

Power Up facts mental math

Building Power Power Up D a. Number Sense: 3 × 30 plus 3 × 2 b. Number Sense: 4 × 20 plus 4 × 3 c. Number Sense: 150 + 20 d. Number Sense: 75 + 9 e. Number Sense: 800 − 50 f. Number Sense: 8000 − 500 g. Measurement: How many yards are in 6 feet? h. Calculation: Start with 1. Add 2; multiply by 3; subtract 4; then divide by 5. What is the answer?

problem solving

Find the next four numbers in this sequence: 2, 3, 5, 8, 9, 11, 14, 15, . . .

Increasing Knowledge

New Concepts rounding whole numbers

When we round a whole number, we are finding another whole number, usually ending in zero, that is close to the number we are rounding. The number line can help us visualize rounding. 667 600

610

620

630

640

650

660

670

680

690

700

In order to round 667 to the nearest ten, we recognize that 667 is closer to 670 than it is to 660. In order to round 667 to the nearest hundred, we recognize that 667 is closer to 700 than to 600.

Example 1 Round 6789 to the nearest thousand. Thinking Skill Verify

Draw a number line to verify the answer is correct.

82

Solution The number we are rounding is between 6000 and 7000. It is closer to 7000.

Saxon Math Course 1

Example 2 Round 550 to the nearest hundred.

Solution The number we are to round is halfway between 500 and 600. When the number we are rounding is halfway between two round numbers, we round up. So 550 rounds to 600.

estimating

Rounding can help us estimate the answer to a problem. Estimating is a quick way to “get close” to the answer. It can also help us decide whether an answer is reasonable. In some situations an estimate is sufficient to solve a problem because an exact answer is not needed. To estimate, we round the numbers before we add, subtract, multiply, or divide.

Example 3 Estimate the sum of 467 and 312.

Solution Estimating is a skill we can learn to do in our head. First we round each number. Since both numbers are in the hundreds, we will round each number to the nearest hundred. 467 rounds to 500 312 rounds to 300 To estimate the sum, we add the rounded numbers. 500 + 300 800 We estimate the sum of 467 and 312 to be 800.

Example 4 Stephanie stopped at the store to pick up a few items she needs. She has a $10 bill and a couple of quarters. She needs to buy milk for $2.29, her favorite cereal for $4.78, and orange juice for $2.42. Does Stephanie have enough money to buy what she needs?

Solution An estimate is probably good enough to solve the problem. Milk and juice are less than $2.50 each, so they total less than $5. Cereal is less than $5.00, so all three items are less than $10. If tax is not charged on food, she has enough money. Even if tax is charged she probably has enough.

Lesson 16

83

Population (in thousands)

Math Example 5 Language According to this graph, about how many more people lived in Ashton in Words such 2000 than in 1980? as about and approximately Population of Ashton 1970–2000 10 indicate that an estimate, not an 8 exact answer, is 6 needed. 4 2 0

1970

1980

1990

2000

Solution We often need to use estimation skills when reading graphs. The numbers along the left side of the graph (the vertical axis) indicate the population in thousands. The bar for the year 2000 is about halfway between the 6000 and 8000 levels, so the population was about 7000. In 1980 the population was about 4000. This problem has a subtraction pattern. We subtract and find that about 3000 more people lived in Ashton in 2000 than in 1980.

Practice Set

Round each of these numbers to the nearest ten: a. 57

b. 63

c. 45

Round each of these numbers to the nearest hundred: d. 282

e. 350

f. 426

Round each of these numbers to the nearest thousand: g. 4387 Estimate

h. 7500

i. 6750

Use rounded numbers to estimate each answer.

j. 397 + 206 l. 29 × 31

k. 703 − 598 m. 29  591

Use the graph in example 5 to answer problems n and o. n.

Estimate

About how many fewer people lived in Ashton in 1980 than in

1990? o.

84

Predict The graph shows an upward trend in the population of Ashton. If the population grows the same amount from 2000 to 2010 as it did from 1990 to 2000, what would be a reasonable projection for the population in 2010?

Saxon Math Course 1

Written Practice

Strengthening Concepts

* 1. What is the difference between the product of 20 and 5 and the sum of (12) 20 and 5? * 2. Walter Raleigh began exploring the coastline of North America in 1584. (13) Lewis and Clark began exploring the interior of North America in 1803. How many years after Raleigh did Lewis and Clark begin exploring North America? Write an equation and solve the problem. * 3. (15)

Explain Jacob separated his 140 trading cards into 5 equal groups. He placed four of the groups into binders. The remaining cards he placed in a box. How many cards did he put in the box? Write an equation and solve the problem. Explain why your answer is reasonable.

4. Which digit in 159,342,876 is in the hundred-thousands place?

(12)

5. In the 2004 U.S. presidential election, 121,068,715 votes were tallied for president. Use words to write that number of votes.

(12)

6. What number is halfway between 5 and 11 on the number line?

(9)

* 7. Round 56,789 to the nearest thousand. (16)

* 8. Round 550 to the nearest hundred. (16)

* 9. Estimate the product of 295 and 406 by rounding each number to the (16) nearest hundred before multiplying. 10. 45 + 5643 + 287 (1)

7308 12 7308 12 14. (5 + 11) ÷ 2

12. (2)

11. 40,312 − 14,908 7308 (1) 100  5367 7308 12 13. 100  5367 12 100  5367 (2) 100  5367

(5)

15. How much money is 12 of $5? (6)

1

1

1 14 4

1 21

1 2 4 16. How much money is 4 of $5? 2 (6)

17. $0.25 × 10 (2)

19. Compare: 1 + (2 + 3) (9)

18. 325(324 − 323) (5)

(1 + 2) + 3

* 20.

Wind chill describes the effect of temperature and wind combining to make it feel colder outside. At 3 p.m. in Minneapolis, Minnesota, the wind chill was −10° Fahrenheit. At 11 p.m. the wind chill was −3° Fahrenheit. At which time did it feel colder outside, 3 p.m. or 11 p.m.? Explain how you arrived at your answer.

* 21.

Your heart beats about 72 times per minute. At that rate, how many times will it beat in one hour? (Write an equation and solve the problem.)

(14)

(15)

Explain

Formulate

Lesson 16

85

22. (8)

Estimate The distance between bases on a major league baseball diamond is 90 feet. A player who runs around the diamond runs about how many feet?

2nd base 3rd base

90

home

ft

1st base

Refer to the bar graph shown below to answer problems 23∙26.

Hay Eaten Daily (in kilograms)

100

Hay Eaten by Elephants

80 60 40 20 0

Baby

Mother

Father

* 23. How many more kilograms of hay does the father elephant eat each day (16) than the baby elephant? * 24. Altogether, how many kilograms of hay do the three elephants eat each (16) day? * 25. How many kilograms of hay would the mother elephant eat in one (16) week? * 26. (16)

Formulate

Using the information in this graph, write a comparison word

problem.

Analyze

Find each unknown number. Check your work.

27. 6w = 66 (4)

28. m − 60 = 37 (3)

29. 60 − n = 37 (3)

* 30. (Inv. 1)

86

Each day Chico, Fuji, and Rolo drink 6, 8, and 9 glasses of water respectively. Draw a bar graph to illustrate this information. Represent

Saxon Math Course 1

LESSON

17 Power Up facts mental math

The Number Line: Fractions and Mixed Numbers Building Power Power Up E a. Number Sense: 5 × 30 plus 5 × 4 b. Number Sense: 4 × 60 plus 4 × 4 c. Number Sense: 180 + 12 d. Calculation: 64 + 9 e. Number Sense: 3000 − 1000 − 100 f. Calculation: $10.00 − $7.50 g. Measurement: How many millimeters are in 2 meters? h. Calculation: Start with 5. Multiply by 4; add 1; divide by 3; then subtract 2. What is the answer?

problem solving

If an 8 in.-by-8 in. pan of lasagna serves four people, how many 12 in.-by12 in. pans of lasagna should be purchased to serve 56 people? (Hint: You may have “leftovers.”) An eight-inch-square pan of lasagna will serve four people. We are asked to find how many 12-inch-square pans of lasagna are needed to feed 56 people. Understand

Plan We will draw a diagram to help us visualize the problem. Then we will write an equation to find the number of 12-inch-square pans of lasagna needed.

First, we find the size of each serving by “cutting” the 8-inch-square pan into four pieces. Then we see how many pieces of the same size can be made from the 12-inch-square pan: Solve

8 in.

12 in.

4 in. 4 in.

4 in. 4 in. 4 in. 4 in.

4 in. 8 in. 4 in.

12 in.

4 in.

Each person eats one 4-inch-square piece.

4 in. One 12-inch-square pan of lasagna can be cut into nine 4-inch-square pieces.

One 12-inch-square pan of lasagna can serve 9 people. We use this information to write an equation: N × 9 = 56. We divide to find that N = 6 R2. Six pans of lasagna would only provide 54 pieces, so we must buy seven pans of lasagna in order to serve 56 people.

Lesson 17

87

Check We found that we need to buy seven 12-inch-square pans of lasagna to serve 56 people. It would take 14 8-inch-square pans of lasagna to feed 56 people, and one 12-inch-square lasagna feeds about twice as many people as one 8-inch-square pan, so our answer is reasonable.

Increasing Knowledge

New Concept Math Language Recall that integers are the set of counting numbers, their opposites, and zero. 1

12

On this number line the tick 1marks show 1 the location of the integers: 4

–3

4

–2

–1

0

1

2

3

There are points on the number line between the integers that can be named with fractions or mixed numbers. A mixed number is a whole number plus 1 1 1 a fraction. Halfway between 20 and 1 is 2. Halfway between 1 and 2 is 12. 1 Halfway between −1 and −2 is�−12. 1

1

1

–3 –2 2 –2 –1 2 –1 – 2

0

1 2

1

1

12

2

1

22

3

We count from zero.

The distance between consecutive integers on a number line may be divided into halves, thirds, fourths, fifths, or any other number of equal divisions. To determine which fraction or mixed number is represented by a point on the number line, we follow the steps described in the next example.

Example 1 Point A represents what mixed number on this number line? A

0

1

2

3

Solution We see that point A represents a number greater than 2 but less than 3. So point A represents a mixed number, which is a whole number plus a fraction. To find the fraction, we first notice that the segment from 2 to 3 has been divided into five smaller segments. The distance from 2 to point A crosses 3 three of the five segments. Thus, point A represents the mixed number 2 5. Why did we focus on the number of segments on the number line and not the number of vertical tick marks? Analyze

88

Saxon Math Course 1

1

12

Activity

Inch Ruler to Sixteenths Materials needed: • inch ruler made in Lesson 7 In Lesson 7 we made an inch ruler divided into fourths. In this activity we will divide the ruler into eighths and sixteenths. First we will review what we did in Lesson 7. We used a ruler to make one-inch divisions on a strip of tagboard. 1

2

3

4

Then we estimated the halfway point between inch marks and drew new marks. The new marks were half-inch divisions. Then we estimated the halfway point between the half-inch marks and made quarter-inch divisions. 1

2

3

4

We made the half-inch marks a little shorter than the inch marks and the quarter-inch marks a little shorter than the half-inch marks. Now divide your ruler into eighths of an inch by estimating the halfway point between the quarter-inch marks. Make these eighth-inch marks a little shorter than the quarter-inch marks. 1

2

3

4

Finally, divide your ruler into sixteenths by estimating the halfway point between the eighth-inch marks. Make these marks the shortest marks on the ruler. 1

2

3

4

Example 2 Estimate the length of this line segment in inches. Then use your ruler to find its length to the nearest sixteenth of an inch.

Solution The line segment is about 3 inches long. The ruler has been divided into sixteenths. We align the zero mark (or end of the ruler) with one end of the line segment. Then we find the mark on the ruler closest to the other end of the line segment and read this mark. As shown on the next page, we will enlarge a portion of a ruler to show how each mark is read.

Lesson 17

89

Thinking Skill Connect

1

inch

How is a number line like a ruler?

1 1 3 5 3 7 9 5 11 13 7 15 16 8 16 1 16 8 16 16 8 16 3 16 8 16 1 4 4 1 2 3 1 2

2

2 11

1

1

12

12

3

3 16

1 8

1 216

4

7

7

28

28 1 12

5 16

1 4

3 8

7 8

7 8

1 1

1 2

7 4 1Estimate We find that the line segment is about 2 8 inches long. This is the 2 nearest sixteenth because the end of the segment aligns more closely to the 14 13 7 15 1314 8 mark (which equals 16 16 mark or to the 16 mark. 1616 ) than it does to the

1

12

1

12

7 8

1

Practice Set

a. 2

5 16

1 4

3 8

51 6846 7 16

1 2

3 8 1 1 16 8 7 15 3 7 1 4 8 16 16 16 2

1 16 3 15 3 16 416 8

11 16 2

1

Continue this sequence to 12:

Generalize

1 1 , 1616

15 16

15 16

13513 16 16848 16

1 8

1 16

1 3 1 51 3 7 , , , 8, , , 8 16 4 16 8 16

7 8

3 16

1 8

13 16 1 3 ,… p 2 16

� 11215

16 3 8

5 16

1 4

1 4

b. What number is1halfway between −2 and −3? 7 c.

16 Represent

31 8 16 5 13 7 28 Connect 16d. 16

line?

1 315 1 3 7 16 16 16 84 8 16 3 16

2

What number is halfway between 2 and 5? Draw a number line to show and 3 1the number15that is halfway 5 3 between 2 3 7 7 5. 1

151 3 7 3 2164 8 16 16

16 4 3 1 7 1 Point 8 2 1616A 1 8 1 4

4 16 8 8 16 16 1 7 1 3 1 represents number on this41 2 16 what 2mixed 16 8

3 5 3 7 16 11 216 4 8 16 5 16

0

15 3 7 2 16 8 16

1

1 4 3 8

5 1 3 A7 16 2 8 16

7 16

1 2

16

2

number 7 16

3 8 1 2

1 2

1 1 3 1 5 3 7 1 , , , , , , , , p 2 8 16 4 16 8 16 2 16

e. f.

1 1 3 1 15 13 37 11 5g. 3 7 1 , , , , ,, ,, ,, ,, p, , , , p 16 8 16 4 16 16 88 16 16 42 16 8 16 2 1 1 1 3 1 5 3 7 1 , p , , , , , , , , p 2 Strengthening Written Practice 16 8 Concepts 16 4 16 8 16 2 1 1 3 1 5 3 7 1 , , , , , , , p 31 73 11 1 5 1 3 3 7 1 5 3 7 1.,18What 16 4is the 16 sum 8 16of2twelve thousand, five hundred and ten thousand, , , , , , , , , p , , , , p, , 16 , , p (12) 6 88 16 2 716 18 16 2 116 124 316 18 516 34 six hundred ten? , , , , , , , , p 16 8 16 4 16 8 16 2 Saxon Math Course 1

1 2

5 16

Use your ruler to find the length of each of these line segments to the nearest 1 1 3 1 5 3 7 1 sixteenth of an inch: 16, 8, 16, 4, 16, 8, 16, 2, p

90

15 16

3 8 7 16

1 2

2. In 1903 the Wright brothers made the first powered airplane flight. In 1969 Americans first landed on the moon. How many years was it from the first powered airplane flight to the first moon landing? (Write an equation and solve the problem.)

(13)

* 3. Linda can run about 6 yards in one second. About how far can she run (15) in 12 seconds? Write an equation and solve the problem. * 4. A coin collector has a collection of two dozen rare coins. If the value of (15) each coin is $1000, what is the value of the entire collection? Write an equation and solve the problem. * 5. (16)

Estimate Find the sum of 5280 and 1760 by rounding each number to the nearest thousand before adding.

480 6. 480 (2) 3 3 8.

(1)

7. 6 � 66 � 6 3 3

(5)

The letters a, b, and c represent three different numbers. The sum of a and b is c. Represent

a+b=c 1 8

3 16

1 5 3 7 1 , , , , , , p 6 4 16 8 16 2

1 4

480 Rearrange the letters to form another addition equation and two 5 3 7 1 3 16 8 2 subtraction equations. Hint: To16be sure you arranged the letters in the correct order, choose numbers for a, b, and c that make a + b = c true. Then try those numbers in place of the letters in your three equations. 52805280 9. Rewrite 2 ÷ 3 with a division bar, but do not divide. 44 44 (2) 10. (8)

* 11. (17)

A square has sides 10 cm long. Describe how to find its perimeter. Explain

Connect Use a ruler to find the length 5280 below to the 480of the line segment 44 nearest sixteenth of an inch. 3

Find each unknown number. Check your work. 12. $3 − y = $1.75

13. m − 20 = 30

14. 12n = 0

15. 16 + 14 = 14 + w

(3)

(3)

(4)

(3)

16. Compare: 19 × 21

20 × 20

(9)

18. 5280 (2) 44

17. 100 − (50 − 25) (5)

19. 365 + 4576 + 50,287 (1)

20. What number is missing in the following sequence? (10)

5, 10,

, 20, 25, …

21. Which digit in 987,654,321 is in the hundred-millions place? (12)

22. 250,000 ÷ 100 (2)

23. $3.75 × 10 (2)

Lesson 17

91

6� 3

* 24. (15)

Estimate An 8-ounce serving of 2% milk contains 26 grams of protein, 1 fat, and carbohydrates. About half the grams are carbohydrates. About 4 how many grams are not from carbohydrates?

25.

1 4 Analyze

What is the sum of the first six positive odd numbers?

26.

Explain

How can you find 4 of 52?

(10)

(6)

1 2

1

1 2

27. A quarter is 41 of a dollar. 1

1 4

(6)

1 4

1 2

2 a. How many quarters are in one dollar? 1

2 three dollars? b. How many quarters are in 1 4

* 28. On an inch ruler, which mark is halfway between the (17) the 12-inch mark?

1 1 2 2 1 -inch mark 4

and

* 29. Point A represents what mixed number on the number line below? 1 2A

(17)

1 2

1 2 1 2

3

4

5

6

* 30. A segment that is 12 of an inch long is how many sixteenths of an inch 1 (17) 2 long?

Early Finishers

Real-World Application

Read the problem below. Then decide whether you can use an estimate to answer the question or if you need to compute an exact answer. Explain how you find your answer. Manny has one gallon of paint. The label states that it covers about 350−450 square feet. Manny wants to paint the living room, dining room and family room of his home. The lateral surface area of all of the rooms totals about 800 square feet. How much more paint should Manny buy?

92

Saxon Math Course 1

LESSON

18 Power Up facts mental math

Average Line Graphs Building Power Power Up B a. Number Sense: 4 × 23 equals 4 × 20 plus 4 × 3. Find 4 × 23. b. Number Sense: 4 × 32 c. Number Sense: 3 × 42 d. Number Sense: 3 × 24 e. Geometry: A hexagon has sides that measure 15 ft. What is the perimeter of the hexagon? f. Measurement: How many days are in 2 weeks? g. Measurement: How many hours are in 2 days? h. Calculation: Start with a half dozen. Add 2; multiply by 3; divide by 4; then subtract 5.What is the answer?

problem solving

In his pocket Alex had seven coins totaling exactly one dollar. Name a possible combination of coins in his pocket. How many different combinations of coins are possible?

New Concepts average

Increasing Knowledge

Here we show three stacks of books; the stacks contain 8 books, 7 books, and 3 books respectively. Altogether there are 18 books, but the number of books in each stack is not equal.

If we move some of the books from the taller stacks to the shortest stack, we can make the three stacks the same height. Then there will be 6 books in each stack.

Lesson 18

93

By making the stacks equal, we have found the average number of books in the three stacks. Notice that the average number of books in each stack is greater than the number in the smallest stack and less than the number in the largest stack. One way we can find an average is by making equal groups.

Example 1 In four classrooms there were 28 students, 27 students, 26 students, and 31 students respectively. What was the average number of students per classroom?

Solution The average number of students per classroom is how many students there would be in each room if we made the numbers equal. So we will take the total number of students and make four equal groups. To find the total number of students, we add the numbers in each classroom. 28 students 27 students 26 students + 31 students 112 students in all We make four equal groups by dividing the total number of students by four. 28 students 4  112 students If the groups were equal, there would be 28 students in each classroom. The average number of students per classroom would be 28. Notice that an average problem is a “combining” problem and an “equal groups” problem. First we found the total number of students in all the classrooms (“combining” problem). Then we found the number of students that would be in each group if the groups were equal (“equal groups” problem).

Example 2

4  1132 students

Use counters to model the average of 3, 7, and 8.

Solution Thinking Skill Connect

What are some other reallife situations where we might need to find the average of a set of numbers?

This question does not tell us whether the numbers 3, 7, and 8 refer to books or students or coins or quiz scores. Still, we can find the average of these numbers by combining and then making equal groups. We can model the problem with counters. Since there are three numbers, there will be three groups of counters with 3, 7, and 8 counters in the groups. To find the average we make the groups equal. One way to do this is to first combine all the counters in the three groups. That gives us a total of 18 counters. 3 + 7 + 8 = 18 Then we divide the total into three equal groups. 18 ÷ 3 = 6

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Saxon Math Course 1

That gives us three groups of six counters. We find that the average of 3, 7, and 8 is 6.

Example 3 What number is halfway between 27 and 81?

Solution The number halfway between two numbers is also the average of the two numbers. For example, the average of 7 and 9 is 8, and 8 is halfway between 7 and 9. So the average of 27 and 81 will be the number halfway between 27 and 81. We add 27 and 81 and divide by 2. Average of 27 and 81 = 27  81 2

=

27  81 2 108 2

= 54 The number halfway between 27 and 81 is 54. The average we have talked about in this lesson is also called the mean. We will learn more about average and mean in later lessons.

line graphs

Line graphs display numerical information as points connected by line segments. Whereas bar graphs often display comparisons, line graphs often show how a measurement changes over time.

Example 4 This line graph shows Margie’s height in inches from her eighth birthday to her fourteenth birthday. During which year did Margie grow the most? Margie’s Height

68 Height (in inches)

Reading Math On this line graph, the horizontal axis is divided into equal segments that represent years. The vertical axis is divided into equal segments that represent inches. The labels on the axes tell us how many years and how many inches.

64 60 56 52 48 44

8

9

10 11 12 13 14 Age (in years)

Solution From Margie’s eighth birthday to her ninth birthday, she grew about two inches. She also grew about two inches from her ninth to her tenth birthday. From her tenth to her eleventh birthday, Margie grew about five inches. Notice that this is the steepest part of the growth line. So the year Margie grew the most was the year she was ten.

Lesson 18

95

Your teacher might ask you to keep a line graph of your math test scores. The Lesson Activity 2 Test Scores Line Graph can be used for this purpose.

Practice Set

a. There were 26 books on the first shelf, 36 books on the second shelf, and 43 books on the third shelf. Velma rearranged the books so that there were the same number of books on each shelf. After Velma rearranged the books, how many were on the first shelf ? b. What is the average of 96, 44, 68, and 100? c. What number is halfway between 28 and 82? d. What number is halfway between 86 and 102? e. Find the average of 3, 6, 9, 12, and 15. Use the information in the graph in example 4 to answer these questions: f. How many inches did Margie grow from her eighth to her twelfth birthday? g. During which year did Margie grow the least? h.

Based on the information in the graph, would you predict that Margie will grow to be 68 inches tall? Predict

Written Practice

Strengthening Concepts

1. Jumbo ate two thousand, sixty-eight peanuts in the morning and three thousand, nine hundred forty in the afternoon. How many peanuts did Jumbo eat in all? What kind of pattern did you use?

(11)

2. Jimmy counted his permanent teeth. He had 11 on the top and 12 on the bottom. An adult has 32 permanent teeth. How many more of Jimmy’s teeth need to grow in? What kind of pattern did you use?

(11)

* 3. Olivia bought one dozen colored pencils for an art project. Each pencil (15) cost 53¢ each. How much did Olivia spend on pencils? What kind of pattern did you use? * 4. (16)

Estimate Find the difference of 5035 and 1987 by rounding each number to the nearest thousand before subtracting.

* 5. Find the average of 9, 7, and 8. (18)

* 6. What number is halfway between 59 and 81? (18)

* 7. What number is 6 less than 2? (14)

8. $0.35 × 100

(2)

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Saxon Math Course 1

9. 10,010 ÷ 10

(2)

10. 34,180 ÷ 17 (2)

11. $3.64 + $94.28 + 87¢

12. 41,375 − 13,576

13. 125 × 16

14. 4 ∙ 3 ∙ 2 ∙ 1 ∙ 0

(1)

(1)

(2)

(5)

Find each unknown number. Check your work. 15. w − 84 = 48 16. 234 n =6 (3) (4) Analyze

1 2

17. (1 + 2) × 3 = (1 × 2) + m

(3, 5)

18. (8)

Draw a rectangle 5 cm long and 3 cm wide. What is its perimeter? Model

19. What is the sum of the first six positive even numbers? (10)

20. (10)

Describe the rule of the following sequence. Then find the missing term. Generalize

1, 2, 4, 500 ÷ 1

21. Compare: 500 × 1 (9)

22. (6)

Generalize

1

What number is 2 of 1110?

23. What is the place value of the 7 in 987,654,321? (12)

Refer to the line graph shown below to answer problems 24–26. Heart Rates During Various Activities 200 Heartbeats per Minute

234 n

, 16, 32, 64, …

140

80

20 Resting

Walking

Running

* 24.

Running increases a resting person’s heart rate by about how many heartbeats per minute?

* 25.

Estimate About how many times would a person’s heart beat during a 10-minute run?

* 26.

Using the information in the line graph, write a word problem about comparing. Then answer the problem.

* 27.

Explain In three classrooms there are 24, 27, and 33 students respectively. How many students will be in each classroom if some students are moved from one classroom to the other classrooms so that the number of students in each classroom is equal? How do you know that your answer is reasonable?

(18)

(18)

(18)

(18)

Estimate

Formulate

Lesson 18

97

1 28. A dime is 10 of a dollar. (6)

1

24

a. How many dimes are in a dollar? b. How many dimes are in three dollars?

1 10

1 Model10

* 30.

Formulate Word problems about finding an average include which two types of problems? (Select two)

combining

Real-World Application

3

Use a ruler to draw 2 4 a rectangle that is 2 4 inches long and 1 4 inches wide.

(17)

(18)

Early Finishers

1

1

* 29.

separating

comparing

equal groups

The cost of one cubic foot of natural gas in my town is $0.67. My meter reading on July 31 was 1518. On August 31 it was 1603. The meter measures natural gas in cubic feet. a. How many cubic feet of gas did I use in the month of August (between the two meter readings)? b. What was my total gas cost for the month?

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Saxon Math Course 1

3

14

6 6

LESSON

19 Power Up facts mental math

Factors Prime Numbers Building Power Power Up D a. Number Sense: 3 × 64 b. Number Sense: 3 × 46 c. Number Sense: 120 + 18 d. Calculation: 34 + 40 + 9 e. Calculation: 34 + 50 − 1 f. Calculation: $20.00 − $12.50 g. Measurement: How many decades are in 2 centuries? h. Calculation: Start with 100. Divide by 2; subtract 1; divide by 7; then add 3. What is the answer?

problem solving

Doubles tennis is played on a rectangularshaped court that is 78 feet long and 36 feet wide. In the drawing of the doubles tennis court, how many rectangles can you find?

New Concepts factors

1 4 Skill

Thinking How are the two definitions of factor connected?

Recall from Lesson 2 that a factor is one of the numbers multiplied to form a product. 2×3=6

Both 2 and 3 are factors.

1×6=6

Both 1 and 6 are factors.

1 4 numbers

We see that each of the 1, 2, 3, and 6 are factors of 6. Notice that when we divide 6 by 1, 2, 3, or 6, the resulting quotient has no remainder (that is, it has a remainder of zero). We say that 6 is “divisible by” 1, 2, 3, and 6.

6 3 3 1  6 2  26 6

6 0

Increasing Knowledge

6 1 6 6 0

3 2 6 6 0

2 2 3  36 6 6 0

1 1 6 6 6 6 0

3 2 6

2 3 6

This leads us to another definition of factor. 6 0

The factors of a given number are the whole numbers that divide the given number without a remainder.

Lesson 19

99

6

We can illustrate the factors of 6 by arranging 6 tiles to form rectangles. With 6 tiles we can make a 1-by-6 rectangle. We can also make a 2-by-3 rectangle. 1

2 6

3

The number of tiles along the sides of these two rectangles (1, 6, 2, 3) are the four factors of 6.

Example 1 What are the factors of 10?

Solution The factors of 10 are all the numbers that divide 10 evenly (with no remainder). They are 1, 2, 5, and 10. 10 1  10

5 2  10

10 1  10

5 2  10

10 2 1 510  10

5 2  10

2 5  10

1 5 10  102  10

2 1 5 10 10 10

We can illustrate the factors of 10 with two rectangular arrays of tiles. 1

2 10

5

The number of tiles along the sides of the two rectangles (1, 10, 2, 5) are the factors of 10.

Example 2 How many different whole numbers are factors of 12?

Solution Twelve can be divided evenly by 1, 2, 3, 4, 6, and 12. The question asked “How many?” Counting factors, we find that 12 has 6 different whole-number factors. Twelve tiles can be arranged to form three different shapes of rectangles. The lengths of the sides illustrate that the six factors of 12 are 1, 12, 2, 6, 3, and 4. 1 12

2

3 6

4

Draw tiles to illustrate the factors of 18. How many different shapes of rectangles can you make? Model

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Saxon Math Course 1

2 5  10 

prime numbers

Here we list the first ten counting numbers and their factors. Which of the numbers have exactly two factors? Number

Factors

1

1

2

1, 2

3

1, 3

4

1, 2, 4

5

1, 5

6

1, 2, 3, 6

7

1, 7

8

1, 2, 4, 8

9

1, 3, 9

10

1, 2, 5, 10

Counting numbers that have exactly two factors are prime numbers. The first four prime numbers are 2, 3, 5, and 7. The only factors of a prime number are the number itself and 1. The number 1 is not a prime number, because it has only one factor, itself. Therefore, to determine whether a number is prime, we may ask ourselves the question, “Is this number divisible by any number other than the number itself and 1?” If the number is divisible by any other number, the number is not prime.

Example 3 The first four prime numbers are 2, 3, 5, and 7. What are the next four prime numbers?

Solution We will consider the next several numbers and eliminate those that are not prime. 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 All even numbers have 2 as a factor. So no even numbers greater than two are prime numbers. We can eliminate the even numbers from the list. 8 , 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 Since 9 is divisible by 3, and 15 is divisible by 3 and by 5, we can eliminate 9 and 15 from the list. 8 , 9 , 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 Each of the remaining four numbers on the list is divisible only by itself and by 1. Thus the next four prime numbers after 7 are 11, 13, 17, and 19.

Lesson 19

101

Activity

Prime Numbers List the counting numbers from 1 to 100 (or use Lesson Activity 3 Hundred Number Chart). Then follow these directions: Step 1: Draw a line through the number 1. The number 1 is not a prime number. Step 2: Circle the prime number 2. Draw a line through all the other multiples of 2 (4, 6, 8, etc.). Step 3: Circle the prime number 3. Draw a line through all the other multiples of 3 (6, 9, 12, etc.). Thinking Skill Conclude

Why don’t the directions include “Circle the number 4” and “Circle the number 6”?

Practice Set

Step 4: Circle the prime number 5. Draw a line through all the other multiples of 5 (10, 15, 20, etc.). Step 5: Circle the prime number 7. Draw a line through all the other multiples of 7 (14, 21, 28, etc.). Step 6: Circle all remaining numbers on your list (the numbers that do not have a line drawn through them). When you have finished, all the prime numbers from 1 to 100 will be circled on your list. List the factors of the following numbers: a. 14

b. 15

c. 16

d. 17

Justify Which number in each group is a prime number? Explain how you found your answer.

e. 21, 23, 25 f. 31, 32, 33 g. 43, 44, 45 Classify

Which number in each group is not a prime number?

h. 41, 42, 43 i. 31, 41, 51 j. 23, 33, 43 Prime numbers can be multiplied to make whole numbers that are not prime. For example, 2 ∙ 2 ∙ 3 equals 12 and 3 ∙ 5 equals 15. (Neither 12 nor 15 are prime.) Show which prime numbers we multiply to make these products: k. 16

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Saxon Math Course 1

l. 18

Written Practice Math Language The dividend is the number that is to be divided.

Strengthening Concepts

1. If two hundred fifty-two is the dividend and six is the quotient, what is the divisor?

(4)

* 2. In 1863, President Abraham Lincoln gave his Gettysburg Address, which began “Fourscore and seven years ago … .” A score equals twenty. What year was Lincoln referring to in his speech? Explain how you found your answer.

(11, 15)

* 3. The temperature in Barrow, Alaska was –46°F on January 22, 2002. (14) It was 69°F on July 15, 2002. How many degrees warmer was it on July 15 than on January 22? * 4. If 203 turnips are to be shared equally among seven rabbits, how many (15) should each receive? Write an equation and solve the problem. * 5. What is the average of 1, 2, 4, and 9? (18)

6.

(10)

Predict

What is the next number in the following sequence? 1, 4, 9, 16, 25,

, ...

7. A regular hexagon has six sides of equal length. If each side of a hexagon is 25 mm, what is the perimeter?

(8)

8. One centimeter equals ten millimeters. How many millimeters long is the line segment below?

(7)

cm

1

2

3

4

5

* 9. What are the whole-number factors of 20? (19)

* 10. How many different whole numbers are factors of 15? (19)

* 11. (19)

Classify

Which of the numbers below is a prime number?

A 25

12. 250,000 ÷ 100 (12)

* 14. 6  18  9 (15) 3

B 27

C 29 13. 1234 ÷ 60 (2)

15. $3.45 × 10 (2)

Find each unknown number. Check your work. 16. $10.00 − w = $1.93 17. w  4 (3) (4) 3 18. (2)

The letters a, b, and c represent three different numbers. The product of a and b is c. Represent

ab = c Rearrange the letters to form another multiplication equation and two division equations.

1 2

Lesson 19

103

w 4 3 * 19. Arrange these numbers in order from least to greatest: (17)

3, −2, 1, 21, 0

20. Compare: 123 ÷ 1 (9)

123 − 1

21. Which digit in 135,792,468,000 is in the ten-millions place? (12)

* 22. (16)

Connect

3

1 2

Round 123,456,789 to the 1nearest million. 8

23. How much money is 21 of $11.00? (6)

3

24. If a square has a perimeter of 48 inches, how long is each side 1 of8 the (8) square? w 4 25. (51 + 49) × (51 − 49) 3 3 1 (5)

* 26. (19)

8

Classify

A 2

Which of the numbers below is a prime number? B 22

C 222

* 27. Prime numbers can be multiplied to make whole numbers that are not (19) prime. To make 18, we perform the multiplication 2 ∙ 3 ∙ 3. Show which prime numbers we multiply to make 20. * 28. The dictionaries are placed in three stacks. There are 6 dictionaries (18) 1 in one stack and 12 dictionaries in each of 2the other two stacks. How many dictionaries will be in each stack if some dictionaries are moved from the taller stacks to the shortest stack so that there are the same number of dictionaries in each stack? * 29. (17)

* 30.

(10, 19)

Early Finishers

Real-World Application

Model

3

Draw a square with sides that are 1 8 inches long.

Explain How can you use the concepts of “even” and “odd” numbers to determine whether a number is divisible by 2.

The school newspaper reported that middle school students own an average of 47 CDs. Yolanda asked seven of her friends how many CDs they owned and got these results. 32, 49, 21, 59, 37, 44, 52 a. What is the average number of CDs owned by Yolanda’s seven friends? b. How does this average compare to the average reported in the school newspaper?

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Saxon Math Course 1

LESSON

20

Greatest Common Factor (GCF) Building Power

Power Up facts

Power Up C

mental math

a. Number Sense: 6 × 23 b. Number Sense: 6 × 32 c. Number Sense: 640 + 1200 d. Calculation: 63 + 20 + 9 e. Calculation: 63 + 30 − 1 f. Number Sense: $100.00 − $75.00 g. Measurement: How many minutes are in 2 hours? h. Calculation: Start with 10. Multiply by 10; subtract 1; divide by 9; then add 1. What is the answer?

problem solving

A card with a triangle on it in the position shown is rotated 90° clockwise three times. Sketch the pattern and draw the triangles in the correct positions.

New Concept

Increasing Knowledge

The factors of 8 are 1, 2, 4, and 8 The factors of 12 are 1, 2, 3, 4, 6, and 12 Math Language We see that 8 and 12 have some of the same factors. They have three factors in common. These common factors are 1, 2, and 4. Their greatest common Recall that a factor is a whole factor—the largest factor that they both have—is 4. Greatest common factor number that is often abbreviated GCF. divides another whole Example 1 number without a Find the greatest common factor of 12 and 18. remainder.

Solution

The factors of 12 are: 1, 2, 3, 4, 6, and 12. The factors of 18 are: 1, 2, 3, 6, 9, and 18. We see that 12 and 18 share four common factors: 1, 2, 3, and 6. The greatest of these is 6.

Lesson 20

105

Example 2 Find the GCF of 6, 9, and 15.

Solution The factors of 6 are: 1, 2, 3, and 6. The factors of 9 are: 1, 3, and 9. The factors of 15 are: 1, 3, 5, and 15. The GCF of 6, 9, and 15 is 3.

Visit www. SaxonPublishers. com/ActivitiesC1 for a graphing calculator activity.

Practice Set

Note: The search for the greatest common factor of two or more numbers is a search for the largest number that evenly divides each of them. In this problem we can quickly determine that 3 is the largest number that evenly divides 6, 9, and 15. A complete listing of the factors might be helpful but is not required. Generalize

What is the GCF of any two prime numbers? Explain your answer.

Find the greatest common factor (GCF) of the following: a. 10 and 15

b. 18 and 27

c. 18 and 24

d. 12, 18, and 24

e. 15 and 25

f. 20, 30, and 40

g. 12 and 15

h. 20, 40, and 60

i.

Analyze

Written Practice

Write a list of three numbers whose GCF is 7.

Strengthening Concepts

1. What is the difference between the product of 12 and 8 and the sum of 12 and 8?

(12)

2. Saturn’s average distance from the Sun is one billion, four hundred twenty-nine million kilometers. Use digits to write that distance.

(12)

3. Which digit in 497,325,186 is in the ten-millions place?

(12)

* 4. (16)

Jill has exactly 427 beads, but when Dwayne asked her how many beads she has, Jill rounded the amount to the nearest hundred. How many beads did Jill say she had? Estimate

* 5. The morning temperature was −3°C. By afternoon it had warmed to (14) 8°C. How many degrees had the temperature risen?

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Saxon Math Course 1

* 6. In three basketball games Allen scored 31, 52, and 40 points. What was (18) the average number of points Allen scored per game? * 7. Find the greatest common factor of 12 and 20. (20)

* 8. Find the GCF of 9, 15, and 21. (20)

9.

(6)

1 410. (2)

Connect

How much money is 14 of $3.24?

28  42 14

11. 28  42 (5) 14

5432 ÷ 10

12. 56,042 + 49,985

13. 37,080 ÷ 12

14. $6.47 × 10

15. 5 × 4 × 3 × 2 × 1

(1)

(2)

(2)

(5)

Find each unknown number. Check your work.

Analyze

16. w − 76 = 528

17. 14,009 − w = 9670

18. 6w = 90

19. q − 365 = 365

(3)

(3)

(4)

(3)

20. 365 − p = 365 (3)

21. (10)

Generalize

Find the missing number in the following sequence: , 10, 16, 22, 28, . . .

22. Compare: 50 −1 (9)

49 + 1

23.

Predict The first positive odd number is 1. What is the tenth positive odd number?

24.

The perimeter of a square is 100 cm. Describe how to find the length of each side.

(10)

(8)

* 25. (17)

Explain

Estimate Estimate the length of this key to the nearest inch. Then use a ruler to find the length of the key to the nearest sixteenth of an inch.

1 8

1 8

26. A “bit” is 18 of a dollar. (6)

a. How many bits are in a dollar?

b. How many bits are in three dollars? * 27. In four boxes there are 12, 24, 36, and 48 golf balls respectively. If the (18) golf balls are rearranged so that there are the same number of golf balls in each of the four boxes, how many golf balls will be in each box?

Lesson 20

107

* 28. (19)

* 29. (19)

Classify

A 5 Explain

Which of the numbers below is a prime number? B 15

C 25

List the whole-number factors of 24. How did you find your

answer?

* 30. Ten billion is how much less than one trillion? (12)

Early Finishers

Real-World Application

Tino’s parents like mathematics, especially prime numbers. So they made a plan for his allowance. They will number the weeks of the year from 1 to 52. On week 1, they will pay Tino $1. On weeks that are prime numbers, they will pay him $3. On weeks that are composite numbers, they will pay him $5. a. How much money will Tino receive for one year? Show your work. b. Would Tino receive more or less if his parents paid him $5 on “prime weeks” and $3 on “composite weeks”? Show your work.

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Saxon Math Course 1

INVESTIGATION 2

Focus on Investigating Fractions with Manipulatives In this investigation you will make a set of fraction manipulatives to help you answer questions in this investigation and in future problem sets.

Activity

Using Fraction Manipulatives Materials needed: • Investigation Activities 4–8 • scissors • envelope or zip-top bag to store fraction pieces Preparation: To make your own fraction manipulatives, cut out the fraction circles on the Investigation Activities. Then cut each fraction circle into its parts. Thinking Skill Connect

What percent is one whole circle?

Model

Use your fraction manipulatives to help you with these exercises:

1. What percent of a circle is

1 2

of a circle?

1

2. What fraction is half of 2? 3. What fraction is half of 14? 4. Fit three 14 pieces together to form 34 of a circle. Three fourths of a circle is what percent of a circle? 5. Fit four 18 pieces together to form 48 of a circle. Four eighths of a circle is what percent of a circle? 6. Fit three 16 pieces together to form 36 of a circle. Three sixths of a circle is what percent of a circle? 7. Show that 84, 36, and 42 each make one half of a circle. ( We say that 48, 3 2 1 6, and 4 all reduce to 2.) 8. The fraction 28 equals which single fraction piece? 9. The fraction 68 equals how many 41s? 10. The fraction 62 equals which single fraction piece? 11. The fraction 46 equals how many 31s? 12. The sum 18  18  18 is 38. If you add 38 and 28, what is the sum?

Investigation 2

109

12

3

3

13. 1 6

6

Form a whole circle using six of the 16 pieces. Then remove (subtract) 16. What fraction of the circle is left? What equation represents your model? Connect

14. Demonstrate subtracting 13 from 1 by forming a circle of 33 and then removing 13. What fraction is left? 15. Use four 41s to demonstrate the subtraction 1 14. Then write the answer. 16. Eight 18 s form one circle. If 38 of a circle is removed from 3 one circle ( 1 8 ), then what fraction of the circle remains? 17. What percent of a circle is 13 of a circle? 18. What percent of a circle is

1 6

of a circle?

Fraction manipulatives can help us compare fractions. Since 12 of a circle is larger than 13 of a circle, we can see that 1 1 7 2 3 Model For problems 19 and 20, use your fraction manipulatives to construct models of the fractions. Use the models to help you write the correct comparison for each problem.

1 1 7 2 3 Reading Math  means is less than; = means is 5 equal to; 4  means is greater than.

3 2 2 19.Compare: 3 4 3 20. Compare: 2 3 Represent 1 14 21.

33

1 5

1 3

 48

3 3 7 5 10

3 8

We can also draw pictures to help us compare fractions. 1

3

7

1 4 2 of one rectangle and Draw two rectangles of the4 same size. Shade 3 1 of the other rectangle. What fraction represents the rectangle that has 5 the larger amount shaded?

22. Draw and shade rectangles to illustrate this comparison: 3 3 7 5 10 Problems 23–29 involve improper fractions. Improper fractions are fractions that are equal to or greater than 1. In a fraction equal to 1 the numerator equals the denominator (as in 33 ). In a fraction greater than 1 the numerator is greater than the denominator (as in 43 ). Work in groups of two or three students for the remaining problems. 1

23. Show that the improper fraction 54 equals the mixed number 1 4 by combining four of the 14 pieces to make a whole circle. 24. The improper fraction 74 equals what mixed number? 25. The improper fraction

110

Saxon Math Course 1

3 2

equals what mixed number?

3

1 12

1 1 1 1 , , , 2 4 3 6

2 12

1 26. Form 1 12 circles using only 1 4s. How many 1 1 1 1 1 2 2 4 4 1 2? 1 3

4 3

1

1 3

12

4 3

1 12

11 6 11 27. 6

1 4

12

1 4 Explain

12

1

1 4

pieces are needed to make1 1 1 1 1 1 12 3 , , , 2 4 3 6

1

4

11

1 2 1 pieces are3needed to make 3 two whole circles? 6 How many 3 How do you know your answer is correct? 1 1 1 1 12 12 4 4 1 2 1 1 1 1 28. The improper fraction 34 12 equals what mixed 12 2 , 4 , number? 3, 6

2 12

1 8

1 1 1 1

11 29. Convert 2 , 4 , 3 , 66 to a mixed number.

30.

1 2 a An analog clock can serve as 12 12 visual reference for twelfths. At 1 o’clock the 11 1 1 hands mark off8 12 of a circle; at6 2 o’clock 1 hands mark off 2 of a circle, and so 1 1 1 the 12 , , , 2 4 3 on. 6 How many twelfths are in each of these fractions of a circle?

Evaluate

1 1 1 1 ,6 2 , 4 , 312 1 11 10 2

9

3 8

7

6

5

4

1 1 1 1 , , , 2 4 3 6 Hint: Try holding each fraction piece at arm’s length in the direction of 1 1 1 the clock (as an artist might1extend a thumb4 toward a subject.) 2 4 After you have completed the exercises, gather and store your fraction 1 1 1 1 manipulatives for 1 later use. 1 2

4

4

a.

Justify

1 8

2. Each of 3 students ate of a new box cereal in the box was eaten?

c.

9 3 5 4

3 3 2 4

1 2

1 2

1 1 2 2

11 of 6 cereal.

1 12

What amount of

3. More than half of the students in the class are girls. What fraction 11 1 8 6 of the students in the class are boys? 1 1 1 1 , , , Estimate Copy this number 2line. 4 3 6 11 1 8

6

1 1 1 1 , , , 2 4 3 61

0

3 4

1 3

2

1 1 1 1 1 2 , , , 12 Use examples and nonexamples to support or 2 4 3 6 1 1 disprove 1 12 1 12 12 4 4 that the Commutative Property can be applied to the addition and subtraction of two fractions. 1 1 1 1 1 2 , , , 1 12 12 2 4 3 6 12 b. Analyze Choose between mental math or estimation and use that method to answer each of the following questions. Explain your choice. 1 1 1 1 1 2 , , , 12 12 2 4 3 6 1. One-fourth of a pizza was eaten. How much of the pizza was not eaten?

extensions

3 4

1

12

3

2

1 8

Then estimate the placement of the following fractions on the number 1 1 1to1help you if you need to. line. Use your fraction pieces , , , 2 4 3 6 9 3 10 33 3 13 3 3 9 3 33 7 39 34103 39 7 7 1 11039 9 1 10 7 9 110 410 1 4 10 103 3 3 3 3 p 3 3 2 3 2 2 p 2 p 2 2 3 5 4 34 5 42 4 84 4 2 5 4 42 7 25 48 3 2 8 345 5 8 3 7 5 83 1 8731 18 18 73 8 32 4 , , , 2 4 3 6

1 2

1 2

1 1 1 4 2 2

11 24

1 2

1 1 2 4 2

1 2

1 4

1 1 2 4

1 4

1 4

Investigation 2

111

LESSON

21

Divisibility Building Power

Power Up facts

Power Up D

mental math

a. Number Sense: 4 × 42 b. Number Sense: 3 × 76 c. Number Sense: 64 + 19 d. Number Sense: 450 + 37 e. Calculation: $10.00 − $6.50 f. Fractional Parts:

1 2

of 24

100  4030

678 6

1

1

1, 2, 0, 2, 4

g. Measurement: How many months are in a year? h. Calculation: Start with 25, × 2, − 1, ÷ 7, + 1, ÷ 2

problem solving

Here is part of a randomly-ordered multiplication table. What is the missing product?

New Concept Thinking Skill Discuss

Without dividing by 2, how can you tell that a number is even?

30

42

32

?

28

56

35

49

Increasing Knowledge

There are ways of discovering whether some numbers are factors of other numbers without actually dividing. For instance, even numbers can be divided by 2. Therefore, 2 is a factor of every even counting number. Since even numbers are “able” to be divided by 2, we say that even numbers are “divisible” by 2. Tests for divisibility can help us find the factors of a number. Here we list divisibility tests for the numbers 2, 3, 5, 9, and 10. Last-Digit Tests

Inspect the last digit of the number. A number is divisible by … 2 if the last digit is even. 5 if the last digit is 0 or 5. 10 if the last digit is 0.

112

48

Saxon Math Course 1

Sum-of-Digits Tests

Add the digits of the number and inspect the total. A number is divisible by … 3 if the sum of the digits is divisible by 3. 9 if the sum of the digits is divisible by 9.

Example 1 Which of these numbers is divisible by 2? 365

1179

1556

Solution To determine whether a number is divisible by 2, we inspect the last digit of the number. If the last digit is an even number, then the number is divisible by 2. The last digits of these three numbers are 5, 9, and 6. Since 5 and 9 are not even numbers, neither 365 nor 1179 is divisible by 2. Since 6 is an even number, 1556 is divisible by 2. It is not necessary to perform the division to answer the question. By inspecting the last digit of each number, we see that the number that is divisible by 2 is 1556.

Example 2 Which of these numbers is divisible by 3? 365

1179

1556

Solution To determine whether a number is divisible by 3, we add the digits of the number and then inspect the sum. If the sum of the digits is divisible by 3, then the number is also divisible by 3. The digits of 365 are 3, 6, and 5. The sum of these is 14. 3 + 6 + 5 = 14 We try to divide 14 by 3 and find that there is a remainder of 2. Since 14 is not divisible by 3, we know that 365 is not divisible by 3 either. The digits of 1179 are 1, 1, 7, and 9. The sum of these digits is 18. 1 + 1 + 7 + 9 = 18 We divide 18 by 3 and get no remainder. We see that 18 is divisible by 3, so 1179 is also divisible by 3. The sum of the digits of 1556 is 17. 1 + 5 + 5 + 6 = 17 Since 17 is not divisible by 3, the number 1556 is not divisible by 3. By using the divisibility test for 3, we find that the number that is divisible by 3 is 1179. Discuss

Is 9536 divisible by 2? by 3? How do you know?

Lesson 21

113

Example 3 Which of the numbers 2, 3, 5, 9, and 10 are factors of 135?

Solution First we will use the last-digit tests. The last digit of 135 is 5, so 135 is divisible by 5 but not by 2 or by 10. Next we use the sum-of-digits tests. The sum of the digits in 135 is 9 (1 + 3 + 5 = 9). Since 9 is divisible by both 3 and 9, we know that 135 is also divisible by 3 and 9. So 3, 5, and 9 are factors of 135. Will the product of any two prime factors of 135 given above also be a factor of 135? Predict

Practice Set

a. Which of these numbers is divisible by 2? 123

234

345

b. Which of these numbers is divisible by 3? 1234

2345

3456

Use the divisibility tests to decide which of the numbers 2, 3, 5, 9, and 10 are factors of the following numbers: c. 120

Written Practice Math Language The word product is related to multiplication, the word sum is related to addition, and the word difference is related to subtraction.

d. 102

Strengthening Concepts

1. What is the product of the sum of 8 and 5 and the difference of 8 and 5?

(12)

2.

In 1787 Delaware became the first state. In 1959 Hawaii became the fiftieth state admitted to the Union. How many years were there between these two events? Write an equation and solve the problem.

* 3.

Formulate Maria figured that the bowling balls on the rack weighed a total of 240 pounds. How many 16-pound bowling balls weigh a total of 240 pounds? Write an equation and solve the problem.

(13)

(15)

Formulate

4. An apple pie was cut into four equal slices. One slice was quickly eaten. What fraction of the pie was left?

(6)

5. There are 17 girls in a class of 30 students. What fraction of the class is made up of girls?

(6)

6. Use digits to write the fraction three hundredths.

(6)

1 7. How much money is 12 of $2.34? 4

(6)

8. What is the place value of the 7 in 987,654,321?

(6)

114

Saxon Math Course 1

9.

(10)

Generalize

Describe the rule of the following sequence. Then find the

next term. 1, 4, 16, 64,

10. Compare: 64 × 1

, ...

64 + 1

(9)

* 11. Which of these numbers is divisible by 9? (21)

* 12. (16)

1 2

A 365

B 1179

Find the sum of 396, 197, and 203 by rounding each number to the nearest hundred before adding. Estimate

* 13. What is the greatest common factor (GCF) of 12 and 16? (20) 678 1 14. 100  4030 48,8401 ÷ 24 6 1,15. 2 , 0, 2, 4 (2)

(2)

678 100  4030 16. 6 (2)

1 2

C 1556

1

1

1, 2, 0, 2, 4

17. $4.75 × 10

1 2

(2)

100  403

Find each unknown number. Check your work. 18. $10 – w = 87¢ (3)

1 2

19. 463 + 27 + m = 500 (3)

* 20. Arrange these numbers in order from least to greatest: 678 (17) 1 1 100  4030 6 1, 2, 0, 2, 4 * 21. What is the average of 12, 16, and 23? (18)

* 22. List the whole numbers that are factors of 28. (19)

* 23. What whole numbers are factors of both 20 and 30? (19)

24. Use an inch ruler to draw a line segment four inches long. Then use a (7) centimeter ruler to find the length to the nearest centimeter. 25. (12 × 12) – (11 × 13) (5)

* 26. (Inv. 2)

* 27.

(Inv. 2)

Represent To divide a circle into thirds, John first imagined the face of a clock. From the center of the “clock,” he drew one segment up to the 12. Then, starting from the center, John drew two other segments. To which two numbers on the “clock” did John draw the two segments when he divided the circle into thirds? Model

11 10

12 1

2

9

3 8

4 7

6

5

Draw and shade rectangles to illustrate this comparison: 3 2 6 3 4

1 8

Lesson 21

115

3 2 6 A “bit” is 1 of a dollar. *3 28. 8 (6) 4 a. How many bits are in a dollar? b. How many bits are in a half-dollar? 29. A regular octagon has eight sides of equal length. What is the perimeter (8) of a regular octagon with sides 18 cm long? * 30. (Inv. 2)

Early Finishers

Real-World Application

Represent Describe a method for dividing a circle into eight equal parts that involves drawing a plus sign and a times sign. Illustrate the explanation.

The 22 sixth grade students at the book fair want to buy a mystery or science fiction novel. Ten of the 15 students who want a mystery book changed their minds and decided to look for a humorous book. How many sixth grade students are NOT looking for mystery books? Write one equation and use it to solve the problem.

116

Saxon Math Course 1

LESSON

22

“Equal Groups” Problems with Fractions Building Power

Power Up facts

Power Up C

mental math

a. Number Sense: 4 × 54 b. Number Sense: 3 × 56 c. Number Sense: 36 + 29 d. Calculation: 359 − 42 e. Calculation: $10.00 − $3.50

1 4

1

f. Fractional Parts: 4

1 2

of48 48 of

1 3

2 3

g. Measurement: How many yards are in 9 feet? h. Calculation: Start with 100, − 1, ÷ 9, + 1, ÷ 2, − 1, × 5 3

1

4 problem solving

New Concept

1 2

Increasing Knowledge

1 4

Here we show a collection of six objects. The collection is divided into three equal groups. 1 1 4 We see that1there are12two objects in 13 of the of 48 1 1 4 of 48 4 4 2 collection. We also see that there are four objects in 23 of the collection.

1 4

This collection of twelve objects is divided into four equal groups. There are three objects in 14 of 3 3 1 4 are 3nine objects in 3 of the 4 the collection, so there 4 4 4 collection.

1 3

of 48

3

Truston4has 16 tickets, 4Sergio has 8 tickets, and Melina has 6 tickets. How many tickets should Truston give to Sergio and to Melina so that they all have the same number of tickets?

3 4

2 3

1 3

1 4

3 4

2 3

1 2

3 4

Example 1 Thinking Skill Infer

480 3

Why do you divide the musicians by 3?

Two thirds of the 12 musicians played guitars. How many of the 66 musicians played guitars? 3 Solution

5280 44

This is a two-step problem. First we divide the 12 musicians into three equal groups (thirds). Each group contains 4 musicians. Then we count the number of musicians in two of the three groups.

Lesson 22

117

of 48

12 musicians 1 did not play 3 guitars. 2 played 3 guitars.

4 musicians 4 musicians 4 musicians

1 2 in two 1 there are 4 musicians 1 Since in each1 third, the number of musicians 3 3 4 4 2 of 48 thirds is 8. We find that 8 musicians played guitars.

Example 2 3 3 1 Cory on the assignment. How many 4 has finished 4 of the 28 problems 4 problems has Cory finished?

Solution First we divide the 28 problems into four equal groups (fourths). Then we find the number of problems in three of the four groups. Since 28 ÷ 4 is 7, there are 7 problems in each group (in each fourth). 28 problems 1 are not 4 finished.

7 problems

3 are 4 finished.

7 problems

7 problems

7 problems

In each group there are 7 problems. So in two groups there are 14 problems, 1 are 21 problems. 3 Cory has finished 2 1 and in three groups there We see that 3 3 4 4 21 problems.

Example 3

1 3

3 4

2 3 3

3 5

2 5

How much money is 5 of $3.00?

3 4

1 4

Solution 3

3

2

5 First we divide5 $3.00 into five 5equal groups. Then we find the amount of money in three of the five $0.60 in each group groups. We divide $3.00 by 5 to find the amount 5  $3.00 of money in each group.

$0.60 in each group group 5  $3.00 1 3

2 3

3 5

118

2 5

We find that 35 of $3.00 is $1.80.

Saxon Math Course 1 $0.60 in each group 5  $3.00

2 of $3.00 5

$0.60 3 $1.80

$0.60 $0.60 $0.60

3 of $3.00 5

Now we multiply $0.60 by 3 to find the amount of 3 3 1 money in three 4 groups. 4 4 ×

$3.00

$0.60 $0.60

3 4

Example 4 3 What number is 34 of 100? 5

1 4

3 4

of 100

of 100

Solution We divide 100 into four equal groups. Since 100 3 are 25 in each 1 ÷ 4 is 25, there 1 group. We will 5 5 find the total of three of the parts.5

100 1 of 100 4

25

3 × 25 = 75

2 5

of 100% Example 5

3 5

3 5

of 100%

3 of 100 34

3 5 3 4

1 a5 b

4 1 4

3 a5 b

a. What percent of a whole circle is 15 of a circle?

1 What percent circle is 353of a circle? 4 of a whole 51 3 5 of 100 4 4 of 100 34 3 Solution 5 4 2b. 3

3 5

3 We divide 100% 3 circle is 100%. 1 A whole into five 5 4 4 of 100 2 equal groups. 5 of 100%

a. One of35the five parts 2Q15of R is100% 20%. 5

1 5

Practice Set

2 5

2 3

3 4

25

of 100

3 5 1 5 1 4

1

25 4 of 100 3 4

of 100

1 5

of 100

3 4

of 100

100% of 100 1 3 a b 25 of 100% 20% 5 3 1 100% of 5a 5 b 20%a 5 b

3 53 of 100% 1 3 5 which 5 1 of 100 20% 3 of b. is 3 × 20%, 4 4 4 34 1 2 20% of 100% 5 55 3 2 4 20% 3 5 3 1 3 a5 b a5 b 100% 3 1 3 11 5 of 100% 3 5 a 5 5b a5 b 2 5 Model Draw a diagram to5 illustrate of 100% of 100% each 5problem. 3 1 a. Three fourths of 3the 12 musicians play thea 5piano. How many of a 5 could b b 2 5 of 100% 5 of 100% the musicians could play the piano? 4 5 4 b. How much money is 23 of $4.50? 5 3 1 3 3 9 a b a5 b 2 5 2 4 5 of 100% 10 5 of 100% 10 c. What number is 5 of 60? 3 1 Three of the five parts5 Q35R 3 1 equals 60%. 5 5

of

25

3 d. What number is 10 of 80?

9 10

e. Five sixths of 24 is what23 number?

4 5

3 9 f. Giovanni answered 10 of the questions correctly. What percent of the 10 questions did Giovanni answer correctly?

Written Practice

Strengthening Concepts

3 10

3 4

2 3

1. When the sum of 15 and 12 is subtracted from the product of 15 and 3 3 2 12, what is the difference?

(12)

3

10

4

2. There were 13 original states. There are now 50 states. What fraction of the states are the original states?

(6)

2 3

3 4

3 10

Lesson 22

w  345 15

119

100

3 10

9 10

3 10

9 10

* 3. 3A marathon race 9 is 26 miles plus 385 yards long. A mile is 1760 yards. 10Altogether, how 10 many yards long is a marathon? (First use a multiplication pattern to find the number of yards in 26 miles. Then use an addition pattern to include the 385 yards.)

(11, 15)

* 4.

3 3 If 23 of the 12 apples were eaten, how many were eaten? Draw a 10 4 diagram to illustrate the problem.

* 5.

3 2 What number is 34 of 16? Draw a10 diagram to3illustrate the 3 2 3 4 problem.

* 6. 23

3 3 How much money is 10 of $3.50? Draw a diagram to illustrate 4 the problem.

(22)

(22)

(22)

Model

Model

3 10

Model

* 7. As Shannon rode her bike out of the low desert, the elevation changed (14) from –100 ft w to 600 ft. What was the total elevation change for her  345 ride? 15 Find each unknown number. Check your work. w  345 8. w − 15 =15 9. w  345 8 (3) (4) 15 10. w36¢ $4.78 + $34.09 + 345 (1) 15 7  9  14 12. 35  1000 335  1000 (2) * 14. (16)

11. $12.45 ÷ 3 (2)

13. (5)

7  9  14 3

Estimate Shannon bought three dozen party favors for $1.24 each. To estimate the total cost she thought of 36 as 9 × 4, and she thought of $1.24 as $1.25. Then she multiplied the three numbers.

a. What was Shannon’s estimate of the cost? 7  9  14 b. To find the cost quickly, which two numbers should she multiply 35  1000 3 first? 1 14 15. 4Which digit in 375,426,198,000 is in the place? 1 ten-millions 1 (12) 5 7  39  14 25 3 2 35 1000 * 16. Find the greatest3 common factor of 12 and 15. (20)

* 17. List the whole numbers that are factors of 30. (19)

* 18. The number 100 is divisible by which of these numbers: 2, 3, 5, 9, 10? (21)

* 19. (22)

1 1 Jeb answered 45 of the questions correctly. What percent of the 3 2 questions did Jeb answer correctly? Draw a diagram to illustrate the problem. Model

4 20. Compare: 5 (9)

* 21. (19)

Classify

1 3

1 2

Which of these numbers is not a prime number?

A 19

B 29

C 39

22. (3 + 3) − (3 × 3)

(5, 14)

23. (18)

120

Generalize

Saxon Math Course 1

Find the number halfway between 27 and 43.

Math Language Perimeter is the distance around a closed, flat shape.

24. What is the perimeter of the rectangle below? (8)

15 cm

1 3

4 5

10 cm

25. Use an inch ruler to find the length of the line segment below. (7)

* 26. (Inv. 2)

Corn bread and wheat bread were baked in pans of equal size. The corn bread was cut into six equal slices. The wheat bread was cut into five equal slices. Which was larger, a slice of corn bread or a slice of wheat bread? Analyze

* 27. Compare these fractions. Draw and shade rectangles to illustrate the comparison.

(Inv. 2)

2 4 * 28. (22)

3 5

1 4

3 2 A quarter of a year is 14 of a year. There are 12 months in a 5 4 year. How many months are in a quarter of a year? Draw a diagram to illustrate the problem. Model

29. A “bit” is one eighth of a dollar. (6)

a. How many bits are in a dollar? b. How many bits are in a quarter of a dollar?

* 30. (1)

Represent The letters c, p, and t represent three different numbers. When p is subtracted from c, the answer is t.

c−p=t Use these letters to write another subtraction equation and two addition equations. Hint: To be sure you arranged the letters in the correct order, choose numbers for c, p, and t that make c – p = t true. Then try those numbers with these letters for your three equations.

Lesson 22

121

LESSON

23

Ratio Rate

Power Up facts mental math

Building Power Power Up D a. Number Sense: 5 × 62 b. Number Sense: 5 × 36 c. Number Sense: 87 + 9 d. Number Sense: 1200 + 350 e. Calculation: $20.00 − $15.50

1 4

f. Fractional Parts:

1 2

of 84

g. Measurement: How many millimeters are in 3 meters? h. Calculation: 10 × 3, + 2, ÷ 4, + 1, ÷ 3, × 4, ÷ 6 13 15 problem

solving

How many different bracelets can be made from 7 white beads and 2 gray ones?

Increasing Knowledge

New Concepts ratio

Math Language 13 and 15 are the terms of the ratio.

1

A ratio is a way to describe a relationship between 4 numbers. If there are 13 boys and 15 girls in a classroom, then the ratio of boys to girls is 13 to 15. Ratios can be written in several forms. Each of these forms is a way to write the boy-girl ratio: 13 to 15

13:15

13 15

Each of these forms is read the same: “Thirteen to fifteen.” In this lesson we will focus on the fraction form of a ratio. When writing a ratio in fraction form, we keep the following points in mind: 1. We write the terms of the ratio in the order we are asked to give them. 2. We reduce ratios in the same manner as we reduce fractions. 3. We leave ratios in fraction form. We do not write ratios as mixed numbers.

Example 1 A team lost 3 games and won 7 games. What was the team’s win-loss ratio? 122

Saxon Math Course 1

1 2

distance time

price quantity

distance fuel used

55 miles 1 hour Solution

28 miles gallon

71 3

The question asks for the ratio in the order of wins, then losses. The team’s win-loss ratio was 7 to 3, which we write as the fraction 73. distance distance $2.89 number number of boy 1 pound time time of games won  7 number of games lost number of girl 3 We leave the ratio in fraction form.

60 miles 4 hour

number of games won 7 Is the win-loss ratio the same as the loss-win ratio?  5 number of games lost 3 6

Discuss

distance fuel used

Example 2 In a class of 28 students, there are 13 boys. What is the ratio of boys to girls in the class?

Solution

7 3

To write the ratio, we need to know the number of girls. If 13 of the 28 students are boys, then 15 of the students are girls. We are asked to write the ratio in “boys to girls” order. number of games won 7  number of games lost 3 Model

rate

2 3

number of boys 13  number of girls 15

Use tiles of two colors to model this ratio.

2 3

A rate is a ratio of measures. Below are some commonly used rates. Notice that per means “for each” and substitutes for the division sign. Common Rates Name Speed

distance distance time time 2 3

miles 5555miles Mileage hour 1 1hour

Rate of Measures Example distance distance 55miles miles 55 distance distance 55miles miles 55 time time 1 hour 1 hour time time hour 1 1hour distance distance fuelused used fuel

miles 2828miles gallon 1 1gallon

Alternate Form distance 28miles miles distance 28 distance 28miles miles distance 55fuel miles per hour28 gallon 11gallon fuel used used gallon fuelused used 1 1gallon fuel 28 miles per gallon

price price distance distance $2.89 distance distance $2.89 price price $2.89 per pounddistance distance distance $2.89 distance $2.89 quantity pound time time quantity 11pound time time quantity pound time time quantity 1 1pound time time distance distance 28 miles 55 miles price price distance distance distance distance $2.89 $2.89 Rate problems are a fuel type of equal groups problem. The rate is the number in time 1 hour 1 gallon used quantity 1 pound time time quantity 1 pound time time each group. The problems often involve three numbers. One number is the rate, and the other two numbers the measures that form the rate. distance 60miles milesare about distance 60 distance 60miles miles distance 60 hour 44hour fuelused used fuel The three numbers are related by multiplication or division as we show below. hour fuelused used 4 4hour fuel price $2.89 distance= distance × timedistance miles 6060miles 5 5 Pattern: distance distance quantity 1 pound time time 6 6 hour 4 4hour fuelused used fuel Unit price

Example: 165 miles = 55 miles per hour × 3 hours

In a rate problem one of the numbers is unknown. We find an unknown by multiplying, and we find an unknown factor by dividing the distance 60 miles product 5 6 the known factor. 4 hour product fuelby used

Lesson 23

123

distance time

Example 3

55 miles 1 hour

distance fuel used

28 miles 1 gallon

On a bike trip Jeremy rode 60 miles in 4 hours. What was his average speed in miles per hour? distance 28 miles price 55 miles distance distance $2.89 1 hour Solution 1 gallon quantity fuel used 1 pound time time

distance time

We are given the distance and time. We are asked for the speed, which is distance divided by time. price quantity

$2.89 1 pound

distance time

distance 60 miles distance = 15 miles per hour time 4 hour fuel used distance distance 55 miles time 1 hour fuel used

28 miles 1 gallon

Example 4 distance Mr. Moscal’s car averages 32 miles per gallon 5on the highway. Predict 6 fuel used about how far he can expect to travel on a road trip using 10 gallons of price distance distance $2.89 fuel. quantity 1 pound time time Solution

60 miles 4 hour

This is a problem about gas mileage. Notice the similar pattern. distance 60 miles × fuel used distance = 4 hour fuel used We are given the rate and the fuel used. We are asked for the distance, which is the product. distance = 32 miles per gallon × 10 gallons Mr. Moscal can 7 3

7 expect 3

= 320 miles to travel about 320 miles on 10 gallons of fuel.

Making a table can help us solve some rate problems. Here is the beginning of a table for example 4. Distance Traveled at 32 Miles per Gallon number games won Fuel Usedof(gallon) 1  72 3 4 5 number of boys  13 number of games lost number of girls 15 3 32 64 96 128 160 Distance (miles)

7 3

number of games won 7 number of boys 13   number of games lost girls 15 that has 19 cats and 3 Practice Set a. What is the ratio of dogs to number cats in aofneighborhood 12 dogs? number of games won 7 number of boys 13   number of games lostb. Analyze numberratio of girls 15 of 30 students with 3 What is the girl-boy in a class 17 boys? c. If the ratio of cars to trucks in the parking lot is 7 to 2, what is the ratio of trucks to cars in the parking lot? d. How long will it take a trucker to drive 400 miles at 50 miles per hour? 2

e. If a four-quart 3 container of milk costs $2.48, what is the cost per quart? 2 3

124 2 3

Saxon Math Course 1

Written Practice 3 5

3 5

1.

(7)

* 2. (22)

2 5 3 10

3 10

2 5

3 2 1 52 5 3 3 3 2 Strengthening Concepts 5 5 2 7 5 10 2 7 How many millimeters 5long is a ruler that is 30 cm long? 10 3 7 23 8 2 Model Dan has finished10 of the 30 problems on an assignment 35 5 3 7 a during class. How many problems did7 Dan finish during class? Draw 7 10 8 10 7 1 diagram to illustrate10the problem. 18 8 7 1 18 8

3. Diego walked the length of 2 a football field in 100 large paces. About 2 7 3 how long was the football3 field?

(7)

3 5

5

7 8 2 3

2 3

7 1 1 1 8245 miles on 7 gallons of gas. 1 8 the 3car traveled *8 4. On the open highway (23) What was the car’s gas mileage for the trip in miles per gallon?

* 5.

3 10

(22)

3 5

3 5

2

10

10

* 6. (22)

What number is 53 of2 25? Draw a diagram to illustrate the 7 1 18 8 3 problem. 3 Model

3 7 How much money is 10 of $36.00? Draw a diagram to illustrate 3 7 8 10 8 the problem.

*7 7. What is the sum of

4 8

4 8

102) (Inv.

3 8

and 84 ?

* 8. The improper fraction

(Inv. 2)

3 5

9 8

9 8

2 3

8

$3.75 $3.75 23 $3.75 ∙ 16 3 11. 25 25 (2) 3 4 12. What is the place value4 of the 6 in 36,174,591?

9 810. (2) (12)

3 4

2 9 * 13. Explain How 8 can you find 3 of a number? (22) 3 4 each unknown number. Check your work. Find (4)

3 16. Compare: 4 (17) * 17. (8)

18. (10)

3 8

$3.75 25

$3.75 25

3 4

15. $10.20 − m = $3.46

14. $0.35n = $35.00 2 3

9 8

7 10

$3.75 25 3 4 9 38 4

equals what mixed number? 2

(Inv. 2)

2 3 4 8

9 8

3 8

7 * 9. Two eighths10 of a circle is 3what 4 percent of a circle?

9 8

7 10

Model

3 8 to Use your fraction manipulatives help answer problems 7–9. 5

7 10

7 10

5

4

2 3

7 8

(3)

1

The length of a rectangle is 20 inches. The width of the rectangle is half its length. What is the perimeter of the rectangle? Analyze

Describe the rule for the following sequence. Then, write the sixth number in the sequence. Generalize

2, 4, 8, 16, . . . 19. Yesterday it snowed. The meteorologist on the radio said that it was 14º (10) outside. What scale was the meteorologist reading? How do you know? 20. Compare: 12 ÷ 6 − 2 (9)

12 ÷ (6 − 2)

* 21. What is the greatest common factor (GCF) of 24 and 32? (20)

22. What is the sum of the first seven positive odd numbers? (10)

Lesson 23

125

1

Use your fraction to help answer28 problems distance 4 miles 23–25. distance 55manipulatives miles time 1 gallon fuel used 1 1 1inhour 1 * 23. a. How many 41 s are 1? 2 Model

1 4

(Inv. 2)

1

4

1

2

1 2

b. How many 4 s are in 2?

* 24. One eighth is what percent of a circle? price of a circle$2.89 distance distance (Inv. 2) quantity 1 1 pound time time * 25. Write a fraction4 with a denominator of 8 that is equal to 12. (Inv. 2)

* 26. There were 16 members of the 2004–2005 men’s national swim team. (23) Five of them competed in the freestyle. What is the ratio of those who distance 60 miles competed in the freestyle to those who did not? 4 hour fuel used * 27. (19)

Classify

Which prime numbers are greater than 20 but less

than 30?

1 28. Which of the figures below represents a line? 4 (7)

1 2

A B C

* 29. (21)

Classify

A 252

Which of these numbers is divisible by both 2 and 5? B 525

C 250

* 30. If a team lost 9 games and won 5 games, then what is the team’s (23) win-loss ratio?

Early Finishers

Real-World Application

126

Mrs. Akiba bought 3 large bags of veggie sticks for her students. Each bag contains 125 veggie sticks. One sixth of Mrs. Akiba’s students did not eat any veggie sticks. The remaining students split the veggie sticks evenly and ate them. How many veggie sticks did each of the remaining students eat?

Saxon Math Course 1

5 6

LESSON

24

Adding and Subtracting Fractions That Have Common Denominators Building Power

Power Up facts

Power Up C

mental math

a. Number Sense: 6 × 24 b. Number Sense: 4 × 75 c. Number Sense: 47 + 39 d. Number Sense: 1500 − 250 e. Calculation: $20.00 − $14.50 f. Fractional Parts:

1 2

of 68

g. Measurement: How many yards are in 12 feet? h. Calculation: 6 × 7, − 2, ÷ 5, × 2, − 1, ÷ 3

problem solving

1 4

Tom followed the directions on the treasure map. Starting at the big tree, he walked five paces north, turned right, and walked seven more paces. He turned right again and walked nine paces, turned left, and walked three more paces. Finally, he turned left, and took four paces. In which direction was 3 5 1 1 Tom facing, and how many paces was he from the big tree? 8 8 4 4

New Concept

1 4

Increasing Knowledge

1 2

1

1 3

5

3 5 3 4 2  28 8to  Using our fraction manipulatives, we see that when we add 8 8 the sum is 8. 1 8

3 8 3 8

1 4

Math Language The denominator tells you into how many parts the whole is divided.

51 88

1 8

1 8

1 2

1 38 3 2 2 5 5   8 8 8 8 8 8

1 2

3 3+ 22 = 55  8 8 8 88 8 Three eighths plus two eighths equals five eighths. 3

3

2

5 5

3

 8 from  8 8, then 8 are left. Likewise, if we subtract 8 8 5 8

2 8

3 8 1 8

11 48

5 8

1 8



2 8



5 8

1 8

1 8

1 8

1 8

3 8

5 8

1 8

1 8

1 8 3 8

 28  58

55 – 282=38 3 88 8 8

Five eighths minus two eighths equals three eighths.

1 1 1   4 4 4

5 8

1 1 1 3    4 4 4 4

1 2

1 2

Lesson 24  127

5 8

2 8

Notice that we add the numerators when we add fractions that have the same denominator, and we subtract the numerators when we subtract fractions of the fractions 1 that have the same denominator. The 5denominators  28  38 8 that do not8 change when we add or 5subtract fractions have the same 3 8 8 5 3 denominator.

Thinking Skill 2 8

Justify 2 Why 8

does the denominator 5 Example 1 stay the 5 3 8 same when 8 8 we 1 1 1 add fractions1 Add: 3   4 4 4 8 8 1with the same 8denominator? Solution 5 8

 28  385

3 8

8

3 8

5 8

8

3 8

 28  38

5 82 8

 28  38

1 1 1 3    4 4 4 4

5 8

 28  38

The denominators are the same. We add the numerators. 3 3 3 3 1 1 2   1 1 1 1 1 1 1 3 4444 1 1  38 25 2 2   3 5 2 3   1  4 4 4 4 4 4 4 28  82  8 1 1 1 1 1 2 1 83 1 1 5 5 2 8 8       8 8 4 4 84 4 4 4 4 2 2 8 1 1 1 3 1 1  1 1 1 3  1 2 41 4 4  4Example 2 2 11   2 2 4 4 4 4 3 3 3 3 12 2 2  1   33 3 3 3 1 1 2 1 3 4 34 4 4   Add:  1 41   5 2 1 1 1 1 1 1 3 2 2 2   4 14 1 2 2  3 3 1 3 43 4  4 4 5 2 3  5 3 2 8 821 8    12   4 4    4 4 4 88 8 84 3 8 4 8 8 2 2 2 4 4 8 4 45 8 8 3 3 3 38 Solution  3 3 3 3 4 4 4 4  4 4 4 One 4 half 2 one half is two halves, which is one 2 5 7 plus 7 whole.   3 3 3 3 8 128 3 3 8 8 8      3 1 3 3 3 3 1 1 2 1 1 4 4 1 4 1  1  3    1 1 3  1  4 34  3  3 43 412   1 1 1 1 1 3 1 1 1 1 1 1 2 2 2 3 4 4 4 4 4 4 4 4 41 4 2 2     4 4 4 4  5  2  3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 8 8 8 8

3

8

Example 3 12 12 7 2 7 2 5 4 4    3 3 3 73 2 5 83  83  83  3  12  3 2 7 2 8 8   1 Add:  1 1 2 1 14 24 4 4 2 8 8 4 4 4 84 81 8 1 1 1 1 3 4 3  3  3  3 3  3 1 3 1 3 1 1 1 2 2 2   2 2 2 4 4 4 4 4 4 4 4   7 2 5 4 4 4 4 4 4 4 2 2   8 8 78  2  5Solution 8 8 8 Thinking Skill The denominators are the same.12We add the numerators. 12 5  3 3 12 74 2 7 2 5 4 Discuss 8   12    3 3 1238 38 3 12 3 3 3 3 12 8 8 8 4 44 4 3       4  4  4 334 3 4 3 4 3 3 1 1 2 Why can we write 4 4 4 4 4   1    12 2 2 2 4 4 4 4 as 3? 4 12 4

Example 4

7 2 5  7 2 8 8 3 38 12 3 3 Subtract:     8  83 4 4 4 4 4 Solution

12 7 24  8 8

7 2 5   8 8 8

The denominators are the same. We subtract the numerators. 12 12 12 12 4 7 2 7 2 5 4 4 4    8 8 8 8 8

12 4 128

1 1  2 2

Saxon Math Course 1

12 4

12

7 24 5   8 8 8

12 4

3 8

1 2











Example 5 Subtract:

1 1  2 2

1 2

1 1  2 2

Solution 1 1  2 2 1 2

43 4  8 8 3 1  4 4 14 1  8 8 3 2  4 4 21 2  4 14 1  2 2 7 4  8 8 1

12 1  2 32 4  8 8 1 2

3

7 4 8 3 8

3

3 4 4 1 4

7

7 8 8



1 4

4 1  8 8



2 4

1 2  4 4



4 8

1 1  2 2

1 1  2 2

1 1  2 2

1 1 0 1 1 0 1 is01zero.0    0    0 If1 we 1start with 12 and subtract 12, then what1 is left      1 1 0 0 2 2 2 2 2 2 1 2 2  3 4 2 22 2 3 2 1 2 28  8 1 1 0 1 4 4   0 2 2 2 2 3 4 3 1 1 0     0 8 8 4 2 2 2 Find each sum 3 or difference: 3 4 Practice 3 Set 3 1 1 1 1 1 4 1 1 1       3 13 1 1 1 1 1    3 4  �3  1  8 8 8 � 8 8 4 4 8 8 4 4 8 8 8 b. 8 � 8  8 3 8 2 4 a. 4  4 4  1 8 8 � �4 4  3 1 1 1 1 8 8 4 4    4 4 8 8 8 3 c. 1  1  1 d. 4  1  8 8 8 8 8 4 3 2 3 2 4 1 1 1 4 1 1 1       3 e. 23 2 f. 1  13  12  4  1 1 42 4 8 8 4 4 8 8 4 4 4 4 4 4 4  4 4 47  4 8 84 4 4 48 8 3 2 1 1   41 g. Connect Use words to write the subtraction problem in exercise d. 4 4 4 4 23 1 2 7 1 1    4 4 8 4 4 Strengthening Concepts 1 1 1 1 2 Written 7 4 1 1 1 Practice 2 7 4 1 1       2 7 47 4 1 12 1 2  1 2  7 4 4 4 81 8 2 2 4 4 8 8 2 2 1   1 8 8 2 2 8 2 1 1 0 1 8 1 and0 was paid 1the 1 1 4 Martin worked in1 1. 4Analyze yard for five 8 hours 71  41 (11, 15)    2   0 2 41  2 2 2 12 1 2 2 21 1 2 2 2 1 1 2 1 1 8 8 2 paid 2 $5.00 for washing the car. was 23  1000  per hour. Then9 he  $6.00 2 12 1 1 2 2 21 2 2 21  earn? What pattern did Altogether, how much money did Martin 2 2 2 2 2 3 3 4 0 1 1 you use to find Martin’s yard-work earnings? What pattern did 1 1 1 1 1 1 1     1 0   1 1 1 8 811 4 2 2 22find his total earnings? 4 2 4 you 1 7 to 1 use 2 2 2 2 3 2 4 2 4  1 3 4 3 1 1 1 1 2 1 81  1 2 3 28 1  1    1 1 3 1 3 3 3 3 8 8 4 4 8 8 81 4 1 4 4 8 8 8 4 1 2 4 Model * 2. Juan used 4 of a dozen eggs to make omelets for his8 family. 9  1000      3 7 2 4 8(22) 818 8 4 44 4 8 How many eggs did Juan use? Draw a diagram to illustrate the 8 8 4 problem. 1 1 13 3 4 1 3 7 1 7 1    3 3 1 8 1 8 8 4 8 8 4 8 8 8 8 4 8 4 87 43 3 1 * 3. 8Explain seven8 hundred 3 42 One mile is one thousand, 1 41 sixty yards. How1 1 34  1 13  2    (22) 3 1 88 4 3 3 8 4 4 4 4 4 4 1 yards 2 4 your answer. 4 4 4 1many 2 1 is 8 of a mile? Explain howyou  found  1   4 3 1 8 88 8 4 44 4 4 4 4 4 Model Use your fraction manipulatives to help with exercises 4–8. Then 1 2 7 1 1 3 3 1 1 1 1  one of the 1 1  1 1 choose exercises to4 write a word problem that8 is solved by the 4 4 4 8 4 4 8 4 4 4 8 43 81 7 exercise. 4 7 4 1 1 1 4 1 11  2 17  4  8  44 88 4 8 2 2 1 21 8 2 8 7 47 2 4 2 1 * 4. * 5. 7    1   4(24) 44 4 88 8 2 (24) 8 7 8

7 8

1 1  2 2

1 2 7 8

7 8

8

8

* 6. 1  1 (24) 2 2 7

* 7. 1  1 (24) 2 2

8

1 * 8. What percent of a circle is 21 of a circle plus 14 of a circle? 1 11 2 1 1 1   2 2 2 22 2 * 9. In the classroom library there were 23 nonfiction books and (23) 41 1 fiction books. What was the ratio 7of fiction to nonfiction books 8 4 in the library? 3 8 710. 7Explain 8(18) 8

3 8

How can you find the

3 4

1 3 4 4

3 number 8

1 3 4 halfway 8

1 4

1 4 3 8

1 4

between 123 and 321?

3 4

1 4 3 4

1 2

(Inv. 2)

3 8 3 4

1 2

1 4

1 4

1 4 1 4

1 1 8 4

Lesson 24

1 8

129

1 8

11. Mr. Chen wanted to fence in a square corral for his horse. Each side (8) needed to be 25 feet long. How many feet of fence did Mr. Chen need for the corral? * 12. (19)

Classify

Which of these numbers is not a prime number?

A 21

B 31

1 14. 22,422 ÷ 32 2 (2) 16. Compare: 1 2 (17)

13. 9  1000 (2)

1 15. $350.00 100 9 ÷ 1000 2 (2)

9  1000

* 17. (16)

3 4

C 41 1 4 1 4

1 4

Conclude Mr. Johnson rented a moving van and will drive from Seattle, 3 Washington, to San Francisco, California.1 On the way to San Francisco 4 8 he will go through Portland, Oregon. The distance from Seattle to 3 1 1 Portland miles, and the distance from Portland to San Francisco 8 4 is 172 8 is 636 miles. The van rental company charges extra if a van is driven more than 900 miles. If Mr. Johnson stays with his planned route, will he be charged extra for the van? To solve this problem, do you need an exact answer or an estimate? Explain your thinking.

* 18. If Mr. Johnson drives 172 miles from Seattle to Portland in 4 hours, then (23) the average speed of the rental van for that portion of the trip is how many miles per hour? 19. (10)

What temperature is shown on the thermometer at right? Connect

50°F

40°F

30°F

20. Round 32,987,145 to the nearest million. (16)

Math Language The abbreviation GCF stands for greatest common factor.

* 21. What is the GCF of 21 and 28? (20)

* 22. (21)

Classify

Which of these numbers is divisible by 9?

A 123

B 234

C 345

23. Write a fraction equal to 1 that has 4 as the denominator.

(Inv. 2)

Find each unknown number. Check your work. w 24. 25. 7x = 84 = 20 (4) 8 (4) 26. 376 + w = 481 (3)

130

Saxon Math Course 1

27. m – 286 = 592 (3)

w 8

Refer to the bar graph shown below to answer problems 28–30.

Population (in thousands)

16

Population of County Towns

14 12 10 8 6 0

28. (16)

29. (16)

30.

(Inv. 1)

Early Finishers

Real-World Application

Estimate

Eaton

Chester

Madison

Which town has about twice the population of

Eaton? Estimate

About how many more people live in Madison than in

Chester? Copy this graph on your paper, and add a fourth town to your graph: Wilson, population 11,000. Represent

Gina’s dance team is performing at a local charity event on Saturday. Some members of the team will ride in vans to the event, while others will ride in cars. Seven-eighths of the 112 members will travel to the event in vans. a. How many members will be traveling in vans? b. If each van can carry 11 passengers, how many vans will they need?

Lesson 24

131

LESSON

25

Writing Division Answers as Mixed Numbers Multiples Building Power

Power Up facts

Power Up F

mental math

a. Number Sense: 6 × 43 b. Number Sense: 3 × 75 c. Number Sense: 57 + 29 d. Calculation: 2650 − 150 e. Calculation: $10.00 − $6.25

1 8

1 f. Fractional Parts: 8

1 2

of 30

g. Measurement: Which is greater, 5 millimeters or one centimeter? h. Calculation: 10 × 2, + 1, ÷ 3, + 2, ÷ 3, × 4, ÷ 3

problem3 34 The digits 1 through 9 are used in this subtraction solving 4  15 problem. Copy the problem and fill in the missing digits. 12 3

New Concepts

Increasing Knowledge 1 1 8

___ − 452 3__

1 2

8

writing division answers as mixed numbers

We have been writing division answers with remainders. However, not all questions involving division can be appropriately answered using remainders. Some word problems have answers that are mixed numbers, as we will see in the following3 example. 34 15 4  Example 1 12 1 1 1 1 1 1 3 A 15-inch length of ribbon was cut into four equal lengths. How long 8 8 2 8 8 2 3 was each piece of ribbon? 3 4

Solution We divide 15 by 4 and write the answer as a mixed number. 3

3 34 4 4  15 12 3 Notice that the remainder is the numerator of the fraction, and the divisor is the denominator of the fraction. We find that the length of each piece of 3 ribbon is 3 4 inches.

132

Saxon Math Course 1

1 3

Example 2

A whole circle is 100% of a circle. One third of a circle is what percent 1 3 of a circle? 1 33 3 % 1 46 1 Solution 3  100% 3 6  25 9 1 24 If we divide 100% by 3, we will find the 10 percent equivalent of 3. 1 933 1 % 1 3 46 1 100% 3  1 6  25 46 9 33 13 % 1 24 4 6 6  25 100% 3 10 1 6  125 24 33 3 % 99 24 11 110 3  100% 46 1 25 25 1 1 9 Connect One third of a circle is 33 3% of9 a circle. 6  25 3 Notice that our 6 6 answer 10 1 1 matches our fraction 24 manipulative piece for 3. 9 1 1

1 3

1 3 1

33 3 % 3  100% 1 4 9 6 6  25 10 1 3324 3% 9 3  100%1 1 9 10 9 1 25 6

1 1 3 3

Example 3

1

1

25

33 3 % 3 6 1 25 25 25 1 1 11 1 4 33 % 1 33 % 6 16 33 3Reading % 4 Math Write 6 as a mixed number. 33 3 6 25 4 46 1 11 6 25 25 1 33 3 % 6  625 33 33% 3  3100%  100% 6 46 6 is read “25 6 25 6  25 6  24 Solution 3  100% divided by 9 6”.9 1 1 24241 33 3 % 9 1 3 25 25 25 1 1 1010 1 The 6fraction bar in 6 serves as a 1 division 33 3 % 104We 3 9 9 6 4 1 symbol. 6 divide 25 by 6 and 6 write the remainder as the numerator of the fraction. 9 1 1 1 6  25 46 1 1 24 4 6 6  25 6  125 24 1 46 24 11 46 6  25 1 6  25 24 25 25 25 1 1 1 1 1 25 25 3333 We fraction6 25 number 4 64. 16 3 3find that the improper 6 6 3% 6 6 6 equals the1 mixed 3% 24 1 25 1 33 3 % 3 6 1 multiples We find multiples of a number by multiplying the number by 1, 2, 3, 4, 5, 6, and so on. 1

4 64 16 6  625  25 2424 1 1

The first six multiples of 2 are 2, 4, 6, 8, 10, and 12. 1

4 6and 18. The first six multiples of 3 are 3, 6, 9, 12, 15, 25 6  The first six multiples of 4 are 4, 8, 12, 16, 20, and 24. 24 The first six multiples of 5 are 5, 10, 15, 20,125, and 30.

Example 4 What are the first four multiples of 8?

Solution Multiplying 8 by 1, 2, 3, and 4 gives the first four multiples: 8, 16, 24, and 32.

Lesson 25

133

25 6 25 6

Example 5 What number is the eighth multiple of 7?

Solution 1 2

The eighth multiple of 7 is 8 × 7, which is 56.

Practice Set

1 2

1 3

a. A 28-inch long ribbon was cut into eight equal lengths. How long was each piece of ribbon? b. A whole circle is 100% of a circle. What percent of a circle is 17 of a circle?

35 6 35 6

49 10 49 10

35 6

49 10

65 35 49quotient 49 65 35 the 49 c. Divide 467 by35 10 and write as a65 mixed number. 10 12 6 10 10 12 12 66 d. What are the first four multiples of 12? 3 1 8 in. 8 in. 65 65 35 49 65 35 49 e. What are12 the first six multiples of 8? 35 49 35 49 6 35 49 10 35 65 65 6512 35 49 49 65 356 4910 12 6 10 6 10 65 6 65 6 12 10 10 12 12of 12 6 is both 10 multiple f. Classify What number the third 8 and the second 6 10 12 12 multiple of 12? 1

1

1

1

1

1

1

123 3 as a 2mixed Write each of2 these improper12fractions number:9 2 2 65 35 49 65 35 49 65 35 49 g. h. 1 1 1 11 11 1i.1 6 10 12 6 10 12 6 10 12 2 2 3 22 22 33

1 9 11 33

1 2

1 7

3 8 3 8

in.

3 8

(12)

1 * 4. in.24) 8 (8,

in. in.

3.

1 7

1 8

in.

1 8

7 1 1 1 1 1 1 11 2 37 2 7 23 from 711 the 2 Earth’s average 33 2distance in. in. 8 in. 88in. 88in.

7 1 1 1 7 3is Sun

1 6

1 3

1 3

1 3

65 12

one 8 in. six hundred thousand kilometers. Use digits to write that distance.

in. * 5.

Formulate

1

1 8

3 in. 8

in.

1 8

in. 1 in. 8

1

1

1 3

A 30-inch length of ribbon was cut into 4 equal lengths. How 2 2 3 long was each piece of ribbon? Write an equation and solve the problem. 3 3 1 3 1 1 8 in. 8 in. 8 in. 8 in. 8 in. 6. Two thirds of the class finished the test on time. What fraction of the (Inv. 2) 1 class did not finish the test on time? 7 (25)

1 1 * 7. Compare: of 12 2 2 (22) * 8.

(Inv. 2)

Evaluate

1 1 of 12 3 3 1

1

1

1 9

12that 3is half of 2? What fraction is half 2 of the fraction 3

1

1 1 1 1 8 in. 8 in. * 9. A whole circle is percent 12 What 2 100% of a circle. 3 2 of a circle is 9 of (25) a circle?

134

1 3

1 3

1 1 1 1 7 1 7 3forty-nine 3 3 million, hundred

Connect 1 What is the perimeter of the 144 rectangle? 5 5 144 1  12 3 3 m 1 m1 1 1 12 100 100in. n  in. in. in. 8 88 3n 3 8 8 8 3 3 13 3 1 1 1 11 in. 1 8 8 in. in. in. in. in. in. 7 7 7 in. in. 8 8 8 8 8 8 8

Saxon Math Course 1

2



2



2

2 2 2   3 3 3 6 5 

1 6

1 3

Strengthening Concepts 5 55 5 21 22 22 2 61 65 5 1 1 1  1   1 3 1  1  2 3 3 6 6 12 33 33 3 6 6 12 1 112 12 1 1 2 2 3 2 2 3 * 1. What1 is the difference of 21 and the321sum 1 between 1 1 1 1 1 1the sum 1 2 and 1 2 (Inv. 2) 1 1 2 1 1 1 3 3 2 2 1 2 21 1 2 3 3 3 3 2 of 3 and 7 77 3 2 3? 3 35 49 2. In three tries Carlos punted the football635 yards, 30 yards, 10 and 1 (18) 37 yards. How can Carlos find the average distance of his punts? 1 1 7

1 2

1 2

65 12

1 6

1 6

Written Practice 1 1 2

65 12

1 6 5 2 6 6

1 6

1 3

12 5



5

5 1 5  12 2 12

12

1 2

2 2  3 3 2 2  3 3

144 n  12

1

1

Math Language 3 2 What operation 1 1 12 do 3 the words 2 1 “are in” tell us to 1 9 perform? 6 1

6 5  6 6 6 5  6 6 5 5  12 12

5 m1 8 5 m1 8

1 3 1 3

1

1

* 10. 9a. How many 6 s are in 1? (Inv. 2)

11 92b.

1

1

1

11 1 ? How 2many s are 1in 22612 23 2 1

1 1 6 2 1 1 1111 11 11 12 12 62 1 33 12 3 3 2229 12 2 1 1 1 12 33 3 % of a1 circle? 3 2 9

1

33 3 % 1 1 11 1 1 33 1 % 1 2 3 3 99 6 3 2 1 9 1 6

11 1 66 6 1 9 1 6

1 9

1 1 * 11. What 6 fraction 2 1 is 1 1 of 2a circle (25) 12 2 3 2 6 5 5 * 12. Divide  365 by 7 and write the answer as a mixed number. (25) 12 12 1 1 1 1 1 11 1 1 1 5 5 55 126 5 5516 55165 12 5 2 2 5 2 5 2227 2 3 662 2 9 656  95  23 2 222622512   2 2 2 * 13. 12  * 14.     666 6 12 12 12 1212 12 312 3 33 6 33 6333 3 663 (24) 126  12 (24) 3 6 1 1 12 63 53 3 5 15 2 2 2 12  3    5 2 5 2 6 5 2 2 2 3 3 3 6 6 12 12 * 16.  15.  30 × 40÷ 60  3 3 (5)1 3 6 6 121 1 1 (24) 12 1 1 12 2 3 2 9 6 6 17. A of6the twenty5 games the rest. What 52 won 5 played and lost 6 team 3 2 seven 2 2 2* (23) 1 5 5 5 2      in.  in.  was 3 3 3 6 6the 12 12 3 8team’s 3 3win-loss 6 8ratio? 6 12 12 6 5 2 2 2  much did 18. Formulate Cheryl bought 10 pens for 25¢ each. How 3 3 3 6 she 6 pay (15) 144 for all 10 pens? Write an equation and solve the problem. 5 5 6 5 2 2 2     n  12 3 3 3 6 6 12 12 144 5 144 1 5 144 144 5(GCF) 144 15 1 1 19. What is the greatest common factor 30? 12 144 1 m1  12 m   n24 1 m of 12n 1 5and 100 n  12 (20) 100 100 100 1 88 n m 8 12 n  1 m 8 n  12 8 5100 144 1    20. What 5number is 100 of 100? m 1 12 n 144 1 8 (22) m1 100 n  12 8 Find each unknown number. Check your work. 5 144 5 1 144 1 21.  m100 1  m  122. n  12 100 n 8 12 8 (Inv. 2) (4) 5 144 1 23. Estimate What is the sum of 3142, 6328, and 4743 to the nearest m1 100 n (16) 8 thousand? 5 144 1 m1 100 n  12 8 * 24. 1Model Two 1 thirds of the 60 students How many 11 11 1 1 liked peaches. (22) 1 1 22 3liked peaches? 33 Draw3a diagram that illustrates the of 2 the students 2 3 1 12 problem. 1 1 2 3 2 3 25. Estimate Estimate the length in inches of the line segment below. Then (17) use 1 1 an1 inch ruler to find 1 the length of the line segment to the nearest 2 3 2 of an inch. 3 sixteenth 1515 1 1 44 15 2 3 4 26.1 Represent To1divide a circle into thirds, Jan imagined the circle was the (Inv. 2) 2 face of a clock. 3 Describe how Jan could draw segments to divide the 15 circle into 15 thirds. 4

* 27. Write (25)

28.

(Inv. 2)

4 15 as 4

a mixed number.

Draw and shade rectangles to illustrate and complete this comparison: Model

33 44 434 545 5 4 * 29. What are the first four multiples of 25? (25)

3 30. (21)4

4 3 Which 4 of these numbers is divisible 2by both 9 and 10? 2 22 11 5 4 5 How do 32 3 32 3 21 2 3 you 4 know? 3 3 2 5 4 B 8910 C 78,910 A 910 Classify

2 3

2 3 2 3

2 3

2 3 2 3

1 2

1 2 1 2

Lesson 25

135

LESSON

26

Power Up facts mental math

Building Power Power Up C a. Number Sense: 7 × 34 b. Number Sense: 4 × 56 c. Number Sense: 74 + 19 d. Calculation: 475 + 125 e. Money: $5.00 − $1.75

3 6

4 8

2 f. Fractional Parts: 4

1 2

of 32

g. Statistics: Find the average of the following: 20, 25, 30 h. Calculation: 7 × 5, + 1, ÷ 6, × 3, ÷ 2, + 1, ÷ 5 2 6

1

problem 6 solving

1 8

1 8

2 3

1

James was thinking of a prime number between 75 and 100 that did not 6 have 9 as one of its digits. Of what number was he thinking?

Increasing Knowledge 3 1 4 2 2 of 32 8 6 4 3 1 4 2 32 fraction8manipulatives to You can 2 of use 6 model these 4fractions:

New Concepts 1 8 1 using 8

manipulatives 1 to reduce 8 fractions 1 2

1 2

4 8

2 6

Using Manipulatives to Reduce Fractions Adding and Subtracting Mixed Numbers

1 82 3

1 2

1 2

3 6

1 8

1 2 1 82 3

3

2

2 3

136

6

2 3

4 8

1 2

2 3

2 6

2 6

3 6

2

3 6

4 8

of 32

1 6

1 6

2 4

2

2 4

1 2

1

1

2

1 Model 2

We can use our fraction manipulatives to reduce a given fraction by making an equivalent model that uses fewer pieces. 1 6

1 2

2 3

2 3

Saxon Math Course 1

2 6

1 2

2 6

1 2

1 2 2 6

2 6

1 6

1 6

1 2

1 6

1 6

2 that each picture 6 6 6 We see 1 6 illustrates4half of a circle.3The model that2uses the 1 2 8 1 2 of 32 3 1 4 2 8 6 4 fewest pieces is . We say that each of the other fractions reduces to 12. 4 2 1of 32 2 8 6 4

2

Why doesn’t 3 1 1 reduce 1to 2?

2 6

2 6 1

2 Thinking Skill

Verify

1 2

1 8

1 2

1 2

1 6

1 6

1 2

2 6

1 3

1 3

2 6 1 33 3 % 1 � 33 3 % 2 66 3 % 1 33 3 % 1 � 133 3 % 6 2 66 3 % 1 6

4

32 3 %

8

1 1 8Example 1 8 1 1 1 3 Use your 3fraction 2 2 1 6 3 Solution 1 2

2 3

2 of 32 1 33 3 %

8 1 33 3 % 1 2

4 8

of 32 1

3 6

1

2 1 33 3 % 33 3 % manipulatives to reduce . 2 6

2 16 13 % 2 � 16 3 % 2 1 First we 2use our manipulatives to4 form 2. 6 32 3 % 6 2 2 1 1623 % 1 6 3 3 2 2 1 1 4 1 2 � 16 33 3 % 16 2 % 3% 6 16 % 6 3 3 3 4 1 32 3 % � 33 3 % 2 search for a fraction 1 Then we piece equivalent 2 2 66 3 3% 1 2 2 4 16 3 % 16 3 % 16 6 3 1 1 2 1 1 13 1 32% 1� 1 3 % 33 3 % 6 2% 6 3 16 6 6 3 2 1 2 The models illustrate that reduces to 13. 6 3 6

1 3

8

1 3 2 4

1 3

2 6

1 33 3 %

1 2

1 3

1 6

1 33 3 %

1 6

6

1 3

1 6

2

16 3 % 1 1 3 2 � 16 32%2 1 1 to the 46 model. We find . 6 3 3 32 3 % 1 13 1

4 33 3 %

2

16 3 %

1

13

3

11

1 3

1 3

1 3

1 1

1 1 3333 % 3333 3% 3%

3333 3 %3 %

1

1 2 1 1 1 1 %% ��3333 32% � 1 3 % 33 3 % 3 6 3 3 3 adding and When adding mixed numbers, we first add the fraction parts, 2and then we 2 3 2 4 1 1 1 1 66 66 % 3 % numbers, � 2subtracting 3 32 % 32% 1 % 33 % 33 3 % add 3the whole-number parts. when subtracting mixed 4 3 Likewise, 6 3 3 mixed we first subtract the fraction parts, and then we subtract the2 whole-number 2 1 1 1 1 16 % 1 1 % 33 % 33 16 3 % numbers 3 3 3 parts. 16 6 6 6 6 1 2 1 2 3 33 % % � � 16 33 % % � � 16 3 3 3 3 24 Example 2 4 2 4 2 1 1 32 3 % 66 3 % 32 3 % 66 3 % 3 � � 2 3 3 4 thirds of a circle is what percent of a circle? Thinking Skill Two 4

32 3 %

Justify

1 6

1 1 6 6

4

4

32 3 % 33 3 % 1 16 2 % 3 1 � 33 3 % 2 3 2 16 % 1 66� % 3 3

3

1 6

2

16 3 %

1 33 3 % 1 3 3% 2 66 3 %1 33 3 %

%

4

32 3 %



1 6

2 1 16 % 33 3 % 3 1 33 3 % 2 66 43 % 32 3 % 2 6 1 6

1 6

3

24

� 3 24



2 6

2

16 3 %

1 One 3

1

33 3 % 2 6

2 4

32 6 3% 1 3

1

1

� 1 3% 32% 3 1 33 3 %

432% 2 � 11 1 % � 32% 1 % 33 1 3 16 3 % 3 3 1 3

1 3

1 � 3332% 3%

2 2 � 1 1 1 16 3 % 33 3 %can be two 3thirds third equals 633 3 %. So 4 1 1 3 1 1 3 32 % 33 3 %32 4 and 33 32% . 3 3 4 1 33 3 % 4 1 2 2 16 3 % 16 3 % 1 6 �3 33 3 % 4 1 13 21 3 2 33 16 33% 66 3 % 2 1 21 1 1 1 �163 % � 16� �3 3% � 3 33 3 33 2 1 4 2 � 16 3 % 32 66 2 3% 61 � 1 13 % 333 13 %4 32%Example 6 3 32 3 %1 1 33 % 3 3 a circle is what percent of Two 2sixths of 1 2 14 16 3 % 616 3 % 63 a circle? 2 2 4 1 16 3 % 16 3 % 13 3 4 32 3 %

4 1 32 3 % 32% � 1 3 % 1 1 32% � 1 3 % 33 3 % 1 3

1

32% 32 3� %1 3 %

1

2 6

4

42 2 32 32 %% 16 316 33%3 %

12

1663 %

Explain how 1 � �1 1 1 1 1 Model to change3 3 to3 Use your fraction 33 3 %manipulatives 33 3 %to 3 1 1 1 11 1 � 33 3 %. 33 3 %represent 33 � 3. 3

2 6

1 3

Solution

1 3 1 33 % 3 1 2 1 23 3 1 1 1 3% 33 % 2642 433 3 % 3 6 3

3

1 1 3

1 3

1 33 3 %

3 11 found 33 1 3

1 1

1

by adding 33 3 3% 1 1 1 1 �� �� 3 3 3 3

33 3 % 16

1

13

� 16 �

1 6 33 1 % 3 12

16 1 33 % 32% �

2 16 3 % 2 16 3 % 24 16 32 33% %

32 2

16 3 %

12

4 3

1663 %

2 16 %

1 3 1 1 3 % 33 3 % 2 1

2 6

16 % 3 6

11

33 3333% %

2 6

1 3

3

24

1 3 3

24

Lesson 26

137

33 3 % 2

33 3 % 1 6 21 33 1 % 33 3 %33� 63 % 3 2 2 66 3� %33 1 % 66 3 % 3

16 3 % 1 16 3 % 1 1 1 33 3 % 33 3 % 3 3 2 2 1 1 % 1 1 1 1 � � 16 3 % 16�233 16 % 33 % 3 % 3 3 3 3 3 3 3 4 4 1 1 2 2 1 1 2 32 % 2 2 32 % 1 2 3�33 3333 % 3 3333 1633% 1616 16 3 % 3%% 3% 3% 3% 3% 3 %1616 3% Solution 2 2 1 3 2 6 2 1 4 2 1 2 2 3 3 3 1 1 2 1 % 16 16 3�%� 3333 �%16 3 %24% � 32 3 % 3333 �� 1616% 66 3 %  1 18  4 28 28 518% 1 8 8 � 33 5 823 % � 16 � 3% 3% 33% 8 3% 8 4 3 3 1 � 16 8 1 8 3 3 1 1 1 1 1 2 2 2 2 4 2 % 2 add2 16 % 2and 16 % 16 4 4 13 . 3% 4 14 3 22 % 4 216 3 � 16% 33 6 6 3% 6 � 16We 366 3 32 3% %2 % 3 32 % %3232 3 %3 3 %66666 32 3 % 6616 % 66 3% 3 3 3 3 3 3 3% 1 2 3 4 1 2 4 1 1 1 16 3 % 4 % 3 % 32 �2316 � 33 3 33 % 32 3 % 16 23 % 16 %3% 13 6 6 3 3 1 2 1 1 4 4 1 1 1 2 12 2 2 24 4 1 4 1 1 1 1 2 4216 % 2 2 2 1 1 % �233 � 32 66 3 16 16 16 %16 16 %3 %31 3 1 31 133 33% 3% 3 %1616 % 6 6 6 6 6 3% 3% 3 3 16 3 1 6 6 6 6 3% 3% 3 3 %1616 6 3 3 3 4 4 1 2 2 2 1 4 2 2 1 16466 16 3 % 1 16 3 % 13 1 2 3 1 2 1 3 6 21 3321 3 % 6 3 %3 % 416 3 % 1 1 1 �31%3 %8 233 3 % 32 3 % 1 32 3 % 32% �11 3 32% % 33 33 3 % 1 33 3 % 8 4 3 26 6 3 Math 6Language We notice that the fraction part of the answer, , is an improper fraction that 16 3 % 16 3 % 13 6 3 4 1 1 1 2 1 1 4 1 2 2 1 � 1 3.% 33 3 %16 % Recall equals 32that 32% 33 3 %1 16 3 % 3 %6 an 6 3 6 3 3 3 improper 2 2 4 4 1 1 2 2 1 1 2 11 1 4 4 1 1 4 11 11 1 1 1 11 1 � � � 32 % 32% 1 % 33 % 3333 So % equals 32% + %, which is %. This makes sense because 3232 % 32% 1 % 33 % 3333 32% 1 % 33 % 32 %3 �331 33% 33 32 %6 � 3 3% 6 6 33 3 3 3 3 6 3% 3 3 332% 3% 3 3% 6 1 3% 3% 3 32% 3 % fraction is a 3 2 1 1 1 1 2 1 1 1 1 3 3 1 2 2 13 � reduces to , which is the same as %. 1 % 33 % 33 % � 132% % 33 33 % 12 %  12 % 3 3 2 1 2 16 %  66 % 3 6 3 with 3 6 3 3fraction 3 a3 2 13 2 31 3 4 4 8 2 4numerator 2 42equal 1 1 4 32 3 % 32% � 1 3 % 33 3 % 33 % 6 3 3 to or greater 2 4 1 1 1 1 3 � 32 % 32% 1 % 33 % 33 3 % 2 Example 4 3 3 6 3 3 4 than the 3 3 3 3 denominator. 3 2 42 3 miles from school. 2 42 4lives 24 Rory He rode his bike from home to school 2 4 mi 4 3 3 1 1 2 2 3 and back to home. How far did  12 12 % % Rory ride? 16 %  66 % 3 2 1 1 1 1 1 2 4 mi 3 � � � � 2 2 3 3 4 4 3 3 3 2 mi 234 3 4 6 Solution 4 4 mi 3 � 2 4 31 � 13  2 34 mi 6 13 1 1an1 addition 1 has This pattern.1 1 2 34 mi 31 � � 13 1 �4�problem � � � 2 4� mi � 1 6 2 1 1 3 3 � 3mi 4 mi 3 234 3 3 3 3 3 3 1 2 (4  1 4  1 2 ) 12 3 1 3 1 2 4 mi mi 2 3 34 3 mi  2 2 4 3 4 2 mi32 4 mi  24  3 3 3 61 1 1 1 6 2 2 mi 1 1  mi 2 mi 2 mi  2 4 mi 6 6 � � 4 mi 4 4 41 2mi ) 12 4 45 2 3 3 1 2 ( 4 41 4mi 6 4 4 3 3 1 1 6 4 4 mi � �3  2 4 mi4 4 mi  2 4 mi mi 3 1 3

1 3

6

6

1 3 3 4 4 mi1 to 11 12 Q( 64  1 241  1 121)R3. So we mi of the 1 6 41part 21 answer 1 11 2 The 2reduces mifraction 2 41( 62 1 4 mi 4    5 1 ( ) 1 5 1 1 1 ) 1 52  1 1 1 6 2 1 12 4 2 4 421 2 22 1 2 6 answer 1 2 8 3 1of1 the 3 3 1 2 to the whole-number 1 2 (4  1 4  1 add (  1 4 and 1 2 ) find that8 Rory81rode 2 ) 2 3 mi  2 part 34 52 2 31 3224 2mi 2 4 2 1 41 4 mi 2 413mi 5 1mi 1  11 2 ) 1 miles. 6 6 1 2 2 1 1 2 1 1 1 1 3 bike 12 1 1 1 his 14286( 4mi 1 41  5 2 1 25 38  1 18  4 285 228 41 241  1 814225 (124 81 4  1 2 ) 5 24 6 2mi 1 2 4568mi 2 1 2)4 8 2 1 2) 3 3 4 4 4  2 4 mi  2 4 mi 3

mi

2 4 mi

3

1

58  18 1

18

3 8

2 8

1 8 1 8 1 12

6 8 6 3 8 8

1

1

52 12

1

1

52

53 2 8

1 4

6 6 3 3 1 1 2 1 6 Example 21 6 5 1 32 4 4 mi 31 2  3 51 1 1 3 311 31 1 2 1 11 1 414 6mi 2 1 12 2 2 2 4 8 82 86  8) 84  51182 ) 1 8 1822 5 1 ( 1 1 1 5 11 2 ( 4  1142  )184  5 ( 4 1235     5 1 4 4 5 1 4 2 4 4 2 2 23 2 83 8 8 8 12 8 82 8 8 3 31 8 1 2118118  4 28 128 8 8 84 4 48 5  1 4 8 3 3 1 155828  8 8 Subtract: 8 84 8 3 2 2 1 1   8 8 1 4 5 1 63 2 1 1 1 6 842 11 38 1 3 35 1 21 211 1 3 4 4 1 2 2 3 4 2 8 3 38 (1 83 21 1 162 1 2 ) 51 81 11 1 281 2 ) 1   5 1 ( 1 2 8 8 4 54 81 14 2 2 8 58 54 841 14 84 8 28 8 5 82  58  18  4 8 8  2 48 2 8  2 4 48 8 5 82  44 88 8 484 88 8 4 4 4 8 8 1 1 52 5 2 Solution 63 21 2 3 1 3 3 112 1 3 2 21 1 2 22 113 2 1 from 36  1 1888538 difference  148 28 1 from , and 6we538 8subtract 5 8  1 8 5 8 We 5 8resulting 48 188  4 8is 48 8. 45448   1subtract 4 85. The 8 8 8 4 8 8 8 6

8

23

48 8

2 8 Practice

1 1 12 %  12 % 2 12 12 % 2 2 12%  12 %  66 % 16 3 23

6 2 3 8 1 8 5 1 1 31 83 1 1 2 8 2 3 3 3 3 2 2 21   8 8 1 4 5 6 58  2 188 8 8 8 86 2  48 8 5 8  418  4 8 8 58  1 8 81 1 8 2 1 3 2 1 28 12 82 1 3 2 1 2 1 1 8 1reduce  543828  the 5 84 81 8 8to448 and 4 8 5 8 4We 4 4 4the answer 8 write 44 4 as 4 4. 8 8 8 1 fraction 6 2 6 6 2 1 1 2 12 %  12 % 16 % 8 8 8 8 8 Model Use your fraction manipulatives to reduce Set these fractions: 2 2 3 6 6 2 2 3 3 1 1 2 2 1 7  12 %  66 12a. % 3 2 1 2 16 % b. 8 8 8 3% 28 2 3 4 4 8 8 Add. Reduce the answer when possible. 3 33 2 %3 66 1 2 1221 %2  1221 % 1 2 %17 1 1 1 16    12  2   122%12 % % 66 % 3 2 1 16 % 66 % 3 2 16 c. 122 %12 % d. 2 2 3 3 3 3 3 13 1 3 7 42 44 2 4 8 3 88 3 1 2 12 %1  2 16 2% 266 2% 2  12 % % 66 % 16 3 42 1 4 72 8 8 3 3 2 23 3 32  66 3 37 33 1 2 1 % 2 7 2 1 12 16 % 37 2 212 13%  12 11%3 4  12 %2 66 % 16 %  66 3  1  32 16 3 2%  42 3%  f.12 81 1%8 e. 66 3  12  32  %22 2 16 2 2 3 3 48 3 3 4 4 8 43 4 8 3 48 4 8 8 8

8

3 58

1 18

3 122 12 3 3 332 31 2 7 11 1 2 11 1 7  2%1  2 %21 66 3  12 % 12 12 % % 12 % 66 %% 12 3  2 316 16 % 12 66 % 3  16 233 23 4 2 3 4 4 443 22 2 48 3 82 28 8 32 3 37 1 1 2 1 1 12 2 3 1 3  1  32  2 16 %  66 % 12 %  12 %2 16 %  66 3 4 12 2 %  12 2 22 % 3 43 4 8 2 23 3 48 138 Saxon Math Course 1 2 2 23 7 3 1 33 7 3 1 3 71 22  66 3% 2 1666% % 3 1 23 2 2 1  21  2 %  6616 %%  3 3 34 8 4 8 44 8 4 8 88 33 4 3 1 3 3 1 1 3 2

3

3

3

3

3

3

3

3

2

2 4 mi 2 4 mi2 4 mi2 4 mi

mi2 4mi2 4 mi  2 4 mi  24  3 1 2 24 6 mi2 4 6 mi4 6 mi4 6 3mi 33 2 17 7 g.163 + 2 % h.32 +24 1 12  2 66% % 34  66 % 16 4 3 34 3 4 4 3 44 4 8 88 883 1 %

1 1 1 1 12% 12 % 12 % 12 % 2 2 2 2

i. Use words1 to6 write f.1 1 1 1 2 6the1 addition 162 1 6 21 problem 21 1in exercise 1  1 2 ( 4 112 4( 412112( 4)41 14 42)114 2 ) 1 2) 1 2 1 2 5 21 2 2 (1

Written Practice

1 1 83 3 %83 3 %

3 34

3 34

1

72

1 62

1 62

1 22

1 22

2

1

62

3

1

52

1

1

52 52

Strengthening Concepts

2 53

3

2 53

3

44

44

1 * 1. Maya rode 3her bike to the2 park and back.1 If the trip was 13 34 miles 1 1each 2 2 (26) 34 7 of 3 2 2 2 2 way, how 3far did1 3she ride 31 1 in3 1all?1 1 3 3 3 1 31 2 2 2 32 13 5 8  158 8  5188 8518 8 81 8 8 8 8 4 4 4 8 5 84 8 158 8 8 8 8 8

2. The young elephant was 36 months old. How many years old was the elephant? 1 7 5 7 3 142 3. Justify Mrs. Ling 2 12 % dozen balloons for the party. 7 bought12 5  5 Is this 8 2 8 2 62 (6, 15) 6 6 86 2 enough balloons for 30 children to each get one balloon? Explain 8 8 8 88 8 8 8 your thinking. (15)

1

72

4. There are 100 centimeters in a meter. There are 1000 meters 3 in a1 12 1 1 4  1 5 13 37 % 2 kilometer. How are in a12kilometer? 8 58 2 many centimeters 2

3 1 5 1 8 8

(15)

1 4

1 4

1 4

31 44

3 4

3 4

3 4

* 5. What is the perimeter of the equilateral 2 in. 2 1 1 1 1 triangle shown? 3 3 3 31 1 34 3 2 2 7 2 of 2 5 51  1 3 31 12 2 8 88 8 5 51 14  4 8 88 88 8 22 1 1 3 3 1 1 2 1 3 1 1 324  4 of3 2  n  1 3 342 1  37  2 2 2 2 3 5 1 5 1 1 13 3 11 1 1 2 4 2 2 2 422 2 22 3 2 332  12 %  66 % * 6. Compare: 12 12 % 12 % 66 66 % 66 % 16 % 16 16% 12 % 12 %12% 12 % %16% % 3  23 plus of % 2 2 2 2 22 2 2 3 3 3 3 33 3 3 4 44 (Inv. 2) 2 1 1 1 1 3 34 7 of 3 2 2 2 2 5 7 3 1 14 * 7. 5  7 12 2 % 5 8 8 8 (26) 3 1 71 15 31 1 5 percent141of 6 4a circle is 3 of 2 1 a circle. 1 5 71 4 3 3 % of  a7 circle is 12 * 8. One 51 5of eighth What 7 12 % 5 2 8 34 7 of 2 8 1 1 1 1 8 (26) 3 28 2 1 2 1 82 2 1 8 2 3 3 3 1 83 % 2 32 36 6 56 5 5 45 4 4 4 a circle? 83 3 % 83 3 %8363 % (8, 24)

2

2

2

32

3

3

43

4

4

4

5 7 3 1 14 7 12 2 % 1 a fraction equal 15 8 to 4 8of 12. 5 9. 13 8that 12 Write 1 has a denominator 37 % 2 5 1 1 2 1 12 1 1 2(Inv. 2) 1 22 1 1 12 1 3 3 3 34 34 of 2 7 2 of 2 2 2 of 2 7 3 34 2 3 2 7 3 2 1 12 1 2 factor 12 1 and 25? 14 10. What is the greatest common of1315 13 37 % 37225% 2 12 13 2 12 3 7 1 5 (20) 1 1 14 2 12 3 3 3 3 1 1 1 1 34 7 of  7 5 12 2 %2 3 2 238 3 3 4 2 23 4 252 2 2 2 2 8 28 4 11. Generalize 4 Describe the rule of the following sequence. Then find 4the 12 1 1 (10) 13 2 37 2 % 2 12 seventh term. 3 3 2 4 2 1 5 1  n1 1 3 3 2 2 5 1 35 51 3 5 7 5 17 3 3 4 40, . . . 4 2 2 7 14 14 16, 24,14 8, 32, 5  7 5  71252 % 712 2 % 8 12 2 % 8 5 5 8 5 8 8 8 8 8 18 3 3 2 4 2 1 1 21 4 23 3 12 1 41 n 2 3 2  n1 1 5 7 3 1 14 51 5 1 13 37 % 2 1 1 3 4 4 2 2 5 5 5 3 4 2 2 12 * 12. Write 5 as 5 7 12 2 % 7 2 a mixed 7 2 number. 72 72 8 8 8 (25) 3 3 2 4 2 * 13. Add and simplify:  n1 3 1 3 3 1 1 1 5 5 3 4 4 (26) 1 64 3 4 12 4 1 4 125 1 1 1 12 1 1 4 2to 13 2 2 512 Remember 13 2 3713 37Find 25 5 check your work. 2 2 % 12 2% 2 % 12 the37unknown number. 3 1 1 1 1 11 1 6 1 54 12 1 1 4 5 3 3 3 2 4 2 1 31 4 4 13 2 37 2 %  12 113 31 3 1 3 1  3 2 2 14.  n 25 13 5 5 5  15 451 3 5 1  134 231 3 12 21 2 22 (Inv. 2) 3  8 88 1 885 88 15 8 451  5 14 1 4 1 4 3 1 8 8 8 8 88 8 88 8 8 8 1 5 greatest factor 64 15. Classify What is the of both 12 and 3 4 18? 2 4  5 5

2 4  5 5

(20)

3 3 23 1 3 3 1 3 1 1 2 42 2 42 1 1   16.n  17. 1 3 1* n1 1 3 3  32 3  2 32 *18. 1n 1 2 2 2 2 5 5 3 15 5 3(Inv. 2) 1 3 4 23 4 2 2 (26) 2 2 4 1 4 (26)4 4 3 3 2 1 1 64 4  n  11 5  1  19. 3 Classify 3  3 2 2 3 4(21) 4 Which of the 2numbers 2 below is divisible by both 2 and 3?

1

1

15 1

15

15 1 3

1 3

1

1 3

15 1 4

A 4671 1

1 1 4 3

4

31

6 44

B 33858 3 64 64

C 6494

3

64

Lesson 26

139

20. List the prime numbers between 30 and 40. (19)

21. (16)

* 22. (26)

Find the difference of 5063 and 3987 to the nearest

Estimate

thousand. Model

Use your fraction manipulatives to reduce 68.

3 5

1 , 0,  2

23. At $2.39 per pound, what is the cost of four pounds of grapes? (23)

* 24. (22)

6 How much money is 35 of $30? Draw 1a, diagram 0,  1, 1to illustrate the 8 2 3 3 problem.

Model

4

25.

(7, 17)

Connect

5

a. How many millimeters long is the line segment below? 3 4

3

mm 10 5

20

30

2 5

 1, 0, 12, 1

b. Use an inch ruler to find the length of the segment to the nearest sixteenth of an inch. 26. Arrange these numbers in order from least to greatest: (17)

6 8

1  1,1 1 , 0, −1, 2

3 5

Adriana began measuring rainfall when she moved to her new home. The bar graph below shows the annual rainfall near Adriana’s home during her first three years there. Refer to this graph to answer problems 27–30. 3 3 2 1  1,Years 0, 2, 1 5 5 4 Rainfall Amounts for First Three Rainfall (in inches)

60 50 40 30 20 10

First Year

Second Year

Third Year

27. About how many more inches of rain fell during the second year than (16) during the first year? 28. What was the approximate average annual rainfall during the first three (18) years? 29. The first year’s rainfall was about how many inches below the average (18) annual rainfall of the first three years? 30. (11)

140

Formulate Write a problem with an addition pattern that relates to the graph. Then answer the problem.

Saxon Math Course 1

2 5

LESSON

27

Measures of a Circle Building Power

Power Up facts

Power Up E

mental math

a. Number Sense: 7 × 52 b. Number Sense: 6 × 33 c. Number Sense: 63 + 19 d. Number Sense: 256 + 50 e. Money: $10.00 − $7.25 f. Fractional Parts:

1 2

3 4

of 86

1 4

g. Geometry: The perimeter of a square is 16 ft. What is the length of the sides of the square? h. Calculation: 8 × 8, − 1, ÷ 7, × 2, + 2, ÷ 2

If

+

= 25, and ×

then

New Concept

c

u

m

i

r

C

f

Diameter

Radius

e

r

e

Why is the diameter of a circle twice the length of the radius?

=?

There are several ways to measure a circle. We can measure the distance around the circle, the distance across the circle, and the distance from the center of the circle to the circle itself. The pictures below identify these measures.

c

Verify

= 5,

Increasing Knowledge

n

Thinking Skill



e

problem solving

The circumference is the distance around the circle. This distance is the same as the perimeter of a circle. The diameter is the distance across a circle through its center. The radius is the distance from the center to the circle. The plural of radius is radii. For any circle, the diameter is twice the length of the radius.

Lesson 27

141

Activity

Using a Compass Materials needed: • compass and pencil • plain paper A compass is a tool for drawing a circle. Here we show two types:

cm

1

in.

2

3

1

4

5

6

7

2

8

9

3

10

11

4

To use a compass, we select a radius and a center point for a circle. Then we rotate the compass about the center point to draw the circle. In this activity you will use a compass and paper to draw circles with given radii. Represent Draw a circle with each given radius. How can you check that each circle is drawn to the correct size?

Thinking Skill Discuss

Why do we use the length of a radius instead of a diameter to draw a circle?

a. 2 in.

b. 3 cm

3

c. 1 4 in.

Concentric circles are circles with the same center. A bull’s-eye target is an example of concentric circles. d.

Represent

Draw three concentric circles with radii of 4 cm, 5 cm, and

6 cm.

Example 1 What is the name for the perimeter of a circle?

Solution The distance around a circle is its circumference.

Example 2 If the radius of a circle is 4 cm, what is its diameter?

Solution The diameter of a circle is twice its radius—in this case, 8 cm.

Practice Set

In problems a–c, name the described measure of a circle. a. The distance across a circle b. The distance around a circle

142

Saxon Math Course 1

c. The distance from the center to the circle d.

If the diameter of a circle is 10 in., what is its radius? Describe how you know. Explain

Written Practice 1.

Strengthening Concepts What is the product of the sum of 55 and 45 and the difference of 55 and 45? Analyze

(12)

* 2. Potatoes are three-fourths water. If a sack of potatoes weighs (22) 20 pounds, how many pounds of water are in the potatoes? Draw a diagram to illustrate the problem. 3.

There were 306 students in the cafeteria. After some went outside, there were 249 students left in the cafeteria. How many students went outside? Write an equation and solve the problem. Formulate

(11)

* 4. a. If the diameter of a circle is 5 in., what is the radius of the circle? (27)

b. What is the relationship of the diameter of a circle to its radius? 5.

Which of these numbers is divisible by both 2 and 3?

Classify

(21)

B 123

A 122

6. Round 1,234,567 to the

(16)

1 1 2 2 in. If ten pounds4of apples costs $12.90, what is the price per pound? Write an equation and solve the problem.

1 1 2 2 in. 8. 4 What is the denominator of

(6)

23 24

1 your5fraction 6  6

5

1 6

1 8

2 5

1 8

2 56 5   8 86 6

7

(Inv. 2)

2 3

2 3

2 3

1 8

3

1 2

1 8

1 3

1 4

1 2 1 8

2 3

1 3

3

1 8

7 6 7 6

432 18 432 18

5 14. 2  8 8

1 27 18 2

1

16

1 1

126

1 6

1 4 1 4

1 4

4 5

7 1 6 4 8 5

7 6

1 4

1 8

4 5

7 8

1

1 2

432 18

7 6

4 4 5

1 2 331 % 1 16 % 4 3 3 1 1 61 22 2 5   33 %33 16 % %16 % 3 3 3 3

4 5

1 16

2 3

(Inv. 2)

What fraction of a circle is 50% of a circle?

1 1 1 37 % 112 % 6 1 2 2 1 1 8 1 1 1 2  12 37 %37 %12 % % 2 2 2 2 1 1 4 16 5 2

1 2

1 2

3

15. 181a. How8many 181s are 1in 1?

(Inv. 2)

1 8

1 8

1 8

2 5

manipulatives 2 5 7  8

2 1 1 12 16 8 b. How 8many 8 s are in 2 ? 1 1 7 7 4 *23 16. Reduce: 2 6 8 8 6 (26) 1 1 1 1 1 17. fraction is2 half of 4? 8 (Inv. 2) 8What 2 1 1 1 1 1 1 1 1 8 8 8 1 8 2 6 2 18. 8

1 8

1 8

3 5

5

1 2 31 7 31 213. 36 57 3      2 in.   6 2) 66 68 8 6 6 62 8 86 (Inv. 1 2

3 5

3 5

Model 3 1

3 5

1 6

1 3

Use to help answer problems 11–18. 7  2 8 8 1 2 63 6 7 31 8 2 8 36 57 3 2 56 5 2 5 1 11. 1  1  1   7  7 12.    268 6 66 86 8 6 26 6 2) 68 8 1 8 86 6 2 8 8 2 (Inv. 2) 326 8(Inv.

1 2 3   6 6 6

1 3

3 5

Model

(22)

1 2

23 ? 24

23 2 4 What number is 35 of 65? Draw a13diagram to illustrate the 24 3 6 6 5 1problem. 2 3 7 3 2 5      6 6 6 8 8 6 6 8 8 6 5 26 5 1 2 31 3 2 37 3 17 3 2 5 42 5           How8much to illustrate the 2 6* 10. 6 6 Model 6 6 68 6 is356 36of $15? 8Draw358 6a8 diagram 8money 6 8 38 (22) 5 5 problem.

* 9.

3 5

1

2 2 in.

Formulate4

(15)

1 4

1

4 ten thousand.

3

7.

3 4

C 132

3 4 nearest

7 6

432 18

7 6

432 18

432 18

1 4

432 18 1

16

Lesson 27

4 5

143

8

8 1

1

1

1

19. Divide 2100 by 52 8 8 and write the2answer with a remainder. 4 (2)

432 18

7 6

20. If a 36-inch-long string is made into the shape of a square, how long will (8) each side be? 5 6

21 5  82 8

432 7 Convert 6 to a mixed number. 6 56 5 3 1 21 32 37 37 18 2 52 5          6 6 6 8 8 6 6 8 88 8 6 6 6 8 8 6 6 432 22. 23. (55 + 45) ÷ (55 − 45) 18 (2) (5)

1 * 21. 4 (25) 7 6

1 4

24.

Classify

(21)

Which of these numbers is divisible by both 2 and 5?

1 1 1 2 C 250 A 502 B 205 37 %  12 % 33 %  16 % 2 2 3 3 25. Justify Describe a method for determining whether a number is (21) divisible by 9. 1 1 433 %

3

2 6 % 3

Which prime 432 number is not an odd number? 2 7 26. (19)  16 % 3 6 1 1 1 1 1 18 1 1 8 8 for 8the perimeter 2 8is the name * 27. What of2 a circle?4 (27)

28. (23)

2 %  16 % 3

Early Finishers

Real-World Application

144

What is the ratio of even numbers to odd numbers in the square below? Explain your thinking.

3 4

1 2

7 6

Explain

1 1 1 1 * 29. 37 %37 % 12 %12 % 2 2 2 2 (26)

1 2

7 6

1 4

1 2

1 2

1 4

3 4

3 1 4 2

1

2

3

4

5

6

7

8

9

1 1 2 2 *33 30.%33 % 16 %16 % 3 3 3 3 (26) 11 1 4 22 2

1 3 4 4

33 3 44 4

1 4

11 1 44 4

While the neighbors are on vacation, Jason is taking care of their dogs Max, 3 1 1 Fifi, and Tinker. Max needs 2 14 cups of food. Fifi needs 34 cup of food, and 4 4 2 1Tinker needs 13 cups. How much 1 food will Jason need to feed all three dogs? 4 4 2 Show your work.

Saxon Math Course 1

432 432 18 18

LESSON

28 Power Up facts

Angles Building Power Power Up F

mental math

a. Number Sense: 8 × 42 b. Number Sense: 3 × 85 c. Number Sense: 36 + 49 d. Number Sense: 1750 − 500 e. Money: $10.00 − $8.25 f. Fractional Parts:

1 2

of 36

g. Measurement: Which is greater, 9 inches or one foot? h. Calculation: 8 × 4, + 1, ÷ 3, + 1, × 2, + 1, ÷ 5

problem solving

Zuna has 2¢ stamps, 3¢ stamps, 10¢ stamps, and 37¢ stamps. She wants to mail a package that requires $1.29 postage. In order to pay exactly the expected postage, what is the smallest number of stamps Zuna can use? What is the largest number of stamps she can use? If Zuna only has two 37¢ stamps, what is the fewest number of stamps she can use?

New Concept

Increasing Knowledge

In mathematics, a plane is a flat surface, such as a tabletop or a sheet of paper. When two lines are drawn in the same plane, they will either cross at one point or they will not cross at all. When lines do not cross but stay the same distance apart, we say that the lines are parallel. When lines cross, we say that they intersect. When they intersect and make square angles, we call the lines perpendicular. If lines intersect at a point but are not perpendicular, then the lines are oblique.

parallel lines

perpendicular lines

oblique lines intersecting lines

Where lines intersect, angles are formed. We show several angles below.

Lesson 28

145

Math Language A ray has one endpoint and continues in one direction without end.

Rays make up the sides of the angles. The rays of an angle originate at a point called the vertex of the angle.

vertex

Angles are named in a variety of ways. When there is no chance of confusion, an angle may be named with only one letter: the letter of its vertex. Here is angle B (abbreviated ∠B):

B

An angle may also be named with three letters, using a point from one side, the vertex, and a point from the other side. Here is angle ABC (∠ABC): A

B

C

This angle may also be named angle CBA (∠CBA). However, it may not be named ∠BAC, ∠BCA, ∠CAB, or ∠ACB. The vertex must be in the middle. Angles may also be named with a number or letter in the interior of the angle. In the figure below we see ∠1 and ∠2. Thinking Skill Analyze

What figure is created when we add the measures of ∠1 and ∠2?

2

1

The square angles formed by perpendicular lines, rays, or segments are called right angles. We may mark a right angle with a small square.

right angles

Angles that are less than right angles are acute angles. Angles that are greater than right angles but less than a straight line are obtuse angles. A pair of oblique lines forms two acute angles and two obtuse angles. obtuse angle acute angle

acute angle obtuse angle

146

Saxon Math Course 1

Example 1 a. Name the acute angle in this figure. R

Q

M

S

b. Name the obtuse angle in this figure. Reading Math When we use three letters to name an angle, the middle letter is the vertex of the angle.

Solution To avoid confusion, we use three letters to name the angles. a. The acute angle is QMR (or RMQ). b. The obtuse angle is RMS (or SMR).

Example 2 In this figure, angle D is a right angle.

D

A

a. Which other angle is a right angle? b. Which angle is acute? C

c. Which angle is obtuse?

B

Solution Since there is one angle at each vertex, we may use a single letter to name each angle. a. Angle C is a right angle. b. Angle A is acute. c. Angle B is obtuse.

Practice Set

a.

Use two pencils to approximate an acute angle, a right angle, and an obtuse angle. Represent

Describe each angle below as acute, right, or obtuse. b. b. b. b.

e.

c. c.c. c.

Connect

d.d. d. d.

What type of angle is formed by the hands of a clock at 4

o’clock? f.

What type of angle is formed at the corner of a door in your classroom? Connect

Lesson 28

147

g. Which two angles formed by these oblique 1 lines are acute angles? 3

1

4 3

2 3

2

1 3

h.

Model

Draw two parallel line23segments.

i.

Model

Draw two perpendicular lines.

3

Refer to the triangle to answer problems j and k.

1 3

1 3

1 3

j. Angle H is an acute angle. Name another acute angle. 1 3

2 3

2 3

k. Name an obtuse angle.

H

Written Practice 1 3

23 3

Strengthening Concepts 2 2 1 1 33 3 2 23 1 1 122 * 1. 3What is3the sum of 3 and 3 3 3 and 3?

11 33

2 23 3 3 (24)

* 2. (22)

23 3

2323 33

3

2

2 3

* 9. (26)

10. 4 8

(22)

Model

1 2 2 2 222 5 4 4 2 5  m31 14 7.1 * 8.  14 3  1 3  43 3  1 3 41*(26) 6 6 3 3 3 333 6 6 (26)

44 88

2 2  1 Draw a diagram to illustrate How much money is 23 of1$24.00? 3 3 4 the problem. Model

8

3 10 2 3

3 b. What is210 of 2100%? 2 3  43 1 1 3 3 12. 4Twenty-five percent4of 4a circle is what fraction of a circle? 8 3 8

(Inv. 2)

1 10

8

10

8

Find each missing 2 number. Remember to check your work. 2 1 3 14.  m  13 15. 423 − w = 297 (Inv. 2) 4 (3)

2 3

4 8 22

148

3  43

3 3

2 3

5 4 3 1 6 6

25 5 2 4 4 3 343 3 4311 3 66 66

2 to drive on 13. At 26 miles 1per  gallon, m  1 how far can Ms. Olsen expect (23) 4 3 11 gallons of gas? 4

1 m1 4

2 3

233 333

2

33 Saxon Math Course3 1

5 3 1 6

Use your fraction manipulatives to reduce 48.

(22)

4 8

2 3

to a mixed number.

1 11. a. What is 10 of 100%? 1 10

3 3

8

23 3

22 22 2 2 2 2 11 11 3  4* 36. 1  1 3  4 3 33 33 (26) 3 3

4 8

G

Seven hundred sixty-eight peanuts are to2 be shared equally 2 1 1 3 3 by the thirty-two children at the party. How many peanuts should each 1 2 23 4 3 3 3 child receive? Write an equation and solve the problem. 8 3 1 4 2 210 2 2 2 2 5 2  1How  Declaration 1 3  4 3 of Independence 3  4 3in 310 3  43 1 4.1 Formulate 1The was signed 1776. 6 6 3 3 3 (13) 3 many years ago was that? Write an equation and solve4the problem. (25)

10

23 3

3 3

Formulate

* 5. Convert 2 2 1  31 3 3

3 3

2 3

2 2 people According to the 2000 census, about 5 of the 202 million 1 1 3About3how who lived in Texas at the time were under the age of 24. 1 3 old in 2000? Explain how you found many Texans were under 24 years 23 23 23 your answer. 3 3 3

3.

2 5

F

Explain

(15)

2 2 1 1 3 3

23 3

2 3

2 3

2

3  43

* 16. (28)

Represent

Refer to the figure below to answer a and b. Q

P

S

R

a. Name an obtuse angle. b. Name an acute angle.

1 10

17. On the last four tests the number of questions Christie answered (18) correctly was 22, 20, 23, and 23 respectively. She averaged how many correct answers on each test? 18. The three sides of an equilateral triangle are of equal length. If 1a m1 (8) 4 36-inch-long string is formed into the shape of an equilateral triangle, how long will each side of the triangle be? 19. What is the greatest common factor (GCF) of 24, 36, and 60? (20)

20. 10,010 − 9909

21. (100 × 100) − (100 × 99)

(1)

22. (22)

(5)

3

1 10

If of the class was absent, what percent 10 of the class was absent? Draw a diagram to illustrate the problem. Model

* 23. Divide 5097 by 10 and write the answer as a mixed number. (25)

24. (22)

Three 1 fourths of two dozen eggs is how many eggs? Draw a m1 diagram to4illustrate the problem. Model

* 25. a. Use a ruler to find the length of the line segment below to the nearest (7, 17) sixteenth of an inch. b. Use a centimeter ruler to find the length of the line segment to the nearest centimeter. * 26. (25)

27.

(Inv. 2)

1

1

of 3 List the first five 2multiples of 6 and the first five multiples of 8. Circle any numbers that are multiples of both 6 and 8. Analyze

Model

1

1

Which fraction manipulative covers 2 of 3?

28. There are thirteen stripes on the United States flag. Seven of the stripes (23) are red, and the rest of the stripes are white. What is the ratio of red stripes to white stripes on the United States flag? 1

1

2 of 3 29. Here we show 24 written as a product of prime numbers: (19)

1 2

Show how prime numbers can be multiplied to equal 27.

1

of 3

2∙2∙2∙3

30. (21)

Classify

A 234

Which of the numbers below is not divisible by 9? B 345

C 567

Lesson 28

149

LESSON

29 Power Up facts

1 8

mental math

Multiplying Fractions Reducing Fractions by Dividing by Common Factors Building Power Power Up B a. Number Sense: 7 × 43 b. Number Sense: 41× 64

1 4

2

c. Number Sense: 53 + 39 d. Number Sense: 325 + 50

1 1 1   2 2 4

e. Money: $20.00 − $17.25 f. Fractional Parts:

1 2

of 70

3 4

g. Measurement: Which is greater, 10 millimeters or one centimeter? h. Calculation: 4 × 5, − 6, ÷ 7, × 8, + 9, × 2 3 4

problem solving

Vivian bought a pizza and ate one fourth of it. Then her sister ate one-third of what was left. Then their little brother ate half of what his sisters had left. What fraction of the whole pizza did Vivian’s little brother eat?

New Concepts 1 1 multiplying 8 8 fractions

Increasing Knowledge

1 11 11 1     2 22 24 4 1 8

1 8 1 8

1 8

3 4

1 2

1 2

1

1

1 8 1 4

We see that 2 of 2 is 4.

Reading Math 1 The 3“of” 3in “2 1 of 2”4 means to 4 multiply.

When we find 2 of 2, we are actually multiplying.

1 2 3 4

1

1 2

1 11 111 1     2 22 224 4 1 11 11 1     2 22 24 4 3 4

1

Below we have shaded 2 of 2 of a circle.

1 2

1

1

1 4

1 4

1 1 1   2 2 4

1 4 1 4

1 4

3 4

3 4

1 2

1 4 1 2

When we multiply we multiply the numerators to3 find the numerator 1 fractions, 1 3 3 2 we 2 and 4 denominator of 4 4 multiply the denominators to find of the product, the 1 1 3 3 the product. 2 2 4 4

3 4

150

1

1 4

Saxon Math Course 1

3 4

3 3 4 4

4

3

2

1

3



2



1 2

4

Example 1 6 12 1 2 2 1 1 2

1 8 4 1 1 2

1 1 1 1 2  1 21 2 4   2 4

1 8

1 3 3  32  1 4  38  3 1 4 2 4 8  4

What fraction is 2 of 4?

1 8

1 4

1 1 11 1  2 2 22 4

31 42

Solution 1 of in The word 2 1 1 3 of . 2 2 4

1 3 3  3 2 44 18 2

1 2

1 4

3 4

1 2 1 14 1 the8question 2

3

1

4 3 4

1 4

4 21 83 3   1 32 34  8 1 2

(1 × 3 = 3) (2 × 4 = 8)

8 3

3 1 3 1 1 1of 3 is 3 Model 4 We find that illustrate this 2 4 8. You can 1 4 1 3 2 2 4 3 3 1 3 fraction 2manipulatives by using three 4 s to 1   3 make 4 of 2 4 8 8 1 2 covering half of that area with three 8 s. 4 1 22

3 2Example 3  3 8 4 33 1 3 8 2 8 1 12  Multiply: 2 3 24

3 4

3

4 means to multiply. We1 multiply 2 and 4 to find

3 2  1 3 3 4 3   2 4 8 1 3 33   2 4 48 3

3 8

3 4

3

with your 1 4 2 a circle, then

1 2

3 4

1 4 3 3 34 44

3 8

2 3

3 8

14 1 1 1   2 6 2 2 4 3 2 6   Solution 4 3 123  2 12 3 3 1 31 211 3 4 1 6 3 2    2 3 2 1 3 6 4 3 2 4   3 the we will find 4 of 3. We 2 42 334 2 2  4 4 33 12 By performing 3 this multiplication, 2 multiply 2  4 3 12 23 4 3 3 1 6 3 2 3 4 3 3 4 34 6 numerators we multiply the 3find 3the numerator of the product, and 12 to 38 4 4 4 3 8 12 3 12 2 4to find 8 the denominator 8 denominators 4 of the product. 6 1 2  6 3 2 6 12 2 1   3 1 63 3 4 3 12 12 3 2 6 1 3 1 6 2 6 6 3 2  6 6 4 4  332212  1 1 12 2 1 68 12 62  4 8 4 3 12 12 The fraction can be reduced to 2, as we can see in this figure: 3 2 6 1 2 4 3 12 12 12 2 3 2 34  2 3 34  32 12 1 8 4 3 4 2 4 4 6 1 2 3 1  3 12 2 12 6 12 2 8 4 3 2 2 4 6 1    4 2  3 4 12 64 2  13 1 2 2 3 1 3 3 1 1 6 12 2 3 1 2 1 12 2 46 34 4 23 612 3 2 6 612 1 1 3 4 4 3 12 12 = 12 2 8 2 2 2 3 4 3 4 3 1 23  2  6 8 2 4 8 2 1 6 can be written 8 4 A whole number as a fraction by writing the whole number as 2  4 2 2 48 4  4 2 3 12 2 4 2 8 12 4  3 1 3 3 3 1 8 8 3 4 4 3  the   the  fraction. 4 3 and 11 as 13 the numerator Thus, 86 1of12the fraction 23 1 of 3 3 denominator 3 3 3 1 6 612 12 2 1 6 2  3 2 can be written as the fraction 1. Writing whole numbers 3 the whole number 2 12 2 12 2 1 as fractions is helpful when multiplying whole numbers by fractions. 8 2 2 3 3 8 26  1 2 Example 3 8 2 2 22 1  2 3 8 12 3 2 3 3 2 3 4 3 Thinking Skill Multiply: 4 2 2 41 3 4  8 Discuss 6 2 1 1 3 1 23 3 12 2 Solution How do you 8 6 2 1 2 4 2 8 12 to 23 8 4 3 2   4  change We write 4 as and multiply. 6 2 1 3 2 ? Explain 1 3 3 3 1 3 2 8 2 12 4 2 8 83  22 your thinking. 4 8   2 3 1 3 3 3 1 3 3 8 2 2 3 3

Lesson 29

151

4

1 8

2 3

4 1

1 2

4 2 8   fraction 8 to a mixed number. Then we convert the 1 improper 3 3 3

4 1

8 2 2 3 3

1 1 1   2 2 4

1 2

Example 4 Three pennies are placed side by side as shown below. The diameter of 3 one penny is 4 inch. How long is the row of pennies?

3 4

1 2

3

3 in. 44 3 4

3 3 1 Solution 3   2 4 8 8 We can find the answer by adding or by multiplying. We will show both ways. 3

Adding:

3 3 3 9 1 in.  4 in.  in.  in.  2 in. 4 4 4 4 4

9 3 3 1  in.  in.  2 in. 1 4 4 4

3 3 23 3 9 3 9 33 3 1 1 12 Multiplying:  in.  in.  2 in. 2 in. in.  in. 2 in. 2 43 1 4 in. 4 4 4 34 4 4 1 4 4 4 3 2 9 9 3 3 1 1 1  in.  2 in.  in.  in.that  2thein. 3 long. 3 9 9 3 3 1 4 4 1 4 We find 4 4 row of pennies is32in. 4 inches   in.  in.  2 in.  in.  in.  2 in. 3   6 6 3 6 2 1 4 4 4 4 4 1 4 4 4    6 2the numerator and the denominator 12dividing 12  2by a6 We can reduce fractions by reducing 6 3 2 8 33 1 6 2 1 1 1 fractions  1 6by 3 the numerator and 12  4 63factor 12 of both2 numbers. To reduce6 12, we 4 6 3 3 a  b will divide both 3 2 12 2 2 4 12 2 denominator 12 dividing by 6 the 3 3 4 by 6. 2 85 3 3 6 1   3 6 2 common   6 3 2 12 6 436 1 9 61  2 3 312  23 6 3 9 1 8  2 3 1  factors 6 6in.2 in.  2 in. in.  in.  in.  in.  2 in.  2 12 2 12 12 6 4 4 4 4 4 1 4 4 4 4 2 8 6  2 1We 4 Math Language  divided both the numerator and2 the denominator by 6 because 6 is the 6 1 1212 6 1 1  2 2largest 6 factor (the GCF) of 6 and 12. If we had divided by 2 instead of by 6, and are 2 12 12 2 called equivalent we would not have completely reduced 8  2the 4fraction.  fractions or equal 8 33 1 9 9 3 3 1 1 2 12 2 6  6 in.    in.  6 3 6 1 1 in.  2 in. 2 in. 2 fractions.  63 2 4 4 1 4  4 4 4 12 12  6 2 12  2 6 2 4 2 8 2 8 4 3   4  The fraction can be reduced by dividing 5 the numerator and the denominator 3 1 3 3 3 1 6 by 3. 18 4

1 2 in. 4 2 7

12 27

1 2

8 2 4 8 33 1  6  2 3 12  2 6 63 2 12 12  2 6 1 2 48  2 2It takes two or more steps to reduce fractions if we do not divide by the 3 3 greatest common factor in the first step.

Example 5 8 33 1 Reduce: 63 2 12

152

Saxon Math Course 1

6 6 1  12  6 2

Solution

6 2 3  12  2 6

We will show two methods. Method 1: Divide both numerator and denominator by 2. 8 2 4  12  2 6 Again divide both numerator and denominator by 2.

84 2  12  4 3

42 2  62 3 Method 2: Divide both numerator and denominator by 4.

84 42 2 2 2 8   62 3 12  4 3 3 12 5 8 84 4 2 2 2 2 2 8 8 2 to  . Since the greatest common factor   Either way, we find that 12 reduces 1 12  3 124 312 12 843 4 32 2 8  of 8 and 12 is 4, we reduced in one step in Method 2 by dividing the 12  4 3 3 12 84 42 2 2   numerator6and by 4.  2denominator 3 12  4 3 5 1 5 2 2   2 12 1 1 12 1 10 102 10 5 10 2 2 5 510 2 2 2 Example 6    12 2  84 4  2 21 1 2 3 1 12 12 12 6 12 1 12 12 2 8 10 5 2   8 84 4 2 2  4 4222 2 10    62 3 12 4 3 3 8 4 12 4 2 2 2  5 1 1 612 2 12 3 12 2   12  4 43 2 4  268  242 3 2  12 224 32 Multiply:   8 8 4 44 2 2 2 42 2 6 4 3 12  8 2 2 8 8  8 12  1     6 4 6  212 3 4 3 12 12  43 3 12 124 3 62 3 3 12  10 22 35 3 12 2  12 Solution 28 6 4 2 3 2  number of boys 8 2 1 4 4 3 3 12 2 2  2 5 of 5 10 12 number of girls 3 3 3 102 2 5   2 5 2 We write 2 as and multiply. 2 12 1 1 212 12 12 2 2 2  5 5 5 10  2 2 3 12 1 2  5 5 5 10 5 2 21 12 1  12 2 2 25 6 12 4  2 2  10 2 3 2  84   2 2 5  10 210  2 2   12 112 1 1 12 12 1 1 12 12 of boys 1number 12 1 12 12 8 212 2 1 4 1   6 2 3 12 4 3   of of 2 of 5girls 10 12 103 number 56 12 are divisible by 2. 4 32 25 and   reduce We can because both 10 8 82 2 1 12 12 12 5 1 12 2 6  5   3 10  2 15 of14of 4 2 23 3 of of 2 4 s2 123   8 2 2 2    8 434 212 5 4 3 5 4 3 2 2 5 10 2   2  8 8 4 4 2 2 2 2 1 8 4 4 1  2 2 2 8 4  2  8 4 4 2 2 2 2      2 8 5 5 2 10 2 10 2 8   2 2  6 32  3 2 342 4 312 of 8 2 of 4 2  2 2   number  412 4 of3boys 3  12  322 63 12 s 4 12 2 2 33 3 22 10 848 12 8  3  3 4 2 3 658 4 12 5  3 2 212 66 33 2 658  12 4 1   12 212 34 4 3  5 10 10 2 12 2 12 2 6 5 2 3 63 4 of2 3 3 3 3 12 62 3 4 3 2of  12  12  number  12 2 4 3 2 2 35 12 32 girls 12  2 6 35 12 6 2 12 3 3 2 62 6 4 12 5  1  2 6 5 3 3 2 2 Example 7 4 4 5  2 2  number of5boys 8 2 3 3 3 34 155the 2 4 1 10 of2boys of 10 boys  There were  and 12ofgirls in was ratio 5 55 5 3410 2 2210 228 5class. 2 the  12  number 52 8 boys of 5 of number 4 22 boys number of22What boys 1010 2 10 23 8 8 of girls 2 3  21of 2     2   52number 2  10 10 5 3 5 5 3 2 2 4 1 1 124 5 6 2 12 1 1 12 12 10 10 2 2     12 1 1 12 12 1the 1 12 12 1212 312 number of girls 3 1 1 12 12 12     2  to girls in class?   4 of 12 1 of 5 2 number of boys number of boys number of boys boys 12 8 8 8 2 2 2 12 12 number of girls number of girls 3 3 8 2 2 15 5 2 1 4 4 1 1 4 1 1 1 12 12 12 1 1 12 12 12 5 2 2 5 6 3  52 of 10 21  1 4   of  2  number of girls  of 3 2 of of  3 2 of of 12 12 12 number of girls number of girls 3 3 3  12 3 girls 5 5 4 3 5 4 3 2 2 2   12 24 63 3 2 5Solution 3 4 3 4 1 2 4 2 3 of  5 4 We3 reduce ratios the reduce fractions. The ratio 5 6 4 8 to 12 3 2 4same way we 5  10 2 5 2 5 56 6 2  5 10 2 5 2 10  2 5  5 10 2  5 6 3 3 5 6 2 2 2  2 5 5  2 reduces to . 2  10 12 212522 66    3 5 5 6 3 2 2  12 2 6 3 3   6 3 52 12 2 5 6 66 25 3 6 2 2 4 43 64  5 3 12  2 6 5  6 3 3  5 number2 of 2  5boys 5 6  5 8 2 5  32  3 1 4 6 5 3 6 5 3  3 3 3 of number of girls 12 3 2 5 2 4  2 3 3 number of boys Practice Set Multiply; reduce if possible. 2 number boys number of 8 then 88 1 8 of boys 2 of boys 12 23 2 of4boys number 22 44boys  8 1 2 41 2 112 of 1 411ofof3 number   8 number 8  2  ofofgirls 2 of 2 of3 of 12 number girls 3 2 of of 2 2 1 1 4 1 4 1 of 5 4 3 2 12 of girls 12 3 3 12 3 number   girls 12 number of girls 3 5 43 3 2 a. b. c. 5 4 3 2  5 4 3 2 3 3 2 2 of 5 of 5 4 of 6 3 12 3 of girls 12 2 3of 5 number 3 2 44 3 4   4 3 2 5   54 3 4 3 6 5 3 Multiply; then convert each answer from an improper fraction to a whole number or to a mixed number. 5 6 2 4 55 66 22 44 2 4  25  2 d. 5  6 e. 5  42  f. 2 55 3 22 3  5 666 55 2 4 5 65  5 3 3 3 3 3  6 2 3 55 23 3 3 6 5 3 3 Lesson 29

153

2 3 7  12 12

2



2 3

2 93   12 3 12

3

2

3

12

12

212 3

12

6 9 Reduce each fraction: 10 12 6 i. 18 6 9 18 h. 6 g. 9 10 24 12 10 10 12 24 6 9 j. In a class ratio of boys to 12of 30, there were 20 girls. What was the10 girls? 6 18 18 16 1 10 24 24 2 24 16 1 3 16 16 1 3 Strengthening Concepts 1 1 Written Practice   6 18 2 24 8 8 8 8 2 24 24 18 1024 24 16 1 1. The African elephant can weigh eight tons. A ton 24 is two thousand 2 (15) pounds. 16 1 3 weigh? 1 3How many pounds can an African elephant  1 2 3  7   8 8 24 8 8 2 3 12 12 3 7 1 2 3 beans weigh 3 ounce, then 2. 1  27 dried 7 6 how many dried beans  9 If9sixteen   one 1 3 6 12 2 312one 12 12 2 3 161  3 (15) weigh 12 =12  pound (1 pound 16 ounces)? 8 8 10 10 248 812 12 1 2 3 7   2 3 1 1 1 3 1 3 12 12 *63. Analyze If6 the product of 2 and 2 is subtracted from 18 18the sum of 2 and 4 2, 4 3 9 7 (Inv. 2, 6  29) what is the 10 10difference? 24 24 12 12 10 12

18 24

1 3  8 8

18 18 24 24

16 What 16 was the team’s win-loss 1 31 3 31 7 1    * 4. A team won 6 games and lost 8 games. 8 8 8 24 24 12 122(29) 2ratio? 1 31 33 3 1 3 3 1 33 3 3       1 3 2 4*26. 814  38 2 8 4 2 88 4 8 *165. Reduce: 161  16 8 8 8 24 24 (29) (24, 29) 8 2 24 1 21 2 * 7. * 8. 7  73  3   2(29) 32 3 12 12 12 (24, 29) 12

1 2

1 2  2 3

9. The Nobel Prize is a famous 3 international award that recognizes 3 3 2 3 3 2 23 2     3 3 7 7 4 43 4 43 3 4 3medicine, economics, important work 1  in 3 2 physics,4chemistry, 7 12 12 12  12  2 and 3 literature. Ninety-six Nobel Prizes12 12 peacemaking, in Literature were awarded from 1901 to 1999. One eighth of the prizes were given to Americans. How many Nobel Prizes in Literature were awarded to 6 6 3 2 3 62 3 62 3 662 Americans from 1901 to 1999?     3123 412 24 18 4 3 4 123 12 12 123  1 4 3 2 10. Predict Find the next three numbers in the sequence below: (22)

(10)

1, 4, 7, 10, 11.

(Inv. 2)

,

, ...

6 have 6 2 16 2 1 16 1 When five months what fraction of the year  1 passed,    1 1 12 2 12 2 12 1 2 12 1 2 remains? 2 a  b 2 2 4 3 Connect

12. $3.60 × 100 (2)

,

13. 50,000 ÷ 100 (2)

3 2 the * 14. Convert to a mixed number. 2Remember to reduce part2of 2  fraction 1 2 (25) 3 2 4  4  4 4 4 4 the mixed number. 3 3 13 13 18 4

15. The temperature rose from −8°F to 15°F. This was an increase of how (14) many degrees? 1 1 1 8 8 8 8 2 2 2 2 b a  number. Find each unknown  2 to2 check your 32work. 2 2 42 Remember 3 3  23  23 3 3 3 5 6 16. m + 496 + 2684 = 3217 (3)

17. 1000 − n = 857 (3)

18. 24x = 480 (4)

19. 7 ∙ 11 ∙ 13 (5)

154

Saxon Math Course 1

2 5

3 6

20. (16)

Estimate

Explain how to estimate the quotient of 4963 ÷ 39.

* 21. Compare: 2  3 (29) 3 2

18 4

22. (8)

18 4

1 1 1 a  b 2 2 4

11

2  3

18 4

The perimeter of the rectangle shown is 60 mm. The width 2 of3the rectangle  1 is 10 mm. What is its length? 3 2 Analyze

23. 12 − 40 (14)

10 mm

* 24. a 1  1 b  1 (29) 2 2 4

2 3

3 figure 2 3 * 25. a. Which angles2inthe at  2 1318 1 18 (28)  4 1 34 21 1 3 2 1 right are acute angles? 3 22 a  b 22 33 2 2 4 3  11 b. Which angles are obtuse angles? 33 22 18 4

D 3 5

C

1 1 1 1 1 a 2 1b  What fraction is 2 3 a2 4b  * 26. 2 2 4 (29)

22 2 35 3 8 3

2 3

of 53 2? 3

(29)

8 3

* 28.

2 58

2

3

23 2 5

32 65

3 3 6 6

Early Finishers

Real-World Application

2 23 Connect

2 2 3diameter the

3 5

2A 3  3 2

32 43

3 5

16 inches, 212

3 6

1 the 2

If of a bicycle wheel 24 what is ratio of the radius of the wheel to the diameter of the wheel? 8 3 2 8 22 6 2 6 3 3 6 1 3 3 12 * 29. Represent What 12 2 type of an angle is formed by the hands of a clock at (28) 2 o’clock?

(27, 29)

30. What percent of 3a circle is 52 of a circle? Explain why your answer is 3 (22) 6 6 correct.

2

1

2

3 6

1

The high school basketball team has 14 players: 7 are 7 2 are forwards, 2 guards, and the rest are centers. Find the number of players in each position on the 2 1 team. Show your work. 7 2 2 1 7

1 3 4

4 3 3 4

2 5

6 is 12

2 3

B

1 1 1 a 3 b  4 2 42 43

3 * 27. What is the536product of43 43 and 43 ?43 5

4 3

3 4

3 5

2

Lesson 29

155

1 2

LESSON

30 Power Up facts mental math

Least Common Multiple (LCM) Reciprocals Building Power Power Up E a. Number Sense: 9 × 32 b. Number Sense: 5 × 42 c. Number Sense: 45 + 49 d. Number Sense: 436 + 99 e. Money: $20.00 − $12.75 f. Fractional Parts:

1 2

1 3

of 72

g. Statistics: Find the average: 120, 99, 75 h. Calculation: 7 × 7, − 1, ÷ 6, × 3, + 1, × 2, − 1

problem solving

A pentomino is a geometric shape made of five equal squares joined by their edges. There are twelve different pentominos, and they are named after letters of the alphabet: F, I, L, N, P, T, U, V, W, X, Y, and Z. (The rotation or reflection of a pentomino does not count as a different pentomino.) Can you create the remaining ten pentominos?

New Concepts least common multiple (LCM)

F N

Increasing Knowledge

A number that is a multiple of two or more numbers is called a common multiple of those numbers. Here we show some multiples of 2 and 3. We have circled the common multiples. Multiples of 2: 2, 4, 6 , 8, 10, 12 , 14, 16, 18 , 20, . . .

Multiples of 3: 3, 6 , 9, 12 , 15, 18 , 21, . . . Math Language Recall that a We see that 6, 12, and 18 are common multiples of 2 and 3. Since multiple is the the number 6 is the least of these common multiples, it is called the product of a least common multiple. The term least common multiple is abbreviated LCM. counting number and Example 1 another number. What is the least common multiple of 3 and 4?

156

Saxon Math Course 1

Solution We will list some multiples of each number and emphasize the common multiples.

Visit www. SaxonPublishers. com/ActivitiesC1 for a graphing calculator activity.

Multiples of 3: 3, 6, 9, 12 , 15, 18, 21, 24 , . . . Multiples of 4: 4, 8, 12 , 16, 20, 24 , 28 , . . . We see that the number 12 and 24 are in both lists. Both 12 and 24 are common multiples of 3 and 4. The least common multiple is 12. When we list the multiples in order, the first number that is a common multiple is always the least common multiple.

Example 2

2 3

2 3

3 2

What 3is the LCM of 2 and 4? 2

Solution

1

We will list some multiples of 2 and 4. 5 6 Multiples of 2: 2, 4 , 6, 8 , 10, 12 , 14, 16 , . . . 5 Multiples of 4: 4 , 8 , 12 , 16 , . . . 6

1 12

12

The first number that is a common multiple of both 2 and 4 is 4.

6 5

6 5

reciprocals

6

Reciprocals 5are two numbers whose product is 1. For example, the numbers 1 1 1 2 and 652 are reciprocals because 2 × 21 = 1. 3 3 1

2×2=1 5 6 30 � � 6 5 30 5 6 30 � � 6 5 30

1 5

8 5

30 say 30 that

1 2

8 5

1 � �1 6

1 �1 6 1 2 1 2

reciprocals 1

1

1

1 3

2 is the reciprocal of 2 and that 2 is the reciprocal of 2. Sometimes We 3 we want to find the reciprocal of a certain number. One way we will practice finding the reciprocal of a number is by solving equations like this:

1 5



30 30

1 3

2 3

5 12

1 number 5

1 5 1 that 2

The reciprocal of 3.

3 × □2= 1 3

goes in the box is 13 because 3 times 13 is 1. One third is the

Reciprocals also answer questions like this: 1 1 1 How many are s2 2 2 4211 2 in21 1? 2 2  1 12  12   3 3 3 101 5 5 5 5 5 5 The answer is the reciprocal of 14, which is 4. 3 1 2 and the denominator. To form the Fractions have two terms, the numerator 4 1 4 1 of a2 fraction, we reverse the terms of reciprocal 3 3 the fraction. 2 7 5 5 2 32 3 42 � �1 1111 5113   4  8 5 45 23 3 7 4 35 1212 12124 � �1 8 3 The new fraction, 43, is the reciprocal of 4. 3

12 12 If we34multiply 4 by 43, we see that the product, 12 1. , equals12

3 24 2 2122  = = 1 4 33 3 3123

11 5 5 1 52 5 1   12 12 2  2 33 3 3 4 3 4

12 12 12 12

5 12 5  12 3 12 2 12

Lesson 30

2 2  3 3

157

2  5

3  4 4 3

2  3 12 12

2 3

2 3

2 3

2 3

2 3

2 3

2 3

2 3 3 2

3 2 2 3 5 6

5 3 6 2 30 30

30 5 30 6 5 1

5 301 30 2

5 1

2 3

Example 3

3

2 3 3 8

3 28 3

3 8

2 3

3 2

3 2

30 330 2

5 51 6

30 3 30

3

58 1

3 2 3 2

3 2 5 5 6 6

5 26 3

2

2How 3

many

Solution 2

2

2 3

2 3

2 23 3s

2

2 3 in are 3 3 2 2 2 3 3

1?

22

2 23 3

2 3

3 2

3

2

2

2

23 3 1 1 22 3 3 3 2 2

3 23

3

5

3

6

56 3 65

5

6 5

3 5 3 3 2 2

66 3 5 530

2

5

5 6

8

3 8

2 3

2

Solution 3

2 3

6 5

3 a. 6 and 83 b. 3 and c. 5 and110 6 5 8 1 7 1 1 7 � � �1 �6 � 3 � � � � 11 � 1 1 1 8 6 8 88 6 1 8 1 1 1 1 � �1 8 8 reciprocal of each 1number:37 8 6 37 1 4 4 2the 1 Write 4 7 3 3 1 5 3 5 5 � �1 � �1� �1 � �1 � 1� 1 � � 1 � � 3 6 2 d. 6 68 8 g.6 1 38 f. 8 85 8 8 5 5 8 e. 2 6  5 5 5 53 1 4 21 3 3 3 3 8 8 1 1 4 Analyze 5 For problems h–k, find the number 8 3 that3goes into the box to4 make 5 true. 1 1 the�equation 7 �7� 7 �1�1�1 � �1 � �1 3 11 � 8 � 1 8 6 h. 6 × □ = 61 8 �= � i. 4 × □ 1 1 8 6 2 12 3 1 13 7 1 1 �1 � 1 � �1 � � � �1 3 12 8 6 6 8 8 6 1 7 1 7 �1 � 1 �1 � 1 j.�□ � k. □ � � 6 8 6 8 2

l. How many 5 s are in 1? m. How many

2 5

5

3 2

5 6

2 3 6 5 3 2

2 3

5 6

3 2

5 31 6 122

5 6

6 5

6 5

3 1 6 6 6 6 5 5 30 30 30 30 12 2 �6 �30 � 5 5 5 30 30 30 3 56 5� 65 30 30 30 5 6 30 666� 56 30 30 30 6 6 � 30� � � 6 30 3 6 5 6 5 5 6 5 1 5 30 5 5 30 30 5 5 6 1 30When by its reciprocal, is 51. So 6the answer 5 30 is the 30 5 5 5 30 5 the6product 1 6 is multiplied 1 5 �6 1�2 1 5 30 1 1 6 6 6 5 6 5 6 6 6 5 30 5 2 2 3 6 5 30reciprocal 6 1 , we 6 . When 30 we 6 6 530get 30 65 30 of 65, which 5 multiply 6 1by 5 5 6 30. 5 5 � �is 5� 2 5 6 30 530 5 1 30 5 530 6 30 5 630 30 5 2 630 30 3 3 1 30 1 30 5 30 6 5 6 � � � � � � 1 1 1 6 6 5 5 30 30 30 30 30 1 2 30 6 5 30 30 30 2 2 2 2 30 30 30 30 � 6 � 5 30 � 5 5 6 30� � 66 61 51 30 30 30 56 51 30 5 5 1 6 5 5 3066 �30 6 5 30 3030 30 5 30 30 12 12 5 1 6 5 � � 1 6 155 165 30 30 30 51 1 6 5 30 30 1 6 6 5 equals 30 6 fraction 5 15The 30 30� 5 130 5 � 1. 5 5 �1 � 5 66 630 30 30 30 1 1 6 66 � 5 � 6 30 630 5 6 � � 85 5 30 6 25 30 30 30 30 5 5 5 56 5 306 6 30 5 5 5 5 51 30 30 5 6 30 30 5 56 6 61 30 5 5 30 61 3030 1 30 � � � � 30 � � 5 �3 5 �5 51� 30 5� 5 30 5 6 56 1 5 6 6 6 6 5 5 301 6 56 30 1 130 65 130 30 5 30 56 2 6 51 30 5 5 1 1 1 5 Example 1 1 5 5 6 6 5 5 5 5 1 5 5 1 5 5 55 5 1 15 53 6 6 6 5 1 5 5 6 30 30 6 5 1 1 8 8 2 2 � � 5 5 5 5 30 5 1 5 What is the reciprocal of 5? 8 2 5 1 5 3 5 3 8 2 8 2 16 5 30 1 1 1 1 1 5 1 1 55 3 5 1 8 1 5 30 5 3 6 5 30 55 3 5315 633 30 30 305 1 5 5 16 � 30 � � 1 � Solution � � � � 2 1 30 5 1 1 55 5 8 1 1 1 4 6 30 30 30 305 1 30 30 6 5 1 6 5 6 30 3 6 18 5 8 30 5 1 1 1 1 5 5 1 1 �5 �5 5 30 5 3 355 16 1 230 5 330 30 4 30 4 30 1 1 55 630 830 5 5 166 830 2 2 30 8 3 5 � � � � � � Recall that a whole number can be written as a fraction that has a 8 8 2 2 1 1 2 1 30 86 30 5 30 4 6 530 530 128 65 3035 30530 3 35 4 330 3 30 1 4 3 6 5 30 6 5 5 5 3 3 3 3 1 3 1 denominator of 1. So 5 can be written as . (This means “five wholes.”) 8 2 1 5 3 3 � � � � 1 30 5 �1 � 1 � 1 30 528 30 6 56 30 3 6� 116 38 5 5 and 1the 1 gives 3 3 1 us the8 7 of the 8positions � 1reciprocal 8 3Reversing the � 1� of � � 1 5, which 1 � �11 1 8 2 16 13 1835 1 53 6 1 15 8is 1. This makes 7 4 8 6 8 4 s make 1,1 5and 5 of 5 is31. � 8 �5 1 �five 5 2 5 5 3 8 5 2 sense because 18 �5 1 8 2 81 3 1 1 8 71 7 1 1 6 1 1 5 1 5 � � � � � � 1 1 1 1 4 4 8 4 5 5 3 1 1 3 7 3 15 81 81 3 3 3 1  711  1 1 5 7 1 35 3 61 65 5 � � � � � � � � � � � 1 3 3 1 1 7 1 1 1 1 1 5 5 1 1 5 5 5 3 1 7 Practice Set Find the least common multiple of each pair of numbers: � � � � � � 10 12 12 1 1 1 8 6 8 8 6 8 6 8 � � 12 1 � � 18 1 1 1 8 8 6 68 3 1 7 8 5 5 5 5 � � 16 � � 18

3 8

2 5

3

3 3 3 3 number of s in 31, we 2 To find the need to find2 the reciprocal of 322. The2 easiest 33 3 2 3 2 3 1 3 1 way to find the reciprocal of is to reverse the positions of the 2 and the 3. The 1 1 3 2 3222 3 2 1 2 12 3 3 1 2 13232 1 write of233 is 52. (We may convert reciprocals 2 2 6 2 3 2 to 1 2 , but2we usually 2reciprocal 5 1 3 3 12 2 6 5 3 as fractions rather than as mixed numbers.) 3 2 3 3 2 2 2 25 6 5 1 1 3 2 3 2 3 2 3 2 3 31 1 2 2 5 5 3 1 6 55 1 1 6 1 1 1 3 1 1 6 5 3 1 2 2 2 2 1 1 6 6 2 12 2 2 6 5 6 5 62 2 Example 4 5 6 1 3 12 6 5 65 5 6 56 6 5 6 3 1 5 5 1 6 1 6 55 3 1 3 1 2 true? 6 6 number to36make 3the equation 2 21 2 6What 2 21 2goes 55 6 30 30 into the box 1 � � 2 2 6 2 5 5 5 2 5 3 2 3 3 2 2 30 5 1 6 61 6 6 5 530 1 3 1 36 6 ∙ = 1 12 1 1 1 3 ∙ 3 52 35 302 6 6 2 26 25 2 2 6 66 6 5 2 25 62 5 6

3 2

3 2

158 2

2 25 5

2 5

Saxon Math Course 1 2 5

2

2 5

5 12 s 2 25

are in 1?

2 3

2 5

5

1

1 2

2 5

11



1

5

7 � � 8 11 � �1 36

1 1 3 41 � �1 6

7 � � 8 7 � �1 8

12

3

2 3

2 3

5

2 2 2   5 5 5

2 2 2   5 5 5

Written Practice

3 5 4 5 5  2 3

2 1 3 12

1 10

12

5

5

12 12

3 4

12 12

2 1 3

11 1  12 12 11 1  12 12

(30)

6.

(29)

13. (19)

2 3

14. (1)

16. (16)

* 17.

11 1  12 12

12 12

(28)

100% 7

12 12

5 5  2 3

12

12 Connect

3 4 3  5 4 3 4

33 44

* 18. Compare: (29)

(29)

2 5

3

3 4

2 3

* 12. (29)

4 12 2 2  3 3

B

C

2 1 3

(29)

2 5

2 2  3 3

5 5 5 5   2 3 2 3 100% 100% 8540 35 35 24  8540 The number has how many different whole-number 757 factors?115 1 3 5 3 5 3 4    5 1 5 2  2  2 1   4 2 3 4 2 3 4 3 5 5 10 15. $5 − $1.50 12 12 $3 + $24 +5 $6.50 (1) 12 2 2 2 2 12 1  1 Estimate 12 3 3 3 the product: 596 3 × 405 12 33 22  11 55 8A8 Which triangle 2 3at right is an 2 3 of the 12 2 2 angle 5 125   1 4  1  5 3 4 12 obtuse 3 3 3angle? 4 3 2 123 * 11. 5 

* 19. 500,000 ÷ 100

35  8540 35  8540 12 12

2 3

There were 12 minutes of commercials during the one-hour 2 2 2 1 11 1 11 1 1  1  program. What was the ratio time   1  of commercial to noncommercial 5 5 5 10 12 12 10 12 12 during the one-hour 12 program? Explain how you found your answer. 1 2 2 2 1 12 11 1 2 11  1  7.  1  8.5 1  * 9. 12 3 12 5 5 5 10 12 10 12 12 (24) (Inv. 2) (24, 29)

* 10.

5 5  2 3 5 5  2 3

1 10

2

5

12 the first four multiples 3 12 4. What are of 12? (25) 5 5  2 common 3 * 5. What is 5the least 5 multiple (LCM) of 4 and 2 6?

(23)

2 3 2 3

1

2 3

5 12

1

 512 5

Strengthening Concepts 2 2 2 1 3 4 11 1     1 5 5 5 10 12 12 4 3 * 1. Analyze If the 3fourth 4 multiple of 3 is subtracted from the third multiple  (12, 25) 4 3 3 3 4 11 1 of 4, what is the difference?   5 4 12 12 4 3 2 2 2 2. Model About 23 of a person’s body weight is water. Albert   weighs 5 5 5 (22) 3 3 4 of Albert’s weight 5 how many pounds 11 1 117 pounds.5About 2 is2 water?     5 4 12 12Draw a diagram 4 3 2 2 to 3 illustrate 3 3 the problem. 2  3 3 2 5 5 2 2 1  3. Formulate Cynthia ate 42 pieces  2 (15,323) 3 of 3popcorn during the 3first 15 minutes 3 11 1 4 of of a movie. If she kept  eating at the same rate, how many pieces  12 12 4 3 she eat during the 2-hour and 2 5 5 popcorn did 12 2 2movie? Write an equation 1   3 12 2 3 solve the problem. 3 3

1 1 10 11 1  12 12

1 0



3  1 100% 100% 8 * 21. 7 7 (25)

45 10

12 12

100 7

20. 35  8540 (2)

4 4 * 22. Reduce: 12 12 (29)

23. What is the average of 375, 632, and 571? (18)

3  8

2

5 24. A regular hexagon has six sides of equal length. If a regular 100% hexagon (8) 8540 35  3 from a 36-inch-long string, what 45 3 made 45length of is is the 7 each  1  1 10 10 8 8 side?

* 25. What is the product of a number and3 its reciprocal? (30)

3 4

3 4

* 26. How many 25 s are in 1? (30)

4

3  1 8

Lesson 30 3 4

159

35  8540

100% 7

7

* 27.

Analyze

* 28.

Connect

(30)

2 5

3 4

100% 35  8540 7 4What number goes into the box to make the equation true? 12 3 45  = 11 10 8 □

What is the reciprocal of the only even prime number? 3 2  1 5 8 45 29. Convert 10 to a mixed number. Remember to reduce the fraction part of (25) the mixed number.

(19, 30)

3  1 8

12

* 30. Four pennies are placed side by side as shown below. The diameter of (29) one penny is 34 inch. What is the length of the row of pennies? 35  8540 100% 4 7 12

40

3  1 8

Early Finishers

Real-World Application

3 in. 4

45 10

2 5

Fernando’s class is going to make cheese sandwiches for their school picnic. 3 4 They want to have at least 80 sandwiches. Each package of bread contains enough slices for 10 sandwiches. Each package of cheese contains enough slices for 18 sandwiches. The class wants to buy the fewest packages of cheese and bread with no slices left over. a. How many sandwiches should the class make? Explain how you found your answer. b. How many packages of bread and cheese should they buy? Show your work.

160

Saxon Math Course 1

INVESTIGATION 3

Focus on Measuring and Drawing Angles with a Protractor

B

100 110 80 12 70 0 60 13 50 0

170 180 160 10 0 20

0 10 20 180 170 30 160 15 0

0 15 30

P

50 0 13

80 90 70 100 90 60 110 0 12

0 14 0 4

4 14 0 0

Reading Math We abbreviate 360 degrees as 360°.

One way to measure angles is with units called degrees. A full circle measures 360 degrees. A tool to help us measure angles is a protractor. To measure an angle, we place the center point of the protractor on the vertex of the angle, and we place one of the zero marks on one ray of the angle. Where the other ray of the angle passes through the scale, we can read the degree measure of the angle.

O

A

The scale on a protractor has two sets of numbers. One set is for measuring angles starting from the right side, and the other set is for measuring angles starting from the left side. The easiest way to ensure that we are reading from the correct scale is to decide whether the angle we are measuring is acute or obtuse. Looking at ∠ AOB, we read the numbers 45° and 135°. Since the angle is less than 90° (acute), it must measure 45°, not 135°. We say that “the measure of angle AOB is 45°,” which we may write as follows: m∠ AOB = 45° Practice reading a protractor by finding the measures of these angles. Then tell whether each is obtuse, acute, or right. Classify

2. ∠ AOE

3. ∠ AOF

4. ∠ AOH

5. ∠ IOH

6. ∠ IOE

G

H 50 0 13

F

80 90 70 100 90 0 0 6 11 0 12

E

D

0 15 30

160 20

O

170 180 0 10

I

100 110 80 12 70 0 60 13 50 0

0 14 0 4

4 14 0 0

In exercises 1 and 3, we found m∠ AOC and m∠ AOF. Tell how to find m∠ COF without using the protractor.

1. ∠ AOC

30 15 0

Extend

0 10 20 180 170 160

Thinking Skill

C B A

Investigation 3

161

Activity

Measuring Angles Materials needed: • Investigation Activity 10 • protractor Use a protractor to find the measures of the angles. To draw angles with a protractor, follow these steps. Begin by drawing a horizontal ray. The sketch of the ray should be longer than half the diameter of the protractor. horizontal ray endpoint

Next, position the protractor so that the center point of the protractor is on the endpoint of the ray and a zero degree mark of the protractor is on the ray. Then, with the protractor in position, make a dot on the paper at the appropriate degree mark for the angle you intend to draw. Here we show the placement of a dot for drawing a 60° angle: 90 90

12 0 60

180 0

Finally, remove the protractor and draw a ray from the endpoint of the first ray through the dot you made.

60°

Represent

162

Use your protractor to draw angles with these measures:

7. 30°

8. 80°

10. 135°

11. 45°

Saxon Math Course 1

9. 110° 12. 15°

13.

Draw triangle ABC by first drawing segment BC six inches long. Then draw a 60° angle at vertex B and a 60° angle at vertex C. Extend the segments so that they intersect at point A.

A

Represent

60° 60° B

6 in.

C

Refer to the triangle you drew in problem 13 to answer problems 14 and 15. 14. Use a ruler to find the lengths of segment AB and segment AC in triangle ABC. 15. Use a protractor to find the measure of angle A in triangle ABC. 16.

Represent Draw triangle STU by first drawing angle S so that angle S is 90° and segments ST and SU are each 10 cm long. Complete the triangle by drawing segment TU.

U

10 cm S

10 cm

T

Refer to the triangle you drew in problem 16 to answer problems 17 and 18. 17. Use a protractor to find the measures of angle T and angle U. 18.

extensions

Use a centimeter ruler to measure segment TU to the nearest centimeter. Estimate

a.

Analyze The building code for this staircase requires that the inclination be between 30º and 35º. Does this staircase meet the building code? Explain your thinking.

b.

Look at the two sets of polygons. Set 1 contains something not found in Set 2. Name another figure that would fit in Set 1. Support your choice. Conclude

Set 1

Set 2

Investigation 3

163

LESSON

31 Power Up facts

Areas of Rectangles Building Power Power Up F

mental math

a. Number Sense: 4 × 25 b. Calculation: 6 × 37 c. Number Sense: 28 + 29 d. Money: $6.25 + $2.50 1 e. Fractional Parts: 8

1 8

f.

1 Number 3

Sense:

1 of 3 600 10

63

600 10

g. Measurement: A minute is how many seconds? h. Calculation: 10 × 10, − 20, + 1, ÷ 9, × 2, ÷ 3, × 5, + 2, ÷ 4

problem solving

Franki has 7 coins in her hand totaling 50¢. What are the coins?

New Concept

Increasing Knowledge

Mr. McGregor fenced in an area for a garden. 20 feet 40 feet The perimeter of a shape is the distance around it.

The number of feet of fencing he used was the perimeter of the rectangle. But how do we measure the size of the garden? To measure the size of the garden, we measure how much surface is enclosed by the sides of a shape. When we measure the “inside” of a flat shape, we are measuring its area.

The area of a shape is the amount of surface enclosed by its sides.

We use a different kind of unit to measure area than we use to measure perimeter. To measure perimeter, we use units of length such as centimeters. Units of area are called square units. One example is a square centimeter.

This is 1 centimeter.

164

Saxon Math Course 1

This is 1 square centimeter.

Math Language We use “sq.” to abbreviate the word square.

Other common units of area are square inches, square feet, square yards, and square meters. Very large areas may be measured in square miles. We can think of units of area as floor tiles. The area of a shape is the number of “floor tiles” of a certain size that completely cover the shape.

Example 1 How many floor tiles, 1 foot on each side, are needed to cover the floor of a room that is 8 feet wide and 12 feet long? 12 ft

8 ft

Solution The surface of the floor is covered with tiles. By answering this question, we are finding the area of the room in square feet. We could count the tiles, but a faster way to find the number of tiles is to multiply. There are 8 rows of tiles with 12 tiles in each row. 12 tiles in each row × 8 rows 96 tiles To cover the floor, 96 tiles are needed. The area of the room is 96 sq. ft. Discuss

Why do we use square feet as the unit of measure?

Example 2 What is the area of this rectangle? 8 cm

4 cm

Solution The diagram shows the length and width of the rectangle in centimeters. Therefore, we will use square centimeters to measure the area of the rectangle. We calculate the number of square-centimeter tiles needed to cover the rectangle by multiplying the length by the width. Length × width = area 8 cm × 4 cm = 32 sq. cm

Lesson 31

165

Example 3 The area of a square is 100 square inches. a. How long is each side of the square? b. What is the perimeter of the square?

Solution a. The length and width of a square are equal. So we think, “What number multiplied by itself equals 100?” Since 10 × 10 = 100, we find that each side is 10 inches long. b. Since each of the four sides is 10 inches, the perimeter of the square is 40 inches. Why can’t we divide 100 by 4 to find the length of each side?

Justify

Practice Set

Find the number of square units needed to cover the area of these rectangles. For reference, square units have been drawn along the length and width of each rectangle. a.a.

b. b.

6 4

7

7

Find the area of these rectangles: c. c.

8m

d.

12 m

5m

Analyze

12 m

The area of a square is 25 square inches.

e. How long is each side of the square? f. What is the perimeter of the square? g. h.

Find the area of Mr. McGregor’s garden described at the beginning of this lesson. Analyze

Analyze

Choose the appropriate unit for the area of a room in a home.

A square inches

166

Saxon Math Course 1

B square feet

C square miles

Written Practice 1.

(12, 25)

Strengthening Concepts When the third multiple of 4 is divided by the fourth multiple of 3, what is the quotient? Analyze

2. The distance the Earth travels around the Sun each year is about five hundred eighty million miles. Use digits to write that distance.

(12)

1 1  8 8

2 3. Convert 10 3 to a mixed number. 3

5 1  6 6 * 4. Generalize How many square stickers with sides 1 centimeter long (31) would be needed to cover the rectangle below? (25)

4 cm

1 4

1

33

2 3

1 3

2 cm

* 5. How many floor tiles with sides 1 foot long would be needed to cover (31) the square below?

10 ft

* 6. What is the area of a rectangle 12 inches long and 8 inches wide? (31)

7.

Generalize

(10)

Describe the rule for this sequence. What is the next term? 1, 4, 9, 16, 25, 36, . . .

8.

(22)

1 1 10  5 1 the What number is 23 of 24? Draw a8diagram to illustrate  8 3 6 6 problem. Model

9. Find the unknown number. Remember to check your work.

(3)

1 4

1

33 10 3

1

33

2 3

1 4

10 3

1

33

24 + f = 42

2 3

Write each answer in simplest form: 1 1 2  11. 5  1 10. 1  13 5 18 8  8 8 6 (24) (24) 6 6 6 2 2 1 2 2 1 * 313. 3 5 * 12.  5  3 2 2 3 (29) 3 (29) 3 10 10 1 2of 387 and 514. 1 14. 2Estimate the product 1 (16)

4

3

15. $20.00 ÷ 10 (2)

1 3

3

1 33

3

1 3

3 4

3 3  10 10

3 4

3

16. (63)47¢ (2)

1 3

17. 4623 ÷ 22

1 2 66 3 %3 3

(2)

1 2

66 3 %

2

1 2

4 3

8 8

* 18. What is the reciprocal of the smallest odd prime number?

(19, 30)

* 19. Two thirds of a circle is what percent of a circle? (26)

5

28

9 100

5

28

4 3

9 100

8 8

Lesson 31

167

20. (9)

Estimate

Which of these numbers is closest to 100?

A 90

B 89

C 111

D 109

21. For most of its orbit, Pluto is the farthest planet from the Sun in our (12) solar system. Pluto’s average distance from the Sun is about three billion, six hundred seventy million miles. Use digits to write the average distance between Pluto and the Sun. * 22. The diameter of the pizza was 14 inches. What was the ratio of the radius to the diameter of the pizza?

(23, 27)

* 23. Three of the nine softball players play outfield. What fraction of the (29) players play outfield? Write the answer as a reduced fraction. 24. Use an inch ruler to find the length of the line segment below. (17)

2 1  3 2

2 5 3

2 1  * 25. 33 2 3 10 (29) 10

3 4

2 5 3

3 3 * 26. How many 34 s are in 1? 10 10 (30)

* 27. Write a fraction equal to 1 with a denominator of 8. (29)

1 3

1

33

28. (22)

Model 12

Five sixths of the 24 students in the class scored 80% or higher 1

1

2

1

663 3the % test. How on many students 66 scored 3 3 % 80% or 2higher? Draw a diagram 23

to illustrate the problem.

9 100

5

28

* 29. a. Name an angle in the figure at right that (28) measures less than 90°. 4 8 9 4 5 2 3 8 8 100 b. Name an obtuse angle in the 3figure at right. * 30. (31)

5 6

1 6

168

P M

8 8

Q

R

Using a ruler, how could you calculate the floor area of your classroom? 5 1 Evaluate 6

Saxon Math Course 1

6

LESSON

32

Expanded Notation More on Elapsed Time

Power Up facts mental math

Building Power Power Up G a. Number Sense: 4 × 75 b. Number Sense: 380 + 1200 c. Number Sense: 54 + 19 d. Money: $8.00 − $1.50 600 e. Fractional Parts: 100

f. Number Sense:

1 of 2 600 100

240 1 2

g. Geometry: A square has a length of 4 ft. What is the area of the square? h. Calculation: 12 × 3, − 1, ÷ 5, × 2, + 1, ÷ 3, × 2

problem solving

The product of 10 × 10 × 10 is 1000. Find three prime numbers whose product is 1001.

New Concepts expanded notation

Increasing Knowledge

The price of a new car is $27,000. The price of a house is $270,000. Which price is more, or are the prices the same? How do you know? Recall that in our number system the location of a digit in a number has a value called its place value. Consider the value of the 2 in these two numbers: 27,000

270,000

In 27,000 the value of the 2 is 2 × 10,000. In 270,000 the value of the 2 is 2 × 100,000. Therefore, $270,000 is greater than $27,000. To find a digit’s value within a number, we multiply the digit by the value of the place occupied by the digit. To write a number in expanded notation, we write each nonzero digit times its place value.

Example 1 Write 27,000 in expanded notation.

Solution The 2 is in the ten-thousands place, and the 7 is in the thousands place. In expanded notation we write (2 ∙ 10,000) + (7 ∙ 1000) Since zero times any number equals zero, it is not necessary to include zeros when writing numbers in expanded notation.

Lesson 32

169

Example 2 Write (5 ∙ 1000) + (2 ∙ 100) + (8 ∙ 10) in standard notation.

Solution Standard notation is our usual way of writing numbers. One way to think about this number is 5000 + 200 + 80. Another way to think about this number is 5 in the thousands place, 2 in the hundreds place, and 8 in the tens place. We may assume a 0 in the ones place. Either way we think about the number, the standard form is 5280. How would the expanded notation change if we added 20 to 5280? Verify

more on elapsed time

The hours of the day are divided into two parts: a.m. and p.m. The 12 “a.m.” hours extend from midnight (12:00 a.m.) to the moment just before noon (12:00 p.m.). The 12 “p.m.” hours extend from noon to the moment just before midnight. Recall from Lesson 13 that when we calculate the amount of time between two events, we are calculating elapsed time (the amount of time that has passed). We can use the later-earlier-difference pattern to solve elapsed-time problems about hours and minutes.

Example 3 Jason started the marathon at 7:15 a.m. He finished the race at 11:10 a.m. How long did it take Jason to run the marathon?

Solution This problem has a subtraction pattern. We find Jason’s race time (elapsed time) by subtracting the earlier time from the later time. Later − Earlier Difference

11:10 a.m. − 7:15 a.m.

Since we cannot subtract 15 minutes from 10 minutes, we rename one hour as 60 minutes. Those 60 minutes plus 10 minutes equal 70 minutes. (This means 70 minutes after 9:00, which is the same as 10:10.) 10:70

11:10 − 7:15 3:55 We find that it took Jason 3 hours 55 minutes to run the marathon.

Example 4 What time is two and a half hours after 10:43 a.m.?

170

Saxon Math Course 1

Solution This is an elapsed-time problem, and it has a subtraction pattern. The 600 1 elapsed time, 2 2 hours, is the difference. We write the elapsed time as 2:30. 100 The earlier time is 10:43 a.m. Later Later − 10:43 a.m. − Earlier 2:30 Difference 600

1

We need to find the later time, 100so we add 2 2 hours to 10:43 a.m. We will describe two methods to do this: a mental calculation and a pencil-andpaper calculation. For the mental calculation, we could first count two hours after 10:43 a.m. One hour later is 11:43 a.m. Another hour later is 12:43 p.m. (Note the switch from a.m. to p.m.) From 12:43 p.m., we count 30 minutes (one half hour). To do this, we can count 10 minutes at a time from 600 1 12:43 p.m.: 12:53 p.m., 1:03 p.m., 1:13 p.m.100 We find that 2 2 hours after 10:43 a.m. is 1:13 p.m. Thinking Skill Discuss

What is another way to solve this problem using mental math?

Practice Set

To perform a pencil-and-paper calculation, we add 2 hours 30 minutes to 10:43 a.m. 10:43 a.m. + 2:30 12:73 p.m. Notice that the time switches from a.m. to p.m. and that the sum, 12:73 p.m., is improper. Seventy-three minutes is more than an hour. We think of 73 minutes as “one hour plus 13 minutes.” We add 1 to the number 600 1 of hours and write 13 as the number of100 minutes. So 2 2 hours after 10:43 a.m. is 1:13 p.m. Write each of these numbers in expanded notation: a. 270,000 b. 1760 c. 8050 Write each of these numbers in standard form: d. (6 × 1000) + (4 × 100) f.

e. (7 × 100) + (5 × 1)

George started the marathon at 7:15 a.m. He finished the race at 11:05 a.m. How long did it take George to run the marathon? How do you know your answer is correct? Explain

600

1

g. 100 What time is 3 2 hours after 11:50 p.m.? h.

600 1 Dakota got home from soccer 100 practice 4 2 hours before she went to sleep. If she went to sleep at 10:00 p.m., at what time did she get home?

Analyze

Lesson 32

171

Written Practice

Strengthening Concepts

1.

When the sum of 24 and 7 is multiplied by the difference of 18 and 6, what is the product?

2.

Davy Crockett was born in Tennessee in 1786 and died at the Alamo in 1836. How many years did he live? Write an equation and solve the problem.

(12)

(13)

Analyze

Formulate

40

1

3 2$2.24. What is100 3. A 16-ounce box of a certain cereal costs the cost per (15, 23) ounce of the cereal?

40 100

* 4. What time is 3 hours 30 minutes after 6:50 a.m.? (32)

40 40 1 1 40 * 5. Forty percent equals 32 3 2 100. Reduce 100.100

40 100

(29)

* 6. A baseball diamond is the square section formed by the four bases on a (31) baseball field. On a major league field the distance between home plate and 1st base is 90 feet. What is the area of a baseball diamond? 2nd base 3rd base

90

home plate

ft

1st base

7. What is the perimeter of a baseball diamond, as described in problem 6?

(8)

8.

(10)

Generalize

Describe the sequence below. Then find the eighth term. 2 5

2 5

1, 3, 35, 7, . . .

3 4

4

* 9. Write 7500 in expanded notation. (32)

10. (9)

Which of these numbers is closest to 1000?

Estimate 2A 5

990

2 5

3 2B 4 5

9093 4

3 4

C 1009

2

D 10905

3 4

11. In three separate bank accounts Sumi has $623, $494, and $380. What (18) is the average amount of money she has per account? 3 * 13. How many 25 s are in 1? 12. $0.05 × 100 4 3 2 (30) (2) 3 3 3 33 5 1 3 13 4 1 3 1 3 72 1 7 3 1 2     1 1 1 1  2  3   5 5 5 5 4 4 4 3 10 10 3 3 4 4 4 3 10 10 3 3 14. Model How much money is 4 of $24? Draw a diagram to illustrate the 5 (22) problem.

Write each answer in simplest form: 3 3 3 1 3  3 1 3 3 1 3 3 1 1 3 1 3 3 71 3 72 1 3 3 3 3 3 2 1 7 1 2 5 5 4 16.  15.     * 17.   1   1 1 4 1 1 1 3 10 10310 3103 3 5 5 5 4 5 4(24)5 4 4 4 4 3 4 4 3 10 (24) 5 (29)104 3 3

172

Saxon Math Course 1

3 3  5 5

3 3  5 35 1  4 4

3 3 3 1 55  4 34 1  4 3

3 1 3 1 44  4 3 7  10 10

3 1 3 74  3  10 210 1 1 1 3 3

1

2 3



3 5

3 13 3   4 45 5 3 3  5 5

37 1 3 13 1 3  * 18.    4 34 4 (29) 10 410 3 3 1  4 4

2 1 2 31 7 * 19. 1  1 1 1  3 3 103 10 (26) 3

20. Connect Three fourths of a circle is what percent of a circle? 3 1 3 7 2 1   1 1 4 each 3 unknown 10 number. 10 3 3to check your work. Find Remember

(Inv. 2)

21. w − 53 = 12

22. 8q = 240

(3)

(4)

23. Fifteen of the three dozen students in the science club were boys. What (23) was the ratio of boys to girls in the club? * 24. What is the least common multiple of 4 and 6? (30)

* 25. (Inv. 3)

Draw triangle ABC so that ∠C measures 90°, side AC measures 3 in., and side BC measures 4 in. Then draw and measure the length of side AB. Represent

* 26. If 24 of the 30 students finished the assignment in class, what fraction (29) of the students finished in class? * 27. Ajani and Sharon began the hike at 6:45 a.m. and finished at 11:15 a.m. (32) For how long did they hike? * 28. Compare: (3 × 100) + (5 × 1) (32)

29. (17)

Connect

350

What fraction is represented by point A on the number line

below? A 0

30.

(15, 23)

1

Explain Some grocery stores post the price per ounce of different cereals to help customers compare costs. How can we find the cost per ounce of a box of cereal?

Lesson 32

173

LESSON

33

Writing Percents as Fractions, Part 1 Building Power

Power Up facts

Power Up G

mental math

a. Order of Operations: (4 × 100) + (4 × 25) b. Number Sense: 7 × 29 c. Calculation: 56 + 28 d. Money: $5.50 + $1.75 e. Number Sense: Double 120. f. Number Sense:

120 10

1 2

1 4

g. Geometry: A rectangle has a length of 6 in. and a width of 3 in. What is the area of the rectangle?

problem solving

h. Calculation: 2 × 3, + 1, × 8, + 4, ÷ 6, × 2, + 1, ÷ 3 60 60%  100 Monica picked up a number cube and held it so that she could see the dots on three adjoining faces. Monica said that she could see a total of 7 dots. How many dots were on each of the faces she could see? What was the total number of dots on the three faces she could not see?

New Concept

Increasing Knowledge

Our fraction manipulatives describe parts of circles as fractions and as 120 percents. The manipulatives show that 50% is equivalent to 12 and that 25% 14 10 1 1 is equivalent to 4. A percent is actually a fraction with a denominator of 100. 2 The word percent and its symbol, %, mean “per hundred.”

120 10

We can use a grid with 100 squares to model percent. 60% 

60 60%  100

50% =

50 1 = 2 100

25% =

60 100

1 25 = 4 100

To write a percent as a fraction, we remove the percent sign and write the number as the numerator and 100 as the denominator. Then we reduce if possible.

Example 1 In Benjamin’s class, 60% of the students walk to school. Write 60% as a fraction.

174

Saxon Math Course 1

120 10

Solution

1 2

1 4

We remove the percent sign and write 60 over 100. 60%  60%

60 100

60  20

3

60 We can reduce 100 in one step by dividing60 and 100 by their 35GCF, which is 4%  Reading Math 40 1002  20 5 1 20. If we begin by dividing by a number smaller than 20, it will take more than GCF stands for 5 100 greatest common one step to reduce the fraction. factor. The 60  20 = 3 3 60 greatest common  4%  100 5  5 20 100 1 = 6 23 4 1 1 factor is the 5 20 4 25 100   3 20 60 4 4 4 3 60 greatest number  We find that 60% is equivalent to the fraction 5 . 4%  100 100  20 5 100 100  4 that is a factor of each of two or more 6 23 4 1 1 Example 2 5 20 4 25 100 numbers.

1 44 the reduced  that 1 fraction 1 equals 4%.  100 4 25 20 4 Solution 4Find 5

6 25

23 100

1 10

We remove the percent 4  4 sign 1and write 4 over 100.   4 25 100 3 60  20 4 44 1 3 60 1  4%  100 5 25 100  20 54  4  1 100 100  4 25 How do you 100  4 25 find the greatest We reduce the fraction by dividing both the numerator and denominator common by 4, which is the GCF of 4 and 100. factor of two  20numbers? 3 3 20 60  60 4 4 4 423 4 4 = 1 1 1 1 1 3 3 1 4 1 1 1   6  4%4% 5 5 4 5 20  205 5 20 1004 4 = 25 2510 25 25 100100 5 25100100 100100 50 3 0  20 4 44 1 3 1   4% is equivalent to the fraction 25 We find that . 4% 5 00  20 5 100 100  4 25 Thinking Skill Discuss

1 20

1 1  425 25

Practice Set

1 20

1

1

4  4 4 14  1001 4 25 4

6 25

6 25

Write each percent as a fraction. Reduce when possible. 6 25a.

80%

23 23 100 100

23 100

1 10

d. 24% g. 20% j.

1

Justify

1

1 5

b. 10 5%10 e. 23% h. 2%

1 5

1 5

1 50

1

1

50 c. 50 25%

3 4

3 4

3 4

3 4

f. 10% i. 75%

Describe the steps you would take to write 40% as a reduced

fraction.

Written Practice 1.

(12)

Strengthening Concepts When the product of 10 and 15 is divided by the sum of 10 and 15, what is the quotient? Analyze

2. The Nile River is 6690 kilometers long. The Mississippi River is 3792 kilometers long. How much longer is the Nile River than the Mississippi? Write an equation and solve the problem.

(13)

Lesson 33

175

3. Some astronomers think that the universe is about fourteen billion years old. Use digits to write that number of years.

(12)

* 4. Write 3040 in expanded notation. (32)

* 5. Write (6 × 100) + (2 × 1) in standard notation. (32)

* 6. (29)

Write two fractions equal to 1, one with a denominator of 10 and the other with a denominator of 100. Connect

* 7. By what number should (30)

5 3

8. What is the perimeter of this rectangle?

(8)

* 9. How many square tiles with sides 1 inch (31) long would be needed to cover this rectangle? 10. (21)

Classify

5 1  8 8

4 4 3 be multiplied for the product to be 1? 5 5 5

5  2

12 in. 5 3

8 in. 5 3

5 8

4 4  5 5

3 5 3 5

4 4  5 5

Which of these numbers is divisible by both 2 and 3?

A 56

B 75

C 83 1 w1 11. Estimate the difference of2 4968 and 2099.

D 48

(16)

5 14 4 5 35 1 3 4 43÷ 25 13. $402.00     5   (2) 5 15w 8 85 5 2 28 8 10 1 5 21 5 3 3 3 4 4 3 14. What is 4 of 20? 1 3 5 5 1 1 23 2 3 5 3 10  100 3 3 4 4 45 5 3 5 5 5 8 8 (29)      1    5 3 5 5 5 55 2 2 8 8 1002 42 4 8 23w8 10 10 5 100 1   Write each answer in simplest form: 5 3 5 5 8 8 53 3 3 5 14 4 5 35 1 3 3 4 4        15.  35 16. 5 10 100 8 85 5 2 (24) 28 8 10 2 100 2 (24) 5 12. $4.30 ×53100

3 5

(2)

5 3 5

5 3 5 3

5 3

3 5

4 43  5 55

5 14 4 5 35 1  * 17.    8 85 5(29) 2 28 8 19. (10)

1 w1 2 1 w1 2

53 3 3   10 2100 2

5 3

* 18. (29)

3 3  10 100

Describe the sequence below. Then find the tenth term. 1 1 w1  8, w 6, 1 ... 2,24, 2

Generalize

1 w  1 1 unknown number. Remember to check your work. 2 Find each w1 2 1 * 21. w  1 20. q − 24 = 23 (3) (30) 2 1 w 22. 1 Here we show 16 written as a product of prime numbers: 2 (19) 2∙2∙2∙2

1

Write 15 as a product of prime numbers. 23. (7)

Estimate A meter is a little longer than a yard. About how many meters tall is a door?

* 24. Five of the 30 students in the class were absent. What fraction of the (29) class was absent? Write the answer as a reduced fraction. How do you know your answer is correct?

176

1

Saxon Math Course2 1

1

62

1 11 1 11 , , , 8 42 16 22

1

62

1

62

1 1 11 1 1 1 , , , , , 8 4 16 8 2 4 16 2

* 25. (17)

To what mixed number is the arrow pointing on the number line below? Connect

0

1

1 1 1 1 1 62 , , , 8 4 fraction: 16 2 * 26. Write each percent as a reduced 1 2

(33)

27. (22)

28. 1 6

1 2

(Inv.22)

* 29. (32)

2

b. 30%

a. 70%

Model Four fifths of Gina’s 20 answers were correct. How many of Gina’s answers were correct? Draw a diagram to illustrate the problem.

1 11 1 at the numerator 1 looking 1 denominator of a fraction, 1 1 1and By , , 6 2, , , , 8 4 16 2 8 4 16 how can you tell whether the fraction is 2greater than or less than 12 ?

1 Explain 2

1 Analyze2

1 1 1 1 1 , ,p.m.? , What time is 6 2 hours after 8:45 8 4 16 2

30. Arrange these fractions in order from least to greatest: (17)

1 2

1 2

1

2 Early6Finishers

Real-World Application

1

62

1 1 1 1 , , , 8 4 16 2

1 1 1 1 , , , The was built in Ancient Greece over 2500 years ago. The 8 4 Parthenon 16 2 Parthenon’s base is a rectangle measuring approximately 31 meters by 70 meters. a. What is the approximate area of the Parthenon’s base? b. What is the approximate perimeter of the Parthenon’s base?

Lesson 33

177

LESSON

34

Decimal Place Value Building Power

Power Up facts

Power Up D

mental math

a. Order of Operations: (4 × 200) + (4 × 25) b. Number Sense: 1480 − 350 c. Number Sense: 45 + 18 d. Money: $12.00 − $2.50 e. Number Sense: Double 250.

1 8

1 f. Number Sense: 8

1500 100

1 10

1 10

g. Measurement: Which is greater, 3 feet or 1 yard? h. Calculation: 3 × 3, × 9, − 1, ÷ 2, + 2, ÷ 7, × 2

problem solving

Jeanna folded a square piece of paper in half so that the left edge aligned with the right edge. Then she folded the paper again so that the top edge aligned with the bottom edge. With scissors, Jeanna cut off the top, left corner of the square. Which diagram will the paper look like when it is unfolded?

New Concept

A

B

C

D

Increasing Knowledge

Since Lesson 12 we have studied place value from the ones place leftward to the hundred trillions place. As we move to the left, each place is ten times as large as the preceding place. If we move to the right, each place is one tenth as large as the preceding place. 1 of 100 is 10. 10

hundreds

1 of 10 is 1. 10

tens

ones

Each place to the right of the ones place also has a value one tenth the value of the place to its left. Each of these places has a value less than one (but more than zero). We use a decimal point to mark the separation between 178

Saxon Math Course 1

the ones place and places with values less than one. Places to the right of a decimal point are often called decimal places. Here we show three decimal places: 1 1 of 1 is . 10 10

ones

tenths 1 1 10decimal point10

1 10

1 10

1 10

1 10

1 10

1 10

1 10

1 1 100 100

1 100

1 10

1 100

1 1000

mill 1

1

9

1

1

9

49 10 49$10do A mill is 1000 1000 of a dollar and not have a coin for a mill. 10 10 of a cent.$We However, purchasers of gasoline are charged mills at the gas pump. A price 9 of $2.29 $ 49 10 per gallon is one mill less than $2.30 and nine mills more than $2.29. Of course, place1values extend beyond the thousandths place. The 1 1 1 1 1 1 decimal 1 1 1 10 100 1000 10,000 100,000 10 100 1000 10,000 100,000 chart below shows decimal place values from the millions place through the millionths place. Moving to the right, get smaller and 1 1 1 the place values 1 1000 10,000 100,000 1,000,000 smaller; each place has one tenth the value of the place to its left. Discuss

What pattern do you see?

millionths

hundred-thousandths

thousandths 1 1000

ten-thousandths

hundredths 1 100

tenths

decimal point

ones

tens

1 1,000,000

1 100,000

1 10,000

1 10

1

10

. 100

, 1000

,

hundreds

thousands

Decimal Place Values

ten thousands

1 100

1 10

10,000

Why is thinking about money a helpful way to remember 1 decimal place 10 values?

thousandths

hundredths

Thinking about money is a helpful way to remember decimal place values.

hundred thousands

1 10

100,000

1 10

Connect

millions

1 1000

Thinking Skill

1,000,000

1 10

1 1 1 of is . 1000 100 10

1 1 1 of is . 100 10 10

Example 1 Which digit in 123.45 is in the hundredths place?

Solution The -ths ending of hundredths indicates that the hundredths place is to the right of the decimal point. The first place to the right of the decimal point is the tenths place. The second place is the hundredths place. The digit in the hundredths place is 5.

Lesson 34

179

Example 2 What is the place value of the 8 in 67.89?

Solution The 8 is in the first place to the right of the decimal point, which is the tenths place.

Practice Set

a. What is the place value of the 5 in 12.345? b. Which digit in 5.4321 is in the tenths place? c. In 0.0123, what is the digit in the thousandths place? d.

Connect

What is the value of the place held by zero in 50.375?

e. What is the name for one hundredth of a dollar? f.

Written Practice 1.

(22)

What is the name for one thousandth of a dollar?

Conclude

Strengthening Concepts Three eighths of the 24 choir members were tenors. How many tenors were in the choir? Draw a diagram to illustrate the problem. Model

2. Mom wants to triple a recipe for fruit salad. If the recipe calls for 8 ounces of pineapple juice, how many ounces of pineapple juice should she use?

(15)

* 3. The mayfly has the shortest known adult life span of any animal on (32) the planet. The mayfly grows underwater in a lake or stream for two or three years, but it lives for as little as an hour after it sprouts wings and becomes an adult. If a mayfly sprouts wings at 8:47 a.m. and lives for one hour and fifteen minutes, at what time does it die? * 4. (33)

Connect

Write each percent as a reduced fraction: b. 40%

a. 60%

100 * 5. Compare: 100 (29) 100 100 * 6. 100 100

10 10

(32)

Analyze

10 10 10 10

Write (6 × 100) + (5 × 1) in standard notation.

100 10 * 7. 100 Which digit 10 is in the ones place in $42,876.39? (34) * 8. If the perimeter of a square is 24 inches, (31)

a. how long is each side of the square? b. what is the area of the square?

* 9.

(Inv. 3)

Represent 1 1 2 inches,

Draw triangle ABC so that ∠C is a right angle, side AC is and side BC is 2 inches. Then measure the length of side AB.

* 10. What is the least common multiple of 6 and 8? (30)

180

Saxon Math Course91

9  10 10

d  144 6

5 5  2 2

2

5 2

3 3  8 8

1 12

9 9  10 10 (2) (29) 1 1 12 13. Estimate the quotient1when 2 898 is divided by 29. 11. $5.60 ÷ 10 (16)

11

11

1122 1 22

7

18

18. (24)

21.

3323) (10,

81 81 100 100

44

6 71 1 30 8 5

81

3

5

4 6 Find each unknown number. Remember to100check your work. 5 5 5 9 9 d 15. 6d = 144 16.   2   144 10 10 2 2 2 (4) (4) 6 55 55 55 dd 33 33 * 17.     22   144   Compare: 144 66 22 22 22 88 88 (29)

5 d 5 5 55 5 5 5   144    2  2  6 2 22 22 2 2 2 2

75

5 5   2 2 2

11

(16)

1 22

1 86

d  144 6 1 22

114. 122 Round 36,847 2222 to the nearest hundred.

99 99  10 10 10 10

5 6

12.

6 30

 15

22. (15)

23. (14)

3 3  8 8

3 3 113  13 11 15 113 15 3 5  3 * 20.    19.  52 81 8 12 12 8 8 128 (24) 4 2 12 4 1223 12 (29) 4 100

4

11 11 11   12 12 12 12 7

6

What is the ratio of77the first term 6to the fifth term of the 55 6 11   1 1 5 5 6 6 8 8 30 30 sequence below?

18

Predict

6 30

6, 12, 18, 24, …

 15

The movie theater sold 88 tickets to the afternoon show for $7.50 per ticket. What was the total of the ticket sales for the show? Write an equation and solve the problem. Explain why your answer is reasonable. Formulate

To what number is the arrow pointing on the number line

Connect

below?

–20

–10

0

10

20

24. (80 ÷ 40) − (8 ÷ 4) (5)

25. Which digit in 2,345.678 is in the thousandths place? (24)

26.

(Inv. 2)

Model

Draw a circle and shade

2 3

of it.

27. Divide 5225 by 12 and write the quotient as a mixed number. (25)

28.

The first glass contained 12 ounces of water. The second 5 glass contained 11 ounces of water. third glass contained 7 ounces 435The 12 of water. If water was poured from the first and second glasses into the third glass until each glass contained the same amount, then how many ounces of water would be in each glass?

29.

The letters r, t, and d represent three different numbers. The 1 12 product of r and t is d.

(18)

(2)

Evaluate

Represent

rt = d Arrange the letters to form another multiplication equation and two division equations. 30. (2)

Instead of dividing 75 by 5, Sandy mentally doubled both numbers and divided 150 by 10. Find the quotient of 75 ÷ 5 and the quotient of 150 ÷ 10. Connect

Lesson 34

181

LESSON

35

Writing Decimal Numbers as Fractions, Part 1 Reading and Writing Decimal Numbers

Power Up facts mental math

Building Power Power Up G a. Order of Operations: (4 × 300) + (4 × 25) b. Number Sense: 8 × 43 c. Number Sense: 37 + 39 d. Money: $7.50 + $7.50 1 of 3 3600 10

e. Fractional Parts: 1

f. Number Sense: 3

360

3600 10

g. Measurement: Which is greater, 10 centimeters or 10 millimeters? h. Calculation: 5 × 5, − 1, ÷ 3, × 4, + 1, ÷ 3, + 1, ÷ 3

problem solving

Copy this problem and fill in the missing digits:

Increasing Knowledge

New Concepts writing 25 decimal 100 numbers as fractions, part 1

Decimal numbers are actually fractions. Their denominators come from the 1 sequence 10, 100, 1000, . . .. The denominator of a decimal fraction is not 4 written. Instead, it is indicated by the number of decimal places. One decimal place indicates that the denominator is 10.

11 43

25 100

3600 10

3 0.3  10

3 0.03  100

Two decimal places indicate that the denominator is 100. 3600 0.25 10

1 3

0.25 

25  100

3 0.3  10

25 100

182

___ × 9 __2

Saxon Math Course 1

3 0.03  100

3 0.003  1000

3 10

0.003 

3 100

3 1000



1 4

Three decimal places indicate that the denominator is 1000. 3600 10

3 0.3  10

25 100

3 0.03  100

3 1000

3 0.003  1000

1 equals the number of Notice that the number of zeros in25the denominator 100 4 decimal places in the decimal number.

Example 1 A quart is 0.25 gallons. Write 0.25 as a fraction. 3600 10

1 3

Solution

0.3 

3 10

0.03 

The decimal number 0.25 has two decimal places, so the denominator is 100. The numerator is 25. 0.25  25 25

25 100

11

The fraction100 100 reduces to 44.

Example 2

6 6 21 6 21 6 A kilometer is about 10 of a mile. Write 10 as a decimal number. 10 33 10010 33 100 33 3600 11 3600    0.003 0.3 0.03 0.003 0.3 0.03 33 10 10 10 100 100 10 100 10 Solution The denominator is 10, so the decimal number has one decimal place. We write the digit 6 in this3place. 2 3 2 13 3 13 3 25 25 6 21 60.25  0.25 0.— 0.6 100 100 10 10 100

reading and writing decimal numbers

We read numbers to the right of a decimal point the same way we read whole numbers, and then we say the place value of the last digit. We read 0.23 as 3 2 “twenty-three hundredths” because the last digit is in the hundredths place. 13 3 To read a mixed decimal number like 20.04, we read the whole number part, say “and,” and then read the decimal part.

Example 3 The length of a football field is about 0.057 miles. Write 0.057 with words.

Solution We see 57 and three decimal places. We write fifty-seven thousandths.

Lesson 35

183

Example 4 An inch is 2.54 centimeters. Use words to write 2.54.

Solution The decimal point separates the whole number part of the number from the decimal part of the number. We name the whole number part, write “and,” and then name the decimal part. two and fifty-four hundredths

Example 5 Write twenty-one hundredths a. as a fraction. b. as a decimal number.

Solution The same words name both a fraction form and a decimal form of the number. a. The word hundredths indicates that the denominator is 100. 6 10

21 100

6 10

b. The word hundredths indicates that the decimal number has two decimal places. 3 3

0.21

2

13

Example 6 Write fifteen and two tenths as a decimal number.

Solution The whole number part is fifteen. The fractional part is two tenths, which we write in decimal form. 15.2

(one decimal place)

fifteen and two tenths Verify

Practice Set

How would you write 15.2 as a mixed number?

Write each decimal number as a fraction: a. 0.1

b. 0.31 1

3600 10

1 3

184

c. 0.321 3600

1

3 10 Write each fraction as3 a decimal number: 3 123 f. 123 3 e. 17 123 17 17 d. 3 10 100 1000 1000 10 100 10 100 1000 3 12317 17 3 10 10010 1000 100

Saxon Math Course 1

3600 10

123 1000

3 123 17 Use words to write each 10 number: 100 1000 3 123 i. 1.2 17 g. 0.05 h. 0.015 10 100 1000 17 as a fraction,123 Connect Write 3each number first then as a decimal number: 10 100 1000 j. seven tenths k. thirty-one hundredths l. seven hundred thirty-one thousandths Write each number as a decimal number: m. five and six tenths n. eleven and twelve hundredths 3 123 17 10 100 1000 o. one hundred twenty-five thousandths

Written Practice

Strengthening Concepts

1. What is the product of three fourths and three fifths?

(29)

2.

(22)

Thomas planted 360 carrot seeds in his garden. Three fourths of them sprouted. How many carrot seeds sprouted? Draw a diagram to illustrate the problem. Model

* 3. Sakari’s casserole must bake for 2 hours 15 minutes. If she put it into (32) the oven at 11:45 a.m., at what time will it be done? * 4. (35)

Represent

Write twenty-three hundredths

a. as a fraction. b. as a decimal number.

* 5. (35)

Represent

Write 10.01 with words.

3 2 as a decimal 5 number. 5 2 3 * 6. Write ten and five tenths    (35) 5 5 8 8 3 4 * 7. Connect Write each percent as a reduced fraction: (33)

2 5

2 5

b. 75%

a. 25%

* 8. Write (5 × 1000) + (6 × 100) + (4 × 10) in standard notation. (32)

* 9.

Which digit in 1.23 has the same place value as the 3 25 in 0.456?5 53 2 2 35 5 2 3 2 2 2       5 5 5 5 5 8 85 5 3 48 8 3 4 10. What is the area of the rectangle below? (34)

(31)

Connect

2 5

20 mm 10 mm

11. In problem 10, what is the perimeter of the rectangle? (8)

Lesson 35

185

12. There are 100 centimeters in a meter. How many centimeters are in (7) 10 meters? * 13. Arrange these numbers in order from least to greatest: 3 1 23 3 23 1 23 1 3 1 (34)

23 100

1 4

14. (7)

3 2  5 5

5 5  8 8

5 2

10 2

5

20 6

100% 6

0 6

1 3

2

16 3 %

60 20

6 2

6 2

21 100

21 00

21 100

3 4

100

Estimate

is a door?

4 100

4 4

4

2 4

4

0.001, 0.1, 1.0, 0.01, 0, −1

1 2

1 2

1 2

A meter is about one big step. About how many meters wide 23 100

3 4

1 4

1 2

5 2 5 5 2 5 3 5 2 3 3 16. 2 3 2 2 15. 3  2 3  2 5     17.25  25  5 5 5 5 5 8 5 8(24)5 8 8 3 8 4 8 3 4 4 (24) 5 (29) 3 3 2 * 18. 2a.  How many 25 s are in 1? 5 4 (30) 3 3 5 5 2 2 3  number of 25 s the answer to part a to find the b. Connect Use 5 5 8 8 3 4 5 in 2. 5 10 5 10   5 10 5 2 2 2 5 2 2 2 100% 4 1 4 20 19. Convert 6 to a mixed number. Remember 1  4 part reduce the 3 to 5 fraction 6 4 4 4 (25) of the number. 5 10 5 2 4 2 100% 4 1 4 100%20 1 20. 21.   34  1 5  4 3 1 5 4 4 4 6 6 4 46 (25) (26) 4 1 4 3 1 2 2 22. 3 Compare: 1 3 35  4 4 16 3 % 24 16 3 % 4 4 (29) 23.

1 33

100

(Inv. 2)

Represent

3

26.

60 20

6 25 25 12 12 60 2 20 4 4 8 3 3 3 3 1 1 1 1 2 2 2 2 28 2 2 2 33 3333 33 16 316 %16 16 316 %16 3 %3 % 2 4 2 4 2 4 2 4 3 %3 % 3 %16 3 %16 25 1 21 21 –20 –10 0 10 20 12 100 8 100 2

25 8

25 8

25 25 25 25 8 8 8 8

Which of these division problems has the greatest 1 21 1 12 2 12 2 100 60 6060 60 6 6 6 6 25 2525 25 12 1212 12 B C D 1 A 12 2 2 2 2 2 20 2020 20 4 4 4 4 8 8 8 8 (2)

* 27. (35)

Conclude

21 21 quotient? 100 100

Write 0.3 and 0.7 as fractions. Then multiply the fractions. What is the product? Conclude

21 write the 1 1 21 21 1 1 fraction 21 21 21 21 21Then * 28. Write 21% as a fraction. as a decimal 12 212 12 2 2 100100100100 100100100100 2 12 number.

(33, 35)

29. Instead of solving the division problem 400 ÷ 50, Minh doubled both (2) numbers to form the division 800 ÷ 100. Find both quotients. 30.

(23, 25)

186

A 50-inch-long ribbon was cut into four shorter ribbons of equal length. How long was each of the shorter ribbons? Write an equation and solve the problem. Explain why your answer is reasonable. Formulate

Saxon Math Course 1

2 5

2 5

One sixth of a circle is what percent of a circle?

2 2 3 25 2 16 3 % 16 3 % 2 16 3 %2 4 8 24. Compare: 3 × 184 100% ÷ 6 3 × (18 ÷ 6) 100% 4 4 41 41 1 1 100% 100% 4 4 4 4 20 20 20 20 (9)  15  1 31 54  4 4 54  3 331 5  3 25 2 6 6 66 6 24 16 3 % 44 44 4 4 6 66068 6 4 4 12 4 4 4 4 25 25. Connect To what number is the arrow pointing on the number line 2 20 4 8 (14) below?

12 4

1 2 16 3 %3 3

2 5

LESSON

36

Subtracting Fractions and Mixed 1  1  1 1 1 1 3 6 2 Numbers from Whole Numbers 3 6 2 Building Power

Power Up facts

81 100

Power Up F

mental math

a. Order of Operations: (4 × 400) + (4 × 25) b. Number Sense: 2500 + 375 c. Number Sense: 86 − 39 d. Money: $15.00 − $2.50

1 3

1 e. Fractional Parts: 6

1 2 of 4800 100

f. Number Sense:

1 1 1   3 6 2

320

81 1000

3 4

3 4

g. Measurement: Which is greater, 6 months or 52 weeks? h. Calculation: 2 × 4, × 5, + 10, × 2, − 1, ÷ 9, × 3, − 1, ÷ 4

problem solving

The playground is filled with bicycles and wagons. If there are 24 vehicles and 80 wheels altogether, how many bicycles are on the playground? How many wagons?

New Concept

1 8

Increasing Knowledge

1 8

4800 100

1 2

Read this “separating” word problem about pies. 4800 100

There were four pies on the shelf. The server sliced one of the pies into 6 1 sixths and took 2 6 pies from the3 6shelf. How many pies were left on the shelf? We can illustrate this problem with circles. There were four pies on the shelf. 1 8

1 8

1

1 2

6

2 6 there were 3 6 pies, The server sliced one of the pies into sixths. (Then which is another name for 4 pies.)

Lesson 36

187

6 The server took 2 16 pies from the3 shelf. 6

5

1

26

16

3

4

4 pies � 2 16 pies

5

2

5

4

16

6 6

6

36

� 2 16

� 2 16 5

126 1 6

5�1

2 3

3 3

2

13

3

4�3 1

6

1

36

26

5

2 3

6 36

1 26 5

We see 1 6 pies left on the 1 6 shelf.

1

26

5 6

� 1 23

�1 3

5 16

1 26

6

3�6 6

Now we show the arithmetic for subtracting 2 16 from 4. 4 pies

4

1 226216 1pies1 6 26

55 1 1� 16 62 61 56

5

1

36

26

16

6

36

6 1 1 5 � 26 1 21 336 6 3 6 116 5 1�5 2 1 226 1 2 1 6 2 6 6 6 6 6 6 6 6 6 36 5 1 1 The server sliced one of the pies into sixths, so we change 4 wholes into 6 6 � 26 s 4 4 pies � 2 16 36 3 wholes plus 6 sixths. Then we subtract. 5 1 � 2 16 � 2 16 pies 16 5 26 3 2 3� 2 6 5 1 1 � 5 � 1 1 4 � 26 16 1 2 3 6 5 5 1 1 1 6 3 3 5 5 3 1 5 6 4 3 26 2464 43 66 3 6 1 6 3 62616 6 1 6 26 26 6 16 16 1 63 � 6� 4 pies 6pies 4 pies 66 3 3 3 6 1 3 5 1 2 1 1 54 3 6 42� 1pies 13 1 �2 1 1 6 � 13 11 5 1 ��2216 � � 26 � 26 � 2 6 pies 263 3 6� 26 266 3�6 63 pies 6 256 1 6 2 6 2 6 5 1 1 1 6 6 5 5 2 1 1 1 6 6 � 1 2 3 2 1 1 2 562 3 2 3� 6 5 1 1 3 5 2 66 2 16 5 5 3 1�3 66 3 � 66 4� 3 6 3 66 2 6 2 66 1 6 1 66 1 6 1 66 � 1 3 2 65 �1 3 1636 1 6 1 2 6 33 Example 3 36 6 6 4 4 pies 4 pies 3 3 641 6 52 5 1 55 5 3 2 3 3� 63 6 443 3 4 3 2 3 2 2 3 61 26 6 1 1 1 2 3 3 2 3 � 1 5 � 1 1 4 1 6 6 6 1 1 6 � 3 3 3 5 � 1 1 4 � � 5 � 1 1 4 Subtract: . Then write a word problem that is solved by the 2 3 � 26 3 � 62 pies 1 3 343 �3 2� 3� 3 1 3 66 6 5 �3� 56 2 63 pies 1 1 61 2 2 22 6 1 2 6236 2 �� 26 3 � 3 2 � 26 35 6 4 piessubtraction. 26 1 1 2 3 1 1 � 1 6 6 6 6 6 6 � 3 3 1 � 3 1 4 4 pies 6 2 3 4 pies 1 64 � 41 1 3 5 3 51 6 5 5 11 6 3 pies 6 � 2 61 pies 16 3 36 1266 16 16 36 26 6 11 �226 6 1 1 1 1 1 333 3 �2 1 1� �2 pies 2 66 pies �2 � 2 16 � 2 66 Solution� 2 1 � 3 2 66 3 5 6 pies 3 1 5 6 3 3 6 5 4 3 � 4 5 2 5 5 2 3 2 3 3 3 2 3 2 6 1 3 3 5 �To 4 pies 2 25 21 1from plus 1 subtract 55,�we 1 first change 1 3 5 to 44 � 4 � 3 43 6 3 3. Then we subtract. 3 �33316136 3 3 6 � 1 2 3 3 2 1 � � 1 23 1 6 3 3 � 12 42 3 � 5 pies 4 pies � 2 6 3 21 3 3 � 3 4 2 � 6 Thinking Skill 4 3 3 3 5 31 3 1 36 2 5 � 1 13 3 3 12 6 5 2 2 3 2 3 4 4 4 pies 3 1 4� 52� � 21 pies 1 5 � 2 63 1 2 5 �3 33 3 4 3 4 33 � 1 3 2 63 4 � 1 23 3336 3 15 � 163 1 3 1 33 4� 2 16 � 3 Discuss 2 3 3 2 1 1 � 1 3 2 2� � 1 � 1 � 26 5 � 1 3 21633 pies Why do we need � 1 23 � 1 33 1 � 26 16 3 2 2 3 1 5 3 to2change 5 to 3 3 1 5 3 3 16 4 33 1 3 3 3 33 1 4�3 3 4 33 before we 2 3 5 � 2 3 2 3 22 3 �1 43 3 1about 5 � 1 subtract 1 3 3from 4 �word 2 Common subtraction problems are comparing or separating. 2 3 3 3 2 �31 3 5 1 122 1 1 11 15 2222 3 � 5? 3a1 2 cuts 4  2a 3 3  3 2 problem. 25 If 4 333 from 2 6 10 3 � Here is a sample word clerk yards of fabric � 1 1 4 � 1 3 1 3 2 4 4 3 3 2 4 4 12 212 3 3 1 3 � 33 5-yard length, how many yards of fabric remain? 1 5 3 1 1 1 1 1 5 3 1 1 1 3 5 3 3 32 2 32 4  22  3 42 6  110  2 1 6 6 4 1 10 32 2 4 4 4 12 2 12 10 2 4 2 10

4

s

Practice Set

2 3

32

1 2

188

Show the arithmetic for each subtraction: 1 1 1 1 1  22  2  3  22 4 b. a. 3  2 2 4 4 4 32

1 1 1 1 2  3  2 c. 4  22  2 4 4 4 Saxon Math Course 1

5 1 3 42 4 12

1 5 10d.32 2 12

5 1 3 42 12 4

1 5 10 32 2 12

3 1 5 6  110  2 16 2 10

3

6 4 1

3 10

3 6  110  2 10 5

16

1 1  32 4 2 2

1 1 4  22  4 4

1 4

42

32

2

1 4

2

3 4  4 63 61 10

1 2

3

5 12

g.

j.

3 4  4 6

31

1 4

Model

5

3 4

16

5 3 12 * 1.

61

3 10

Formulate

answer.

(33)

1 2

1 1 33  2 4 44

5

16

5 3 1 1 5 3  whole pies the 4 i. 2There were3four 2 shelf. The 6 server 1 10on took 1 6 pies.4How 4 12 2 10 many pies were left on the shelf?

1 4

1 42 4

10  2

3 3 1 5 3 6  110  2 1 6f. 6  4 1 10 10 2

Select one of the exercises to model with a drawing. 3 1 5 3 61 16 10  2 4 2 10 h. Formulate Select another exercise and write a word problem that is solved by the subtraction.

Written Practice

1 2 4 3 4  4 6 5 12

1 2

5 1 5 1 3  4  2 e. 10 32 12 2 12 4

Write a word problem similar to problem i, and m then find the 6 12

12 n 6

Strengthening Concepts 3 1 5 3 61 16 10  2 4 2 10 Analyze Twenty-five percent of the students played musical instruments. What fraction of the students played musical instruments?

5 3 * 2. 1About of the Earth’s surface is covered with water. What fraction of 6 4 (36) Earth’s surface is not covered with water? 5 1 1 1 2 2 2  a yard. How 4 many 3 a 3. A mile is 5280 feet.3There 2 are 3 feet in 4 4 yards are in 12 (15) mile?

1

* 4. Which digit in 23.47 has the same place value as the 6 in 516.9? (34)

* 5. Write 1.3 with words. (35)

* 6. Write the decimal number five hundredths. (35)

* 7. Write thirty-one hundredths (35)

a. as a fraction. b. as a decimal number.

* 8. Write (4 × 100) + (3 × 1) in standard notation. (32)

3 is in 4 * 9. Which digit in 4.375  the tenths place? (34) 4 6 * 10. Analyze If the area of a square is 9 square inches, (8, 31)

a. how long is each side of the square? b. what is the perimeter of the square?

11. Name two obtuse angles in the figure below. (28)

B

1 11 1 1 21 1 1 23 3  23  2A 3  1 3  1 3  23  2 4 4 44 4 3 33 4 4 3 M

3 4

3 43  4 64

C

D

1 1 1 1 1 1 23 1 3 2 1 11 211 24 3  213  14. 12. 3  2 3  2 3  1 323– 1 3 3 +2 3* 13. 4 4 4 3 4 4 4 3 3 4 4 4 3 43 (26) (36) (26)3

m 12 m 12  6  6 n  6n  6 12 12

33 4 3 4   44 6 4 6

Lesson 36

189

5 6

3 4

2

1 2 4 16. 3  1 1 What 2 3 3 5 5 4 2 3 3is 34of228? 3 115. 4 46 6 4 3 3 3 3 (29) (29) 4 6 6 3 4 1 1 Chen 2 3  He spent 56 of his money in the 3  1 17. Model 2 went 4to the mall with $24. 3  4(22) 3 3 4 6 music store. How much money did he spend in the music store? Draw a 1 1 11 2 1 3 1 1 1 2 3 43   2 3  32  1 3  2 to illustrate 3  1the 3 problem. 3 2 diagram 4 4 44 3 4 4 4 3 3 3 4 64 m 12 6 6 12 18. What is the average of 42, 57,nand 63?

1 11 3 321 4 44 1 1 3 2 4 4

1 4

3 4  4 6

5 6

(18)

19. The factors of 6 are 1, 2, 3, and 6. List the factors of 20. (19)

a. What is the least common multiple of 9 and 6? 12 (20,20. 30) n 6 b. What is the greatest common factor of 9 and 6? 12  6 n Find each unknown number. Remember to check your work. m 12 m 12 6 21. 22. n  6 6 6 n  12 (4) 12 (4)

m 12 6 6 12 n m 6 12

6

23. Round 58,742,177 to the nearest million. (16)

24. Estimate the product of 823 and 680. (16)

25. How many millimeters long is the line segment shown below (7) (1 cm = 10 mm)? cm

1

2

3

4

5

6

7

8

1 1 1   3 6 2 26. Model Using your fraction manipulatives, you can find that the sum 1 181 1 81 13 1 111 1 1 181 11 1(Inv. 2) 1 1 1 of 13 and 16 is 12.  1000  6  1000  4  23 2 3 2 6 1000 6 3 6 2 3 6 23 6 2 1 1 1 81 3 1 1   6 2 1000 4 3 6 2 1 3

1 3 1 3

1 6

81 1000

1 2

3 4

3 4

1 3 4 3

3 4



13 1  64 2

3 4

Arrange these fractions to form another addition equation and two subtraction equations. Represent

1 6

* 27. Write 0.9 and 0.09 as fractions. Then multiply the fractions. What is the 1 1 1 (35) 81 1 1 1   product? 3 6 2 1000 3 6 2 1 1 1 1 1 1   1 1 1Write 81 3 3 6 2 3 in 6 * 28. Represent as a decimal number. (Hint: Write a zero the 2   3 1000 4 4 (35) 3 6 2 tenths place.)

1 2

1 2 1 6

1

4800

1 2

100 Math6 Language 4800 The radius is the 100 distance from the center of a circle to a point on the circle. The diameter is the distance across a circle through its center.

190

1 1* 29.1 Analyze 81 3 a. How many 34 s are in 1?   4800 1000 4 3 (30)6 2 100 1 1 1 81 3 b. Use part a to find4the number of 34 s in 3. theanswer to 1000 3 6 2 4800 1 1 1 81 3 3 1 4800 4800   100 2 30. Connect 1000 4 4 100 100 3Freddy 6 drove 2 a stake into the (23, 27) ground, looped a 12-foot-long rope over it, and walked around the stake to mark off a circle. What was the ratio of the radius to the diameter of the circle?

Saxon Math Course 1

4800 100 4800 100

12 ft

3 4 81 1000

LESSON

37

Adding and Subtracting Decimal Numbers Building Power

Power Up facts

Power Up G

mental math

a. Order of Operations: (4 × 500) + (4 × 25) b. Number Sense: 9 × 43 c. Number Sense: 76 − 29 d. Money: $17.50 + $2.50 1 2 of 2500 10

e. Fractional Parts: f. Number Sense:

1 2

1 2

2500 10

520

g. Geometry: A rectangle has a length of 10 in. and the width is as the length. What is the perimeter of the rectangle?

1 2

1 2

as long

h. Calculation: 6 × 8, + 1, ÷ 7, × 3, − 1, ÷ 5, + 1, ÷ 5

problem solving

If a piglet weighs the same as two ducks, three piglets weigh the same as a young goat, and a young goat weighs the same as two terriers, then how many of each animal weighs the same as a terrier?

New Concept

Increasing Knowledge

When we add or subtract numbers using pencil and paper, it is important to align digits that have the same place value. For whole numbers this means lining up the ending digits. When we line up the ending digits (which are in the ones place) we automatically align other digits that have the same place value. 23 241 + 317

Lining up the ones place automatically aligns all other digits by their place value.

Lesson 37

191

Thinking Skill Analyze

In the number 2.41, in what place is the digit 4? the digit 1?

However, lining up the ending digits of decimal numbers might not properly align all the digits. We use another method for decimal numbers. We line up decimal numbers for addition or subtraction by lining up the decimal points. The decimal point in the answer is aligned with the other decimal points. Empty places are treated as zeros. 2.3 2.41 + 31.7 .

Lining up the decimal points automatically aligns digits that have the same place value.

Example 1 The rainfall over a three-day period was 3.4 inches, 0.26 inches, and 0.3 inches. Altogether, how many inches of rain fell during the three days?

Solution We line up the decimal points and add. The decimal point in the sum is placed in line with the other decimal points. In three days 3.96 inches of rain fell.

3.4 0.26 + 0.3 3.96

Example 2 A gallon is about 3.78 liters. If Margaret pours 2.3 liters of milk from a one-gallon bottle, how much milk remains in the bottle?

Solution We subtract 2.3 from 3.78. We line up the decimal points to subtract and find that 1.48 liters of milk remains in the bottle.

Practice Set

Find each sum or difference. Remember to line up the decimal points. a. 3.46 + 0.2

b. 8.28 − 6.1

c. 0.735 + 0.21

d. 0.543 − 0.21

e. 0.43 + 0.1 + 0.413

f. 0.30 − 0.27

g. 0.6 + 0.7

h. 1.00 − 0.24

i. 0.9 + 0.12

Written Practice * 1. (33)

2.

(22)

192

3.78 − 2.3 1.48

j. 1.23 − 0.4

Strengthening Concepts Sixty percent of the students in the class were girls. What fraction of the students in the class were girls? Represent

Analyze Penny broke 8 pencils during the math test. She broke half as many during the spelling test. How many pencils did she break in all?

Saxon Math Course 1

3.

Analyze What number must be added to three hundred seventy-five to get the number one thousand? 10 1 * 4. 3.4 + 0.62 + 0.3 100 * 5. 4.56 − 3.2 10 (3)

1 10

(37)

(37)

6. $0.37 + $0.23 + $0.48

(1)

3 3  4 4

8.

(10)

7. $5 − m = 5¢

(3)

What is the next number in this sequence? 3 100, 1 1000, . . . 1, 3 10, 5

3 2  3 2

1 sides 1 foot long to Harriet used 100 square floor tiles with 10 cover the floor of a square room. 10 of each side of the room? 10 1 1 1 1 a. What was the length 100100 100 10 1 3 1 3 1 2 3   3 3 3 7 b. What was the perimeter4 of the 4 room? 4 4 3 3  10. Which digit is in the ten-millions place in 1,234,567,890? 4 4 * 9.

(8, 31)

1 10

Predict

10 100

Connect

(12)

3 2 3 Three of the shown below are equal. Whichnumber 3 numbers 1 5 3 2 1 is not equal to 10 the others? How do you know? 10 1 10 1 1 10 10 10 100 10 100 10 10 1 1 C 100100 1 A 10 1 D 0.01 1 10 100 100 B 0.1 100 10 1 2 3 10 3 3 3 1 10 11 10 1 12. Estimate the product of 29, 42, and 39. 10 100 10 100 10 10 10 1 (16) 110 1 1 10 1 10 1 100 1 3 10 1 3 1 2 3 10 100 100 100 10   3 3 3 100 10 10 1 13. 3210 ÷ 3 3 27 3 3 43 4 4 14. 4 32,100 ÷ 30    3 1 (2) (2) 100 3 100 3 33 2 3 3 10 2 3 5 3 2 4 4    31 31 3 3  15 4 5 3 2 5 15. 4 3 2  16. $10,000 − $345 4 (1) (24) 4 33 22 33 33 33    11  33 5 33 22 4 4 5 4 4 3 2 3 2 3 3 3 3 3   * 317. * 18.   1Analyze 33   −1 5 3 2 4 4 4 3 (29) 2 3 2 3(36) 5 1 1 1  100  2  3 3  15 3 3 1 1 1 3 2 1 100 1 1  1  3 3 3 7 100   19. 3 1 3 1 3 1 1 3 2 3 3 34 4 1 3 1 1 1 3 1 1  3 4 47   3  4 2 4 3 4 4 (26) 3 7 1 4 4 4 4 4 4 4 4 3 3 3 11 11 11 100 100 11 33 11 33 22  33 11   331 33 33* 20. Analyze Compare:    100 77 1 1 1 1 100 44 3 44 44 44  2  3 1 1 2  3 1 1 13 3 1(29) 3     3 3 3 3 3 7 7 1 2 44 improper 4 4 fraction 4 4 3 4 the 100 3 1 3 421. 1 to a mixed number.  (25) Convert  3 7 4 4 4 4 22. What is the average of 90 lb, 84 lb, and 102 lb? 3 3  4 4

* 11. (35)

Classify

10 10 100 100

(18)

23. What is the least common multiple of 4 and 5? (30)

24. The stock’s value dropped from $38.50 to $34.00. What negative (14) number shows the change in value? 25. (17)

To what mixed number is the arrow pointing on the number line below? Connect

10

1 2

11

12

Lesson 37

193

10 100

3

1  4

* 26. (35)

Write 0.3 and 0.9 as fractions. Then multiply the fractions. Change the product to a decimal number. Represent

27.

Write three different fractions equal to 1. How can you tell whether a fraction is equal to 1?

28.

Instead of dividing 6 by 12, Feodor doubled both numbers and divided 12 by 1. Do you think both quotients are the same? Write a oneor two-sentence reason for your answer.

(Inv. 2)

(2)

Explain

Justify

* 29. The movie started at 2:50 p.m. and ended at 4:23 p.m. How long was (32) the movie? 30. (22)

Three fifths of the 25 students in the class were boys. How many boys were in the class? Draw a diagram to illustrate the problem. Model

1 2

Early Finishers

Real-World Application

Chamile is making a shirt and she needs to determine how many yards of fabric to buy. After calculating her measurements, Chamile determines she needs 90 inches of fabric (1 yard = 36 inches). a. How many yards of fabric does it take to make the shirt? b. If the fabric store only sells fabric in full yards, how much fabric will Chamile have leftover?

194

Saxon Math Course 1

LESSON

38 Power Up facts mental math

Adding and Subtracting Decimal Numbers and Whole Numbers Squares and Square Roots Building Power Power Up A a. Order of Operations: (4 × 600) + (4 × 25) b. Number Sense: 875 – 125 c. Number Sense: 56 – 19 d. Money: $10.00 – $6.25 $40.00 10

1 2 of 150 $40.00 10

e. Fractional Parts: 1 2

f. Number Sense:

g. Geometry: A regular octagon has sides that measure 7 cm. What is the perimeter of the octagon? h. Calculation: 10 + 10, – 2, ÷ 3, × 4, + 1, × 4, ÷ 2, + 6, ÷ 7

problem solving

Andre, Robert, and Carolina stood side-by-side for a picture. Then they changed their order for another picture. Then they changed their order again. List all possible side-by-side arrangements of the three friends.

New Concepts adding and subtracting decimal numbers and whole numbers

Increasing Knowledge

Margie saw this sale sign in the clothing department. What is another way to write three dollars? Here we show two ways to write three dollars: $3

T-shirt close-out $3 each

$3.00

We see that we may write dollar amounts with or without a decimal point. We may also write whole numbers with or without a decimal point. The decimal point follows the ones place. Here are several ways to write the whole number three: 3

3.

3.0

3.00

As we will see in the following examples, it may be helpful to write a whole number with a decimal point when adding and subtracting with decimal numbers.

Lesson 38

195

Example 1 Paper used in school is often 11 inches long and 8.5 inches wide. Find the perimeter of a sheet of paper by adding the lengths of the four sides.

Solution When adding decimal numbers, we align decimal points so that we add digits with the same place values. The whole number 11 may be written with a decimal point to the right. We line up the decimal points and add. The perimeter is 39.0 inches. 11. 8.5 11. + 8.5 39.0

Example 2 Subtract: 12.75 ∙ 5

Solution We write the whole number 5 with a decimal point to its right. Then we line up the decimal points and subtract. 12.75 − 5. 7.75

squares and square roots

Recall that we find the area of a square by multiplying the length of a side of the square by itself. For example, the area of a square with sides 5 cm long is 5 cm × 5 cm, which equals 25 sq. cm.

Area is 25 sq. cm Side is 5 cm

From the model of the square comes the expression “squaring a number.” We square a number by multiplying the number by itself. Reading Math An exponent is elevated and written to the right of a number. Read 52 as “five squared.”

“Five squared” is 5 × 5, which is 25. To indicate squaring, we use the exponent 2. 52 = 25 “Five squared equals 25.” An exponent shows how many times the other number, the base, is to be used as a factor. In this case, 5 is to be used as a factor twice.

Example 3 a. What is twelve squared? b. Simplify: 32 + 42

196

Saxon Math Course 1

Solution a. “Twelve squared” is 12 × 12, which is 144. b. We apply exponents before adding, subtracting, multiplying, or dividing. 32 + 42 = 9 + 16 = 25 What step is not shown in the solution for b?

Analyze

Example 4 Reading Math We can use an exponent of 2 with a unit of length to indicate square units for measuring area. 1 cm2 = 1 square centimeter

What is the area of a square with sides 5 meters long?

Solution

Reading Math The square root symbol looks like this: 2 .

We multiply 5 meters by 5 meters. Both the units and the numbers are multiplied. 5 m ∙ 5 m = (5 ∙ 5)(m ∙ m) = 25 m2 We read 25 m2 as “twenty-five square meters.” If we know the area of a square, we can find the length of each side. We do this by determining the length whose square equals the area. For example, a square whose area is 49 cm2 has side lengths of 7 cm because 7 cm ∙ 7 cm equals 49 cm2. Determining the length of a side of a square from the area of the square is a model for finding the principal square root of a number. Finding the square root of a number is the inverse of squaring a number. 2100 2100  10 6 squared is 36. The principal square root of 36 is 6.

2 21  1

We read 2100 as “the square root of 100.” This expression 2100  10 means, “What positive number, when multiplied by itself, has a product of 100?” Since 24  2 10 × 10 equals 100, the principal square root of 100 is 10.

2100 21  1 29  3

2

24  2

29  3

2

2100  2 10

2100

21

A number2is a 2perfect square if it has a square root that is a whole number. 4 Starting with 1, the first four perfect squares are 1, 4, 9, and 16, as illustrated 216  4 below. 1 21 2 100

2 216  4

100 212 1 3×3=9

4 × 4 = 16  2 24 2 100

10

242

2×2=4

1×1=1 21  1

216  4

21  1

Generalize

29  3

29  3

 23 249  2

2 292 43

216  4

21

What is the fifth perfect square? Draw it.

216  4

216  4

Lesson 38

197

Example 5 Simplify: 264

264

Solution The square root of 64 can be thought of in two ways: 1. What is the side length of a square that has an area of 64 square units? 281 2100  249 2. What positive number multiplied by itself equals 64?

2

264 With either approach, we find that 264 equals 8.

Practice Set

Simplify: b. 4.3 − 2

a. 4 + 2.1 281

2100  249 d. 43.2 264− 5

c. 3 + 0.4 e. 0.23 + 4 + 3.7

f. 6.3 −6 264

g. 12.5 + 10

h. 75.25 − 25

264 264

i. 92

264

2k. 8162 + 82 2 2 100 249 m. 15

2100  249

2144

o. 6 ft ∙ 6 ft q.

264

2

j. 281

2100  2

l. 2100  249

2144

n. 2144

264 m2

p. 264 m2

2144

Starting with 1, the first four perfect squares are as follows:

Predict

1, 4, 9, 16 What are the next four perfect squares? Explain how you know.

Written Practice

Strengthening Concepts 29  1.2

3.6  216

1. What is the greatest factor of both 54 and 45?

(20)

2.

Formulate Roberto began saving $3 each week for a bicycle, which costs $126. How many will it take him to save that amount of 2 weeks 2 1 2   1 2 1 3 money? Write an equation 3 3and solve the problem. 4

3.

Formulate Gandhi was born in 1869. About how old was he when he was assassinated in 1948? Write an equation and solve the problem.

(15)

(13)

3.6  216

3.6* 16 − 1.2 4 6. 2 5.63

 216 * 8. 4.75 − 0.6 * 7. 5.376 +3.6 0.24

* 9. 216  29

29  1.2 * 4. 29  1.2 (38)

29  1.2

(37)

3* 5. 4 (38)

5

(37)

(37)

216  29

216 

7 11  10 10

72 11  10 10 3

(38)

* 10. Write forty-seven hundredths

2 2 1 2 3 3

198 3 4

2 2 1 2 3 3

3

(35)

a.2 as a2fraction. 1 32  1 1 2 4 3 3 b. as a decimal 1 number. 2 3 1 4 3

4 5

Saxon Math Course4 1 4 4 5

32  1

1 4

4 5

7 11  10 10

2 3

3.6  216 3.6  216

2

3.6  216

2 22 2 1 12  2 3 33 3 1 9  1.2 9  1.2 2  2 1 3 41 2  1 3  .6 216 4 1.2 3 3

2 2 2 3 3 1 2 1 4 2 2 3

4

4

4 5

2 2 1 2 3 3

4 5

3 4

216  29

 216 3.6 3.6 216 2 2 1 1 2 32  1 3 3 4 1 2 21 2 1 2 11. Write 3(92   ×1 1000)1+1(4 × 10) + (3 × 1) in standard notation. 3  1 3 4 4 3  23 (32) 4     3.6 2 16 2 16 2 9 2 9 1.2 3.6 2 16   1 3.6 2 16 2 16 2 9 Which digit is in the hundredths place in $123.45? 32  1* 12. 2 1 22 2 1 4(34)   32  312  1 216  1 239 1233 2 3 4 4 3 4 * 13. The area of a square is 81 square inches. (8, 38) 5 4   3.6 22 16 216 2 9 4 1.2is4the length of each side? 2 9What 3.6 9 3.6  2 16721611 21 7 11 a.1.2 2 2 3   4 10 10 10 105 5 3 5 3 4 2 1 1 2 b.1What perimeter of the 71is2 the 7 square? 11 11 1 322 2 2 3  14 4 32  1   4 3 3 4 3 1073 1011 3 10 10 5 43 4 2  14. What4is the least common multiple of 2, 3, and 5 4? 5 4 10 10 3 (30) 1 22 2 1 1 12 1 2 32 115. 2 * 16. 32 312  1 34 3 3 3 4 4 (26) (36, 38) 7 11 7 11 2 3 4 4 2  417. What is of ? 18.  10 10 5 3 5 4 (29) 10 10 5(29) 3 7 11 2  How many s are in 1? 19. 10 a.10 3 (30) 7 4 11 4 7 4 3 3  number of 2 s in 2. b. Use the answer to part a to find 10 the 5 3 10 5 10 5 4 4 20. Six of the nine players got on base. What fraction of the players got on (29) base? 21. List the factors of 30. (19)

* 22. Write each percent as a reduced fraction: (33)

b. 65%

a. 35%

23. Round 186,282 to the nearest thousand. (16)

22 � 23 � 24 25. 22 � 23 � 24 3 (5) 3

124. 1 m � 1 m�1 3(30) 3 26. Compare: 24 ÷ 8 (9)

1 1 m�1 m�1 3 3

1 5

216  29 2 2 292 1.2 9 1.29 16

1 m�1 3

* 27. Write 0.7 and 0.21 as fractions. Then multiply the fractions. Change the (35) 1 product to a decimal number. Explain2 why2 your answer is reasonable. 1 5 5 5 (Hint: Round the original problem.) 5 �� 22 � 23 � 24 2423 � 24 22 � 23 � 24 22 � 12322 m�1 3 3 for each 3 3 3 28. Peter bought ten carrots for $0.80. What was the cost (15)

1 5

1 5

240 ÷ 80

carrot?

* 29. Estimate Which of these fractions is closest to 1? (38) 3 1 2 2 3 4 2 2 A B C 5 5 5 5 5 5 5 5 * 30. (36)

3 5

D

4 5

3 5

4 5

Justify If you know the perimeter of a square, you can find the area of the square in two steps. Describe the two steps.

Lesson 38

199

LESSON

39

Multiplying Decimal Numbers Building Power

Power Up facts

Power Up G

mental math

1 8

a. Order of Operations: (4 × 700) + (4 × 25) b. Number Sense: 6 × 45 c. Number Sense: 67 − 29 d. Money: $8.75 + $0.75

1 8

e.

1 2

f.

1 Fractional Parts: 2 of 2500 Number Sense: 100

5 75 375   2500 100 10 1000 100

350

g. Statistics: Find the average of 68, 124, 98, and 42 h. Calculation: 8 × 5, ÷ 2, + 1, ÷ 7, × 3, + 1, ÷ 10, ÷ 2

5 75 375   100 10 1000 375 1000

problem solving

375 1000

Sarah used eight sugar cubes to make a larger cube. The cube she made was two cubes deep, two cubes wide, and two cubes high. How many cubes will she need to make a new cube that has three cubes along each edge?

New Concept

Increasing Knowledge

Doris hung a stained-glass picture in front of her window to let the light shine through the design. The picture frame is 0.75 meters long and 0.5 meters wide. How much of the area of the window is covered by the stained-glass picture? To find the area of a rectangle that is 0.75 meters long and 0.5 meters wide, we multiply 0.75 m by 0.5 m. 0.75 m

0.5 m

One way to multiply these numbers is to write each decimal number as1 a 1 8 2 proper fraction and then multiply the fractions. 1 8

1 2

5 75 375   100 10 1000

200

0.75 × 0.5

5 75 375   100 10 1000

2500 100

375 1000

375 The product 1000 can be written as the decimal number 0.375. We find that the picture covers 0.375 square meters of the window area.

Saxon Math Course 1

Thinking Skill Discuss

Why can we count the decimal places to multiply 0.75 and 0.5?

Notice that the product 0.375 has three decimal places and that three is the total number of decimal places in the factors, 0.75 (two) and 0.5 (one). When we multiply decimal numbers, the product has the same number of decimal places as there are in all of the factors combined. This fact allows us to multiply decimal numbers as if they were whole numbers. After multiplying, we count the total number of decimal places in the factors. Then we place a decimal point in the product to give it the same number of decimal places as there are in the factors. 0.7 5 0.5 0.3 7 5

∙×

We do not align decimal points. We multiply and then count decimal places.



Three decimal places in the factors

Three decimal places in the product

Example 1 Multiply: 0.25 ∙ 0.7

Solution We set up the problem as though we were multiplying whole numbers, initially ignoring the decimal points. Then we multiply.



0.25 3 places × 0.7 0.175

Next we count the digits to the right of the decimal points in the two factors. There are three, so we place a decimal point in the product three places from the right-hand end. We write .175 as 0.175. Predict How could we have predicted that the answer would be less than one?

Example 2 Example2

Simplify: (2.5)2

Solution We square 2.5 by multiplying 2.5 by 2.5. We set up the problem as if we were multiplying whole numbers. Next we count decimal places in the factors. There are two, so we place a decimal point in the product two places from the right-hand end. We see that (2.5)2 equals 6.25. Verify



2.5 2 places × 2.5 125 50 6.2 5

Why is 6.25 a reasonable answer?

Lesson 39

201

Example 3 A mile is about 1.6 kilometers. Maricruz ran a 3-mile cross-country race. About how many kilometers did she run?

Solution



1.6 1 place × 3 4.8

We multiply as though we were multiplying whole numbers. Then we count decimal places in the factors. There is only one, so we place a decimal point in the product one place from the right-hand end. The answer is 4.8 kilometers.

Practice Set

Simplify: a. 15 × 0.3

b. 1.5 × 3

c. 1.5 × 0.3

d. 0.15 × 3

e. 1.5 × 1.5

f. 0.15 × 10

g. 0.25 × 0.5

h. 0.025 × 100

i. (0.8)2 k.

Conclude

Written Practice

j. (1.2)2 How are exercises a–d similar? How are they different?

Strengthening Concepts

1. Mount Everest, the world’s tallest mountain, rises to an elevation of twenty-nine thousand, twenty-nine feet above sea level. Use digits to write that elevation.

(12)

2. There are three feet in a yard. About how many yards above sea level is Mount Everest’s peak?

(7, 25)

* 3. (6)

If you had lived in the 1800’s, you may have seen a sign like this in a barber shop: Connect

Shave and a Haircut

1

22

six bits

A bit is 81 of a dollar. How many cents is 6 bits? * 4. 0.25 × 0.5 (39)

1 * 8

6. 63 × 0.7

(39)

5 75 375 * 8. 12.34 −100 5.6  10  1000 (37)

202

Saxon Math Course 1

1 2

5. $1.80 × 10

(2)

� 0.5 * 7. 1.23 � + 216 + (38)

* 9. (1.1)2 (39)

375 1000

* 10. Write ten and three tenths (35)

a. as a decimal number. b. as a mixed number.

11. (17)

Think of two different fractions that are greater than zero but less than one. Multiply the two fractions to form a third fraction. For your answer, write the three fractions in order from least to 20 1 greatest. 2 Evaluate

8

2

* 12. Write the decimal number one hundred twenty-three (35) thousandths.

20 8

13. Write (6 × 100) + (4 × 10) in standard form. 1 2 (32) 3 2 a  b1  3 3 5 3 * 14. Connect The perimeter of a square is 40 inches. How many square (38) tiles with sides 1 inch long are needed to cover its area? 1 1 20 1 22 22 22 8 15. What is the least common multiple (LCM) of 2, 3, and 6?

20 8

(30)

1 16. Convert 20 2 2 Remember to 8 to a mixed number. Simplify your answer. (25) reduce the fraction part of the mixed number.

1 2 a  b1 3 3

1 2 2 a  b  117. a 1  2 3 1 b 3 3 3 35 3 (24)

8 9 3 19. 2 8 9  (29) 9  8 9 8 5 3 * 20. Represent 1 A pie were eaten. 2 was cut into six equal slices. Two3 slices (36) 2   b 1 a  a reduced What fraction of 20 3 1 as 3 the pie was left? Write the answer 2 25 3 8 fraction.

1

22

20 8

(29)

3 2  5 3

21. What time is 2 12 hours before 1 a.m.? (32)

22. On Hiroshi’s last four assignments he had 26, 29, 28, and 25 correct (18) 1 2averaged how many correct answers 3 on 2 answers.  b1 a He  these papers? 3 3 5 3 Explain how you found your answer. 3 2 8 9   8 9 2 5 23. 3 Estimate the 3 9 8by 39.   quotient of 7987 divided 5 3 9 8 (16)

1

22

2 b1 3

18.

1 2 a  b1 3 3

24. Compare: 365 − 364 364 − 365 (14) 8 9  * 25. Which 9 8 digit in 3.675 has the same place value as the 4 in 14.28?

3 2  5 3

(34)

26. (17)

Use an inch ruler to find the length of the segment below to the nearest sixteenth of an inch. Estimate

27. a. How many 35 s are in 1?

1

22

(30)

3

b. Use the answer to part a to find the number of 5 s in 2. 28. (2)

3 5 3 5

3 5

1

Connect Instead of solving the division problem 390 ÷ 15, Roosevelt divided both numbers by 3 to 1form the division 130 ÷ 5. Then he 2 2 by 2 to get 260 ÷ 10. Find all multiplied both of those numbers 1 three quotients. 2 2

Lesson 39

203

22

* 29. (39)

Analyze

Find the area of the rectangle below. 0.5 m

0.3 m

30. (23)

Conclude In the figure below, what is the ratio of the measure of ∠ABC to the measure of ∠CBD?

C

60° 3

2 16

Early Finishers

Real-World Application

B

D

5 3

On Samuel’s birthday, his mother wants to cook his favorite meat, turkey. She purchased a frozen 9-pound turkey from the grocery store. The turkey takes 3 2 days to thaw in the refrigerator, 30 minutes to prepare and 2 12 hours to 5 cook in the oven. If the turkey starts thawing at 9 a.m. on November 3, when will the turkey be ready to eat? Show your work. 1

22

204

A

120°

Saxon Math Course 1

LESSON

40

Using Zero as a Placeholder Circle Graphs

Power Up facts mental math

Building Power Power Up D a. Order of Operations: (4 × 800) + (4 × 25) b. Number Sense: 1500 + 750 c. Number Sense: 74 − 39 d. Money: $8.25 − $1.50

480 10

e. Number Sense: Double 240. f. Number Sense:

480 10

$0._4 into hours. g. Measurement: Convert 180 minutes 3  $0.12 h. Calculation: 4 × 4, − 1, ÷ 5, × 6, + 2, × 2, + 2, ÷ 6

problem solving

$0._4 Using five 2s and any symbols or operations, write an expression that is 3  $0.12 equal to 5.

New Concepts using zero as a placeholder

480 10

Increasing Knowledge

When subtracting, multiplying, and dividing decimal numbers, we often encounter empty decimal places. 0.5_ – 0.32

0.2 × 0.3 _._6

$0._4 3  $0.12

When this occurs, we will fill each empty decimal place with a zero. In order to subtract, it is sometimes necessary to attach zeros to the top number.

Example 1 Subtract: 0.5 ∙ 0.32

Solution We write the problem, making sure to line up the decimal points. We fill the empty place with zero and subtract. 0.5_ − 0.32 Discuss

41

0.50 − 0.32 0.18

How can you check subtraction?

Lesson 40

205

Example 2 Subtract: 3 ∙ 0.4

Solution We place a decimal point on the back of the whole number and line up the decimal points. We fill the empty place with zero and subtract. 21

3.0 − 0.4 2.6

3._ − 0.4

When multiplying decimal numbers, we may need to insert one or more zeros between the multiplication answer and the decimal point to hold the other digits in their proper places.

Example 3 Multiply: 0.2 ∙ 0.3

Solution We multiply and count two places from the right. We fill the empty place with zero, and we write a zero in the ones place. 0.2 × 0.3 _._6 Verify

0.2 × 0.3 0.06

Why must the product of 0.2 and 0.3 have two decimal places?

Example 4 Use digits to write the decimal number twelve thousandths.

Solution The word thousandths tells us that there are three places to the right of the decimal point. ._ _ _ We fit the two digits of twelve into the last two places. _2 _ ._ 1 Then we fill the empty place with zero. 0.012

circle graphs

206

Circle graphs, which are sometimes called pie graphs or pie charts, display quantitative information in fractions of a circle. The next example uses a circle graph to display information about students’ pets.

Saxon Math Course 1

Example 5 Brett collected information from his classmates about their pets. He displayed the information about the number of pets in a circle graph.

Cats 8 Birds 4

Dogs 16 Fish 4

Use the graph to answer the following questions: a. How many pets are represented in the graph? b. What fraction of the pets are birds? c. What percent of the pets are dogs?

Thinking Skill Discuss

4 32

What information would we need to find the number of 1 8each kind of pet if we were given the percentages?

Solution a. We add the number of dogs, cats, birds, and fish. The total is 32.

1 8

2 3 1 3

3 5

1

4

2 3

4 b. Birds are 4 of 32 the 32 pets. The fraction 32 reduces to 8 . (The bird portion 1 1 of the circle is 8 of the whole circle.) 2 1 c. Dogs are 16 of the 32 pets, which means that 12 of the pets are 8 1 1 dogs. From our fraction manipulatives we know that 12 equals 50%. 8 8 Circle graphs 1 often express portions in1percent form.2 Instead of 3 3 could have showing the3number of each kind of animal, the graph 1 5 labeled each portion with a percent.

Example 6

1 5

2 3

1 5 3 5

3 5

A newspaper polled likely voters to survey support for a local bond measure on the November ballot. Display the data shown below with a circle graph. Then compare the two graphs. Support of Bond Measure 1 2 3 1 3

Oppose

Favor

Lesson 40

207

32

1 8

4 32

4 32

1 3

1 3

Solution

1 8

1 8

1 3

8

1 2

1 8

1 8

1 8

1

2 2 measure3and

3 5 Support

of Bond Measure

Oppose

3 5

3 5

Favor 66

2 % 3

1 33 % 3

The bar graph helps us visualize the quantities relative to each other, and the circle graph helps us see their relationship to the whole.

Practice Set

Simplify: a. 0.2 × 0.3

b. 4.6 − 0.46

c. 0.1 × 0.01

d. 0.4 − 0.32

e. 0.12 × 0.4

f. 1 − 0.98

g. (0.3)2

h. (0.12)2

i. Write the decimal number ten and eleven thousandths. j.

Connect

In the circle graph in example 5, what percent of the pets are

cats?

Written Practice

Strengthening Concepts

1.

Model In the circle graph in example 5, what percent of the pets are birds? (Use your fraction manipulatives to help you answer the question.)

2.

Formulate The U.S. Constitution was ratified in 1788. In 1920 the 19th amendment to the Constitution was ratified, guaranteeing women the right to vote. How many years after the Constitution was ratified were 2 2 women guaranteed the right to vote? Write an equation  4 solve the 6 and  3 3 0.3 2 9 problem.

3.

Analyze White Rabbit is three-and-a-half-hours late for a very important date. If the time is 2:00 p.m., what was the time of his 5 2 5 2 date?  2 5 8 3 6 6

(40)

(13)

(32)

208

1 2

The bar graph shows us that of the voters oppose the bond 2 are in favor of it. Using the fraction circle15 manipulatives from Investigation 2, 3 we can draw a circle and divide it into 3 sectors. We label one sector “Oppose” and two sectors “Favor.” We know from our fraction manipulatives 2 1 2 1 2 1 that 13 is 33 3% and 3 is 66 3%, so we add these32 percentages to labels. 5 our 5 3

3 5

3 5

1 8

4 32

4 32

1 8

Saxon Math Course 1

1 8

1 8

1 5

1 5

Look at problems 4–9. Predict which of the answers to those problems will be greater than 1. Then simplify each expression and check your predictions. 2 2 6 4 * 4. 29  0.3 * 5. 1.2 − 0.12 3 3 * 6. 1 − 0.1 Predict

(38, 40)

* 9. 4.8 × 0.23 2 (39)2 2 6  46  4   3 3 0.3 2 9 3 0.3 2 9 * 10. Write one and two hundredths as a decimal number. (40) 5 5 2 2 2 3 5 3 2 8 23 6  4 (6 5×10,000) 5  36form. 6 3 11. Write + (8 × 100) in standard 3 3 8 8 (32) (40)

2 2 6 4 3 3

(40)

* 8. (0.1)2

* 7. 0.12 × 0.2

29  0.3

(40)

12.

Connect A square room has a perimeter of 32 feet. How many square 5 25 2 5 25 2  with  52thefloor  5 of the floor 8tiles 1 foot long are needed to2cover 3 8 sides 6 6 3 6 room? 2 22 2 5 2 1 3 1 2 61 624  41  3   2 9 50.3   3 3 2 3 0.3 2 9 6 What 6 is 2 2 2of 2, 4, 23and 3 8? 2 2 3 multiple (LCM) 13. the least common (30) 3 3 3 2 22 2 5  38 614.64  4 15. 5  35  3 3 8 33 3 8 (26) (36)

5

5 25 2   16. 8(29) 38 3

1  2

(8, 31)

5 2 2 5 6 6 2 2 6 4 3 3 9 20.3 9  0.3





(40)

2 3

5 2 2 5 6 6

25 2  38 3

5 52 2 17. 2 25  5 6 66 6

(26)

1 2 18. 5 Compare: 52 2 22 2(29) 25  5 6 66 6 19. 1000 − w = 567

1 3  2 3

1 21 21 31 3     2 2 22 32 3

(3)

20. (19)

Classify Nine whole numbers are factors of 100. Two of the factors are 1 and 100. List the other seven factors.

* 21. 92  29 (38)

22. Round $4167 to the nearest hundred dollars. (16)

* 23. The circle graph below displays the favorite sports of a number of (40) students in a recent survey. Use the graph to answer a–c. Basketball 15 students Softball

a. 1 8

4 32

2 3

1 3

3 5

Soccer 10 students

92  29

4 5 students 32

How many students responded to the survey?

Analyze

1 8

1 1 favorite sport? b. What 1 1 fraction 1of the students named softball as their 8

8

2

8

2

c. What percent of the students named basketball as their favorite 2 1 sport? 3 3 24. Jamal earned $5.00 walking his neighbor’s dog for one week. He was (35) 2 1 given 15 of the $5.00 at the beginning of the week and 5 in the middle of the 3 week. How much of the $5.00 was Jamal given at the end of the week? 3 Express your answer as a fraction and as a dollar amount. 5 Lesson 40

209

* 25.

Formulate Write a ratio problem that relates to the circle graph in problem 23. Then answer the problem.

* 26.

Arrange the numbers in this multiplication fact to form another multiplication fact and two division facts.

(23, 40)

(2)

Represent

0.2 × 0.3 = 0.06 92  29

27. (2)

* 28.

(23, 33)

29. (17)

To solve the division problem 240 ÷ 15, Elianna divided both numbers by 3 to form the division 80 ÷ 5. Then she doubled 80 and 5 to get 160 ÷ 10. Find all three quotients. Connect

Forty percent of the 25 students in the class are boys. Write 40% as a reduced fraction. Then find the ratio of girls to boys in the class. Analyze

What mixed number is represented by point A on the number line below? Connect

A

5

6

7

30. Make a circle graph that shows the portion of a full day spent in various (40) ways. Include activities such as sleeping, attending school, and eating. Label each sector with the activity name and its percent of the whole. Draw a circle and divide it into 24 equal sections. Use the circle below as a model. 24 24 20 24

4 24

16 24

8 24 12 24

210

Saxon Math Course 1

INVESTIGATION 4

Focus on Collecting, Organizing, Displaying, and Interpreting Data Statistics is the science of gathering and organizing data (a plural word meaning information) in such a way that we can draw conclusions from the data. For example, Patricia wondered which of three activities—team sports, dance, or walking/jogging—was most popular among her classmates. She gathered data by asking each classmate to select his or her favorite. Then she displayed the data with a bar graph. Favorite Activities 15 Frequency (number of students)

Math Language The term data is plural because it refers to a collection of individual figures and facts. Its singular form is datum.

10

5

0

Team Sports

Dance

Walking/ Jogging

Analyze Which type of exercise is most popular among Patricia’s classmates?

Roger wondered how frequently the residents on his street visit the city park. He went to every third house on his street and asked, “How many times per week do you visit the city park?” Roger displayed the data he collected with a line plot, which shows individual data points. For each of the 16 responses, he placed an “x” above the corresponding number. Number of Visits to the City Park per Week x x x

x x x x

x x x

x x

x

0

1

2

3

4

5

x

x x

6

7

For most of the residents surveyed, the number of visits to the city park are between what numbers? Analyze

Investigation 4

211

In this Investigation we will focus on ways in which statistical data can be collected. We also will practice collecting, organizing, displaying, and interpreting data. Data can be either quantitative or qualitative in nature. Quantitative data come in numbers: the population of a city, the number of pairs of shoes someone owns, or the number of hours per week someone watches television. Qualitative data come in categories: the month in which someone is born or a person’s favorite flavor of ice cream. Roger collected quantitative data when he asked about the number of visits to the park each week. Patricia collected qualitative data when she asked about the student’s favorite sport. In problems 1–5 below, determine what information is collected. Then decide whether the data are qualitative or quantitative. Classify

1. Jagdish collects 50 bags of clothing for a clothing drive and counts the number of items in each bag. 2. For one hour Carlos notes the color of each car that drives past his house. 3. Sharon rides a school bus home after school. For two weeks she measures the time the bus trip takes. 4. Brigit asks each student in her class, “Which is your favorite holiday— New Year’s, Thanksgiving, or Independence Day?” 5. Marcello asks each player on his little league team, “Which major league baseball team is your favorite? Which team do you like the least?” 6. Write a survey about television viewing with two questions: one that collects quantitative data and one that collects qualitative data. Conduct the survey in your class. For question 6 above, organize your data. Then use a line plot to display the quantitative data and a bar graph to display the qualitative data. Interpret the results. Represent

This question will give you quantitative data: “How many hours a week do you spend watching television?” We record the number of hours a person watches TV each week. Then we organize the quantitative data from least to greatest. 0

212

Saxon Math Course 1

1

1 2

2

2

2

3

3

3

3

3

4

4

5

6

6

7

7

7

7

8

8

8

10

5

4

4

4

12 15

4

We display the data with a line plot. (We may use an “x” or dot for this.) Number of Hours of TV Watched per Week

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

The line plot illustrates that the 32 students surveyed watch between 0 and 15 hours of TV each week. Many students watch either 2–4 or 7–8 hours each week. This question will give you qualitative data: “Which type of TV shows do you prefer to watch–sports, news, animation, sit-coms, or movies?” We organize the qualitative data by category and tally the frequency of each category. Category

Tally

Frequency

Sports

11

News

3

Animation

6

Sit-coms

9 3

Movies We can display the data with a bar graph.

Types of TV Shows Preferred Sports

Category

News Animation Sit-coms Movies 0

5 10 Number of Students Who Prefer

15

The most popular type of TV show among the students surveyed is sports. Surveys can be designed to gather data about a certain group of people. This “target group” is called a population. For example, if a record company wants to know about teenagers’ music preferences in the United States, it would not include senior citizens in the population it studies.

Investigation 4

213

Often it is not realistic to poll an entire population. In these cases, a small part of the population is surveyed. We call this small part a sample. Surveyors must carefully select their samples, because different samples will provide different data. It is important that the sample for a population be a representative sample. That is, the characteristics of the sample should be similar to those of the entire population. In order to do this, researchers often randomly select participants for a survey from the entire population. In problems 7 and 8 below, explain why each sample is not representative. How would you expect the sample’s responses to differ from those of the general population? Validate

7. To determine public opinion in the city of Miami about a proposed leash law for dogs, Sally interviews shoppers in several Miami pet stores. 8. Tamika wants to know the movie preferences of students in her middle school. Since she is in the school orchestra, she chooses to survey orchestra members. Often the results of a survey depend on the way its questions are worded or who is asking the questions. These factors can introduce bias into a survey. When a survey is biased, the people surveyed might be influenced to give certain answers over other possible answers. For problems 9 and 10, identify the bias in the survey that is described. Is a “yes” answer more likely or less likely because of the bias? Verify

9. The researcher asked the group of adults, “If you were lost in an unfamiliar town, would you be sensible and ask for directions?” 10. Mrs. Wong baked oatmeal bars for her daughter’s fundraising sale. She asked the students who attended the sale, “Would you have preferred fruit salad to my oatmeal bars?”

extensions

Thinking Skill Verify

Why would a good survey for a and b include data from the same number of young people and adults?

214

Represent For extensions a–c, conduct a survey and organize the data with a frequency table. Then display the results with a circle graph. Interpret the results.

a. One might guess that young people prefer different seasons than adults. As a class, interview exactly 24 people under the age of 15 and 24 people over the age of 20. Ask, “Which season of the year do you like most—fall, winter, spring, or summer?” b. One might guess that young people drink different beverages for breakfast than adults. As a class, interview exactly 24 people under 15 and 24 people over 20. Ask, “Which of the following beverages do you most often drink at breakfast—juice, coffee, milk, or something else?” If the choice is “something else,” record the person’s preferred breakfast drink.

Saxon Math Course 1

c. For ten consecutive days, count the number of students in your class who wear something green (or some other color of your choosing). Count the same color all ten days. d.

e.

With a friend, construct a six-question, true-false quiz on a topic that interests you both (for example, music, animals, or geography). Have your classmates take the quiz; then record the number of questions that each student answers correctly. To encourage participation, ask students not to write their names on the quiz. Formulate

Analyze

Use the menu to answer the questions below.

Seafood Cafe Appetizers Shrimp Cocktail..................$7.00

Main Course Halibut .............................. $15.75

Zucchini Fingers ................ $5.00

Swordfish ......................... $18.00

Soup Seafood Gumbo ................ $4.50

Flounder ........................... $13.75

Lobster Bisque .................. $4.50

Dessert Sorbet.................................$3.25

Crab Cakes ...................... $12.50

• What is the most expensive item on the menu? The least expensive? • What is the average price of the main course dinners? • What is the range of prices on the menu? f.

A number of bicyclists participated in a 25-mile bicycle race. The winner completed the race in 45 minutes and 27 seconds. The table below shows the times of the next four riders expressed in the number of minutes and seconds that they placed behind the winner. Evaluate

Rider Number

Time Behind Winner (minutes and seconds in hundredths)

021

–1:09.02

114

–0.04.64

008

–1:29.77

065

–0:13.45

The least time in the table represents the rider who finished second. The greatest time represents the rider who finished fifth. Write the four rider numbers in the order of their finish. g.

The average rate of speed for rider 008 was 33 miles per hour. At that rate, did this rider finish more than 1 mile or less than 1 mile behind the winner? Show your work. (Hint: Change the average rate in miles per hour to miles per minute, and round the rider number 008’s time to the nearest second.) Analyze

Investigation 4

215

LESSON

41

Finding a Percent of a Number Building Power

Power Up facts

Power Up G

mental math

a. Number Sense: 4 × 250 b. Number Sense: 625 + 50 c. Calculation: 47 + 8 d. Money: $3.50 + $1.75

1 8

1 2 of 600 10

e. Fractional Parts:

1 2

f. Number Sense:

600 10

700

g. Measurement: How many feet are in 60 inches? h. Calculation: 5 × 3, + 1, ÷ 2, + 1, ÷ 3, × 8, ÷ 2

problem solving

Tennis is played on a rectangular surface that usually has two different courts drawn on it. Two players compete on a singles court, and four players (two per team) compete on a doubles court. The singles court is 78 feet long and 27 feet wide. The doubles court has the same length, but is 9 feet wider. What is the perimeter and the area of a singles tennis court? How wide is a doubles court? What is the perimeter and the area of the doubles court?

New Concept Math Language Recall that a percent is really a fraction whose denominator is 100.

Increasing Knowledge

To describe part of a group, we often use a fraction or a percent. Here are a couple of examples: Three fourths of the students voted for Imelda. Tim answered 80% of the questions correctly. We also use percents to describe financial situations. Music CDs were on sale for 30% off the regular price. The sales-tax rate is 7%. The bank pays 3% interest on savings accounts. When we are asked to find a certain percent of a number, we usually change the percent to either a fraction or a decimal before performing the calculation.

1 4

216

1 4

5 5 100 100 Saxon Math Course 1

25 100

25 25% means 100 , which reduces to 41. 5

5

1 5% means 100, 100 which reduces20 to

1 20.

1 4

25 100

25 100

5 100

5 100

15

15

75

75

25 100

25 100

1 4

1 4

A percent is also easily changed to a decimal number. Study the following changes from percent to fraction to decimal: 25 25% 100

15 100

25 100

0.25

5 100

5%

0.05

75 100

15 25 100 100

5 100

1 75 4100

We see that the same nonzero digits are in both the decimal and percent forms of a number. In the decimal form, however, the decimal point is shifted two places to the left. 3 3 3 3  20  15 5 20  15 25 4 4 4 4 Example 1 100 100 Write 15% in decimal form. 25 100

Solution 1 4

25 100

5 100

1 4 15

Model

Use a grid to show 15%.

25 100

15 100

3 4

3  20  15 4

3 75  100 4

75 We write 75% as 100 and reduce. 75 100

25 100

20

5 Example 2 100 3 5 4 fraction. Write 75% as a reduced 100 Solution

15 100

3 4

75 100

5 be written 0.15.1 Fifteen percent means 100, which can 100

1 4

3 75  100 4

5 100

3  20  15 4

Example 3

1 20

1 20

3  20What  15number is 75% of 20? 4 5 Solution 100 We can translate this problem into an equation, changing the percent into either a fraction or a decimal. We use a letter for “what number,” an equal sign for “is,” and a multiplication sign for “of.” 3 75 Percent To Fraction Percent To Decimal 75  100 4 100 What number is 75% of 20? What number is 75% of 20?

n

3  15 = 4  20 × 20

n

= 0.75 × 20

We show both the fraction form and the decimal form. Often, one form is easier to calculate than the other form.

Lesson 41

217

80 100

3  20 =  15 × 4

3 Thinking Skill 4 Conclude

Since 75% of 20 is 15, what is 25% of 20?

0.75 × 20 = 15.00

We find that 75% of 20 is 15. Discuss

Which is easier to compute,

3 4

× 20 or 0.75 × 20? Why?

Example 4 Jamaal correctly answered 80% of the 25 questions. How many questions did he answer correctly?

Solution 80 4 can change 80% to a fraction (  45 ) or to We want to find 80% of( 80 25. We 100 5) 100 3 a decimal number (80% = 0.80). 2 Percent To a Fraction Percent To a Decimal

Thinking Skill 3Verify 4

How do you change 80% to 80  45 ) To a a(fraction? 100 decimal?

80% of 25

80% of 25

4  25 5

0.80 × 25

2 3

44  25 5 5  25  20

Then we calculate. 4  25 5

4  25 =  20 5

0.80 × 25 = 20.00

6 3 ( 100  50 )

3 50

We find that Jamaal 2correctly answered 20 questions. 3

3 2 2 3

Example 5

 45 )

The sales-tax rate was 6%. Find the tax on a $12.00 purchase. Then find the total price including tax.

Solution

4 3  25 We 20can change 6% to a fraction ( 6  3 ) or to a decimal number Math Language 5 100 50 50 4 4 80 80 4 4 6 3 3  25 Sales tax is a tax   25 20   ( ) ( ) (6% = 0.06). It seems easier for us to multiply $12.00 by 0.06  ( 100 50 ) than by 50, 5 5 100 100 5 5 charged on the so we will use the decimal form. sale of an item. 6% of $12.00 percent80 80It is some 4 4 80 4  ( 100  5 ) ( 100  5 ) ( ) 5 100 of the item’s 0.06 × $12.00 = $0.72 purchase price. 4 4 4 So the tax on the $12.00 $0.72. To find the total  25 purchase was  25  25price  20 5 5 5 including tax, we add $0.72 to $12.00. 4  25 5

4 4  25 25  20 5 5

4  25 5 6 3 ( 100  50 )

20

Practice Set

218

3 50

$12.00 4 25  20 +5  0.72 $12.72

4 6 3  25 ( 1002050 ) 5

Write each percent in problems a–f as a reduced fraction: a. 50%

b. 10%

c. 25%

d. 75%

e. 20%

f. 1%

Saxon Math Course 1

4  25  20 5 36 ( 100 50

3  50 )

5

Write each percent in problems g–l as a decimal number: g. 65% j. 8% m. n.

h. 7%

i. 30%

k. 60%

l. 1%

Explain Mentally find 10% of 350. Describe how to perform the mental calculation.

Mentally find 25% of 48. Describe how to perform the mental calculation. Explain

1

9 2% o. How much money is 8% of $15.00? 1 9 2%

p.

1

The sales-tax rate is 9 2 %. Estimate the tax on a $9.98 purchase. How did you arrive at your answer? Estimate

q. Erika sold 80% of her 30 baseball cards. How many baseball cards did she sell?

Written Practice

Strengthening Concepts

* 1. A student correctly answered 80% of the 20 questions on the test. How (41) many questions did the student answer correctly? * 2. Ramon ordered items from the menu totaling $8.50. If the sales-tax rate (41) is 8%, how much should be added to the bill for sales tax? Explain why your answer is reasonable. 3.

(8)

* 4. (41)

Analyze The ten-acre farm is on a square piece of land 220 yards on each side. A fence surrounds the land. How many yards of fencing surrounds the farm? Explain

Describe how to find 20% of 30.

5. The dinner cost $9.18. Jeb paid with a $20 bill. How much money should he get back? Write an equation and solve the problem.

(11)

1

9 2%

6.

(15)

3 4

Two hundred eighty-eight chairs were arranged in 16 equal rows. How many chairs were in each row? Write an equation and solve the problem. 3 Formulate

4

7. Yuki’s bowling scores for three games were 126, 102, and 141. (18) What was her average score for the three games? * 8. What is the area of this rectangle?

2.5 m

(31, 39)

2m

9. Arrange these numbers in order from least to greatest:

(14, 17)

3 2

3 2 , 0, −1, 3 2

2 3 3 2,

2

0, 1, 3

Lesson 41

219

10. List the first eight prime numbers. (19)

11. (21)

Classify

By which of these numbers is 600 not divisible?

A 2

B 3

C 5

D 9

12. The fans were depressed, for their team had won only 15 of the first (27) 60 games. What was the team’s win-loss ratio after 60 games? 13. To loosen her shoulder, Mary swung her arm around in a big circle. If her (27) arm was 28 inches long, what was the diameter of the circle? (28)

The map shows in town. 3 three streets 2 2 , 0, 1, 3 a. Name a street parallel to Vine. Connect

2 3

IVY

b. Name a street perpendicular to Vine.

VINE

MAIN

14.

2 n1 3 15. Rob remembers that an acute angle is “a cute little angle.” Which of the (28, Inv. 3) following could be the measure of an acute angle? 2 3

A 0° 2 * 316.

(38, 39)

(2.5)2

B 45° 2 3

C 90° 4 3 17. 281 4* (38) 3

2 * 18. Write 40% as a reduced fraction. 3 (41)

3 3 2 n1 3

5 3  6 5

5 3  6 5

2 n1 3

* 19. Write 3 2 9% as a decimal number. Then find 9%3of $10. (41) n  14 3 3  25 3 3 4 4  4 reciprocal 3 6 5 20. What is the of 23? (30) 5 3 4 3 2   3Find each unknown to 81 check your 5 4 3 number. Remember 2 6 work. 21. 7m5 =33500 22. $6.25 + w = $10.00 3 (4) (3) 2 2 2 65 n 21n34  1 3 2 3 3 4 3 3 =76 * 23. n  1 2  24. x3 − 37 4 4 (30) 3 4 3 (3) 3 3  2 3 3 3 4 4 3 2 2 3 3 4 425. n1 26. 3  2 3(37) 6.25 + (4 − 2.5) 4 4 (26)

3 4

220

D 135° 2 n1 3

3

2 n1 3

3 4

5 3  6 5

5 3 4 34 3 27. 28.  53  3  5 3 56 5 4 3 (Inv. 2) 4  3 (29) 6 6 5 4 29. What is 34 of 48? 5(29) 3 3  5 6 4 * 30. Justify Fran estimated 9% of $21.90 by first rounding 9% to 10% (41) and rounding $21.90 to $20. She then mentally calculated 10% of $20 and got the answer $2. Use Fran’s method to estimate 9% of $32.17. Describe the steps.

Saxon Math Course 1

3 3 3 323  2 4 44

3 4

3 4

LESSON

42

Renaming Fractions by Multiplying by 1 Building Power

Power Up facts

Power Up G

mental math

a. Number Sense: 4 × 125 b. Number Sense: 825 + 50 c. Calculation: 67 + 8 d. Money: $6.75 + $2.50

580 10

1 2 of 580 10

e. Fractional Parts: f. Number Sense:

1000 1 2

g. Measurement: How many millimeters are in 10 centimeters? h. Calculation: 3 × 4, − 2, × 5, − 2, ÷ 6, + 1, ÷ 3 3

2 4

problem solving 1 1

5 10

4 8

6

3

2

6 4 ball and found that each bounce was half Victor dropped a rubber as high as the previous bounce. He dropped the ball from 8 feet, measured the height of each bounce, and recorded the results in a table. Copy this table and 3 4 complete it through the2 fifth bounce. 2 3 4 Heights of Bounces 1 21 1

First

1 10

4 ft

Second 1 1 Third 1 2 2 Fourth 1

1 2

2

New Concept

10 2 3 10

3 10 9 10

5 5

1 1 1 2 2

Fifth

Increasing Knowledge

With our fraction manipulatives we have seen that the same fraction can be named many different ways. Here580 we show six ways to name the fraction 12: 10

1 2

2 4

3 2 6 4

4 8

5 10

6 3 12 6

In this lesson we will practice renaming fractions by multiplying them by a fraction equal to 1. Here we show six ways to name 1 as a fraction: 1 1

1 1

2 2

3 3

1 2

2 2

4 4

5 5

6 6

Lesson 42

221 1 1 1 2 2

1 1

4 8

1

1 2

?1 2

0

1  2 22

4 8

2 2 2

2

10

20

23, 32 3

4 4

1 1 2 2 1 2 2 133 We 1know that when we a number by2 1, the product equals the 3 6 3 1 multiply 2 2 2 1 1 1 1 2 1 2 31 1   number multiplied. So if we multiply by 1, the answer is . 1 3 2 4 4 2 3 3 12 13 2 ? 2 2 1 12 ? 2 3 2   2 2, 3 4  4 2 , 3 2, 3 2 6 3 6 2 2 4 2 3 6 1 1 441 442 3 111 22 33 221 133  1  222 333 ,, 442 442 ,23 3 222 233 2 222,, 2 222,,, 333 10 1 22 33 10 2, 3 10 10 2 1 1 1 3 2 2 1 However, if we multiply 2 by  fractions names  equal to 1, we find different 1 2 4 1 2 1 1 6 6 by 62, 3, and 41:  1  1   for 22,. 3Here we show 2 multiplied 4 1 4 2 3 24 1 4 2 3 41 2 3 4 21 2 22 3 4 3 1 3 2 2131 2 3 23 1         2 42,  3 82  26 4  8 63,4 3 2 4 6 2 3 26 2 4 28

Math Language 3 The Identity 1 Property of 2 11 1 Multiplication 22 2 states that if one 2 2 of two 1factors   1 3 2 is 1, the product62 1 2 2 equals  the other 2 2 4 factor. 22 11 22

1 2

33 33 11 222  111   222       22 1 22233× 66222 = 444

332 2 332 2 111  111442  442333 333 111 1               =  × × = 2221 2  332 2  664 4 1 222 222442 884333 666

5 5

2 3 2, 3

443 443 3 1  111 1     3  2 2 443 3  886 6 2 2 2 1 2 3 1 2 , 3 2 3 3 2 2 3 23 ,4 1 21 2 3 1 4 1 3 same value but different 4 called 4 equivalent 1 3 are  Fractions  with   2 3 the names 6  4 2 2 2 4 3 4 6 2 3 2 4 8 4 1 2 3 2 2 2 2 4to 1.8 11 3 2 3 8 fractions. The8 fractions 4, 6, and 18 are ?all equivalent    2  22 6 4, 6, 3 2 6 6 6 4 1 20 2 3 442 44 4 112 3 444 114 22 33 2 3 222 333 4 2 , 4 3 2 2 3 8 , 2 8 8 2 , , 8 8 8 8 , , 2 ,, 66Example ,, 2 , , 8 8 , , , , 4 6 4 6 4 4 4 6 4 6 4 6 11 ? 8 , , 1 ? 1 1 3 3 14 6 2 2 1 1  1  4    3 1 2 2 1 6 3 5 2 3 2 3 3 6 3 6 3 3 2 2 1 1 2 2 4 2 3     , 4   Write a fraction equal to 2 that has4 a denominator of 20. 1 2 3 6 3 1 2 2 4 2 3 6 6 62, 3, 1 4 8 4 1 8 4 6 32 2 310 2 1 ?2 11 1 10 ?2, 10 1 10 1 4 , 13 ? 2 1 ? 1  2 3 2 2     10 2  10 2 2 20 2 20 2 20 3 10 3 2 4 4 1 41 20 10 10 11 1 10 112 12 21 3 1 ??2 20 ?111 1 1 ?? 111 ???  10 1  ,3 42  22131?6?    4  10  222   10 22 2 10 10 24 2 2 8 22  2 20 20 20Solution 2 22 20 2 2 2020 203 1 2 3 203 1 222 3 20 220 4 2 3   4 2 2 2 3 3 1 4, 6, 4 2 3 2  3  6 1 8 2 3 6 4    8 10 1 8 4 , 6 ,10 6 1 2 3 1 6 ?To rename 2 4 the equal to 1. 1 10 3 1 3 2 fraction 3 1 2 2 a2fraction,1 we1multiply 4 4 2 1 2 10by3a fraction 10 101      5   3 2 10 1 2 1020 1 10 10 1 ? 11 1 ?3 2 6 4  8 2 3 1 with a  1 to 1 The denominator We want make an equivalent fraction  2 2 4 1 of 2 is 2. 2 3 6 2 2 4 2      2 4 10 22 20 2, 3 2 1 64 10 20 20 23 43 12 6 6 106 33 2 6 310 denominator 12, 3, 1132 10 4 18 2 of 10 10 10 10 10 10 1 10 10 10 10 10 10 1 10 10 10 10 20. 1 1 1 2 1 1 1 1 1 1 1     1 1 1 1 1 1 10 10 1 ,  2   4     6     2 4 6 1 8 20 102 2 20 10 20 2 2 10 10 20 20 223 10 2 2 10 332 10 20 2 20 20 3 22 2 2 23222 10 3321 20 224 222 310 10 2 1 1 ? ? 20     4 1 1  = 1 2 ? 6 6 6 6 66 6 2 20  8 2 2 2 20 10 10 1 2 20 10 1 1  1 4 1 4 1  2 3 20 1 2 ? 2,2131, 22 ? 8 3 1 3 ? 8 32 3 20 2 1 2 ? 10 , , 2 1 1 1 1 ? ? 1   1 4 6   2 64 36  13 6 21 6   1 10 1 2 6 3 6 3 2 2 2 4 2 3 6 2 6 3 6 3 2 4 1 ? 1 3 the denominator 1?? 2 2 1??111to multiply Since we11need by 10,111we1multiply ?111 1? 1 ?4 11 ?? 111 1 ??? ? ??3 ? ? by4 . 111 1 ? ? 11 2  2  482  20 2   2              3  622  14 2 8 10 6 3 6 3 6 3 6 2 6 3 2 3 2 2 6 3 6 2 6 3 6 2 6 3 6 2 6 3 6 3 3 2 6 3 6 2 6 3 6 2 2 ? 2 6 3 6 10 10 1 1 10 1   10 10 0 1 2 1 10 ?1 20   2 10 2 1 ? 1 ? 2 10 20 1 2 1 1  1  1 10 1 1 ? 6 ? 6 2 1 2 6 3 2 1 2 4 3 3   1 23 1023 3 2 1 3 3 2 3 1 2 1 2 2 3      2 2 20 2 12  202  2  8 , 6, 3 3 3 2 4 4 3 2 6 62 3363 3366 132 33 1 6 1 2 10 10 10 11 Example 2 2 3 1 3 3 1 1 1 1 22 1 2 1 2 1 2 1 2 1 2 1 2 1 2     8 1 2  10 22 2      10 4, 6, 20 3 2 2 2 22 2 33 223 24 3 333 223 33222 66 2 2 3 6 6 1 1 1 1 5 3 2 1 ? 1 ?  Write and as fractions with denominators of 6. Then add the renamed   2 3 2 6 3 1 1 2 6 1? 3 1 6 31 3? fractions. 1 2 2 16 3 3 3 2 2  3 6   10 1 2 10 10 1 2 10 3 2 2 31 6 1 3 1 2 2 2 1 1 1     2 3 3 2 6 3 1 2 2 2 2 3 1 2 2 1 2 2? 2 ? 20 1   2 , 10 3  Solution 2 10 20  2  2    2  2 3 3 1 2 ? 266 20 6 6 6 6 6 2 6 1 ? 3 2 6 10 126 26 2 6 20 32 1 2 2 1223 2? 12 3 3 3 2 2 2 1 2 2 3 3 3 3 1 2 2 2 2 3 3 1 2 2 2 2 1 2 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1   1 , , 3 3 2 2 2 1 2    2     10             2    3 6     4 6 2 633  22 222 20 Skill We multiply a fraction have 66 each 66333 66fraction 3 form 2666 fractions 6 66 Thinking 6666 666 that 22 66 663 3  662 2equal 66  6 6 6 6 222 666 3by 666 6 to 1 to 1 2 1 ? 1 2 Represent denominators of 6. 3 2 1 1  2 33 23 2 1 2 3 2 2 2 3 6 2 2 Draw a diagram 3  21 6?  12 ?62 61 6? ? 1 1 ? 1 31 3 1 1  61 ? 2 66 6 6 1 10 ? show that 1 36 2 35 6  10  3 6 1 2 3 2    to   2 20 3 1 26 46 6 2 10 20 2 1 2 1? 332 3 6 4 2 1061 4 . 1 ? 112 ??201 1??21 10202  ? 1 ? 1 ? 1 ? 3 1 2 21 3 ? 1 1 1 1       1 by? ., 2 multiply 2 2 10 20 33 6   4 , and we    2We multiply 2 by 4 2 3 6 2 20 2 3 22, 3 20 2 20 20 2 320 20 20 22 20 212 320 22 20 3 2 2 220 2  3 1 3 3  1 2 2 2 6 6 6 2  2  3 2 6   2 ? 1 2 3 3 2 6 6 6 6 6  1 1 3 3 2 1 5 2 4 2 20 3 2 1   3 2?  5 renamed 1fractions 3 2 3The 3 2 5 1 are 1 8? 1 1  21. We are2 told to add these 3 2 and fractions. , , 2 3 6       2 64636 6 2 626 2 3 2 3 6 6 6 6 6 2 3 6 6 10 10 1 1 2 2 122 1 ?55 2 2 2 33 22 155 ? 222 3 555 1 33 1133 2 15 1 33 3 3 2 2 4 1 4 1 1 4 4 1 5 3 2 3 1 1 1 1 111 1  5 1   1 111       20 2                 3 2  332 2 22 43 66 26 6 3 336 626222 6 243 10 2 4 2 8 26 6 666 3 666 266 36226 6666 6  8 66 2 666 2 234 266 666 266 2 ? 1  3 2 2 ? 1   3 2 5 2 1 20  1 6 6 6 2 20 5 1  1 1 2 2 3 23 2 1 22 2 3 3 2 2 2 6 6 6 1 2 2 2       ?   1 6 1 2 6  3 2 3 2 6 2 66 6 6 6 6 6 6 6 2 13 21 2 20 ? ? 1 4 42 1 1 1 3 3 2 3 2 22 2 2 1 2 3 3 2 22 4 2 3 2    Saxon Math Course 1 8 86 2 6 66 6 6 3 4 , 6 , 222 62 2 66 2 2 3 62 6 620 4, 6, 6 3 2 5 1  3 5 1 2 6 6 6 2  2

   01 2210 44  02 20 2  4 2 3 4, 6,

22 ? 1  2 20

 

22

 

22 44

8176Practice 41

1 ? � 3 12 2 ? � 3 6

1 2

Set  36

2 ? 53 � 6 4  20

e. 2 3

1 22

2 3

1 4

1 2

1? ?1 ? � 1 � ?3 � 8 12 3 � 12 3 10 2 ? � 3 6



1  36 2 ? ? 1 1  � 8176 3 � 8 2 3 10 41 3 20 each 2 In problems a–d, multiply by a fraction equal to 1 to complete 5 3 4 equivalent fraction. 1 1? ? 2 ?2 ? � � a. b. � � 3 12 3 12 3 63 6 3 ? 1 ? 3 2 ?6 ? �5 7 3 � � � 4 2 3 6 4 8 4 12 2 8 3 ? 33 4 ? 3 ? 236  4 � c. � 4  20 5 5  � d. � 3 4 8 44 512 8 4 12 3 6

1 2

1 6

6

1 ?7 � 2 2�? 8 3 6 1 2

3 4

2 3

3 4 5 4 5 5

1

1

22

3

? 1 � 3 10 6 7

? 1 � 3 10 1 ? � 2 8

1 2 ? 4 3�6

1 6

2

2. According to some estimates, our own galaxy, the Milky Way, contains 41 3 (13) about two hundred billion stars. Use digits to write that number of 2 2 1 3 3 stars. 1 ? 23 6 2 � ? 1 1 ? 7 2 8 � � 3. Explain The 3 rectangular 8 3 10 school yard is 120 yards long and 40 yards 5 (31) 1 1 wide. How many square4 yards 20 is its area? Explain why your answer is 4 4 reasonable. 1 ? 1? 5 1 2 5 1 � � ? 1 1 ? 4.32 4.32 6 � 0.6��0.6 3 2� 8112 281 3 10 3 10 6 � � * 4. Analyze 6What number is2 40% of 30? 3 8 3 10 1 ? (41) � 7 2 8 5 1 In problems and 2 2 by a fraction equal to 1 to complete each � 6, 4.32 � 50.6 2multiply 81 6 equivalent fraction. ?1 ? 3 7 1 1? ? 1 ? *1 5. 6. 1 � 1? � ? � � 1 � 6 *(42) � � 2(42) 82 8 2 10 2 10 4 8 3 8 3 10 7 4 2 8 � * 7. 4.32 * 8. 6.3 − 0.54 0.6 � 281 ? 1 (37, 38) (37) � 2 10 2 * 9. (0.15)

(38, 40)

5 1 ? 10. What is the reciprocal12of 67? � 2 8 (30) 1 ? 1 � 4 10digit 11. 2 Which in 12,345 has 5 1 the same place value as the 6 in 67.89? (34)

1 4

1

2 3

8176

2 3

1 ? � 2 8

5 6

3 3 4 1 Write and 4 as fractions with denominators 2 8176 236  4 renamed fractions. 3 41 2

f.

1 4

of 12. Then add the 8 1 2  3 2 3 1 ? 2 ? � � 3 12 3 6 g. Describe how the rectangles you 3 4 drew for exercise e can help you add 2 5 4 236  4 ? 1 1? 1 ?the 1 ? 5 5 3 � �fractions in f. � � 4.3 3 83 8 1 3 10 3 10 3 3 2 ?2 ?1 � ? ? ? 3 3 ? ? 5 5 10 6 1 1 the ?denominator. � � 3 as 10 � � 4.32 �the � 5 with � 40.6 � a fraction Subtract renamed 2 81 11 12 as  ? 3 1 3 h. 6Write 12 3 66 42 12 5 2 � ? 4 3 � 124 84 812 �5 5 310 2 20 4  3 6 �2 �the � 281 answer. 4.32 � 0.6 81 0.6 4.32 3 10 fraction from . Reduce subtraction 12 3 ? 3 ? � � 6 1 4 8 4 Concepts 12  3 1 1 ?1 ? 1 6 6 Strengthening 5 4 Written Practice � � � 7 7 2 8 2 8 2 1 ? ? 1 3 � 7 � 20 2 ? 5 8176 2 1 ? 8 2 12 and 2 10 of * 1. 1Analyze with denominators 6. Then 5 ?42 Write 1 41 ? ? ?fractions 1 13 as � 14 ?� 8 6 � � (42) 3 � ? � �3 6 12 7 2 1 32 3 10 ? 38 Write 2 � the 10 renamed 2 8 3 10 add fractions. the answer as a mixed number. � 8 3 2 4 8 4 12 2 3

5 4  20

?1 ?4 � 823 8

1 2

On your paper draw two rectangles 2 that look like this one. Shade 3 of the 1 squares1 of one rectangle and shade 4 of the 6 22 1 ?  3 rectangle. squares of the55other Then name � 3 12 each shaded 6rectangle as a fraction with a denominator of 12. 2 2 1 1 Model

3 � 4

6

12

12. What is the least common multiple of 3, 4, and 6? 3 4

(30)

7 8

1 4 3 4

7 8

Lesson 42 1 4

223

176 41

4 5

2

76 1

5 22. 5 Explain 3 3 4 4the quotient of 1 Estimate 5 5 4 4 20 4(16)   204 6 5 5 5 5 estimate.

5 20 34 4 54  4 55 5 5 6

3 4 5 4 5 5

8176 41

6 3 5 236  4 2 35

12

8176 41

1

1

22 5 2 4  20 3

1 4

7 8

4 1

22

5 12 2 32 4 36 2362 4  Model How many 5 4  eggs 4

1

1 4

(30)

5 12

1 6 5 6

224 1 6

8176 41 .

Describe5 how you performed the 2 2  4 4 236 236 12 3 3

2 3

1 1 1 the 6answer 622 1 52 2 1 12 6 3 4

Saxon Math Course 1

7 8

7 8

3 4

1

2 3

5 6

5 6

1

22 5 5 to part a to find the number of1256 s12in 3.

1 6

3 41

5 6

1 6 7 8

1 6 1 63 4 5 12

2

8

* 30. a. How many 56 s are in 1? b. Use

3 4

8 18 1   3 23 2

5 1 8 318 14 22 20 6 2 2 5   are in of a dozen? Draw a diagram to   5 3 53 5 3 523 425236 4 3 3 4 3 4 2 illustrate the problem. 5 8 5 1 5 4 5 236  4 5 5 5 5 1  3 61 6 1 1 2 23 2 22 3 6 6  7 2 4  620 6the renamed  a3 denominator 1 fraction 3 5 3  4 4of 8. Subtract 1 * 24. Write5 4 with  2 36 4 8 5 5 4 4 5 5 3 (42) from678. 3 4 2 5 10 6 10 5 36  1 1 4 5 1 14 3 1  3 2 1  1  56 5  35 1 5 10 5 4 10 3 4 4  2025. What is the perimeter 5 5 0.4 m 4 3 of 2this 6rectangle? 8 1 1 (8)   236  4 1 5 3 3 14 2 10 5 1 5 3 3 2 4  5 242 2366 4 0.2 8176 3 4 6 8176 2 2 1 m 3 2 5 5 3 10 5 3 5 41 4 3 36  4 1  41 32 5 5 5 3 4 3 6 4 5 1 12  4 3 4 20 8176 20236 2 2 5 5 8176 5 1   1 5 3 2 6 12   5 5 2 2 26. What 5 41of this rectangle? 4 3 is 3the area 6 4 414 3 (31) 10 5 1 8176 5 20 5 the sale 23 11 8 1 price 641 1 regular Represent The * 112 27. 12 r minus the discount d equals 5 1 5 5 10 4 3 1 3 5  (1) 5 5 38176 1 2 5 12 4 6 46 12 2 price5s. 4620 3 1 4 20 41 3  5 4 12 6 8176 1 7 r − d =3s 3 1 2 5 5 34 4 8 41 5 Arrange these 4letters subtraction equation and two 4  20  20 to form another 5 20 2 7addition equations. 7 5 5 1 7 8176 2 3 10 5 7 38 4 20 4  1 83 5 5 1 1  6 12 41 3 5 2 4 8 28176 2 4 8 5 10 3 2 2 6 6 7 28. Below we 41 show the4same  20 division problem written 3three different ways. 8 (2) 8176 2 5 5 5 Identify which number is the divisor, which is the1dividend, 41 3 5 1 5 and which is 1 6 220 6 22 4  6 12 2 6 the quotient. 5 1 2 2 53 6 7 20 5 20 ÷ 4 = 5 8 5 5 4  204 42 12 6 4  20 51 5 29. What time is202 2 hours after 11:45 3 7 a.m.? 6 4  (32)

8176 23. 41 (22)

2 3

5 5 04  20

2 3

11 1 1 310 110 45 15 *6 18. 1 117. 6 54 2 5 410 10 5 1 210 36  5 3 (36) (Inv. 2) 5 5 10 3  4 4  3 321 1 5 5 5 4 53 5 10 10 4 5 10 5 10 21 6 1 5 1 1  are equal to .   1  * 19. Form15three that 5 8176 5 Multiply 3 by 10 3 (Hint: 4 (42) 10 65 4 different fractions10 2 1  different fraction 3 8 13 4 three 2 names for21). 3  4 5  4 10 5 36 41 5 5 3 3 2 20 to form 35 are 5 and 7. 20 220. 2 Analyze The prime numbers that 6multiply 201 20 8176 2 3245  5  1 3(19) 38176 4 5 5 Which prime numbers can be multiplied to 5 41 3 form 34? 41 3 44 4 20 20 2 2   5 5 3 3 4 scores5were 12,143; 4 Estimate 21. 20 In three games Alma’s 9870; and 14,261. 10 5  5 6 1 (18) 4  3  20 4 1  Describe how to estimate her average 5 10 5 4 score per game.

 1 3

6 3 3 5

236  4

8 1 8 1  3  22 2 36 2 2362 4 4 * 14. 3 8(36, 38) 83 13 1   3 2 6 3 2 16.  3 (29) 5

2 2  2 336 3362 44 44 3 513.54 34 5 5 52 55 (26) 436  20 4 2 236  4 2 3 8 1 3 15.  2 (29) 3

3 4 4 5 52 5  20 4 3

4

20 5 3 4 4 5 4 5 5

2 3

3 4

5 12 7 8

5 12

5 12

5 12 5 127 8

7 85

7 8

LESSON

43 Power Up facts mental math

Equivalent Division Problems Finding Unknowns in Fraction and Decimal Problems Building Power Power Up C a. Calculation: 4 × 225 b. Number Sense: 720 − 200 c. Number Sense: 37 + 28 d. Money: $200 − $175 $70.00 10

1 2 of 1200 $70.00 10

e. Fractional Parts: 1 2

f. Number Sense:

700 350  7 14

problem solving

g. Probability: How many different three digit numbers can be made with the digits 3, 5, and 9? 700 350 h. Calculation: 8 × 4,14 − 2, ×72, + 3, ÷ 7, × 2, ÷3 Teresa wanted to paint each face of a cube so that the adjacent faces (the faces next to each other) were different colors. She wanted to use fewer than six different colors. What is the fewest number of colors she could use? Describe how the cube could be painted.

New Concepts equivalent division problems Math Language Recall that equivalent numbers or expressions have the same value.

Increasing Knowledge

The following two division problems have the same quotient. We call them equivalent division problems. Which problem seems easier to perform mentally? a. 700 ÷ 14 b. 350 ÷ 7 We can change problem a to problem b by dividing both 700 and 14 by 2. 700 ÷ 14 Divide both 700 and 14 by 2. 350 ÷ 7

Lesson 43

225

$70.00

1

By dividing both the dividend and divisor by the same number (in this 2 10 case, 2), we formed an equivalent division problem that was easier to divide mentally. This process simply reduces the terms of the division as we would reduce a fraction. (700 ÷ 2 = 350) (14 ÷ 2 = 7)

700 350  7 14

We may also form equivalent division problems by multiplying the dividend and divisor by the same number. Consider the following equivalent problems: 1 1 1 c. 7  72 2 2 d. 15 ÷ 1

1 2

We changed problem c to problem d by doubling both 712 and 12; that is, by multiplying both numbers by 2. 1 72 2 15 1 1 1   7  72 1 2 2 2 1 2

16  1200

Multiply both 712 and 12 by 2. 11 1 7 72 2 15 2 2  1 15 ÷2 1 1

350 350 700 700   7 14 147

$70.00 $70.00 10 10 1

72

2

This process forms an equivalent division problem in the same way we would 350 350 700 700   form an equivalent fraction. 7 147 14 75 1 1 1 72162 1200 7  8  600 15 2 2   1 56 2 1 2 40 40 Example 1 0 1 75 72 2 below. 16  1200 15 Then Form an equivalent division for the division 1 1 problem 1   8 600  7  1 2 1 72 2 2 calculate the quotient. 56 2 ∙ 16 1200 40 40 Solution 0 75 16  1200 1 72 by 8  600 number 16, we can divide both the 2 the15 Instead of dividing 1200 two-digit   1 2 1 56the equivalent division of 600 ÷ 8. We dividend and the divisor 2 by 2 to form40 then calculate. 40 16  1200

0 both Divide numbers by 2.

75 8  600 56 40 40 0

Both quotients are 75, but75dividing by 8 is easier than dividing by 16. 8  600 56 40 40 0

226

Saxon Math Course 1

1

72

Notice that several equivalent division can be formed from the 1 problems 1  2 7 original problem 1200 ÷ 16: 2 2 1200 ÷ 16

600 ÷ 8

300 ÷ 4

1

72

150 ÷ 2

All of these problems have the same quotient. ƒ

Example 2

1 4  5 5

Form an equivalent division problem for the division problem below. Then calculate the quotient. 1 1 7 2 2 2

Solution Instead of performing the division with these mixed numbers, we will double both numbers to form a whole-number division problem. 1 1 7 2 2 2 1 1 7 2 2 2

1 1 7  271 2 22

1

72

Multiply both 7 12 and 2 12 by 2.

1

22

1 4 ƒ  5 15 ∙ 5 = 35 ƒ

1 4  5 5

1 1 7 2 2 2

1

4

 finding ƒ Since 5 5Lessons 3 and 4 we have practiced finding unknowns in unknowns whole-number arithmetic problems. Beginning with this lesson we will find 1 1 in fraction 7 2 unknowns in fraction and decimal problems. 2 2 If you are unsure how to find and decimal the solution to a problem, try making up a similar, easier problem to help you problems how to find the answer. 1 determine 1 7 2 2 2 Example 3

Solve: d ∙ 5 = 3.2

Solution This problem is similar to the subtraction problem d − 5 = 3. We remember that we find the first number of a subtraction problem by adding the other two numbers. So we have the following: 5 + 3.2 8.2 We check our work by replacing the letter with the solution and testing the result. d − 5 = 3.2 8.2 − 5 = 3.2 3.2 = 3.2

Lesson 43

227

7 2 2 2

72

22

Example 4 Solve: ƒf 

1 4  5 5

Solution This problem is similar to f + 1 = 4. We can find an unknown addend by subtracting 1 the1known addend from the sum. 7 2 2 2 4 1 3   5 5 5

ƒ

3 5

3 1 4 ƒ  3 ƒf4 1 3 5 5 5   ƒ 5 5 5 5 We check the solution by substituting it into the original equation. 3 1 4 4 4 1 4 3 4 1 3 ƒ ƒ   35 5  5 15  45 5 5 5 5 5 5 ƒf   ƒ 5 5 5 3 1 4 4 4    3 5 5 5 4n 41 53 51  4 5  35 5 4 15 35 5   ƒ 5 5 53 54 4 5 3 1 4ƒ  1  4 5 3    3 5 5 4 45 5 5 33 1 5 3 5 4 1 5 3 5 1 34 5 n  1 3 4 4 1 n  1  3    ƒ ƒ ƒ     5 5 5 5 55 55535 5 5 31 5 4 5 5 4 13 3 1 5 5 3 ƒ ƒ  ƒƒ  5 55 5 5 55 5 5 3 3 1 4 4 4 Example 5 3 1 4 ƒ  ƒ   5 5 55 5 5 5 3 555 15 5 3  3 1 n 5 5 Solve: 3 n  1 5 3 15 3 5 3 13 4 3 1 4 4 34 1 4        5 3 44 5 53  55  15  3 1 4 5 5 5 5 5 5 5 5 n1  n 4 4 4 4 1 5 5 5 5 135   5 3 15  3 3 5 3 Solution 3 15 55 55 5    15 n 5 1 n n 5 3 3 15 5 3 5 5 4 4  3 5 are multiplied, and the product In this problem5two5numbers n is11. This3 can 5 3 5 15 5 want to find the 1     n 1 only happen when the two factors are reciprocals. So we 5 53 1 3 5 3 n  5 3 15 5 3 5 3 3 reciprocal of the known factor, 3 5 3 5. Switching the terms of 5 gives us the 3 fraction53 53. We check our answer by substituting 53 into the original equation. 4 1 3   5 5 5

3 5

1 3  5 5 4 1 3   4 5 5 5 5 1 4  5 5 3 1 4   5 5 5

5 3

5 3

3 5

3 5 15   1 5 3 15

5 15  1 3 15 3 5 15    1 Set 5 3 Practice 15



5

3 5 15    1 check 5 3 15

1 3

n

5 3

n 35 5  15  1 53 3 15

5 3 5 15   5 1n13 3 51 5 3 15 3 5 5 15 1 n   n5 5 3 3 15 33 5 1  n a. Connect Form 3 an equivalent division problem for 5  3 by multiplying both the dividend and divisor by 3. Then find the quotient.

Form an equivalent division problem for 266 ÷ 14 that has a one-digit 1divisor. 1 find 4 the quotient. 1 4 Then 1 ƒ  ƒ  m  1 m4 1  4 5 5 5 5 5 5 Solve: 1 4 c. 5 − d = 3.2 d. ƒf   5 5 b.

4 5

Connect

1 1 m me. 1 14  4 5 5

228

1

53

Saxon Math Course 1

2 ?  3 6

f.

3 w 8

1 m1 4 5

3 3 w w 1 1 8 8

2 ?  3 6

1 ?  2 6 2 ?  3 6

1 ?  2 6

2 n 3 1 ?  2 6

Strengthening Concepts Written Practice 3 3 1 1 4 13 1 w1 w1 ƒ  m  1  4 m  1  4 m  1 w4 1 5 8 5 5 5 8 rate was 8%. What was the 5 8 * 1. Analyze The bike cost $120. The sales-tax (41) total cost of the bike including sales tax?

1 4sit at11the ? If ?one Round Table and ? hundred fifty knights ?  ƒ could m  1   5 5 1 4 1 4 1 3 6 2 6 3 6 2 6 only one hundred twenty-eight knights were seated, how many empty ƒ  m1 4 ƒ  m1 4 5 problem. 5 5 5 3 1 41 1were 4 5 5 1solve the and 31 1   m    at the table? Write an equation  w ƒ  places 4 m ƒ 4 1  w 1 4 m 1 5 55 5 5 5 8 5 8 3. During the 1996 Summer Olympics in Atlanta, Georgia, the American (32, 37) 3 athlete Michael Johnson 3men’s 4 set an Olympic and world record11in the w  1 66  w  114  44   663 w m m 8 3 5 4 200-meter the race in 19.32 2 seconds, breaking 1w ? 1 2 run. He finished 52 aa 4 44the  n1 n1 n1 8 3 of 19.73 seconds. 2 6 3Olympic record 3 By how much did Michael previous Johnson break the previous Olympic record? 2.

(11)

ƒ

1 4  5 5

1 m1 4 1 2  ?1 5 1 4 ? m   3 6 52 6

1 ?  2 6

Formulate22

In problems 4 3 and ? to 1 to complete 3 5, multiply by a fraction2 equal 77  133 each    55 18 33 8? 3 6 2 ? 1 2 ? 1 ? equivalent fraction. 4 3 5 3 5 1 1 1 4 1 1 3 5 4 1 8 8     66 6  w m1 4  6 m  4  6 m  4 3 2 6 35 6 2 6 2 ? ? 1 2 ? ? 1 5 5 a 4 4 8 a 4 4 8 5 a 4 4 8 2 ? ? 2 1 * 5.   * 4.     n 1 3(42) 63 62 (42) 2 3 6 2 6 6 3 6 1 ?  23 ? 6 1 3 7  24  65 3 8  1 8 4 6w1 5 3 1 m4 6 4 4a 3 1 m4 6 4 4a 3 5 4 3 7 3 1 8 8 3 7 3 1 8 8

3 7 3 1 8 8

1  2 2 n 3

Find each unknown number. Remember to check your work. 4 * 6. 2 n  1 * 7. 6  w  1 m 5 37 (43) 2 (43) 4 3 1 3 1 3 w  14        6 m 4 6 6 w 1 m 4 6 n 1 5 4 4a 5 4 4a 33 8  4 8 4 3 5 3 1 1 54  146  6 w  1 11 3   m−2.45 6 m = *6 8. * 9. c m w 45 6 34a 4a 5 4 8 4 4 4a 8 5 (43)

(43)

* 10. 12 − d = 1.43 3 1 (43) 5 5  4 5 8 3 11. 5  1 12. 3  5 13. 3 7  1 3 5 (29) (29) 5 4 8 8 4 3 3 7 73(26) 38  5 75 3 31  1 3 1 Which of these numbers is not 3Classify 414. 4 8 a 8prime 8 8number? 8 8 (19)

A 23

B 33

15. Compare: 22 (29) 22

3 7 3 1 8 8

3

7 8

C 43

22 22  22 22

16. In football a loss of yardage is often expressed as a negative number. (14) If a quarterback is sacked for a 5-yard loss, the yardage change on the play can be shown as −5. How would a 12-yard loss be shown using a 11 11 11 11 33 11 55 33 77 negative number? 442222 16 16, ,88, ,16 16, ,44, ,16 16, ,88, ,16 16, , 2 2 2  2 twelve 2 2hundredths. 17. Write the decimal number for nine and (35)

18. Round 67,492,384 to the nearest million. (16)

* 19.

0.37 × 102

* 20.

Analyze 0.6 × 0.4 × 0.2 1 1 42  2 21. The perimeter of a square room is 80 feet. The area of the room is how (38) many square feet?

(38, 39)

Analyze

(40)

22. Divide 100 by 16 and write the answer as a mixed number. Reduce the (25) fraction part of the mixed number.

Lesson 43

229

1 16 ,

2 2 2   2 2 2 * 23. Connect a. Instead of dividing 100 by 16, Sandy divided the dividend (43) and divisor by 4. What new division problem did Sandy make? What is the quotient? 1 1 b. Form an equivalent division problem for 4 2  2 by doubling both the dividend and divisor. Then find the quotient.

1 1 16 , 8 ,

24. What is the least common multiple (LCM) of 4, 6, and 8? (30)

25. (17)

Predict

4

1111 1111 3333 1111 5555 3333 7777 ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, 16888816 164444 16 16888816 16 16 16 16 16 16 16 16 16 16

1111111111111133333331111111555555533333337777777 ,1616 ,1616 ,1616 ,, ,88,8,88,88,16 ,, ,44,4,44,44,16 ,, ,88,8,88,88,16 ,, ,, 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16

1 2

What are the next three numbers in6 1this sequence? ,

,

, ...

26. Find the length of the segment below to the nearest eighth of an inch. (17) 5 18 9999999

1111111 2222222

5555555

111111 4444 666146   25 25 25 25 16 16 16 16 16 16 16 8888888 444444 27. What mixed number is indicated on the number line below? 7 (17) 4 10

7

4 10

3

5

6

* 28. Write 21 and 15 as fractions with denominators of1510. Then add the (42) renamed fractions.

1 2

29.

(22, 33)

5 10

4

2 7  10  10

Forty percent of the 20 seats on the bus were occupied. Write 40% as a reduced fraction. Then find the number of seats that were 5 3 2 22 2 occupied.  Draw  7a diagram to illustrate the problem. 5; 5 5 10 105 ; 10 Model

30. Describe each angle in the figure as acute, right, or obtuse.

(Inv. 3)

A

B

a. angle A b. angle B c. angle C d. angle D

230

Saxon Math Course 1

D

C

2 5

LESSON

44 Power Up facts mental math

Simplifying Decimal Numbers Comparing Decimal Numbers Building Power Power Up G a. Calculation: 4 × 325 b. Calculation: 426 + 35 c. Calculation: 28 + 57 d. Money: $8.50 + $2.75

1 8 1 8

1 8

1 e. Fractional Parts: 8 1 f. Number Sense: 2 of 1400

1 2 of 1400 $15.00 100

$15.00 100

g. Probability: How many different one-topping pizzas can be made with 2 types of crust and 4 types of toppings? h. Calculation: 6 × 8, − 3, ÷ 5, + 1, × 6, + 3, ÷ 9

problem solving

Jeanna folded a square piece of paper in half from top to bottom. Then she folded the paper in half from left to right so that the four corners were together at the lower right. Then she cut off the lower right corner as shown. Which diagram will the paper look like when it is unfolded?

New Concepts simplifying decimal numbers

A

B

C

D

Increasing Knowledge

Perform these two subtractions with a calculator. Which calculator answer differs from the printed answers below? 425 − 125 300

4.25 − 1.25 3.00

Calculators automatically simplify decimal numbers with zeros at the end by removing the extra zeros. Many calculators show a decimal point at the end of a whole number, although we usually remove the decimal point when we write the whole number. So “3.00” simplifies to “3.” on a calculator. We remove the decimal point and write “3” only.

Lesson 44

231

Example 1 Multiply 0.25 by 0.04 and simplify the product.

Solution We multiply.

Thinking Skill Generalize

How many decimal places are in the product when each of two factors has two decimal places?

0.25 × 0.04 0.0100 If we perform this multiplication on a calculator, the answer 0.01 is displayed. The calculator simplifies the answer by removing zeros at the end of the decimal number. 0.0100 simplifies to 0.01 In this book decimal answers are printed in simplified form unless otherwise stated.

comparing decimal numbers

Zeros at the end of a decimal number do not affect the value of the decimal number. Each of these decimal numbers has the same value because the 3 is in the tenths place: 0.3

0.30

0.300

Although 0.3 is the simplified form, sometimes it is useful to attach extra zeros to a decimal number. For instance, comparing decimal numbers can be easier if the numbers being compared have the same number of decimal places.

Example 2 Compare: 0.3

0.303

Solution When comparing decimal numbers, it is important to pay close attention to place values. Writing both numbers with the same number of decimal places can make comparing easier. We will attach two zeros to 0.3 so that it has the same number of decimal places as 0.303. 0.3

0.303

0.300

0.303

We see that 300 thousandths is less than 303 thousandths. We write our answer like this: 0.3  0.303

Example 3 Arrange these numbers in order from least to greatest: 0.3

232

Saxon Math Course 1

0.042

0.24

0.235

Solution We write each number with three decimal places. 0.300

0.042

0.240

0.235

Then we arrange the numbers in order, omitting ending zeros. 0.042

Practice Set

0.235

0.24

0.3

Write these numbers in simplified form: a. 0.0500

b. 50.00

c. 1.250

d. 4.000

Compare: e. 0.2

0.15

h. 0.31

f. 12.5

g. 0.012

1.25 i. 0.4

0.039

0.12

0.40

j. Write these numbers in order from least to greatest: 0.12

Written Practice 1.

(12, 25)

0.125

0.015

0.2

Strengthening Concepts What is the sum of the third multiple of four and the third multiple of five? Analyze

2. One mile is 5280 feet. How many feet is five miles?

(15)

The summit of Mt. Everest is 29,035 feet above sea level. The summit of Mt. Whitney is 14,495 feet above sea level. Use this information to answer problems 3 and 4. 3. Mt. Everest is how many feet taller than Mt. Whitney? 3 4 1 4 1 3  5  w  4 m  6 5  1 w 4 m6 1 5 than5 5 3 * 4. Analyze The summit of Mt. Everest 5is3how5 many feet higher (13)

(13)

miles above sea level? (Refer to problem 2.)

Find each unknown number. Remember to check your work. 3 3 1 41 4  1w  *4 6. m  6  1 * 5. 5  w  4 m  65  53 5 (43) 5 5 3 (43) * 7. 6.74 + 0.2853 + f = 11.0251 (43) 67 4 14 * 8. 0.4 − d = 0.33

1 6

1 1 82  2

5 8

(43)

1 9. Wearing shoes, Fiona stands 67 34 inches tall. 1If14the heels of her 6 5 1 1 1 1 stand8 without 67 4 shoes are 1 4 inches thick,6 then how tall does 8 2 Fiona 2 shoes? (26) 3

5 3 1 1 1  12 by doubling *6710. division both the 14 4 Form an equivalent 6 problem for 8 2 8 (41) dividend and divisor. Then find the quotient.

5 8

5 8

11. Write thirty-two 3 ? thousandths ? a decimal number. 2 as (35)   4 12 3 12

3 ?  4 12

? 2  3 12

3 ?  4 12

? 2  3 12

Lesson 44

233

1 82 

3

1 1 12. What 1 4 number is 6 of 24,042?

67 4 3

(29)

* 13. Compare: 1

1

67 4 1 14

(44)

14

6

1 6

a. 10.251 82  2

5 8

1 1 82  2 0.125 5 8

5 8

1 1 82  2 5

b. 25% 8

5 8

5 8

12.5%

14. 1Write the standard numeral3for1 (6 × + 3 1 1 1 (4 × 1).1 1 1 5 1 1 100) (32)1 14 67 4 8 2 67 1 4 58 82  2 6 6 4 6 2 4 8 * 15. Justify A $36 dress is on sale price. 3 1regular 1 5Mentally 1 3 5 1 for 10% off the 1 1 67 8 2  2 14 8 14 (43) 67 4 6 calculate 10% of $36. Describe the method4you used to arrive at your 6 8 answer. 3 5 2 ? 5 1 1 ? 16. 8 a. 8 24 3 How 12 many 8 s are in 1? 212 (30) 3

67 4

1 6

3 ?  4 212 ?  3 12

1 1 1 1 b. Use part a to find8the of 58 s in 3. number 1 4 the answer to 6 2 2

3

?67 4 2  3 12

5 8

17. What is the least common multiple of 2, 3, 4, and 6?

(30)

3 ?  4 12

3 ?2 ? ? 3 ? 2 ?2 18.2  19.     20. (1.3) 4 12 3 12 4 12 3 12 (39)3 (42) (42) 12 3 ? ? 2 3 ? ? 2     21.4 Find 3 12 12the average 3 of1226, 37, 42, and 43. 4 12 (18) 22. Round 364,857 to the nearest thousand. (16)

23. Twelve of the 30 students in the classroom were girls. What was the (23, 29) 3 ? ? to girls in the classroom? ratio2ofboys  4 12 3 12 24. a. List the factors of 100. (19)

b.

3

Classify

2

3 Which of4the factors of 100 are prime numbers?

25. Write 9% as a fraction. Then write the fraction as a decimal number.

(33, 35)

2 3

3 2 3 2 * 26. with denominators of 12. Then add the 4 Write 4 and 3 as fractions 3 (42) renamed fractions.

27.

(Inv. 2)

Estimate Which percent best describes the 3 shaded portion 2of this rectangle? Explain why. 4

A 80%

3

C 60% D 20%

B 40%

3 4

2 3

28. Shelby started working at 10:30 a.m. and finished working at 2:15 p.m. (32) How long did Shelby work? * 29. (44)

30. (17)

Estimate

A 0.1

Which of these numbers is closest to 1? B 0.8

C 1.1

D 1.2

Connect What mixed number corresponds to point X on the number line below?

X 9

234

Saxon Math Course 1

10

11

1 15 8 2  28

1

82

LESSON

45

Dividing a Decimal Number by a Whole Number Building Power

Power Up facts

Power Up F

mental math

a. Calculation: 4 × 425 b. Number Sense: 375 + 500 c. Calculation: 77 + 18 d. Money: $12.00 − $1.25 e. Fractional Parts: 1 2

f. Money:

1 2

$40.00 10

of 1500

$40.00 10

g. Geometry: A square has a length of 9 cm. What is the area of the 0.45 square? $0.45 3 � 4.2 5 � 2.25 5 � $2.25 h. Calculation: 4 ×0.45 8, − 2, ÷ 3, 3 +� 4.2 2, ÷ 3, × 5, + 1.4 1, ÷ 3 $0.45 3 � 0.24 5 � 2.25 3 � 4.2 5 � $2.25 3 problem Copy this problem and fill in the missing digits: 1 ___ 2 + solving 12 _ 0 ___8

New Concept

Increasing Knowledge 1 1 2

2

$40.00 10

1.4 3 � 4.2 3 12 12 0

3

$40.00 10

Dividing a decimal number by a whole number is similar to dividing dollars and cents by a whole number.

1 2

$0.45 5 � $2.25

$40.00 10

0.453 � 4.2 5 � 2.25

3 � 4.2

1.4 3 � 4.2 3 Notice that the decimal point in the quotient is directly above the decimal 12 point in the dividend. 12 0 Example 1 $0.45 $0.45 5 � $2.25 5 � $2.25

1

0.45 Divide: 3 �24.2 5 � 2.25 Solution

$40.00 10 1.4

0.45 5 � 2.25

3 � 0.24 3 � 4.2 3 12 The decimal $0.45 point in the quotient is0.45 directly above 12 3 � 4.2 $2.25in the dividend. 5 � point 0 5 � 2.25 the decimal

0.08 3 � 0.24 24 0 1.4 3 � 4.2 3 12 12 0

Lesson 45

1.4 3 � 4. 3 12 12 0

3 � 0.24

235

1 2

Example 2

3 � 4.2

$40.00 10 1.4

0.08 Divide: 3 � 0.24 3 � 0.24 3 � 4.2 3 Solution 24 12 0 The decimal3 �point 1.4 is directly 0.08 1 20.45 4.2 in the quotient 3 �above 0.24 50� 2.25 3 � 0.24 3 � 4.2 We fill the empty the decimal point in the dividend. 3 24 place with zero. 12 0 12 0 Decimal division answers are not written with remainders. Instead, we attach zeros to the end of the dividend and continue dividing.

$0.45 5 � $2.25

Example 3

0.12Divide: 5 � 0.6 5 � 0.60 Solution 5 10 0.12 Thinking Skill 5 � 0.6 10The decimal point in the quotient is directly above 0.60 5 � 0the decimal point in the dividend. To complete Verify 5 0.12 5 � 0.6 the division, we attach a zero to 0.6, making Why are 0.6 and 5 � 0.60 10 0.12 we 0.12 decimal 5 � 0.6 5 � 0.6 the equivalent number 0.60. Then 0.60 equivalent? 10 5 0.60 5 � 5 � 0.60 continue dividing. 0 10 5 5 10 0.12 0.125 � 0.6 5 � 0.6 10 10 0 0.60 5 0.60 5 � � Practice Set a. The 10 distance from Margaret’s house10 to� school and back is 3.6 miles. 4.5 0.012 0.7 5 0.6 4 � 0.30.12 2 0.14 10 � 1.4 5 � 5 06 5 3 How far from school? 0 does Margaret5live 6 � 0.60 10 10 5 10 10 b. The perimeter of a square is 6.4 meters. How long is each side of the 10 0 0 4.5 0.12 square? How 10that your answer is reasonable? 5 � can 0.6 you check 4� 2 � 0.14 3 5 � 0.60 2 0 5 4.5 3 � 4.5 8 � 0.24 5 � 0.8 3 Divide:5 3 2 � 0.14 3 4.5 0.012 10 4.5 0.012 0.7 � � 0.12 0.125 � 0.6 0.12 5 0.6c. 5 0.6 d. 0.6 ÷ 4 4 � 0.3 e. 2 � 0.14 4 � 0.3 2 � 0.14 10 � 1.4 3 56 310 6 5 � 0.60 5 � 0.60 5 � 0.60 0 5 2 5 5 5 4.5 0.012 0.012 0.7 3� 4.5 0.7 103� 1.4 10 � 1.4 3 g. 4 � 0.3 4 � 0.3 h. 10 10 103 f. 0.4 ÷ 5 2 � 0.14 2 � 0.14 5 6 5 3 6 5 10 10 10 4.5 0.012 2 3 4 � 0.3 3 2 � 0.14 10 � 1. 0.7 0.7 0 00.012 00.012 5 6 5 2 i. 10 � 1.4 10 � 1.4 2 j. 3 k. 0.1 ÷ 4 14 4 � 0.3 4 � 0.3 3 8 � 0.24 3 � 4.5 8 � 0.24 5 � 4.5 0.8 6 5 6 3 3 3 35

Written Practice 5 4.55

Strengthening Concepts 2 2 0.012 0.7 3 � 4.54 � 0.3 3 � 4.5 8 � 0.24 8 � 0.24 5 � 0.810 �51.4 � 0.8 2 � 0.14 3 3 5 3 6 2 1. By what fraction must 53 be multiplied to get a product3of 1? 8 � 0.24 � 4.5 3 (30) 0.24 8 � 0.24 5 � 0.8 5 � 0.8 0.012 0.012 0.012 34.5 � 4.5 8 � 4.5 0.7 0.7 0.7 4 2 � 0.14 2 � 0.14 4 � 0.3 10 4 � 0.3 � 20.3 � 1.4 � 1.4 510 � 1.4 5 � 0.14 2. How many $20 bills equal 6 one 10 thousand dollars? 5 6 3 3 6 (15) 3

34.5 � 4.5 3

3

5 3. Cindy made 23 of her 24 shots the basket.8Each worth 3 �at 4.5 5 � 0.8 � 0.24basket was 2 points. How many points did she score?

3 (29) 5 3

5 3

2 5 3 3

2 4. 3 *(45)

2

3 � 34.5

3 � 4.5 8 � 30.24 5. 8 � 0.245 � 80.8 * 6. 5 � 0.8 �*4.5 � 0.245 � 0.8 (45)

(45)

7. What is the least common multiple (LCM) of 2, 4, 6, and 8?

(30)

236

Saxon Math Course 1

5 � 0.8

10

10

5

5

85

8

5

8 85 5  31 832  52 8 5 10 3 318 3 384 245 25  2m  2  25 45  42  2  * 8. 236  9. 2g  2m 36 g*  5 3 3 10 1 10 8 5 (43) 5 8 8 3 2 3 2 2 4 85  g(43) 2  5 4 2 236  m  2 5 5 10 8 8 3 2 * 10. m − 1.56 = 1.44 * 11. 32 − n = 5.39 (43) (38, 43) 3 3 8 2 4 1  g2 5 4 2 3 3 82536  m  2 2 4 1 5 5 8 8 3 g223  512. 4  g2  2 2  54 4 3  2 1 10 36  m  2 13. 8  5  2 36 m 5 5 10 8 8 3 2 5 5 (26) 10 8 2 38 (29) 5 3 3 8 5 2 75 4 1 3  5 g2  4 2 7 236  3m  2 5 126 5 10and 412. 8 8 12 3 2 8 of 694 68 14. Estimate the product 3 32 3 each unknown 3 2 2 check 4 8 35 3 work. 1 4 1  5 5 44 your 2 1 236  m  2g  number. 236  m  2 Find 2  5 Remember 4 g g 2 2to 5 5 3 824 8 2 5 10 236  m 102 5 8 8

(16)

15. 0.7 × 0.6 × 0.5 3 236  m  2 5 10

3 8

* 16. 0.46 × 0.17

(39)

(40)

53 3 75 5 3 7 7 3 2 4 1 5 8 gallons g  miles 2  on 4  2 of gas. 36 m  2 car12traveled 8 8 Formulate2 6  12 * 617. Mrs. 177.6 6 Lopez’s 12 5 5 10 8 8 5 3 8 3 5 7 3 5 2 4 1 7 (15, 45)  many 5 g  car 28  4 average 2 of 12 how miles per gallon? Use a 8Her 5 traveled 5 6 an 8 6 8 3 2 12 7 12 multiplication pattern. Write an equation and solve the problem.

3 8

6

8 5  3 2

3 8 5 2 4 1  g2 5 4 2 5 5 5 8 8 18. What 3 2 number is 3 of 6? What operation 7 did you 12 use to find your answer? 5 7 3 5 7 8 6 (29)3 5 3 6 8 2 4 1 12 8 6 12 m2  g2 5 4 2 33 2 5 7 5 5* 19.8 Justify 10 8 A shirt regularly priced at $40 is on sale for 25% off. Mentally (41)

3 8

6

12

3 5 7 * 20. Write a fraction equal to 6 that has 12 as12the denominator. Then subtract 8 (42) 7 12 from the fraction. Reduce the answer.

5 6

21.

7 12

(38)

5 6

8

calculate 25% of $40. Explain the method you used to arrive at your answer.

7 12

Analyze

The area of a square is 36 ft2.

a. How long is each side of the square? b. What is the perimeter of the square?

22. Write 27% as a fraction. Then write the fraction as a decimal number.

(33, 35)

23. Use a ruler to find the length of this rectangle to the (17) nearest eighth of an inch.

24.

(22, 33)

1 25. 2 (29)

Seventy-five percent of the 20 answers were correct. Write 75% as a reduced fraction. Then find the number of answers that were correct. Draw a diagram to illustrate this fractional-parts problem. Model

2 1 The product of 12 and 23 is 13. 3 2

1 2

2 3

1 3

2 3

1 21 1 1 2 11 2 1       2 33 3 2 3 32 3 3 1 2 1   2 3 3 1 3

Arrange these fractions to form another multiplication fact and two division facts. 26.

(Inv. 2)

1 2

1 2

1 2

2 3

2 3

2 3

5

Which percent best describes the shaded portion of this circle? Explain why. Estimate

A 80% 1 211 121 112 1      5  5 60% 2 332 332 333 3

1 3

1 2

1 1 1 53 3 B3

C 40% D 20%

1 3 1

53 Lesson 45

237

2

3

27. Write nine hundredths (35)

a. as a fraction. b. as a decimal number. 1

* 28. Form an equivalent division problem for 5  3 by multiplying both the (43) dividend and divisor by1 3. Then find the quotient. 53 29. The average number of students in three classrooms was 24. Altogether, (18) how many students were in the three classrooms? * 30. (27)

Early Finishers

Real-World Application

238

Coach O’Rourke has a measuring wheel that records the distance the wheel is rolled along the ground. The circumference of the wheel is one yard. If the wheel is pushed half a mile, how many times will the wheel go around (1 mi = 5280 ft)? Analyze

Decide whether you can use an estimate to answer the question or if you need to compute an exact amount. Explain how you found your answer. Emily and Jacob are equally sharing the $13.65 cost of a lunch. Tax is 5% and they want to leave a 15% tip. What is each person’s share of the cost?

Saxon Math Course 1

3

LESSON

46

Power Up facts mental math

Writing Decimal Numbers in Expanded Notation Mentally Multiplying Decimal Numbers by 10 and by 100 Building Power Power Up G a. Calculation: 4 × 525 b. Number Sense: 567 – 120 c. Number Sense: 38 + 17 1 2 of 2000 100

e. Fractional Parts: 1 2

f. Number Sense:

8 100

8 100

d. Money: $5.75 + $2.50 950

2000 100 1 10

$100. 1 2 100

11 is the g. Geometry: The perimeter of a 1regular hexagon11 is 36 mm. What 8 88 28 length of the sides of the hexagon? 1 1000

1

100 h. Calculation: 9 × 7, + 1, ÷ 8, × 3, + 1, × 2, – 1, ÷ 7 1 1000

1 100

problem solving

Copy this problem and fill in the missing9 _digits: 9_ 9 _ 9 _ 9 _ 9 _ __ _ _1 1 (4  1)  a2  _ _ b  a5  _ _ b 100 1000 1 1 __ __ b  a5  b (4  1)  a2  100 1000 0 0

New Concepts writing decimal numbers in expanded notation 2000 100

Increasing Knowledge

We may use expanded notation to write decimal numbers just as we 1 1 2000 have used expanded notation to write whole values of some 2 2 numbers. The 100 1 decimal places are shown in this table: 2000 2

1 10

1

1 100

ones



1 1 b  a5  b 100 1000

1 1000

10 1 1000

1 100

1 1000

100

Decimal Place Values 1 100

tenths

hundredths

20 1

thousandths

1 1 1 1 We write 4.025(4inexpanded way: b this a5  1)  a2 notation b  a5  b (4  1)b  a2  100 1000 100 1000 1 1 b  a5  b (4  1)  a2  100 1000

Lesson 46

239

The zero that serves as a placeholder is usually not included in expanded notation.

Example 1 Reading Math Write 5.06 in expanded notation. We say the Solution word and when we see a decimal The 5 is in the ones place, and the 6 is in the hundredths place. point. Read 5.06 as “five and six 1 b (5  1)  a6  hundredths.” 100

1 10

Example 2 1

5  1)  a6 10 

1 b 100

1

1 Write (4 ∙ 10 ) + (5 ∙ 1000 ) as a decimal number.

1 1000

Solution We write the decimal number with a 4 in the tenths place and a 5 in the thousandths place. No digits in the ones place or the hundredths place are indicated, so we write zeros in those places. 0.405

mentally multiplying decimal numbers by 10 and by 100 Thinking Skill Predict

How many zeros are in the product of 600 × 400?

When we multiply whole numbers by 10 or by 100, we can find the product mentally by attaching zeros to the whole number we are multiplying. 24 × 10 = 240 24 × 100 = 2400 It may seem that we are just attaching zeros, but we are actually shifting the digits to the left. When we multiply 24 by 10, the digits shift one place to the left. When we multiply 24 by 100, the digits shift two places to the left. In each product zeros hold the 2 and the 4 in their proper places. 1000s

100s

2

10s

1s

2

4

24

2

4

0

24 × 10 (one-place shift)

4

0

0

24 × 100 (two-place shift)

When we multiply a decimal number by 10, the digits shift one place to the left. When we multiply a decimal number by 100, the digits shift two places to the left. Here we show the products when 0.24 is multiplied by 10 and by 100. 10s

2

240

Saxon Math Course 1

1 s 10

1 s 100

0

2

4

2

4

1s

4

0.24 0.24 × 10 (one-place shift) 0.24 × 100 (two-place shift)

Although it is the digits that are shifting one or two places to the left, we get the same effect by shifting the decimal point one or two places to the right. 0.24 × 10 = 2.4 ⤻

0.24 × 100 = 24. = 24

one-place shift

two-place shift

Example 3 Multiply: 3.75 ∙ 10

Solution Since we are multiplying by 10, the product will have the same digits as 3.75, but the digits will be shifted one place. The product will be ten times as large, so we mentally shift the decimal point one place to the right. 3.75 × 10 = 37.5 (one-place shift) We do not need to attach any zeros, because the decimal point serves to hold the digits in their proper places.

Example 4 Multiply: 3.75 ∙ 100

Solution When multiplying by 100, we mentally shift the decimal point two places to the right. 3.75 × 100 = 375. = 375 (two-place shift) We do not need to attach zeros. Since there are no decimal places, we may leave off the decimal point.

Example 5 Multiply:

1.2 10  0.4 10

1.2 10 12   0.4 10 4

Solution

1.2 10  0.4 10 1.2 10 12   0.4 10 4 12 3 4

1.2 10 1.2 10 12    10 decimal 1.2 each Multiplying both 0.4point10one place. 4 0.4 1.2 10and 0.4 by 10 shifts  0.4 10 10 1.2 12 12 12  1.2 10 1.2 10 12  3 4    4 40.4 10 0.4 10 4 0.4 10

12 4

1.2 10  0.4 10

12

12 3 4

Practice Set

The expression 4 means “12 divided by 4.” 12 3 12 4 3 4

(2) 

Write these numbers in expanded notation: (2) 

a. 2.05 b. 20.5 (2) 

(2) 

(2) 

c. 0.205

Lesson 46

(7  10)  a8 

1 b 10

a6 

241

1 1 b  a4  b 10 100

Write these numbers in decimal form:

0)  a8 

d. (7  10)  a8 

1 b 10

e. a6 

1 b 10

a6 

1.5 10  0.5 10

1 1 b  a4  b 10 100

Mentally 100 each product: 2.5 calculate  0.05 100 f. 0.35 × 10

100 100

1.5 1  0.5 1

1 1 b  a4  b 10 100

1 1 (5  10)  (6  10 )  (7  1000 ) g. 0.35 × 100

1 1 h. 2.5 10  (6  10 (5 ×10) )  (7  1000 ) i. 2.5 × 100

j. 0.125 × 10

k. 0.125 × 100

For the following statements, answer “true” or “false”: 1  a8  b (7isa10) l. If 0.04 is multiplied by 10, the product whole number. 10

Conclude

m. If 0.04 is multiplied by 100, the product is a whole number. Multiply as shown. Then complete the division. 1.5 10 100 2.5   n. o. 0.5 10 0.05 100

1 1 b  a4  b 0 100

Written Practice

1 1 0)  (6  10 )  (7  1000 )

1.

(25)

2.

1 7  10)  a8  b 5  6.35 10

(13)

(5  10)  100 2.5  0.05 100

Strengthening Concepts When a fraction with a numerator of 30 and a denominator of 8 5  6.35 is converted to a mixed number and reduced, what is the result? Analyze

1 10 5  6.35

5  6.35

1 1 1  a8  b is 98.6° on the Fahrenheit (7  a6  scale. b  a4  b 10) temperature Normal body 10 10 100 A person with have a10 temperature 1 a temperature 1 of 100.2°F would 1.5  b  a4   a6 b how many degrees normal? Write an equation and solve the 5 0.5 10 1 104  0.5 above 100 8  1.0 x3 9 y  16  8 4 problem.

100 2.5 3. Four and twenty is how  many dozen? 0.05 100

1 1 (5  10)  (6  10 )  (7  1000

1 4. Write (5  10)  (6  10   1 in decimal form. 3 ? 1 ) 3 (7 1000 ) 1  y  16  4 4 4 4 12 10 5. Twenty-one percent of the earth’s atmosphere is oxygen. Write 21% as (33, 35) a fraction. Then write the fraction as a decimal number. (46)

6. Twenty-one percent is slightly more than 20%. Twenty percent is 5  6.35 5(33)  6.35 3 5 1 4 reduced fraction?4  0.54  0.5 equivalent to what  a ba b 2 5 4 3 * 7. 5  6.35 4  0.5 * 8. 4  0.5 * 9. 8  1.0 4  0.5 8  1.0 (45)

(45)

(45)

Find each unknown number: 5 5 * x10.  3x 39  9 8 8 (43) 5 x3 9 8

3 1 5 1 − q 5= 0.235 4  x  3  9* 12. (43) x  3 8 y 9 16 4 4 8

3 13 1 * 11. y  16 y 164  4 44 4 4 (43) 3 ?3 1 3 1  426.9 y+1216+ w=4 49.25 y  16* 13. 4 12 4 4(43) 4

1 10 242

1 10

Saxon Math Course 1

1 4  2 5

1 4  2 5

1 4  2 5

3 5 a ba b 4 3

1 4  2 5

8  1.08  1.0 8  1.0 3? 3 ?   4 12 4 12

3 ?  4 12

1 41 4   2 52 5

1 1 10 10 1 10

4  0.5

Formulate

(15)

100 2.5  5 .05 100 x3 9 8

1 a6  b  101 (7  10)  a8  b 10

3 5 a ba b 4 3

3 ?  4 12

3 53 5 a b aa bb a b 4 34 3

3 5 a ba b 4 3

3 1 y  16  4 4 4

14.

(31, 41)

3 ?  Fifty4 percent 12 of the area of this rectangle is shaded. What is the area of the shaded region? Explain your thinking. Verify

4 cm

1 4  2 5

6.35

3 5 a ba b 5  6.35 4 3

4  0.5 15. (23)

2.5 cm

8  1.0

5 56.35  6.35

Connect

4  0.5 4 40.5  0.5

What is the ratio of the value of a dime to the value of a

quarter?

5 3 1 3 ? y  16  4 1x  33  9  0.25  16 * 17.  y16. 4 8 3.7 × 4 4 4 4 55 12 (39) (42) 4 11 33 x 3 3 99 44 y 1616   x y 8 common multiple of 3, 4, and 8? 4 4 4 4 18. What is the8least

5 3 9 8

(30)

* 19. Compare: 1 1 4 2 1 4 a. (0.1) 0.1 b.3 0.15 0.54  0.5 b a8 b1.0 a8  1.0 10 2 5 4  2 5 4 3 11 1 14 4   20. Which digit is in the thousandths place in 1,234.5678? 10 2 25 5 10 (34)

(40, 44)

5  6.35 6.35

21. Estimate the quotient when 3967 is divided by 48. (16)

22. The 3 is 3its ? 13of a3square is 100 cm2. What ? perimeter? 1 area   y 164  4 y(38) 16 44 4 4 12 4 4 12 23. John carried the football twice in the game. One play gained 6 yards. (14) The other play lost 8 yards. Use a negative number to show John’s total yardage for the game.

5 5  3x 39  9 8 8

1 10

3 53 5 25. a b aa bb a b 4 34 3 (29)

1 41 4 ⋅  24. 2(29) 5 2 5 26. (27)

Laquesha bought a 24-inch-diameter wheel for her bicycle and measured it carefully. Arrange these measures in order from least to greatest: Connect

circumference, radius, diameter * 27. (41)

The chef’s salad cost $6.95. The sales-tax rate was 8%. What was the total cost including tax? Explain how to use estimation to check whether your answer is reasonable. Estimate

28. Use a ruler to find the width of this (17) rectangle to the nearest eighth of an inch.

29. a. How many 83s are in 1?

1 2

(30)

3b. 8 3 8

1 2 3 8

3

1 Use the answer to part a to find the number of 8s in 3. 2

1 * 30. Rename 21 and 13 so that the denominators of the renamed fractions are 3 (42) 6. Then add the renamed fractions. 1 1 2

1 2

3

1 3

Lesson 46

243

LESSON

47 Power Up facts mental math

Circumference Pi (𝛑 ) Building Power Power Up E a. Calculation: 4 × 925 b. Calculation: 3 × 87 c. Number Sense: 56 − 19 d. Money: $9.00 − $1.25 e. Fractional Parts:

1 2

1 4

f. Money:

$25.00 10

1 2

$25.00 10

of $12.50

g. Probability: How many different outfits can be made with 3 shirts and 5 pairs of pants? 1  2, × 2, − 10, ÷ 9, + 5, ÷ 3, + 1, ÷ 612  0.5 4 + 0.25 h. Calculation: 6 × 8, 1 2

 0.25

problem solving

3 4

 0.75

Radley held a number cube so that he could see three adjoining faces. Radley said that he could see a total of 8 dots. Could he be correct? Explain your answer.

Increasing Knowledge

New Concepts circumference

 0.5

Laquesha measured the diameter of her bicycle wheel with a yardstick and found that the diameter was 2 feet. She wondered whether she could find the circumference of the tire with only this information. In other words, she wondered how many diameters equal the circumference. In the following activity we will estimate and measure to find the number of diameters in a circumference.

Activity

Circumference Materials needed: • 2– 4 different circular objects (e.g., paper plates, pie pans, flying disks, bicycle tires, plastic lids, trash cans) • Lesson Activity 11 • string or masking tape • scissors • cloth tape measure(s) • calculator(s) 244

Saxon Math Course 1

This activity has two parts. In the first part you and your group will cut a length of string as long as the diameter of each object you will measure. (A length of masking tape may be used in place of string.) Then you will wrap the string around the object and estimate the number of diameters needed to reach all the way around. To do this, first mark a starting point on the object. Wrap the string around the object, and mark the point where the string ends. Repeat this process until you reach the starting point, counting the whole lengths of string and estimating any fractional part. Do this for each object you selected. Model

2

1

1 2 $25.00 Thinking 10

Skills

Connect

Remember that: 1 4 1 2 3 4

 0.25  0.5  0.75

Begin wrapping here.

$25.00 ? 10

Estimate In the second part of the activity, you will measure the circumference and diameter of the circular objects and record the measurements on a recording sheet. If you have a metric tape measure, 3 a customary 1 record the measurements to the nearest centimeter. If you have  0.5  0.75 2 4 tape measure, record 3 your answers to the nearest quarter inch in decimal  0.75 4 form. Using a calculator, divide the circumference of each circle by its diameter to determine the number of diameters in the circumference. Round each quotient to the nearest hundredth.

Record your results on Lesson Activity 11, as shown below. Part 1: Estimates Object

Approximate number of diameters in the circumference

plate

35

trash can

34

1 1

Part 2: Measures Object

Circumference

Diameter

Circumference Diameter

plate

78 cm

25 cm

3.12

trash can

122 cm

38 cm

3.21

Lesson 47

245

pi (𝛑 )

1

34 Thinking Skill Verify

Why can you use the formula π 2r or 2π r and get the same answer?

If we know the radius or diameter of a circle, we can calculate the approximate circumference of the circle. In the previous activity we found that for any given circle there are a little more than three diameters in the circumference. Some people use 3 as a very rough approximation of the number of diameters in a circumference. The actual number of diameters in a circumference is closer to 3 17, which is approximately 3.14. The exact number of diameters in a circumference cannot be expressed as a fraction or as a decimal number, so mathematicians use the Greek letter 𝛑 (pi) to stand for this number. To find the circumference of a circle, we multiply the diameter of the circle by π. This relationship is shown in the formula below, where C stands for the circumference and d stands for the diameter. C = πd Since a diameter is equal to two radii (2r), we may replace d in the formula with 2r. We usually arrange the factors this way:

Reading Math Symbols

We read ≈ as “is approximately equal to.”

C = 2π r Unless otherwise noted, we will use 3.14 as an approximation for π. We may use a “wavy” equal sign to indicate that two numbers are approximately equal, as shown below.

π ≈ 3.14 To use a formula such as C = π d, we substitute the measures or numbers we are given in place of the variables in the formula.

Example Sidney drew a circle with a 2-inch radius. What is the circumference of the circle?

Solution The radius of the circle is 2 inches, so the diameter is 4 inches. We multiply 4 inches by π (3.14) to find the circumference. C = πd C ≈ (3.14)(4 in.) C ≈ 12.56 in. The circumference of the circle is about 12.56 inches. Why is the answer about 12.56 inches reasonable for the circumference of the circle? Justify

Practice Set

246

a.

In this lesson two formulas for the circumference of a circle are shown, C = π d and C = 2π r. Why are these two formulas equivalent? Explain

Saxon Math Course 1

Find the circumference of each of these circles. (Use 3.14 for π .) b.b.

c. c.

1

35

2 in.

3 cm

3 4

d.

The diameter of a penny is about 43 of an inch (0.75 inch). 3 Find the circumference of a penny. Round your answer to4 two decimal places. Explain why your answer is reasonable.

e.

Roll a penny through one rotation on a piece of paper. Mark the start and the end of the roll. How far did the penny roll in one rotation? Measure the distance to the nearest eighth of an inch.

f.

The radius of the great wheel was 14 8 ft. Which of these numbers is the best rough estimate of the wheel’s circumference? Explain how you decided on your answer. 7 14 8 ft A 15 ft B 60 ft C 90 ft D 120

Estimate

Model

7

Justify

g. Use the formula C = 2π r to find the circumference of a circle with a radius of 5 inches. (Use 3.14 for π.)

Written Practice

Strengthening Concepts

1. The first positive odd number is 1. The second is 3. What is the tenth positive odd number?

(10)

2.

(15)

A passenger jet can travel 600 miles per hour. How long would it take a jet traveling at that speed to cross 3000 miles of ocean? Write an equation and solve the problem. Formulate

3. José bought Carmen one dozen red roses, two for each month he had known her. How long had he known her?

(15)

* 4. (47)

Conclude

If A = bh, what is A when b = 8 and h = 4?

5. The Commutative Property of Multiplication allows us to rearrange factors without changing the product. So 3 ∙ 5 ∙ 2 may be arranged 2 ∙ 3 ∙ 5. Use the commutative property of multiplication to rearrange these prime factors in order from least to greatest:

(2)

3∙7∙2∙5∙2∙3∙3∙5 $25.00 10

6. If s = 12, what number does 4s equal?

(29)

* 7. (46)

Generalize

Write 6.25 in expanded notation.

8. Write 99% as a fraction. Then write the fraction as a decimal 1 1 number.  0.25  0.5

(33, 35)

4

2

* 9. 0.18 12  0.18 12

* 10. 10  12.30 10  12.30

12  0.18

10  12.30

(45)

12  0.18

(45)

10  12.30 1.25 1.25 10 10   0.5 0.5 10 10

w

w  236  62 Lesson 47 247

2

2

Find each missing number: 11. w  236  62 2 23 3 3 611 m 3 m 611 5  * n13. 5n 5 5 10 5 510 12 12 (43)

30 11 10 10 12

(4, 38)

3 2 3 5 5

15. 8 3  5 3 4 4 (26)

5 4  10  10  12

10 10

11  10 n5 12 * 12. 5y = 1.25 (43) 33 3 3 523 3 8 8542  * 14. m4 m 6  4634 3 5 55 5 (43) 16. (29)

2 m6 3 5

3 33 8 85  5 4 44

5 5  3 4

3 ?  4 20

3? ? 3 317. 3?  ? *5 * 18. 3  3?  ?  55 5 5  20 4  20 4(42) 20 (42) 5  20 3 43 4 4 20 4 20 19. Bob’s scores on his first five tests were 18, 20, 18, 20, and 20. His 1.25 10 (18) 3 ?   average score is closest to which of these 0.5 numbers? 10 5 20 1 3A2 17 B 18 C 19 D 20

3 3? ?   5 20 5 20

1.25 1.25 10 10   3 11 2 * 20. What is the 0.5 Robert’s 10 10bicycle tires are 20 inches 0.5  10 n  5 in diameter. m6 3 (47) 5 5 12 circumference of a 20-inch circle? (Use1 3.14 for π.) Explain why your 1 4 12 14 2 answer is reasonable. 1 1 1 1 21. factors of 20 are also factors of 1412 14 2 1230? 12 Which 1 2 121 4(19) 14 4 14 12 2 12 2

1

1 2 2 5 5 3 ? 12 12   * 22. Explain Mentally calculate the of 6.25 and 10. Describe how 4 20 3 product 4 (46) 1 you 2 12 performed the mental calculation.

* 23. Multiply as shown. Then complete the division. (46)

1.25 10  0.5 10

* 24. Shelly answered 90% of the 40 questions correctly. What number is (41) 90% of 40? 1 12 Refer to the chart shown below to 4answer problems 25 and 26.

Planet

Number of Earth Days to Orbit Sun

Mercury

88

Venus

225

Earth

365

Mars

687

25. Mars takes how many more days than Earth to orbit the Sun? (13)

26. (15)

In the time it takes Mars to orbit the Sun once, Venus orbits the Sun about how many times? Estimate

27. Use an inch ruler to find the length and width of this rectangle. (17)

248

Saxon Math Course 1

1

14 2

1.25 10  0.5 10

28. Calculate the perimeter of the rectangle in problem 27. (8)

9 29. Rename 52 so that the denominator of the renamed10 fraction (42) 9 2 is 10. Then subtract the renamed fraction from 10. Reduce 5 the answer. 9

2 5

* 30. (46)

10 When we mentally multiply 15 by 10, we can simply attach a zero to 15 to make the product 150. When we multiply 1.5 by 10, why can’t we attach a zero to make the product 1.50?

Verify

9 10

2 5

Early Finishers

Real-World Application

Decide whether you can use an estimate to answer the question or if you need to compute an exact amount. Explain how you found your answer. A music store is having a sale. Hassan wants to buy 3 CDs. One CD costs $11.95. The other two CDs cost $14.99 each. Hassan has $50 but needs to use $10 of it to repay a loan from his brother. Does Hassan have enough money to buy the CDs and repay his brother?

Lesson 47

249

LESSON

48

Subtracting Mixed Numbers with Regrouping, Part 1 Building Power

Power Up facts

Power Up G

mental math

a. Calculation: 8 × 25 b. Number Sense: 630 − 50 c. Number Sense: 62 + 19 d. Money: $4.50 + 75¢

1 8

e. Fractional Parts:

1 2

f. Money:

1 2

$25.00 100

of $15.00

$25.00 100

g. Geometry: What is the relationship between the radius and the diameter of a circle? 2 1 1 6− 1, ÷ 3, × 4, ÷ 6, × 3, ÷ 2 46 h. Calculation: 4 × 7,

1

46 2

1

16

46

problem solving 7

36 1

46

Josepha is making sack lunches. She has two kinds of sandwiches, three kinds of fruit, and two kinds of juice. If each sack lunch contains one kind 7 1 of sandwich, one kind of 36 4 6 fruit, and one kind of juice, how many different sack-lunch combinations can Josepha make? 7 36 1 1

New Concept

8 1 8 Increasing Knowledge 1 8

2

1 2

1 2

$25.00 100 1 2 $25.00 1 on the restaurant shelf. The server1 sliced There were 4 6 pies one of the 6 2 100 1 2 4 6 into sixths. Then the server removed 1 6 pies. How many pies whole pies 1 2 46 16 were left on the shelf?

1 “separating” problem about pies: Here is another

1 8

2

1 8

1

46

Thinking Skill 1 Discuss 46

7

36

2

1

We may illustrate 1 6 this problem with circles. There were 4 6 pies on the 1 shelf. 7

What words or phrases in the problem give clues about which operation 7 3 we 6should use to answer the question?

36

$25.003 7 6

2

1

16

100

7

46

1 46

1 8

46

1

36

46 7

36

1

46 1 46

The server sliced one of the whole pies into sixths. This makes 1 equals 4 6 pies.

1

7 36

46 pies, which

7

36 7

36

250

Saxon Math Course 1

1

46 5

2

5

1 6 Then the server removed 1 26 pies. So 2 6 pies were left on the shelf.2 6 2

5

2

16

16

26

2

2

16 7

16

36

5

5

26

26

1 46 2 16 2 5 2 5 5 2 2 5 16 226 2 1 2 2 2 1 2 16 2 1 5 5 1612 1 1 6 4 616 2 6 6 62 6 6 16 6 26 6 6 2 5 1 5 2 26 2 2 6 2 5 2 16 21 1 62 5 11626 65 5 52 6 2 21 6 2 6 6 2 2 2 5 22 1 2 1 5 2 16 2 26 26 1 16 16 16 5 2 1 65 2 46 2161 122 255 6 26 6 16 166 26 6 6 1 266 5 11666 22666 1 26 1 6 2 5 5 2 2 1 2 1 416 2 5 26 25 22 6 11 6 2 5 1 6 1 2 5 4 61 16 46 26 6 6 6 26 46 16 6 6 6 6 5 2 2 16 26 2 1 16 2 1 5 5 2 6 2 2 51 1 26 5 6 41 5 26 26 5 61 16 2 2 2 21 6 1466 6 62 5 5 6 6 7 1 6 2 1 2 1 56 1 4 62 16 16 46 26 26 26 22 11 55 36 1 4 1 4 2 2 1 1 the arithmetic for subtracting from 6 6Now 6 6 1 4 64 66. 6 6 2 2566 2 6 we show 6 6 6 16 46 26 2 1 1 6 2 2 1 1 1 2 1 1 2 1 4 16 4 6 pies6 2 6 76 16 4 61 46 6 6 6 6 6 36 6 6 6 6 6 6 2 1 5 46 6 − 1 6 pies 1 26 2 1 1 5 6 2 1 2 1 10 2 1 4 16 6 1 10 1 1 6 6 22 6 1 4 6 6 116226 10 2 1 2 6 1 6 14 6 6 61 0.1442 8the 0.144  2 20 1 2 2 20 6 6 1 14 8  2 6 1 2 20 2 1 1 1 1 We cannot subtract from , so we rename . Just as server sliced 4 Thinking Skill 2 1 1 16 5 6 6 6 62 2 62 6�5 � 6 2 � � 3 5 � 2 3 5 4 2� 4 266 1 1 4 3 4 6 6 6 3 3 6 6 3 3 6 6 16 16 1 6the pies 66 into sixths, 2 66 one6 of so6 we one6 2 of6 the 4four Verify 6 6 6 wholes into 66. 6 7 1 change 1 1 6 7 1 1 6 subtract. 106 is 13 7 . Now we can 10 plus 1 1 10 6 6 1 6 4 62 1 , which 16 23 6 6 1 3 65 1 How 2can you 86 0.144 6 three whole pies 6 86 60.144 3 � 6 6� 66This 2makes 5 3 6�plus 2 3 6 3 �206 � 20 26 20 3 3 2 1 1 check the answer6 4 1 2 66 1 6 53 � 23 1 1 3�6 �6 7 2 1 the 6subtraction 1 6 7 6 1 to 3 16 4 6 6 4166 6 4  0.9 61 6 0.9 4 0.9   6 6 6 6 6 6 6  0.9 1 73 6 pies 1 6 7 6 6 6 0.9 4 0.9   3 3 7 3 1 1 3 3 1 problem? 66 77 6 � 1 6 11 16 2 2 16 6 6 66 5 65 � 13 3 3 3� 66 1 3 6 3 36 − 34 6 −3 16 6 10 6 13 26 1 10 33766 6 6 1 6 2 6 1 66 10 6 1 6 pies 6 �6 3 �286 0.144 6 6 6 8  0.144 1 2 6 26 20 2 11 22 1 32 � 61� 163 � 6 �2 12 5 1 � 5203 �� 2736 � 52 202 3 6 11 1 3 3 1 � � 5 � 3 2 3 5 1 2 pies 6 66 4 630.9 3 6  0.9 4  0.9 6  0.96 6 0.9 3 3 � 63 � 66 60.9 6 4 3 36 �6133 � 233 33 3 3 3 6 7 1 36 3 1 10 1 1 6 6 10 1 110 2 2 1 6 8 0.144 1 1 8 0.144 3 3 7 1  1 2 6  1 2 62 1 � 3� 54320� 26 1 2 1 20 2 20 2 52 3 � 2 3 32 3� 6 6 61 6Example 611� 6 3� 6 1 2 33 � 6 � 6 26 1 6 6 � 1 6 1 4 1 3 6 6 143 1 15 31 � 22 3 3 1 3 1 1 � � � � 1 2 2      1 2 2 1      63 � 16 � 6 (566 10) 3 5 � 2 5 � 2 3 5 � 2 10) (6 (4 (5 ) ) (5 10) (6 ) (4 )     (6 (4 ) ) 2 22 2 151 � 100 2 1 36 3 1 6410 � 6 64 5 � 3 31 � 35 3 100 106 � 4� 4 43� 10 �1 �6 � �6 3� 1 � 22 2 3 3 100 3 � 2 33233 3232� 3 3 3 3 5 31 6 3 16 3 3 6 33� � 1666 6 6 33 �66660.9 2 333 Subtract: 6 6 6 50.9 6 6 0.9 4 0.9 4 4 0.9    30.9  3 3 3 1 1 3 1 1 2 2 � 3 3 3 3 1 1 3 1 2 5 � 16 5 � 2 1 34 3 3 3 � 31 3 1 3 2 6 1 3 4 3 3 1 1 3 �13 33) 3� 133 4 43 1 6 3 � 3 3  ) (6  10 ) 10) 6) 10 ) 1 (5 (4 10) )  (4 4 100 (510 4 3(4(6Solution (5  10)  (6  1 2 2 100 100 3�6�6 53 � 23 3 4 1 43 3 3 14 � 3 3� 3 3 14 1 126  0.9 63 0.94  0.9 4 6  0.9 7  0.9 1 2 13 0.9 1 1 change � 2 2 2 1 1 2 2 We cannot subtract from , so we rename 5 . We one of the five 1 5 � 326 33 3 33 � 3 6 4 3 51 1 3 1 3 35 13 4 43 3 4 3 3 5 3 1 3 3 �3 1 5 4 3 5 13 43 3 3 3 133 � 3 34 11 11 4we 3 1 6 into33 .3Then 5 6 1 7 3 3 3 3 3 3 3 3subtract. 4 4 4 wholes we combine 4 and to get . Now � � 1 3 33� 1 3 3 3 5 3can 3 33 2 133 3 33 2 2 3 3 3 3 33 4 1 31  2 3 3 31 3 �  6 6 6    2  2 1  6 6 1 43 23 1 1 3 3 5 3 1 3 3 4 1 2 14 3 3 1 4 51 � 423100 � 2 23 � 253 �  310)  (6  410) (6 (5 ) (4 )  10 2 10) )122 (4 )1 ) (4 ) � 3 1 4 12 3  4 (5 1 (6 3 1 1 433� 4 4 2 101(5 100 2 � 100 3 10 1 4 2 2 1 3 3 � 4 34 4 � 35 � 3 43 4 35 4 54 5335 2 4 33 43 � 34 4 33 4 32 6 4 3 4 3�5 3 � 3 6 3335 3 3 3 3 1 2 2 13 �1 43 3 2 2 3 23 3 3 2 3 2 23 3 2 23    5   1 4 �  2 2 � 1 1  �  1 � 2 3 2 4 1 2 1 2 34 3 1 1 13 23 1 3 4 345 �1 3 5 1 3 1 1 43 5 1 4 �   (4 3310) (5 (6) 3 10 )1  (6)  10 10) (41 13 �) 42(4 (5  310) 33 (6 410 ) 53(5 4 32 32 4) � 100 3 421� 3 � 3 3 3 � 34 � 2 3 2 100 4 3 34 3 3 3 100 44 3 4 � � 2 3 4 34 4� 13 � 3 4�3� 43 43 6 2 41� 3 �22 3 2 3 1 2 33 �1413 1�333 � 444 � 44� 4 5�31 3��2 32 24 3 2 4 3 6 � 3 � 6 � 3161 4 � 34� 5 3 � 2 3 22 1 3 �2 133 33 � 3 44333 4 1 4 6 6 3 3 4 3 5 24 325 35 2 54 443 43 3 35 3 1 3 1 3 3 � 1 1 1 2 4 4 5 4 4 6 1 4 3 3 324 � 3 �4 3 5 43 2 3 543 3 453232 1 3 36 3  1 2 2 3 � 6  2 35  Set  1 3 Practice  123 4 Subtract. 2 14 2  25 3 3 3 4 5 3 1 � 23 2 23 � 2 34 2 2 �3 4223� �22223 2 � 2123 231 �2 2 2 � 2 1 2 3 � 2� 3 � 1 3 4 3 2 3c.54 5 4 35 4 1 1 a. 4 3 3 5 3 3 1 1 4 43 3 53 3 b.3 �43145 4 43 2 53 53 43 1 31 1 2 32 14 � 3 � 15 32 2 14 3 2 36 1 44 3 3 3 3 425213 333 7 2 6 7 3 21 5 3 3 7 1 5 5 5 4 4 4 3  223445 8 2 4 4 2 8 12 4 8 2 12 3 3 1  3 2 3  31 1  14 2 1 344 � 12 3�2 52  1 4 � 2 33 3  22 2 3133 45 3 � 32 2 145143511333 2 45 433 4 � 2 23 3 � 22 3 2 3  102 3 � 2 3 3 �  10 5 3 103 22 3� 2 35 2 3 3 53 2 4 2 2 3 3 2 3 3    4  2 4  2 3 4 �2 � � �2 4 2� 2 3 12 4 8 2� 2 3 2 12 2 122233 2 �2222221623 18 2 � �222223 8 213 35�13 2 3 � � 2 3 5 1 2�2 2 23 1 3 � 1 3 2 33 1 3 3 3 � � 3 3 � 2 2 22 3 �5 2 2 3 6 5 82 2 d. 5386 6 7 12 3 f. 7 12 36 4 4 2 23 2 3e. 7 121 8 322 4 2 3 2 53 43 234 3 3 4� 2 3 10 3 10 42233 10 2  24  2 8  4 12  28 3 1   4 12 4 122 4  28 3 1 2 53 4 � 3 � 3 � 22 � 2 3 3 3 4 1 3 4 1 2 � 2 23 1 43 4�3�3 3 43 4�3�3 3 1 1 23 1 1 3 64 58 7 58 7 126 4 58 7 12 12 3 1 3 10 3 3 3  2 48 24  4 10  2 48  4 10   2 48 4 122 4 Lesson 48 251 12 12 1 1 3 1 1 3 1 3 1 64 58 7 126 4 58 7 12 58 7 12 53 3 4 10 3 4 10 4 10 5

26

5

26

2

2

16

16

2

16

2

16

m5

3 7 12

g. h.

Model

Select another exercise and write a word problem that is solved by the subtraction. Formulate

Written Practice 1.

(18)

Select one of the exercises to model with a drawing.

Strengthening Concepts The average of two numbers is 10. What is the sum of the two numbers? Analyze

* 2. What is the cost of 10.0 gallons of gasoline priced at $2.279 per gallon? (46)

3. The movie started at 11:45 a.m. and ended at 1:20 p.m. The movie was how many hours and minutes long?

(32)

* 4. (44)

Three of the numbers shown below are equal. Which number is not equal to the others? Classify

A * 5. (44)

1 2

1 2

Analyze

B 0.2

C 0.5

10 20

D

10 20

8  0.

Arrange these numbers in order from least to greatest: 1.02, 0.102, 0.12, 1.20, 0, −1

6. 0.1 6+ 0.9 7. (8)(0.125) + 0.9 0.4 4  0.9 4  0.9 10 10.2 + 0.3 6 (39) 5 7 2 2 20 5 w7 12  m  5 12 hiking 12 to a waterfall 3 miles away. After hiking 3 8. Formulate Juan was 10 10 1 1 (11) 2 how many more 20 to reach the 20 2 2.1 miles, 5 did he have to hike 7 2 miles 5 w7 12  m  5 12 12 solve the problem. 3 waterfall? Write an equation and 3 2 1 1 5 1 1 1  n(4 12  m  5 10 10) (62 (4  1004)  0.91 (5 6 0.9 10) (6 )  ))  3(5 2 39. Estimate 4100 46890. the sum of 4967, 8142, and 10 of ? (16) 3 4 3

8

(37)

* 10. 8  0.1446  0.9

* 311. 6  0.9 * 12. 4  0.9 1 (45) of 4(45) ? 3 * 13. What is the cost of 100 pens priced at 39¢ each? 1 2 (46) 2 1 1 35 35 43 43 4 1 1 3.5 10 * 14. Write (5 2 10)  (6  )  (4  100 ) in standard3 form.  2 35  25  1 23 10  1 3 0.7 10 (46) (45)

1 3

15. What

2 3

1 1 multiple of1 6 and is(5 the common 10  (4   100  least   (5 (6 ) 10 ) 8? 10)3.5 (6 10 ) 10) 

4  0.9

5

1

1 100 )

(4  0.7 10 Find each unknown number:7 5 2 7 451 5 w  7 12  5 12 7 23 25 2 12  m  5 3 3 w m 7 5 5 7  m 5 5 * 17. 12 12 *w 16. 12 12122 12 3 3 3 (43) (43)   13 1 25 2 2 1 35 35 43 3 4 31 * 18. n  2  5 2 * 19. x + 3.21 = 4 4  14 (43)  2 35  2 35  1 23 (43) 3 1 3 1 3 20. What fraction is 32 of5 134? 3 7 12 7 12 3 58 1 2(29) 32 1 8 of ? of ? 4 3 3 43 3 4 4  4 10  4 10 2 8 three times during the game. 28 21. Sam carried the  football 12 of 12 He had gains 2 (14) 5 1 4 3 yards and 5 yards and a loss of 12 yards. Use a negative number to 4  1 34 Sam’s overall yardage for the game. show 3.5 10  3 Justify 10 If5 1a spool 10 * 3.5 22.  3.5 thread is 2 cm in diameter, then7 one wind of 0.7for 10  8 12 (47) 0.7 0.7 thread 10 10is about how many centimeters long? (Use 3.1410for π.) Explain  2 48 1  4 12 3 1 3 5 8 your answer is reasonable. 58 7 12 7 12 how to mentally check whether (30)

2 3

 2 48

252

Saxon Math Course 1 1

64

 2 48

 4 10 12

 4 10 12

3 3 n  2n 25 4 4 

1 4

1 4

6

2



3 2 3

4

3

4

1 3

of 4?

1 4

23. If a rectangle is 12 inches long and 8 inches wide, what is the ratio of its length to its width? 1

(23, 29)

1 3

4

24. The perimeter of this square is 4 feet. What is (8) its perimeter in inches?

3.5 10  0.7 10

25. The area of this square is one square foot. (31) What is its area in square inches? * 26.

5 2 7 m w  712  5 5 12 12 3 1

36 2 3

of

3 1 4 ?3

1

36 5

1 ft

3

1 3

3 of 4 ? * 28. Rename 31 and 14 as fractions with denominators of1412. Then add the (42) renamed fractions.

(46)

Analyze

Multiply as shown. Then simplify the answer. 3.5 10  0.7 10

3.5 10  0.7 10 * 30. (48)

Real-World Application

12 in.

If d = rt, and if r = 60 and t = 4, (47) 5 7 what does d equal? 12  m  3 1 32 1 w  7 12  5 12  25  5 n2 5 12  m n 4 4 43 4 27. Seventy-five percent of the 32 chairs in the room were occupied. Write (22, 33) 5 75% as a reduced find the number of chairs that were 5 1 6 fraction.1Then 6 occupied.

* 29.

Early Finishers

1 ft

Evaluate

2

16

12 in.

1

1

5

5

16 16 There were 3 6 pies 3 6 on the shelf. How can the server take pies from the shelf? Explain

Leonardo purchased 4 pounds of grapes. Leonardo gave his neighbor half a pound of grapes so he could make a fruit salad. Next, he gave his friend 0.09 lb of grapes for a snack. Lastly, Leonardo gave his brother 1.25 lb of grapes so his brother could make grape juice. What is the weight of the grapes Leonardo has left? Show your work.

Lesson 48

253

LESSON

49

Dividing by a Decimal Number Building Power

Power Up facts

Power Up H

mental math

a. Order of Operations: (8 × 100) + (8 × 25) b. Number Sense: 290 + 50 c. Number Sense: 58 − 19 d. Money: $5.00 − $3.25 1 2 of 4000 100

e. Fractional Parts:

1.24 0.4 g. Geometry: A circle has a radius of 3 cm. Using 3.14 for π, what is the circumference of the circle? 0.4  1.24 h. Calculation: 5 × 10, ÷ 2, + 5, ÷ 2, + 5, ÷ 2, ÷ 2

1 2

f. Number Sense:

0.4  1.24

problem solving 2 4 8

4000 100

$30.00

M Ned walked from his home to school following the path from H to I to J to K to L to 2 M to S. After school he walked 2K home from S to C to H.4Compare the distance of Ned’s walk40  80 8 L 2 of his walk home. to school to the distance I 40  80 J

S

C

H

4000 100

1 2

Increasing Knowledge

New Concept

When the divisor of a division problem is a decimal number, we change the problem so that the divisor is a whole number. 1 2

4000 100

1.24 0.4

1 2

0.4  1.24

4000 100

4000 100

The divisor is a decimal number. We change the problem before we divide.

0.4  1.24 0.4  1.24 2 2 One way to change a division problem 4  8 is to multiply the divisor and the 40  80 dividend by 10. Notice in the whole-number division problem below that multiplying both numbers by 10 does not change the quotient.

2 40  80

2 4 8

2 4 8

2 40  80

2 40  80

Multiplying 4 and 8 by 10 does not change the quotient.

The quotient is not changed, because we formed an equivalent division problem just as we form equivalent fractions—by multiplying by a form of 1. 8 10 80 � � 4 10 40 254

Saxon Math Course 1

1.24

1.24 0.4

1.24 0.4

4

24 0

8 10 80 1.24 1.24 � � 0.4 4 10 40 0.04use this method to change a division by a decimal number to a We can also 8 10 80 1.24 8 10 80 10 1.24 1.24� � � division � by a whole number. 0.4 � 4 10 40 0.4 4 10 40 0.4 10 8 10 80 1.24 1.24 � If � we multiply the divisor and dividend in 0.4 by 10, the new problem has a Thinking Skill 4 10 40 0.4 whole-number divisor. Summarize 1.24 8 10 80 1.24 3.1 � 10 � 12.4 0.04 0.4 How can we 1.24 1.24 4 10 40 4 12.4 � � � decimal 1.24 4 whole-number 0.4 10 mentally0.4multiply 1.24 divisor divisor 0.04 a decimal 0.04 1.24 number by 10 We divide 12.4 by 4 to find the quotient. 0.04 8 10 80 1.24 or 100? � � 0.4 3.1 1.24 4 10 40 0.04 4 � 12.4 3.1 1.24 100 124 3.1 � � 4 12.4 � 0.04 100 4 1 4Example � 12.4 3.1 4 � 12.4 1.24 Divide: 0.04 3.1 4 � 12.4 1.24 100 124 Solution � � 0.04 100 4 100 1.24 124 places. To make the 100divisor, 1.24 The 124 0.04, is a decimal number with � two decimal � � 10� 80 1.24 100 4 0.041.24100 0.04 8divisor 100 4 3.1 number, we will multiply � � a whole by , which shifts each decimal 0.4 100 100 124 31 41.2410 40 0.4 1.44 � 4 �places 12.4 06.  14.4 point�two 1.2 0.04 100 4 124 4 to the right.



12 1.24 100 124 � � 04 0.04 100 4 4 1.24 This forms an0equivalent division problem in which the divisor is a whole 31 0.04 number. Now we perform the division. 0.24 9 4 � 124 0.4 31 0.3 1.24 100 124 � � 12 31 0.04 100 4 4 � 124 04 4 � 124 31 124 12 4 � 124 04 0 04 3.1 12 310.3 12 4 44 � 12.4 0.05  0.4  04 4 � 124 0 0 4 12 0 04 4 Example 2 31 0 Divide: 4 0.6 06. � 14.4 � 1.44 � 124 1.24 100 12 124 � � Solution 0.04 100 04 4 0.6 � 41.44 The divisor, 0.6, 0 has one decimal place. If we multiply the divisor and dividend by 10, we will shift the decimal point one place to the right in both 0.6 � 1.44 06. � 14.4 numbers. 0.6 06. � 14.4 � 1.44 0.6 � 1.44 06. � 14.4 0.6 � 1.44 0.6 06. � 14.4divisor, which we solve � 1.44 This makes a new problem with a whole-number 31 0.6 � 1.44 0.6 � 1.44 4below. � 124 0.6 � 1.44 06. � 14.4 12� 1.44 2.4 06. � 14.4 0.6 04 6  14.4 4 12 0.6 � 1.44 0 24 0.6 � 1.44 06. � 14.4 24 0 0.6 � 1.44 Lesson 49

255

1.44 1.2

06.  14.4 06.  14.4 1.44 1.2

0.12  0.144

9 0.3

2.4 6  14.4 12 Practice Set 24 4 0.052 0.4 0 .4 6.  14.4 Thinking Skill Discuss

What do you notice about the answers in exercises a and b?

24 .4

0.12  0

1.44 1.2

Some people use the phrase “over, over, and up” to keep track of the decimal points when dividing by decimal numbers. 2.4 0.24 0.24 9 9 6 14.4  up 0.4 0.4 0.3 0.3 12 . 1.44 06.  14.4 0.05  2.5 24 1.2 over over 24 0 06.  14.4 1.44 0.3would 120.3 06.  0.4 a. We multiply the divisor and dividend0.05 of 1.2 by what  14.4 0.05 0.4 number to  12 make the divisor a whole number? 1.44 1.44 0.24 9 06.  14.4 06.  14.4 1.44 1.2 1.2 1.44 0.12  0.144 0.4 0.3 1.2 b. We would multiply the divisor and dividend of 0.12  0.144 by what 1.2 0.12 0.24  0.144 number to make the divisor a whole number? 0.4 0.24 9 0.4problem so that the divisor is a whole0.3 Change each number. Then divide. 9 0.24 0.24 0.3 d. 12 9 0.05  0.4 c. 9 0.3 0.4 0.4 0.3 9 0.05  2.5 0.05  2.5 0.3 2.5 e. 0.05 0.3 f. 0.3  12 0.3  12 0.05  0.4 g. 0.24 ÷ 0.8 h. 0.3 ÷ 0.03 0.3  12 0.3  12 0.4 i. 0.05 0.05  0.4

3  12 .4

0.05  2

1.44 1.2

0.

0.12 0.12 0.144 0 9 0.3

0.

0.05 0.05 2.5  2 0.05  0.4

0.05 0.05 0.4  0.4 j. 0.2 ÷ 0.4

k. Find how many nickels are in $3.25 by dividing 3.25 by 0.05. 2.4 2.4 Strengthening Concepts 6  14.4 6  14.4 12 12 4 product * 1. When2 the of 0.2 and 0.3 is subtracted100 from the sum of 0.2 and 24 (12, 39) 2 4 100 4 difference? 0.3, what is2the 0 0 2. Model Four fifths of a dollar is how many cents? Draw a diagram to (22) illustrate the problem. 2.4 0.07  3.5 0.5  12 6  14.4 3. The rectangular, 99-piece “Nano” puzzle is only 2.6 inches 12jigsaw2.4 (31, 39) 0.07 3.5 0.5  12  4 6 area long and 2.2 inches wide. What is2 the  14.4of the puzzle in square 2.4 2 4 12 inches? 6  14.4 0 24 3 x 4 1 2.4 12 n of6thepuzzle 4 4. 2.4 Find the perimeter described in problem 3. 5  100 2 4 8 8 6  14.4 2 4 6(39)  14.4 0 24 12 12 x 4 * 5. Compare: n  6 1  4 3  0 (44) 2 4 2 4 5 100 8 8 0.301 b. 31% 30.1% 24 2 a. 4 0.31 0 0 5 5 6. 0.67 + 2 + 1.335 7  4 9 7. 12(0.25)  (38) (39) 10 10 2 3 0.5 0.07 3.5  3.5 0.5 12 12 *0.07 8. 0.07 * 9. 0.5 0.07 3.5 0.5 12 12 3.5 9 5 5 7 (49) (49) 4  5 10 10 2 3 * 0.5 10. 12 * 11. (0.012)(1.5) 8  0.14 8  0.14 (45) (39)

Written Practice

100 100

2.4 14.4 12 24 24 0

2

Find each unknown number: 1 3 133 4 3 116 4 *n 12.  66 n n    n 6 4 100 8 (43) 100 4 8 8 88 888 100

1100 x 4  15  5m−m7 = * 14. 4 1.37 5(43) 100

x3 00 8

256

7 9 9 9 77 5   744 9 4 5 4 10 55 10  10 10 10 10 10 10 Saxon Math Course 1 5 5  2 3

xx x 4 4 * 13. 4   4x  5 100 5 100 (42) 5 100 5 100 1  15 * 15. m  7 = 4 (43) 5 5 5 55  555  32  3 2 2 32 3 100 100

8 0.14 8 0.14 0.14 88 0.14

1 11 7 1 m m 714 m m 77 4 4 4

n 6 4 4 n6 8 88 8

5



m

5 100 100

16. Write the decimal number one and twelve thousandths. (35)

9 9 7 7  4 4 517. 5 10 10 10 10 (26)

18. (29)

5 55  5  2 32 3

19. How much money is 40% of $25.00? (41)

20. There are 24 hours in a day. James sleeps 8 hours each night. (29)

a. Eight hours is what fraction of a day? b. What fraction of a day does James sleep? c. What fraction of a day does James not sleep?

21. (19)

* 22.

(18, 45)

23. (41)

3 What factors 4 do 12 and 18 have in common (that is, the numbers that are factors of both 12 and 18).

List

Analyze

What is the average of 1.2, 1.3, and 1.7?

Estimate Jan estimated that 49% of $19.58 is $10. She rounded 49% to 50% and rounded $19.58 to $20. Then she mentally calculated 50% 23257 of $20. Use Jan’s method 2  5  to 7 estimate 51% of $49.78. Explain how to perform the estimate.

24. a. How many 34 s are in 1? (30)

3

b. Use the answer to part a to find the number of 4 s in 4. 25. (17)

Connect

Refer to the number line shown below to answer parts a–c. x y z

2.5 2 0.5 3 02  5  7 257

3 4 1

2

a. Which point is halfway between 1 and 2? 3 4

b. Which point is closer to 1 than 2?

3 4 3 4

2.5 100 250   0.5 100 50 23257 257

c. Which point is closer to 2 than 1?     26. Multiply and divide as indicated: 2 3 2 5 7 257 (5)

3 4

* 27.

We can find the number of quarters in three dollars by 2.5 2323 52 7  5  7dividing $3.00 by $0.25. Show this division using the pencil-and-paper 2.5 100 250   0.5 252 7  5  7 method taught in this lesson. 0.5 100 50 2.5 23257 0.5  7 Connect 2  5 28. Use a ruler to find the length of each side of this square 23257 (8) to the nearest eighth of an inch. Then calculate the perimeter of the 257 square. 2.5 2.5 3  0.5 4 0.5 (49)

2.5 0.5 2.5 0.5

2.5 0.5 2.5 0.5

Represent

2.5 2.5 100 100 250 250     0.5 0.5 100 100 50 50 2.5 100 250   0.5 100 50 2.5 100 2250    7 3 2 5   0.5 100 502  5  7

Lesson 49

257

257

* 29. A paper-towel tube is about 4 cm in diameter. The circumference

2  3  2  5 (47) 7 of a paper-towel tube is about how many centimeters? 257

(Use 3.14 for π.)

* 30. (49)

Explain

Sam was given the following division problem: 2.5 0.5

Instead of multiplying the numerator and denominator by 10, he accidentally multiplied by 100, as shown below. 2.5 0.5

2.5 100 250   0.5 100 50 Then he divided 250 by 50 and found that the quotient was 5. Did Sam find the correct answer to 2.5 ÷ 0.5? Why or why not?

Early Finishers

Real-World Application

After collecting tickets for three years, you are finally promoted to night manager of the local movie theater. You want to look good in your new job, so you try to increase profits. The cost for a bucket of popcorn is as follows: the popcorn kernels and butter used in each bucket cost 5¢ and 2¢, and the bucket itself costs a quarter. Each bucket of popcorn sells for $3.00. a. In one night you sold 115 buckets. What was your profit? b. You really want to please your boss, so you decide to increase profits by charging more per bucket. How much must you charge for a bucket of popcorn to make a $365.70 profit from selling 115 buckets?

258

Saxon Math Course 1

2.5 100 250   0.5 100 50

1 2

LESSON

50

Decimal Number Line (Tenths) Dividing by a Fraction

Power Up facts mental math

Building Power Power Up G a. Order of Operations: (8 × 200) + (8 × 25) b. Number Sense: 565 − 250 c. Calculation: 58 + 27 d. Money: $1.45 + 99¢ 1 of 2 5000 10

e. Fractional Parts: 1 f. Number Sense: 2

$25.00

5000 10 1 10

3

1 10 3 1 10

3 1

1 10 3 10

g. Statistics: Find the average of 134, 120, 96, and 98. h. Calculation: 8 × 9, + 3, ÷ 3, − 1, ÷ 3, + 1, ÷ 3, ÷ 3

problem solving

Half of a gallon is a half gallon. Half of a half gallon is a quart. Half of a quart is a pint. Half of a pint is a cup. Into an empty gallon container is poured a half gallon of milk, plus a quart of milk, plus a pint of milk, plus a cup of milk. How much more milk is needed to fill the gallon container?

New Concepts decimal number line (tenths)

Increasing Knowledge

We can locate different kinds of numbers on the number line. We have learned to locate whole numbers, negative numbers, and fractions on the number line. We can also locate decimal numbers on the number line.

0

1

2

On the number line above, the distance between consecutive whole numbers 3 1 has been divided into ten12 equal lengths.5000 Each length is 10 of the distance 1 10 10 between consecutive whole numbers. 5000 10

5000 10

1 10

1 10

The arrow is pointing to a mark three spaces beyond the 1, so it is pointing 3 3 3 3 to 1 10 . We as the decimal 0.3, so we can say that the arrow is 1 10can rename 10 10 pointing to the mark representing 1.3. When a unit has been divided into ten spaces, we normally use the decimal form instead of the fractional form to name the number represented by the mark.

Lesson 50

259

Example 1 What decimal number is represented by point y on this number line? y

7

6

8

Solution The distance from 7 to 8 has been divided into ten smaller segments. Point y is four segments to the right of the whole number 7. So point y 1 1 1 4 14 44 represents 7 10 . We write 7 10 number 7.4.14 3  7 10as the decimal 7 10 3 4 4 4 4

dividing by a fraction

The following question can be answered by dividing by a decimal number or by dividing by a fraction: 4

7 10 4

7 10 1

1 3 4

 41 4 1

3 1

1

1 How many1quarters 1 are1in 1three dollars? 4 14 4  1 3 3 1  4 1 4 141 4 11 1 4 4 4 4 1 4 If we think 7of10 a quarter as 14 of a dollar, we3have problem:  this division 4 4 1 4 1 1 4 7 10 3 4 4 1 4

4 14

1  4 1 4 1   4 1 1

1

We solve this problem in two steps. First we answer the question, “How 1 4 4

4 4 1   4 is the  3 Language 1 in one dollar?” Math many quarters1 are The answer 1 14 4 7 7 10reciprocal of 4, which 41 10 1 4 4      3 4 1 1 4 1 1 1 4is 4, which equals 4 14.  4 14 Two numbers 4 1 4 4 1 1 1 1 4 whose product is  1 3 3 3 1 3  4 4 14 1 64 3 4 31 1 1 4  44  1 are reciprocals. 4 4 4 3 1 4 4 1 4 4 1 4 4 1 1 1 4 1 14 4 1 1 4 1 × = 1, so            4 4 3 1 1 4 1 4 41 1 4 4 4 4 11 4 4 1 1 1 to the question 1 above to1find4the For the second step, we use the answer and 41 are 11 3 1 4 4 4 number of quarters in three dollars. There are four quarters in one4dollar, and reciprocals.

4 1

 1  4 41 14 

4 4 1  1

4 1

1

there are three times as many quarters in three dollars. We multiply 3 by 4 3 4 4 1 4 4   there 1 3 dollars. and find 1that 3 quarters in three 3 3 4 3 are 12 4 number of quarters in one dollar 7

number of dollars

10

3 � 4 � 12

4

4

7 10of quarters in three 7 10 dollars number

3 3 3 We will review the steps6we the problem. 1  3  4  took to solve 4 4 4 4 3 1 Original problem: How many quarters are in $3? 3 4

Thinking Skill Model

Draw a diagram to represent the number of quarters in $3.

14 41

Step 1: Find the number of quarters in $1. 3  1 4

1

Step 2: Use the number of quarters in $1 to find the number in $3.

3 × 4 = 12

1 4 4

3

1 4 3 4

4 1 4 3

7 10

4

1 4

1 4

3

3 4

1 1 4

1 4 4

 41  1

3 4 1 4

 41

4 1

Example 2 1

1

This row4 of pennies is 2 4 inches long.

3 3 3 3 The diameter ofin.a penny is 34 of an inch. How many pennies 6 are needed4 4 4 4 to make a row of pennies 6 inches long?

260

Saxon Math Course 1

1

3 4  4 3

4 3

1

13

4 3

1 4

3 4

4 3

Solution 3 3 3 In effect, this problem asks, “How many 34 -inch segments  in 6 inches?” 6 are 4 4 4 We can write the question this way: 3 4

3 4

6

3 4

3 3 4 4

3

3

3 4 4 3 6 3 3 3 3 4 4 4 6 4 4 3 will 3 4 4 We will take3two4 steps. First we find the number of pennies (the number 4 1 4 3 3 3 3 3 3 3 3 33 3 13  3 3 33 6  3 3 3 4 4 6  1 34 3 4 4 of 33, 4 3 4 4 4 4 3 of 3-inch segments) in31 inch. The of 34 s43in 1 is the reciprocal 343number 6 46  36  4 3 4 4 4 4 4 4 4 34 4 4 4 3 4 4  3 3 3 4 6 4 4 4 4 3 4 which is 3. 3 4 3 34 34 3 3 4 4 63 4 4 4 1 4 4 4 4 3 4 4 4 1  4 1 33 43 3 4 3 3 3 3 33 4 3 13 3  4 3 6  6  4 4 3  24 1 34 4 44 3 4 14 1 4 4 4 13 6  4 3 8 3 3 3 3 44 3 13 3 4 3 3 4 4 33 43 4 3 34 3 3 3 4   64  3 1 14 we 3 3 We43 3 4 use 4 in 1 31 44 4 convert will not mixed number .44Instead, will 4 24 4 4 1 4 333 4 4 1334 431tothe 1633 4 4 14 33 4 3 3 3 3 3 6        1 1 1 1 8 1 6 4 4 3 4 3 4 44 3 4 44 3 44 3 4 3 44 4 3 4 4 4 1 4 3 3 3 3 3 3 3 3 3 4 4second 3 step 4 Since 3 there are pennies 3 3 1 inch, there 4 of 24 the the solution. in   4 3 4 4 1  1 6 3 3 3 3 3    8 6 4 3 3 3 4 4 24 43 1 3 4 3 3 3 in463 inches. 3 3 times 3 3 3  8 many So 6 by 43.3 4 4 1 3  1 3 3as 3 we 3 3 6 3 3 4multiply 6 are six 43 3 4  3 3 3 6 1 4 1 4 3 4 3 3 44 4 4 4 4 1 3 3 3 4 4 4 4 3 3 3 3 3 3 4 3 3   4 4 3 3 3 1 1 4   1 1 3 3 3 1 34 3 4 6 3 4 24 4 34 6 3 4 6 4 344 4 3 4 6 3 4 4 3 4 4 8  3 = 4 4 1 4  4 4 4 4 3 3 3 3 3 4 24 3 3 4 4 3 3 3 4 4 4 3 3 3 3 44 33 34 3 43 4 4 3 336 4 364 13 338 6  13 33 44 3 24 44 3  3 1 3 3 4 3 3133 4 4 3 46 4   1433We 3 4  1 4 6 1 34  413are 6 1 3will 6 3 1find 64  4steps 3 4of41the 33the 3334 4 4 4 3 4 We there 8 pennies in a 6-inch row. review 3 3 3 4 4 4 4 4 4 4 4 3 4 4 4 4 33 3 4 6 3  43 3 4  4 3 4  424   3 6 1 4 4 4   1 4 36 3 3 4 4 3 34 Skill4 31 solution. 4 24 3 3 4 33 4 34Thinking 3 4 24 4 4 4 6 1 1  33  6833  3 3 4 4 1   4 Connect 43 33 3 4 6 3 3 34 3 3 3 34 143 43 24 4 1 Original 4 4 424 6  4 6 4 4 4How many problem: s are in 6? 4 6  84 1  4  1  13 6 44   34  8 4 1 3 3 3 3 3 3 3 4 3 How many 3 3 3 3 1 1 3 4 are3 in a 3 3 4 3 2 443 3 6  44 324 3 34 4 3 33 3 4  1  pennies    Step 1: Find the number of s in 1.  6 134  113 4 4 4row3that is 1 foot 4 3 43 4 1 4 341  43  3 46 44 3 4 3 4 long? 3 3 41 4 24 3 3 3 3 3 1 Step 142: Use1the 2 4 of 34 s in 11 to find the  number 6 6 3  4 4 4 6 4 4 3 3 3 3 3 4 4 24 3 3 3 3 3 3 4 4 24 4 4 3 3 3 2 4 46  4 6  1  4 1  of number s in 6. Then simplify 4 3 6  3 3 3 4 3 3 the 3 4 4 4 4 4 33 3 4 4 4 3 3 6 41 4 4 answer. 14 1 1 8 4 1 1 24 1 3 3 1 3 24 3= 8 4 4 21 2   24 6 1 1 1 4 4 4 4 4 4 4 4 2 1 4

4

4

1

24

Practice Set

Connect

1 4

3 4 1 4 To which decimal number is each1arrow 1 1 pointing? 214 4  3 2 4 3 4 4 4 a. b.3 d. f. c.4 1 e. 4 5 1  3  2  w  21100 3  12 w 3 4 3 1 1 1 7.25 12 2 2 4 4 4

1

1 4 1 4

24 1 24

3 4

0

1

23

3 4

6

4 3 3 4 3 3 g. Formulate Write and solve a division problem to find the number   of  6 1 4 4 4 4 3 quarters in four dollars. Use 14 instead of 0.252 14for a quarter. Follow this pattern: 3

3

8 Original Problem 8 3 8

Step 1 3 3 ? 8 4 Step 2 24

1

13

1 43 x 6  8

3 4

3 4

3 8

3 8

3 3 3 5 and solve 1 fraction 7division h. Formulate Write for this 1 problem 1 question:6  0.138  128 6 a 7.25  2  w8  2100 x1 0.12 7.2 0.4 8 w 2 4 7 12 8 8 4 Pads of writing 1paper were stacked 12 inches high on a shelf. The 1 3 2 4 pad 4 inch. How many pads were in a of 5thickness 1 each 7 was 8 of an  12 6  x  1 w 7.25  2  w  2100 0.12  7.2 0.4  7 6  0.138 1212-inch stack? 8 8

 w  2100

w

5  12 12

1 7 6 x1 8 8

0.12  7.2

0.4  7

6  0.138

Lesson 50

3



?

261

Written Practice * 1.

The first three positive odd numbers are 1, 3, and 5. Their sum is 9. The first five positive odd numbers are 1, 3, 5, 7, and 9. Their sum is 25. What is the sum of the first ten positive odd numbers? What strategy did you use to solve this problem?

* 2.

Jack keeps his music CDs stacked in plastic boxes 38 inch thick. Use38 the method taught in this lesson to find the number of boxes in a stack 6 inches tall.

(10)

(50)

7.25  2  w  2100

Strengthening Concepts Predict

Connect

3 8

3 8

3 in. 8

3 8

3 8

3. The game has 12 three-minute rounds. If the players stop after two 5 of the 1 7 minutes how 7.2 many minutes  12 twelfth  1 for0.12 w 6  xround, 0.4  7did they play? 6  0.138 3 3 3 3 12 8 8 8 8 8 8 4. Compare: (44) 3 3 3 3 53 1 38 57 1 8 0.60 8 8 0.600 a. 3.4 3.389 b.  8 2 100  2  w 8 2100 2  1 0.12 7.25 w 12 6  w x 12  7.2 7.25 w 6 x 12 8 8 12 8 (15)

Find each unknown number:

5 1 5 1 7 2 0.144 0.4  6w =1  100 w 12  7 6  x 7.25 2 w 6. 1 6 wx 2 0.12  7.2 12 8 8 8 (43) 12 3 3 3 Symbols 57 5 1 7 8 1 8 2 2 8  1 0.12  2 100 1  70.12  7.2 6  0.138 7.25 7.2  6  x  0.4 0.4  7 6  0.13 7.25  2  w  2100 ww 1 6 w x *1 8.  7.2 12 8 8 12 8 3 8 1 2 means (43) (43) 9 3 3 3 3 3 square root. 8 8 8 8 The 8 8 9. The book cost $20.00. The sales-tax rate was 7%. What was the total 3 ? (41) square root of a 4 24 cost of the book including sales tax? number is one of two equal factors 5 1 7  2  w  2100 312 ?6  x  *111. 30.12 ?7.2 10. 1 − 0.97 w 7.25 0.4  7 6  0.138 4 3 of the number. 2 12 8 8 (38) (49)   4 24 4 24 57 1 7 1 2  1 0.12 * 12. 13. 6  0.138 12  7.2 x  0.4 1  70.12  7.2 6  0.138 00 6  0.4  7 x 63  w 12 8 (49) 8 8 8 45) 8 3 ? 14. (3.75)(2.4) 15. 3  ?  24 (39) 4 (42) 4 24 3 3 ? ? 3 3 is in the same place as the 2 in 85.21?  16. Which digit in 4.637  8 8 4 24 4 24 (34) Math Language

3 8

3 ?  4 24

3 8

5. 7.25  2  w  2100

(15)

17. One hundred centimeters equals one meter. (31) 3 How many square centimeters equal3one 8 8 1m square meter? 5 8 3 5 5 2 3 5   4 6  6 4 8 8 8 3 1 3 4 8

100 cm

8 223  3 71

100 cm

2 3  3 4

22 7

5 6

6 10

1m

18. What is the least common multiple of 6 and 9? (30)

3 8

85 35 5 5 5 85 3 2 83 3 2 3 22 2 3 19. 6  4 6  4 6 20.   4   21.  8 8 8 38 18(29) 38 1 3 34 1 3 4(29) 7 3 4 (26) * 22.

22 7

22 7

The diameter of a soup can is about 7 cm. The label wraps around the can. About how many centimeters long must the label be 8 3 5 5 2 3 go4all the way around the soup for π.) Explain how to  can? (Use 22 6 to  7 8 8 3 1 3 4   7    2 3 5 2 3 5 7 5 1 1 7 mentally check whether your answer is reasonable. 3 6 10 3 6 25 25 * 23. Find the average of 2.4, 6.3, and 5.7. (47)

Justify

(18)

* 24. Find the number of quarters in $8.75 by dividing 8.75 by 0.25. (49) 2  3  5 27 3  15 2 7 3  5 5 75 7 1 7 5 1 7 6 106 6 10 6 10 6 3 3 6 25 25 2  35 262

Saxon Math Course 1

2357

1

5

6

7

7

8 3 5 5 2 3 22   6 4 7 83 31 58 58 23 34 22   6 4 7 8 8 3 1 3 4 * 25. Connect What decimal number corresponds to point A on this number (50) line? A

4

26.

2357 25

85 3 2 3   3 1 3 4 5 7 27. 0.375 × 100 6 10 6

5 5 6 4 8 8

1 3

6

22 7

(46)

5 5  3  58  73 2 3 22 5 1 7   6  4 2* 28. Rename as a fraction with 6 as7 the denominator. Then subtract the 6 10 8 8 3 1 3 4 3 6  2 5 (42) 2357 5 7 1 renamed fraction from . Reduce the answer. 6 3 6 10 25 * 29. (50)

Points x, y, and z are three points on this number line. Refer to the number line to answer the questions that follow. Estimate

x

2357 6 25

y

z 5 67

1 3

7

6 10

a. Which point is halfway between 6 and 7? 2357 25

5 7 1 b. Which point corresponds to 6 10 ? 3 6

c. Of the points x, y, and z, which point corresponds to the number that is closest to 6? 30. (21)

Early Finishers

Real-World Application

Clarify

A 552

Which of these numbers is divisible by both 2 and 5? B 255

C 250

D 525

Alex is planning a bicycle trip with his family. They want to ride a total of 195 miles through Texas seeing the sights. Alex and his family plan to ride five hours each day. They have been averaging 13 miles per hour on their training rides. a. How many miles should they expect to ride in one day? b. Should Alex’s family be able to complete the ride in three days?

Lesson 50

263

INVESTIGATION 5

Focus on Displaying Data In this investigation we will compare various ways to display data.

part 1: displaying qualitative data

We have already displayed data with bar graphs and circle graphs. Now we will further investigate circle graphs and consider another graph called a pictograph. There are four states that produce most of the cars and trucks made in the United States. One year’s car and truck production from these states and others is displayed in the pictograph below. U.S. Car & Truck Production Michigan Ohio

Key = 1,000,000 cars or trucks

Kentucky Missouri Other States

In a pictograph, pictured objects represent the data being counted. Each object represents a certain number of units of data, as indicated in the key. The two cars by Ohio, for example, indicate that 2,000,000 cars and trucks were produced in Ohio that year. Conclude How many cars and trucks are produced in Michigan? In the four named states? In the nation?

1.

Conclude

Display the car and truck production data with a horizontal

bar graph. 2.

Conclude

What fraction of U.S. car and truck production took place in

Michigan? Another way to display qualitative data is in a circle graph. In a circle graph each category corresponds to a sector of the circle. Think of a circle as a pie; a sector is simply a slice of the pie. We use circle graphs when we are 3 1 1 interested in the fraction of the group represented2 by each category and 4 12 12 12 not so interested in the particular number of units in each category. Another name for a circle graph is a pie chart. 3  360°  90° 12

1 4 264

Saxon Math Course 1

3 12

3 12

5 12

We have used a template with equal sectors to help us sketch a circle graph. We can also sketch circle graphs with the help of a compass and protractor. We can calculate the angle of each sector (section) of the circle if we know the fraction of the whole each part represents. The sectors of a circle graph form central angles. A central angle has its vertex on the center of the circle and its rays are radii of the circle. Since the central angles of a full circle total 360º, the number of degrees in a fraction of a circle is the fraction times 360º. For example, if each sector is 14 of the circle, then each central angle measures 90º. 1× 1× 360º 360°==90º 90° 4 4

3 12

90° 90° 90° 90°

3  360°  90° 12 To construct a circle graph, we need to determine how many degrees to assign each category. Our bar graph shows that a total of 12 million cars and 5 of cars and trucks produced 3 we2find the 1 fraction 1 trucks were produced. First 12 12 12 12 12 in each state. Then we multiply by 360º. 3 1 4 12 Category Count (millions) Fraction Michigan

3

3 12

Ohio

2

2 12

Kentucky

1

1 12

Missouri

1

1 12

Other States

5

5 12

Total

12

12 12

12 12

The sector of the circle graph representing Michigan will cover 90º. 3  360°  90° 12 3. Determine the central angle measures of sectors for each category. 4. Sketch the circle graph by following these instructions: Use a compass to draw a circle, then mark the center. Position the center of the protractor over the center of the circle and draw a 90º sector for Michigan. Continue around the circle drawing the appropriate angle measure for each category. 5. Compare the pictograph, bar graph, and circle graph. What are the benefits of each type of display?

Investigation 5

265

part 2: displaying quantitative data

We now turn to quantitative data. Quantitative data consists of individual measurements or numbers called data points. When there are many possible values for data points, we can group them in intervals and display the data in a histogram as we did in Investigation 1. When we group data in intervals, however, the individual data points disappear. In order to display the individual data points, we can use a line plot as we did in Investigation 4. Suppose 18 students take a test that has 20 possible points. Their scores, listed in increasing order, are 5, 8, 8, 10, 10, 11, 12, 12, 12, 12, 13, 13, 14, 16, 17, 17, 18, 19 We represent these data in the line plot below.

Reading Math Each x in a line plot represents one individual data point.

Test Scores x x x x xx x x x xxxxx xxxx 0

5

10

15

20

When describing numerical data, we often use terms such as mean, median, mode, and range which are defined below. Mean: The average of the numbers. Median: The middle number when the data are arranged in numerical order. Mode: The most frequently occurring number. Range: The difference between the greatest and least of the numbers. To find the mean of the test scores above, we add the 18 scores and divide the sum by 18. The mean is about 12.6. To find the median, we look for the middle score. If the number of scores were an odd number, we would simply select the middle score. But the number of scores is an even number, 18. Therefore, we use the average of the ninth and tenth scores for the median. We find that the ninth and tenth scores are both 12, so the median score is 12. From the line plot we can easily see that the most common score is 12. So the mode of the test scores is 12. We also see that the scores range from 5 to 19, so the range is 19–5, which is 14. That is, 14 points separate the lowest and highest score.

266

Saxon Math Course 1

6.

Explain The daily high temperatures in degrees Fahrenheit for 20 days in a row are listed below. Organize the data by writing the temperatures in increasing order and display them in a line plot. Explain how you chose the values for the scale on your line plot.

60, 52, 49, 51, 47, 53, 62, 60, 57, 56, 58, 56, 63, 58, 53, 50, 48, 60, 62, 53 7. What is the median of the temperatures in problem 6? 8. The distribution of the temperatures in problem 6 is bimodal because there are two modes. What are the two modes? 9. What is the range of the temperatures? (In this case, the range is the difference between the lowest temperature and the highest temperature.) 10. Quantitative data can be displayed in stem-and-leaf plots. The beginning of a stem-and-leaf plot for the data in problem 6 is shown below. The “stems” are the tens digits of the data points. The “leaves” for each stem are all the ones digits in the data points that begin with that stem. We have plotted the data points for these heights: 47, 48, 49, 50, 51, 52, 53. Copy this plot. Then insert the rest of the temperatures from problem 6. Stem 4 5

Leaf 789 0123

11. Compare the stem-and-leaf plot from problem 10 to the line plot of the same data. Discuss the benefits of each type of display.

extension

Consider a problem to solve by gathering data. State the problem. Then conduct a study. Organize the collected data. Select two ways to display the data and explain your choices. Compare the two displays. Which would you use to present your study to your class? Interpret the data. Does the gathered data help to solve a problem or is it inconclusive?

Investigation 5

267

LESSON

51

Rounding Decimal Numbers Building Power

Power Up facts

Power Up H

mental 8 100 math

a. Calculation: 8 × 125 b. Calculation: 4 × 68 c. Number Sense: 64 − 29 d. Money: $4.64 + 99¢

1 8 1 8

1 e. Fractional Parts: 8 1 2

f. Money:

1 2

of $150.00

$100.00 100

$100.00 100

g. Measurement: Convert 36 hours to days. h. Calculation: 8 × 8, − 4, ÷ 2, + 2, ÷ 4, + 2, ÷ 5, × 10 9_ 9 _ 9 _ problem_ _ The smallest official set of dominos uses only the numbers 0 through 6. Each solving _ _ domino has two numbers on its face, and once a combination of numbers is __ used, it is not repeated. How many dominos are in the smallest official set of 0 dominos? (Note: Combinations in which the two numbers are equal, called “doubles,” are allowed. For example, the combination 3-3.)

9_ 9_ _ __ __ 0

New Concept

Increasing Knowledge

It is often necessary or helpful to round decimal numbers. For instance, money amounts are usually rounded to two places after the decimal point because we do not have a coin smaller than one hundredth of a dollar.

Example 1 Dan wanted to buy a book for $6.89. The sales-tax rate was 8%. Dan 8 calculated the sales tax. He knew that 8% equaled the fraction 100 and the decimal 0.08. To figure the amount of tax, he multiplied the price ($6.89) by the sales-tax rate (0.08). $ 6.89 ∙ 0.08 $ 0.5512

1 8

How much tax would Dan pay if he purchased the book?

Solution _ Sales tax is rounded to the nearest cent, which is two places to the right9 of 9 _ 9 _ the decimal point. We mark the places that will be included in the answer. __ __ $0.55 12 __ 0 268

Saxon Math Course 1

1 8

Next we consider the possible answers. We see that $0.5512 is a little more than $0.55 but less than $0.56. We decide whether $0.5512 is closer to $0.55 or $0.56 by looking at the next digit (in this case, the digit in the third decimal place). If the next digit is 5 or more, we round up to $0.56. If it is less than 5, we round down to $0.55. Since the next digit is 1, we round $0.5512 down. If Dan buys the book, he will need to pay $0.55 in sales tax.

Example 2 Sheila pulled into the gas station and filled the car’s tank with 10.381 gallons of gasoline. Round the amount of gasoline she purchased to the nearest tenth of a gallon.

Solution The tenths place is one place to the right of the decimal point. We mark the places that will be included in the answer. 10.381 Visit www. SaxonPublishers. com/ActivitiesC1 for a graphing calculator activity.

Next we consider the possible answers. The number we are rounding is more than 10.3 but less than 10.4. We decide that 10.381 is closer to 10.4 because the digit in the next place is 8, and we round up when the next digit is 5 or more. Sheila bought about 10.4 gallons of gasoline.

Example 3 Estimate the product of 6.85 and 4.2 by rounding the numbers to the nearest whole number before multiplying.

Solution We mark the whole-number places.

Thinking Skill Generalize

If you are rounding a whole number with two decimal places to the nearest whole number, what place do you look at to round?

6.85 4.2 We see that 6.85 is more than 6 but less than 7. The next digit is 8, so we round 6.85 up to 7. The number 4.2 is more than 4 but less than 5. The next digit is 2, so we round 4.2 down to 4. We multiply the rounded numbers. 7 ∙ 4 = 28 We estimate that the product of 6.85 and 4.2 is about 28. Explain in your own words how to round a decimal number to the nearest whole number. Summarize

Lesson 51

269

Practice Set

Round to the nearest cent: a. $6.6666

b. $0.4625

c. $0.08333

Round to the nearest tenth: d. 0.12

e. 12.345

f. 2.375

Round to the nearest whole number or whole dollar: g. 16.75 j. k.

1.

i. $73.29

If the sales-tax rate is 6%, then how much sales tax is there on a $3.79 purchase? (Round the answer to the nearest cent.) Estimate

Estimate

Written Practice (12, 25)

h. 4.875

Describe how to estimate a 7.75% sales tax on a $7.89 item.

Strengthening Concepts When the third multiple of 8 is subtracted from the fourth multiple of 6, what is the difference? Analyze

2. From Mona’s home to school is 3.5 miles. How far does Mona travel riding from home to school and back home?

(37)

3.

(13)

Napoleon I was born in 1769. How old was he when he was crowned emperor of France in 1804? Write an equation and solve the problem. Formulate

* 4. Shelly purchased a music CD for $12.89. The sales-tax rate was 8%.

(41, 51)

a. What was the tax on the purchase? b. What was the total price including tax?

* 5. Malcom used a compass to draw a circle with a radius of 3 inches.

(47, 51)

a. Find the diameter of the circle. b.

* 6.

Find the circumference of the circle. Round the answer to the nearest inch. (Use 3.14 for π.) Estimate

How can you round 12.75 to the nearest whole number? 4 4  0.5 3.25  1 0.25 7. 0.125 + 0.25 + 0.375 8. 0.399 + w = 0.4 (37) (43)4 5 7 45 0.5 7 3.25? 5 11007 3 531 ? 4 5 4 0.25 2 12  12   * 9. 10.  1 5  17 4  1100 0.5 1100 3.25  3 4  0.50.25 3.25 3 1 12 8 4 24 12 12 8 2412 (49) 0.25 (45) 5 3 ? 3 5 5 4 72 ? 5 7 2  1 5  17 3.25  3 4  1100 0.5 5  17 11. 3.25  * 12. 4  0.5 3 1  1100 0.25 (45) 12 8 4 24 4 12 12 8 (48) 24 12 (51)

4 0.25

4  1100 0.5 3 3.25 

Explain

5 3 ? 5 5 3.257 ? 5 1 72   1 13.1100  3 5  17 12 8 4 24 12 12(42) 8 2412 15. (0.19)(0.21) (46)

18. (38)

270

Analyze

square?

Saxon Math Course 1

(48)

3 4

16. Write 0.01 as a fraction.

(39)

17. Write (6  10)  (7 

* 14. 52  17 (35)

1 100 )

3 9 6 as a decimal 5 number. 10 10

10 1  3 2

The area of a square is 64 cm2. What is the perimeter of the 1 (6  10)  (7  100 )

  (7 5 100 ) 6 (6  10)  (7  (6 100 ) 10) 10 10

5  6 3 10 2 10

3

19. What is the least common multiple of 2, 3, and 4? (30)

(6  10)  (7 

1 100 )

3 9  61 ) 5 (7  10)  (6  20. 10 100 10 (26) * 22.

10 1 3 9 10 1 5  6 21.  3 2 10 10(29) 3 2

A collection of paperback books was stacked 12 inches high. 22335 Each book in the stack was 34 inch thick. Use the method described in 2235 Lesson 50 to find the number of books in the stack. 2 3 2  3  3  5 2  2  3  3  52 3 42  2 4 2235 3 23. Estimate the quotient 3  5 by 98. when 4876 is divided (50)

Connect

(16)

3 4

24. What factors do 16 and 24 have in common? (19) 3 9 10 1 1 6  5 (6  10)  (7  100 ) * 25. 10 10 3 and 2 3.89 by rounding Estimate Find the product of 11.8 the factors to 5 2  2 3 3  3 (51) 2  2  3  23  5 3 3 2 3 9 10 1 1 3 5number 45 multiplying. before Explain  3 2  24  3  5 the 2 2whole 3 6  4 how you arrived at 10) (7 (6 nearest 100 ) 10 10 3 2 your answer.

1 1  10)   ) (7  100 10)100 )5 (6 (7

3 9 3 10 9 1 10 1 Analyze Find  6 5* 26.  the average of the decimal numbers that correspond to (50) 6  10 1010 3 2 10 3 2 points x and y on this number line. x

y

0 3 4

27. (5)

28. (46)

3 4

2  2  3 2325 3  3  5 2  2  3 25 2  3  5

22335 2235

1 2 3

2 3 4

22335 3 2 Mentally2calculate at 3 pounds of bananas 4  2  3  5the total price of ten $0.79 per pound. Explain how you performed the mental calculation. 3 4Justify

3 29. 23Rename 23 and 34 as fractions with 12 as the denominator. Then add the 4 (42) renamed fractions. Write the sum as a mixed number.

* 30. a. (Inv. 5)

Jason’s first nine test scores are shown below. Find the median and mode of the scores. Represent

85, 80, 90, 75, 85, 100, 90, 80, 90 b. Sketch a graph of Jason’s scores. The heights of the bars should indicate the scores. Title the graph and label the two axes.

Lesson 51

271

2 3 32 43



LESSON

52

Mentally Dividing Decimal Numbers by 10 and by 100 Building Power

Power Up facts

Power Up G

mental math

a. Number Sense: 4 × 250 150 b. Number Sense: 368 – 150 12

(1  0.2) (1  0.2)

5 4  2 1

1 2 5 m1 3 3

c. Number Sense: 250 + 99 d. Money: $15.00 + $7.50 e. Fractional Parts:

1 2

of 5

f. Number Sense: 20 × 40 g. Geometry: A rectangle has a width of 4 in. and a perimeter of 18 in. What is the length of the rectangle? 1.25 .125 3 2 h. Calculation: 5 × 10, 4,5 ÷ 6,100 × 8, + 3,4÷ 3 12.50 10 +  12.500

problem solving

The monetary systems in Australia and New Zealand have six coins: 5¢, 10¢, 20¢, 50¢, $1, and $2. The price of any item is rounded to the nearest 5 cents. At the end of Ellen’s vacation in New Zealand she had two $2 coins. She wants to bring back at least one of each of the six coins. How many ways can she exchange one of the $2 coins for the remaining five coins?

New Concept Thinking Skill Verify

How can we check if the quotients are correct?

Increasing Knowledge

1 2

When we divide a decimal number by 10 or by 100, the quotient has the same digits as the dividend. However, the position of the digits is shifted. Here we show 12.5 divided by 10 and by 100: 1.25 1.25 .125 .125 10  12.50 100  12.500 10  12.50 100  12.500 When we divide by 10, the digits shift one place to the right. When we divide by 100, the digits shift two places to the right. Although it is the digits that are shifting places, we produce the shift by moving the decimal point. When we divide by 10, the decimal point moves one place to the left. When we divide by 100, the decimal point moves two places to the left.

Example 1 Divide: 37.5 ∙ 10

272

1 2

Saxon Math Course 1

Solution Since we are dividing by 10, the answer will be less than 37.5. We mentally shift the decimal point one place to the left. 37.5 ÷ 10 = 3.75

Example 2 Example2

Divide: 3.75 ∙ 100

Solution Since we are dividing by 100, we mentally shift the decimal point two places to the left. This creates an empty place between the decimal point and the 3, which we fill with a zero. We also write a zero in the ones place. 3.75 ÷ 100 = 0.0375

Practice Set

Mentally calculate each quotient. Write each answer as a decimal number. a. 2.5 ÷ 10

b. 2.5 ÷ 100

c. 87.5 ÷ 10

d. 87.5 ÷ 100

e. 0.5 ÷ 10

f. 0.5 ÷ 100

g. 25 ÷ 10

h. 25 ÷ 100

i. A stack of 10 pennies is 1.5 cm high. How thick is one penny in centimeters? In millimeters?

Written Practice

Strengthening Concepts

1. What is the product of one half and two thirds?

(29)

2. A piano has 88 keys. Fifty-two of the keys are white. How many more white keys are there than black keys?

(13)

3. In the Puerto Rico Trench, the Atlantic Ocean reaches its greatest depth of twenty-eight thousand, two hundred thirty-two feet. Use digits to write that number of feet.

(12)

* 4.

(46, 52)

Mentally calculate each answer. Explain how you performed each mental calculation. Justify

a. 3.75 × 10

b. 3.75 ÷ 10

5. At Carver School there are 320 students and 16 teachers. What is the 0.2) 150 5 4 1 2 student-teacher ratio (1 at  Carver?  5 m1 12 2 1 3 3 (1  0.2) Simplify: (23)

6. 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2

(5)

8.

(29)

(1  0.2) 150 12 (1  0.2)

150 12

7. (4)(0.125) 5 4(1  0.2) 1 5 4 2  9. 5  m 1 2 (49) 1(1  0.2) 3 2 1 3 (39)

Lesson 52 2 5

3 4

1 2 5  m m1  3 3 273

(1  0.2) (1  0.2) 1 2 5 m1 3 3

150 12 (1  0.2) 5 4 10.  (1  0.2) (29) 2 1

150 12

Find each unknown number: (1  0.2) (1  0.2) 5 45 4 1 1 2 2   11. 5  5m   1m  1 3 3 3 (1  0.2) (1  0.2) 2 12 1(43) 3

5 4  2 1

1 2 5 m1 3 3 1 2 m5 1 3 3

1 m5  3

1 12 2  5m   51  1 m12. 3 33 3 (43)

13. $10 − w = $0.10 (43)

* 14.

(41, 51)

At a 6% sales-tax rate, what is the tax on an $8.59 purchase? 3 to the nearest cent. 2 Round the answer Estimate 5

3 4

2 5

3 4

* 15.

(47, 51)

* 16.

3 4

(44)

4

Estimate The diameter of a tire on the car was 24 inches. Find the circumference of the tire to the nearest inch. (Use 3.14 for π.)

Arrange these numbers in order from least to greatest:

Analyze

1.02, 1.2, 0.21, 0.201(1  0.2) 150 12 (1  0.2) 17. What is the missing number in this sequence?

5 4  2 1

1 2 5 m1 3 3

(10)

1, 2, 4, 7, 11, , 22, (1  0.2). . . 150 5 4 1 2 (1  0.2) 150 5 4  1 2 1  12 5 m   0.2) 5(1  m1 m3 5  1 3 12 2 1 12 18. The perimeter 2 room 1 3 How many 3 floor tiles 1 foot 3 3 (1  0.2)of a square is 80 feet. (38) square would be needed to cover the area of the room? 19. (22)

Model One foot is 12 inches. What fraction of a foot is 3 inches? Draw a diagram to illustrate the problem.

3 1  0.2) 4 2 20.150Model How (1 many cents is 255 of a diagram  a dollar? Draw 45  m  1 to illustrate (22) 12the problem. (1  0.2) 2 1 3 3 2 5

* 21. (50)

1 2 m5 1 3 3

3

2 The diameter 5 of a penny is 4 inch. How many pennies are needed to form a row 12 inches long? Explain how you found your answer.

Connect 3 4

22. What is the least common multiple of 2, 4, and 6? (30)

23. a.

(29, 38)

2

4 2  4 2

4 22 24  2  3

4

2

22233 b. 24  22 232

2  2  2  31  3 2  3  22

24.5

4 About how many meters above the floor is the top of the chalkboard?

25.

To what decimal number is the arrow pointing on the number line below?

(7)

(50)

Estimate

0

(8)

274

1

24  22

2

2 26. 12Rename 12 and 23 as fractions with denominators of 6. Then add the 3 (42) renamed fractions. Write the sum as a mixed number.

27.

1 2

Connect

4 2  4 2

2  2  2 23 23 2  3  3 4  22   2 3 2 232

2 3

Draw a square with a perimeter of 4 inches. Then shade 50% of the square. Model

Saxon Math Course 1

2

Use the data in the table to answer the problems 28–30. Gases in Earth’s Atmosphere

4 2  4 2

4 2  4 2

Gas

Percent Composition

Nitrogen

78.08%

Oxygen

20.95%

Argon

0.93%

0.04% 1 233 2  2Other 2 2 3 232 28. a. Analyze Nitrogen makes up what percent of Earth’s atmosphere? (33, 35) Round to the nearest whole-number percent. 24  22

24  22

b. 2 Write fraction2 and as a decimal  2 the 2  answer 3  3 to a as an unreduced 1 2 3   2 3 2 number.

29. a. About what percent of Earth’s atmosphere consists of oxygen? Round to the nearest ten percent.

(16, 35)

b. Write the answer to a as a reduced fraction. 30.

(40, Inv. 5)

Sketch a graph to display the data rounding to the nearest whole percent. Label the graph and explain why you chose that type of graph. Model

Lesson 52

275

LESSON

53 Power Up facts mental math

Decimals Chart Simplifying Fractions Building Power Power Up D a. Calculation: 8 × 225 b. Calculation: 256 + 34 c. Number Sense: 250 − 99 d. Money: $25.00 − $12.50 1

e. Number Sense: Double 2 2. 1

22 f. Number Sense:

800 20

800 20

g. Geometry: A circle has a radius of 5 ft. What is the circumference of the circle? h. Calculation: 10 × 10, − 20, + 1, ÷ 9, × 5, − 1, ÷ 4

problem solving

You can roll six different numbers with one toss of one number cube. You can roll eleven different numbers with one toss of two number cubes. How many different numbers can you roll with one toss of three number cubes?

New Concepts decimals chart

Increasing Knowledge

For many lessons we have been developing our decimal arithmetic skills. We find that arithmetic with decimal numbers is similar to arithmetic with whole numbers. However, in decimal arithmetic, we need to keep track of the decimal point. The chart on the facing page summarizes the rules for decimal arithmetic by providing memory cues to help you keep track of the decimal point. Across the top of the chart are the four operation signs (+, −, ×, ÷). Below each sign is the rule or memory cue to follow when performing that operation. (There are two kinds of division problems, each with a different cue.) The bottom of the chart contains two reminders that apply to all of the operations.

276

Saxon Math Course 1

1 1 2

5 12

Decimal Arithmetic Reminders Operation

Memory cue

+ or −

×

÷ by whole (W )

÷ by decimal (D )

line up

×; then count

up

over, over, up

+ –

×

D ⤻ ⤻

W

You may need to … • Place a decimal point to the right of a whole number. • Fill empty places with zeros.

simplifying fractions

We simplify fractions in two ways. We reduce fractions to lowest terms, and we convert improper fractions to mixed numbers. Sometimes a fraction can be reduced and converted to a mixed number.

Example Simplify:

4 5  6 6

4 6

4 5  6 6

4 56 6

9 96 3  6 2

5 96 6

Solution Math Language 4 5 A fraction4 is  6 6 in lowest6 terms if the only common factor of the numerator 4 5 3 1and denominator  1  1 is 1. 6 6 6 2 4 6

4 5 10 5 6 6 12 12

5 6

9 6  10 10

43 61 169   124 6 10 10 6 5  6 9 Practice 10 Set 5  6 12 12 8 7  12 12

10 5  12 12

9 6  10 10

3 1 1 1 6 2

9 39 9 43 3 4 54 4 5 9 By the fractions 46 and 56, we get 56the 96 4   1  adding  6 6 2 6 2 6 6 6 66 9 6 56 6 9 3 9 3 3 1 3 1 improper fraction fraction.6  1 1 3 16. We can1simplify  16 this 6 5 2 6 6 2 2 1 1 6 2 5  We may reduce 6 first 2 and then convert the fraction  6 6 to a mixed number, or we may convert first and 9 49 4 then reduce. We show both methods6below. 6 6 6 3 1 1 43 1 4 1 1  1 5 59 3 9 3 3 1 95 6 6 2 First 2 6 Reduce 1 10 5 1. Reduce: 9 6 6 8 7 1 1 6 9 66 6 8  76 2 1 6 6 2 2 510  56 4 12 12 9 10 9 4 10 5 4 12 12  12 12 5 10 10 12 12 4 6 5 9 94 3 9 4 9 5 34 5 4 9 43 9 94 5 3  51 2.1Convert:     6 66 62  6162 6 66 6 6 6 9 66 2 6 6 6 6 26 6 6 6 6 66 6 59 68 10 5 10 6 49 7 7 8 1    14 9    3 9 3 3 1 6 9 5 12 10 10 12 12 12 68 12 12 10 10 12 12 1. Convert:  1 4 1 7Convert61First 6 2 6 6 2  2 5 4 4 4  12 12 6 3561 9 6 3 9 3 9 4 5 4 35 44 1 35 4 1 5 9 1 3 9 4 5 2. Reduce:   1 96 1   61  6 6 1 2 6 6 6 5 2 6 516 6 2 6 6 6 6 166 661219 66 2 2   6 6 6 4 9 9 6 a. Discuss how in the decimals chart apply to each of 9 Connect 6 8 7 the rules 1  5  4 6 4 6 4 4 10 these 12 12 10 problems: 6 6 6 106 8 5 6 7 1109 516 13 1 9 10 6 9 0.12 9 8 6 8 3 7 5 1 13  1113 5 −14.2  1510 10.45 × 0.2 ÷ ÷50.4 10 10 9     5 5 12 12 6 612 21210 1010 1012 1212 12412 124 6 2 12 2 12 4 1012 101 6 6 6 6 b. Draw the decimals chart on your paper. 9 9 9 8 7 1 6 6 6  4 12 12 Simplify: 69 78 59 68 7 10 510  5 10 9 6 1  8  7 e. 14  14 c.  d. 4 12 10 1210 1012 10 12 1012 1212 12 12 1212 1210

Lesson 53

277

4 5  6 6

Written Practice

4 6

5 6

Strengthening Concepts

3 1 1 2 2

9 3 1 6 6

9 3  6 2

9 6

* 1. 3 1 1 1 6 2

Explain The decimals 4 chart in this lesson shows that we line up the decimal points when we 6 add or subtract decimal numbers. Why do we 5 do that?  6 2. A turkey must cook for 94 hours 45 minutes. At what time must it be put (32) into the oven in order to6 be done by 3:00 p.m.?

(53)

10 5 * 3. 9 6 8 7 Billywon the contest   by eating 14 of a berry pie in 7 seconds. At this rate, 12 12 (50) 10 10 12 12 how long would it take Billy to eat a whole berry pie? * 4. In four games the basketball team scored 47, 52, 63, and 66 points. What is the mean of these scores? What is the range of these scores? 36 0.15 3 m 2 3 1  1.25  6 3 10 number: 0.12 4 4 4 3 5 Find each unknown 36 0.15 5. 0.375x = 37.5 * 6. m  1.25 0.12 4 (43) (52) 10

(Inv. 5)

0.15 m the fraction as36 7. Write 1% as a fraction. Then write a decimal  1.25 10 0.12 4 number. 36 m 1 35 m 1 36 0.15 3 0.15 2 3 7 2  311 3  3 * 9. 36  6 0.15 35  7 8. 3.6 +41.25 +40.39 0.12 64 1.25 10 4 4 4 5 0.12 4 3 8 8 3 6542 3 4 10 4 (38) (49) 0.12 (33, 35)

m  1.25 10 m  1.25 10

0.15 36 1.25 4 0.12

36 m 0.15 36  10. 1.25 0.12 10 (45) 4 0.12

3 1 0.15 6 3 4 4

3 5 27 3 3 2 3 1 1 0.15 12.  6  3 5  7  6 3 4 (28) 3 5 4 4 8 38 5 4

2 3 1  3 5 2 7 3  6 * 11. 3 5 7 4 8 38 5 3 (48) 5 4 1 2 13. (26)

53 7 5 8 7 8 8

1 2

5

1 6 3 4 3 1 6 3 4 4

3 5 85 8

53 7 5 8 7 8 8

1 2 3 8

(34)

* 15. The items Kameko ordered for lunch totaled $5.20. The sales-tax rate was 8%.

(41, 51)

Estimate

Find the sales tax to the nearest cent.

b. Find the total price for the lunch including sales tax.

m  1.25 0

* 16. Which number is closest to 1? (50) 0.15 3 5 2 3 1 7 3 A 1.2 C 0.15  7 D 21  6  3 B 0.9 36 4 0.154 5 2 83 13 53 7 8 4 8 1  6 3 5 7 2 0.12 4 4 4 3 5 8 8 17. Estimate The entire class held hands and formed a big circle. If the (47) circle was 40 feet across the center, then it was how many feet around? 36 Round the answer 0.15 to the nearest 3 5 m 2 3 for π.) 1 7  1.25  6  3foot. (Use 3.14 5 7 10 0.12 4 4 4 3 5 8 8 18. What is the perimeter of this square? 3

36 m0.12  1.25 10

(8)

8

19. A yard is 36 inches. What fraction of a yard is 3 inches? (29)

20. a. List the factors of 11. (19)

b. What is the name for a whole number that has exactly two factors?

278

Saxon Math Course 1

in.

3 8

1 2

7 2 7 3 3 85 1 2

14. Which digit in 3456 has the same place value as the 2 in 28.7?

a.

5 8

3 8

2 3 3 2

3 2

2 3  1 3 2 3 2 3 2 3 2 3 2 3 2  1 3   1 the 1 squared is2how 2much greater 3 3 2 21. Four 3 2than 3 2 square root of 4? 3 2

2 3

2 3  1 3 2

(13, 38)

22. What is the smallest number that is a multiple of both 6 and 9? (30)

2 23. The product of 3 (30)

2 3

and

3 2

2 3

3 2

is 1.

2 3 2 3  1  1 3 2 3 2

3 2

2 3  1 3 2 3

2 3  1 3 2 Use these numbers to form another multiplication fact and two division facts. 232525 5 ? 3 1 1  4 6 4    24 2  32 2  53  2  5 3 2 25 325 5 2? 3 2  3122 365 22  55 21  5 5 3?5 ? 1 1 1 1      24. 25.   1 1 2 3 2 6 4 6 24 3 26 3 (5) 2 241 5 22 55 2  563 6 (42) 424 6 24 4 2525255 ? 1  4 6 4 5 6 24 26. Represent Copy this rectangle on your 2 3

2

3 4

(29)

paper, and shade two thirds of it. 2  3  2 253225 5  2  5 5 ? ? 5 1 1  1  4 6 2  5  2 25 5  2  5 6 24 6 244 232525 5 ? 1 1  2  5 4 6  3  2 6 5 5 to?Thompson 1 2  5  Thirty 2  5 percent2 of 24 27. Model the 350 students ride the bus  4 (22, 33) 2525 6 24 School. Find the number of students who ride the bus. Draw a diagram to illustrate the problem.

332  2 253225 52  23 51 2  5  2 25 5  23  52

5 ?  6 24

5 28.?1Rename 1 and 1 as fractions 3 3 1 with denominators of 12. Then add the  6 4 4 6 4 6 (42)244 renamed fractions. * 29. (50)

Connect The number that corresponds to point A is how much less than the number that corresponds to point B?

A

B 8

7

5

5 ?  6 24

* 30. 1 4

Early Finishers

Real-World Application

(50)

9

Connect The classroom encyclopedia set fills a shelf 1 long. Each book is 34 inch thick. How many books are 6

that is 24 inches in the classroom set? ( To answer this question, write and solve a fraction division problem using the method shown in Lesson 50.)

At an online auction site, a model of a 1911 touring car was listed for sale. The twelve highest bids are shown below. $10

$15

$11

$10

$12

$10

$13

$13

$11

$13

$11

$10

Which display—a stem–and–leaf plot or a line plot—is the most appropriate way to display this data? Draw the display and justify your choice.

Lesson 53

279

1 6 3 4

LESSON

54 Power Up facts mental math

Reducing by Grouping Factors Equal to 1 Dividing Fractions Building Power Power Up G a. Number Sense: 6 × 250

1

1 8

8 b. Number Sense: 736 − 400

2235 223

1 2

c. Number Sense: 375 + 99 d. Money: $8.75 + $5.00 1 8

1 e. Fractional Parts: 8

1 2

2235 223

of 9

f. Number Sense: 30 × 30 g. Geometry: Can you think of a time when a figure could have the same value for both its perimeter and its area? h. Calculation: 8 × 8, − 1, ÷ 9, × 7, + 1, ÷ 5, × 10

problem solving

A The PE class ran counterclockwise around the school block, starting and finishing at point A. Instead of running all 100 m the way around the block, Nimah took what she called her “shortcut,” shown by the dotted line. How many meters did Nimah save with her “shortcut?”

1 8

70 m

50 m 30 m

200 m

Increasing Knowledge

New Concepts reducing by grouping factors equal to 1

30 m

The factors in the problem below are arranged in order from least to greatest. Notice that some factors appear in both the dividend and the divisor. 1 8

2235 223

1 2

Since 2 ÷ 2 equals 1 and 3 ÷ 3 equals 1, we will mark the combinations of factors equal to 1 in this problem. 2 2

· ·

2 2

· ·

3 3

·

5

Looking at the factors this way, the problem becomes 1 ∙ 1 ∙ 1 ∙ 5, which is 5. Verify

280

Saxon Math Course 1

Which property helps us know that 1 ∙ 5 = 5?

Example 1 2225 2235

Reduce this fraction:

2 3

2

1113

3 4

1 2

1 2

1

Solution We will mark combinations of factors equal to 1. 1 2

2 2

· ·

3 4·

2 2

·

2 3

· ·

3

1

a4  2 b

5 5

2  2 22 25 2  5 2 22 2 By grouping factors equal to 1, the problem becomes 1  1 11 13, which 1  33 is 3. 2  2 23 25 3  5

3 4

When we divide 10 by 5, we are answering the question “How many 5s are dividing 5 2  2  2  52  2  2 3 1 3 3 2 2 1 1 3 2 2 1 1 When we divide 34 by 12 we are answering the same type of question. 1 fractions 1  1  31  1 3 in 1 10?” 2 4 2 4 4 2 2 2 2  2  3  52  2  2 35 223 2  232  54 3 3 3 1 1 11 2 2 1    case the question is “How many s are in ?” While it is easy to see 1  1  In 1 this 1 1 1 3 4 2 2 4 2 3 4 2 2 3 3    3 3 1 1 1 2235 2 2 3 5 3 3 1 2225 2225 3 1 2 2 a 4 1 2ab4  12 b1 2  23 it is23not as easy how many 51s are s4 are in 34. We 1  in 1 10, 1 2 1to 4see 12 3how 3many 4 2 2 2 2235 2235 remember from Lesson 50 that when the divisor is a fraction, we take two 25 2 find 2 22 25answer. 2  2 steps 2  5 to 3 divisors 2 1 22 3 many 1 1 3 of the are in1 1.1 This  123find 11first  1  2 1  21We 1  33how 1  1the 4 2 4 2 2 2 2 41 3 3 32  2 3 3  15 2  23 231125 33  5 13 1 1 1 reciprocal use a 4isthe 1 2Then we 1 a 4 22 bof the divisor. 2 2 the reciprocal to answer the 4 4 2 2b 2 3 3 3 3 1 1 1 1 1 1 1 1 original division a 4  2 bproblem a 4  21b 2 12 2by multiplying. 2 4 4 2 2 3 3 3 3 1 1 1 1 1 1 1 1 a4  2 b 4 a 4  21b 2 12 2 2 4 2 2 Example 2

225 235

25 35

3

3

1

13

3

1 3 How many s are in 4 ? aQ34 4 12 Rb 2 4 2

1 2

1

1 2

1 a4  2 2ab4  2 b

Solution

11

1

1

12

12

122

Before we show the two-step process, we will solve the problem with our fraction manipulatives. The question can be stated this way: How many 111

2 3

2 3

1 2

s are needed to make

1 4

1 4 1 4

?

We see that the answer is more than one but less than two. If we take one 3 3 3 1 1 1 1 and cut first 412 and 4 another 2 into 2 two equal parts 2 ( ), then we4can fit the 2  5 of the smaller2 parts2 ( ) together to 2  2  2 one 3 make three fourths. 1 1 1113 4 2 2 3 2235

1 2

3 4

1 2 3 4

3

1 2

22 2  22  5 1113 2  2  33  5 3 4

 12 2225 2235

1

1

1

a 4  2 b We see 2 that we need 1 2 of the 1

1 2

pieces to make

3 3 1 Now we4 will use arithmetic a 4 to 2 bshow 3 1 1 2 2  341 “How many s are in 1asks, 2 2 3 4?” 3 3 4

1 2

1

12 2

2

3

1 3 that 2 4

3 1  4 2 1

1 4

1 4 1 4

. 1

1 1 original problem  12 equals 1 2.1The 2 2 1 2

3 4

11 22

1 2

3 4

3 4

1 1

3

1 2 2 1

 3 first step is to find4the number of 2 s 1in 1. 1  1  1 The 2 1 4 3 2 3 2 4 3 1 1 3 1 1 3 1 1 12 3 1 1 1 1 1 1 3 3 3 3 1 1 1 1 1 1 1 3 1  11 11 2 1  2 2 2 12 2 a3424  2 b4 4 b2 2 2 1 22  1221  22 2 12  2 244 2 4 a 42 21 42 2 24 2 3 6 1 3 3 3 6 1 12225 5 3 3 3 2 2 12 2 1 1 1 2 3 11 2 4 1  1 of 4 4 4  The number s in 1 is 2, which is the reciprocal of . So the number 1  1 2 1  3 1 4 4 2 4 2 3 2 4 4 2 2 4 2 3 3 2 5 2  2  5 2 222235 5 3 11 3 2 2225 2 1 23 1 1 2 1 13 3 1  1  23  1  1  3 1  21  1  2of 1 s12in  13 should be of 2. We find43 3412of 122 1 1 by 2 multiplying. 4 22 4 1 2 24 4 1 2 3 2 1 1 2235 2  3  5 34 2  323  3  53 2 3 a43  1 b3 2 34 1 2 3 12 2 4 4 2 2 1 21 6 2 3 63 1 6 26 3 3 3 6 1 3 6 3 3 1 3 6   1  3 61 3 3 6 equals 2 1 , which 2 2 4 4 4 4 4 2 4 4 4 2  4 4 1 4 4 4 4 2 2 4 4 4 4 4 4 4 2 3 1 1 3 3 1 1 1 1 1 2 3 3 3 3 1 1 1 12 1 1 1 2 2 4 2 4 2 2 a 4  2 b2 12 a4  2 b 12 2 2 4 4 31 3 3 3 3 1 1 1 13 1 1 1 13 1 1 1 1 1 a  b 1 a  b a  b a  1 b 1 1 2 24 2 24 2 2 2 2 42 4 4 4 4 2 2 2 2 2 1 1 Lesson 54 3 281 2 2 4 3 1 1 3 1 1 3 1 1 1 3 1 1 1 34 1  1 1   21   2 32 12 1  2 11 2 2 4 2  4 2 24 2 1 22 2 2 4 4 2 2

1 2 1 2 3 4

6 3 4 4 33 44

12

2

4

4



2

1

2

2

2

2

31 6 3 3 3 6 3 3 2 16 3 6 6 61 4 3 3 3 4 3 2 3 3 4 4 4 4 4 4 4 44 2  4 2 4 1 24424 3 3 3 33 33 1 4 34 4 1 1 1 1 1 23   to and then converting to . We will review 1 1 12 2 2 2 2 4 2 2 2 2 4 2 4 4 4 4 4 4 3 6 1 6 3 6 1 2 1 the4steps we took 4to solve the problem. 2 4 4 4 4 4 2 3 1 1 3 3 1 1 1 1   12 1Original 1 2 problem: 4 2 2 4 2 2 4 2 2 3 1 3 3 1 1 1 3 1 22 2 3 22  53 31 1 1 1 3 3 3 1 3 3 3 1 11 2 2 1 1 1 2 13 How 2 1 3 1 4 3 3 3 1 2 1  12 1 1 1 in 1 ? 1 3 many2 2 s are 1113 23 1  11 4 4 3 21 13 4 1 1 2 42 2 4 21 31 4 2 2 21  4  3 4 4 4 2 4 42 22   3  52 2 21 1 2 1 2 4 21 2 3 6 232 1 212 4 12 243 23 1 2 2 3 1 3 33 1 1 1 1 12     2 1 2 1 2 2 2 2 4 2 2 4 2 4 4 4 1 22 4 2 2 4 2 3 3 12 1 1 2  21 45 2 3 1 4 2 2 2 1 3 3 1 1  1 1   24 Step of 2 s in 1. 1  1 2 1  3 4 1: Find4the number 2 2 2 4 3 4 2 2 2  32  5 3 6 1 3 6 3 3 6 1 1   2 1 2 4 4 25 3 3 4 3 4 2 2 2 1 4 1 4 1 4 2 Step of 2 s in 13 64 3 1 2 1  1 11  3 3 to find 62 1 2: Use4the number 1 4 3 1 3 1    2 1  533  5 3 3 21  2 2 23 5 1 1 1 1 1 1 3 1 3 3 21 2 1 4 3 21 3 3 1 1 64 3 31 3 1 13 1 3 1 1number b 4  1 22 21 12 2334the a  4221bof 1 2  14 2 41262122161  . Then 1  1 2 1  32 a 4  4 simplify 22 2 4s in 4 12 4 2 4 3 22 1 4 24  2 42 44 22 2 4 2 234  5 431 26  2  13  5 4 4 2 4 4 2 3 1 1 23 31 3 3 1 1 1 23 3 1 2 1 the answer. a  b a  b a  4 32 4 2 2 34 4 42 6 3 2 2 2 3 4b 4 1 1 42 3 3 1 1 4  2  1 1 a  b 1 2 4 4 42 1 2 2 4 4 2 3 1 1 3 3 1 1  1 1 2 2 2 2 4 2 4 2 2 3 3 13 1 1 Example a 4  213b 12 3 3 1 1 1 2 2 4 3 31 1 1 2 a 2 1 42b 3 1 1 3 33 2 3 2 3 3 1 1 1 1 4 3 11 3 3 4 3 1 3 1 2 2 1 1 1 3 33 1 a  b 3 4 4 4 a  b 1 a  b 1 2 2 2 ab12  b 3 14 3 3 2 4Rb 1 a32  2 2 4 s 4are 2in 2? aQ22 2 2 3 4 4 4 2 How many 2 4 4 4 4 4 2 3 2 4 2 31 1 23 (2  4 ) (2  4 ) (2  4 ) 3 4 4 2 2 3 3 1 1 1 2 a 2  4 b Solution 3 4 2 6 3 2 1 3 1 1 2 1 2 2Using our fraction 4 manipulatives, the question 4 4 2can be stated this way: 3 1 3 1 1 3 1 4) 3 1 ( 2  4 ()1  3 ) ( 2  4 ( 2  4()2  4What ) fraction of 14 14 is needed to make 12 ? 4 2 1 2

3 4 2 3

1 2

1 1 2 2

3 3 4 4

1 3 2 1 3 1 1 4 2 2 6 6 61 4 26 6 We simplified 1 2  4 by reducing 1 4 4 3 4 2 4 3 42 1 32

1 12

3 64

3 4

3 43

1 4 1 4

The answer is less than 1. We need to cut off part of to make 12 . If we cut off one of the three parts of three fourths ( ), we see that two 2  2 2 2 2 512  5 3 3 3 1 1 1 2 1 23 1 1 2 12 3 4 b 23 . So 3 of 4 is needed 1 3a12 equal of the 1 three 1 1 1parts 4 2. 2 2 2 4to make 2 2  2 2 3 2 523  5 2225 3 2 2    Now we will show the arithmetic. The original problem asks, “How many 1 1 1 3 41s 33    2 2 3 5 3 3 3 1 1 1 4 2 are in 2?” 4 2 4 2 4 1 4

1 4 1 4

1 4

3 4

3 1 4 2 1 21

2

1 2



3 4

3 4 1 3 3 3 3 1  1  3 1 4 2 4 4 2 4 4 3 ( 2 14 ) 1 3 3 3 1 1 1 1 13 1 1 a 42 1  2a2 b4 3 2 1 2 2 4 2 4 2 2  2  22  5 2 2 25b 2 3 3 3 1 11  3 find3the number of14 s in 1. The 321  23 is the number reciprocal 1  First 1  1 we 11 1 of 4. 14 1 23 22 3 235 235 2 2 a  b 1442 1 1 33 4 4 4 3 1 33 4 1 3 3 3 1 2 3 3 44 2 3 2 3   1 1 1   3 3 1 4 1 1 3 3 3 3 4 1 2 3 2 3 41 4 3 4 4 4 2 a 4 2 2 b 4 2 44 4 13 44 3 34 4 2 2 4 4 3 4 2 24 4 3 4 1 4 4 222 25131223235 2 4  22 3 35232312 224 43 1 1312 313 44 1 3 31343 111 4 1 3 331  1  2232 22  5 3 12 3 2 31 2  5 13 2 1 32 33 233  31 3                       The number of s in 1 is . So the number of s in is of . We find of by 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 2 42 3  5 4 22 42 3 2 43 44 344 2  2243 32  5 3  33 4 453434 333 34 2 24 424 33 2 4 424 222 3 2 4 44 24 2 6 233 2524 34245 2 33 41 1 3 3 multiplying. 3 3 3 3 3 31 1 3 1 1 1 1 1 4 1  a  b  12 b 3 4 32 3 3 1 2 33 4 2 1 3a44  4 2 4 4 4 2 321 1 4 3 2 4 4 411 2 134 4 4 42 412 3 3 4 1 3 3 1 4 1 3 2 4 1 13 1   4 1 4 44 4 424equals 1 ,2which 2 4 32 4 4 1 24 4 3 1 2 1 24 4 32 4 4 3 2 4 2 3 3 2 4   2 3 6 2 3 6 3 3 2 2 4 2 3 3 2 3 6 3 3 4 1 3will 1 4 4 3 to solve 12121problem. 4 review 4 the 4 4 2 3the 2 43114we 1 11 1     3 331steps 3 4 114 31  1 13 114 134 1 3we 1 3 1 Again, 11 4took 4 3 1 12a4343  1 b1 11 1 43 1 1 121  2 4 3 2 3 6 a b  b a  1 1 a  b a 1  b 2 22 2 2 4 2 4 2 3 3 3 3 232 23 2 6 2 3 2 4 2 2 2 4 2 22 243 262 2 42 4 2324 6 4 2 3 4 4 42 Original problem: 4 4 2 3 1 4 1 1 4 1 1 3 3 3 3 1 113  6 1 22  22  25  2  5 2 3 4 3 4 341 2 22 3 21 1 2 31s are 1in 1? 2 41 1 2 331 3 14 4 2 3 2 4 1 4 3     3 454 4   3 5 2 2 2 4 How many 1  1 1 1 3 4 4 2 1 3 4 4 4 2 1 3 31  1 1 3 1 3 4 2 2 4 2 4 3 4 2 2 4 2 4 3 3      23 2 3 25 4 3 5 3 4 3 2 3 3 1   12 3 24 1 34 24 2 2 1 22 4 2 4 3 13 42 33 3 62 33 6 3 3 4 2  32  5 4 2  23 3 2 5  35 6 12 3 34 14 4 3 2 3 6 3  22  32 5 4 2  2  33 3 4 5 1  1 2  22  2  5 3 3 1 1 4 4 3 Find 121 31121: 3 3 of 44s in 1.1 34 424 21 384 4 3 42 42 12 3 4 4 2( 12  38 ) 4 4 1 Step 4 3the 2 number 33 3 1 13 1    2 2 3 5                 1 1 1 2 24 44 4 4 3 2 42 32  26  5 3 2 2  24  3 3 5  5 2 43 36 32 3 6 3 3 4 1 3 1 4 41 21 2  23 2  5 3 3 3 3 2 1 1 4 4 1 1     23 Use the number of s 13in121131  12 1 3Step 14 1112: 4 4 3 3 31 41 43 3 231 1 4 4 433 2 1 3 3 3 2 3 1 1 1 3 3 23 3 21 1 2 4 4 3 2 6    22 523 25 53 a 2  a3b2  3  2232  252  3224 35 3 3 11 21b 2 1 3 11 3 1 31 3 13 1         1  1  1 1 ) ( 2 the1 number 24 2  4284s) in1 (242.2 3 82)4 2  2  3  3  5 41  1 8 1  431  31283241 to3422 find (22 of 34 88 2 2 4 42 2 34 644 42 2 3 2 81 22 24 3  52  3  5  252  252  33  55  2 3 ( ) 8 8 12 4 4 2 3 1 2  2  3  5  51 48 12 4 2 1 Then simplify the answer.   3 1 2 2 4 2 3 3 3 3 21 1 1 1 2 3 6 3  ) (          3 3 22 22 33 53 152 2 3 3 51 3 83 3 3 3 1 1 1 1 2 a 4 1 23b1 2 4 2  38 ) 1 2 (2  (2  8 ) 8 8 22 22 35 5 252  2  3  5  5 8 8) 2 82 8 2( 2 2 8  3  33  5Reduce: 2  2  3  Practice 51 2  2 Set 31 3 3 1 1 3 4 1 3 3 1 1 1 3( 1  3 ) 1  1  31 b 24 2 232 23 4235 5 1 82 83   2 2 432  33 8 5 2 1 225 2 2  2  3  54  5 a. 2  2  3a 4 52 2 4 3 2 b.  ( ) 3 3 3 3 1 1 1 1 1 1 1 2 2 82 8 a 42 22 b 5 a 4  22b 2  32 252  55 1 2 2  2123  85  5 4 4 2 3 4 1 23 1 4 4 2 3 3 1     1  4 4 2 2 3 4 1 4 3 2 3 6 3 ) ( 8 2282 Saxon Math Course 1

3 1 (8  2 )

3

1

2235   3

1

22335    

3 8

1 2

(



2

4

( 8 4 2 )

4

2

2 2 3 5 1 2 2 3 3 5   225 2 4 2  33  5 25 3 6 3

2  22  3 23 3 5 5 2  22  22  325  3  5 3 1 1 13 1 c. How 34many 1  1  1  3  3 8 8 s are1in 2? Q( 2  2 8 R) 2  352  5 5 5 2 2 3 5 5 1 2  22  232 2 25 22 2  23  5 2  2  33  3 2 5 d. How many 12 s are in 38? Q( 38 12 12 R) 1113 4 3 225 22355 35

1 4

1 2

3 1 (8  2 ) 3 4

Written Practice

3 4 3 (8



1 3 2 )a 4

1

 2b

3 a4



4 3

1 33 ( 2 88 )

1 1 21 2 12

3 1 41 2 1 21 2 (2

3 4

2

11 22



33 8 )4

3 8

33 4 8 44 3 13 2

1 2

3 8

Strengthening Concepts

1 291 1 216 3 4 1 1from3 Lesson 3 3 33 2 2 1 Summarize Draw34the decimals chart 53.   1 4  4 2 4 (53) 2 4 2 4 4 3 1 1 29 216 12 2 2. If 0.4 is the dividend and 4 is the divisor, what is the quotient?

1 2*b1.

1 2

(45)

3.

Estimate In 1900 the U.S. population was 76,212,168. In 1950 the population 2  2  3 was 5 151,325,798. 2  the 2  increase 3  5 3 2 2inpopulation 2  2  3  3 Estimate 5 2  3  3 1 5 1 (2 8 2 1000 4 4 2 1000 4 4 2             3 1 1 2 2 5 2 2 3 5 5 2 2 5 2 2 3 5 5 between 1900 and nearest million. 6 a b  3  59 4 19502to9 the m m 3 61 29 6 a b 2 61 2 165116 5 3 5 324 4 1000 4 5 2 4 3 4 1 24   6 b 59 4 m 3 a 249  61 4 1000 1 3 24 3 1 1 5 5 3  4. Formulate Marjani was 59 4 inches tall when turned 11 and m 3  61 4 she59 61 4 4 5 24 (48) inches tall when she turned 12. How many inches did Marjani grow 216 3 3 1 1 during the problem. ) year? Write an equation and ( 8  2the ( 8 solve 2) 2 216 2 4 1000 4 2 4 3 1 1000 4 2   1 5. 1000 59 6. 6 b m 3 a 61 −61(100 − 1) 4 4 m 5 2 24 3 5  6 5 a 3 b 5 3 9  116 (5) (25) 4 24 7. What number is halfway between 37 and 143? (16)

3

59 4 29

216 29

3

59 4

(18)

2

1000 24

4 2 4 number: m  3  6 a b Find each unknown 29  116 5 5 33 1000 1 59 4 8. $3 − 61 n =4 24¢ 24 (3)

2

4 2 4 9. m  3  6 a b 5 5 3 (43)

* 10. 4.2 ÷ 102 (45)

1000 24

29  116

* 11. (1.2 ÷ 0.12)(1.2) (53)

2

4 2 4 9 6 a 2 b m  3  12. 5 (29, 38)5 3

29  116

13. 29 + 216

21

(38)

14. Which digit is in the hundred-thousands place in 123,456,789? (12)

15. The television cost $289.90. The sales-tax rate was 8%. How much was the sales tax on the television?

(41, 45)

* 16. (9)

Connect

Use rulers to compare: two centimeters

* 17. (10)

one inch

Isadora found the sixth term of this sequence by doubling six and then subtracting one. She found the seventh term by doubling seven and subtracting one. What is the twelfth term of this sequence? Predict

1, 3, 5, 7, 9, . . .

5 ?  24 of a 18. How many square feet of tile would be needed to cover 6the area (31) room 14 feet long and 12 feet wide? 5 5 ? ? 1 2   2 3 6 24 8 24 19. Nine of the 30 students played basketball on the school team. (23)

5 ?  8 24

2 a3 

a. What fraction of the students played on the team? b. What is the ratio of students who played basketball to students who did not play basketball?

20. a. 5  ? (42) 6 24

5 ? 5 ?   1 8 24 6 242 23532 2325

b. 5  ? 23 8 24

1 2 1 2 23  5 a33  22 b 2325

54 2  3  3  Lesson 5 22235

283

2 3

2 3



5 ?  6 24

?1 242

5 ?  6 24 1 2

5 ?  8 24

1 2

6



2

24 3

8

 a2  1 b 243 2

2

21. What is the least common multiple of 3, 4, and 6? 5 ? 2 2 1 1 1 (30) 5 5 5 12 ? ? ? 1 1  5014b  1 2 5 1 ? 2 3 2 21 2 1 23 a 3 2 221b1 2 a123  50 426 4 1 65 524?5? 6? 8 24 5 2 ? 5 5 ? ? 2 1 1 1 1 1 1 1 5 5 ? ? 1 2 2 1 24  How 8 24  b 50   *22. Represent  ab  2 b a 50 many 2s are in2 3? aQ 323 2Rb a 50 2614 4 5026 26a 50 4 4 3 3 342 3 3 4 24b3 42 6 8 24 6 8 24624 (54) 248 2424 82324 24 68224 2

2 1 1 1 2 1 1 23. aModel  12 b Eighty of50 the correct. Write 80% as a a 3  2percent b 4 26were 50 26answers 4 30 4 3 4 reduced fraction. Then find the number of answers that were correct. Draw a diagram to illustrate the problem. 23532 2  3  3 2 5 3  5  3  2 2 3  2  long 5 is 3  2  5 2.54 centimeters. 2Aribbon 24. Connect One inch 2  3equals 2  2  100 3 2 5inches (46) long? 5 5 centimeters ? how many ? 1 2 2 1 1   a3  2 b 50 4 2 3 6 24 8 24  2    2  3  5  3Reduce: 2335 2335 2 1 1 2 3 5 3 2 2 1 1 3 2 2        2 2  3  5  3 2 2 1 1 1 2 3 2 5 2 2 2 3 5 22 3  23 235 232 2 3 21 21 1 1 1 33 252250 5 23 5 334 2 25 5 333 352 5326 35 3 25 1 2 12 1 3 a23 3 2 b25* 25. * 26.3 3 3 3 4 1 1 1 3 3 2 3 2 2 2 2 3 2 22  35  22 2 53 222 235 32 255 2  22  23  252(54) 2  3  2  5 2  3  22 (54) 32 522 23 25 3  5 2  3  3 25 3  3  5 1 1 1 11 1 1 27. 2Rename 23 and 12 as fractions with add 1 2 denominators 1 21 2 2 2 6. Then 2 2 1of 2  2  2 2325 2  3  5 (42) 3 the renamed fractions, and convert the answer to a mixed number. 2 3

3 23) (22,

2 22 1

26 4 1

1 2

12 11 11 22

28.

Justify The diameter of a wheel is 15 inches. How far would your hand move in one full turn of the wheel if you do not let go of the wheel?  3  2the answer to the 2  3  3 inch.  5 (Use 3.14 for π.)2 Describe how 1 nearest 2  3  5Round 3 2  5 whether your answer 2  3 to2 tell 2  2is reasonable. 235 (47)

5 35

2 3

1

12

1 1 1 1 29. Draw a rectangle1that is 1 2 inches 2 2 2 long and 1 inch wide.

(8, 31)

a. What is the perimeter of the rectangle? b. What is the area of the rectangle?

5 5 5? 5? ? ? 2 2 12 2 1 30.   Insteadof dividing  Analyze 212 by 12 , Sandra formed an equivalent a 3  a23b  2 b 3 3 24 8 24 6 24 8 24 (43) 6 division problem with whole numbers by doubling the dividend and the divisor. What equivalent problem did she form, and what is the quotient?

Early Finishers

Choose A Strategy

284

Brooke is planning to plant tulips around a circular flowerbed. The diameter of the flowerbed is about 4 feet. The tulips should be planted about 6 inches apart. She has 12 tulip bulbs. Her friend Jenna said she should buy 50%  3 3  correct?  3  2 Explain why 2or why  not.  3You  5 may use tiles2 to help 2 Is3 Jenna 3 2 3 more. 2 5 5 2 3 5 2 3 3  the  problem.  2 3  2 5 35 2  3 2 2 2  2 2 2 3 5 25 you visualize

Saxon Math Course 1

1

1

50 4 50 4

1 2

1 2

LESSON

55

Common Denominators, Part 1 Building Power

Power Up facts

Power Up I

mental math

a. Calculation: 8 × 325 b. Number Sense: 329 + 50 c. Number Sense: 375 − 99 d. Money: $12.50 − $5.00 1

e. Number Sense: Double 3 2 .

1 32

1 2

600 20

f. Number Sense:

gallon

600 20

g. Measurement: How many yards are in 60 inches? h. Calculation: 8 × 5, + 2, 3÷ 6, × 7, + 7, ÷ 8, × 4, ÷ 7 3 5

problem solving

2

5

_4_ 5 fill in the missing digits: Copy this problem and 6 _ _ 6 _ __ __ __ 3_ 0 1

32

New Concept

_4_ 6 _ _ 6 _ __ __ __ 3_ 0 1 32

3 4

2 5

1 2

600 20

Increasing Knowledge

When the denominators of two or more fractions are equal, we say that the 3 fractions have common denominators. The fractions 53 and 52 have common 5 denominators. 600 600 20 20

1 2

1 gallon 2 gallon

1 5 1 5

1 5

1 5

1 5

3 2 5 5 The common denominator is 5. 2 5

2 5

3 3 The fractions and 12 do12not have common denominators because the 4 4 denominators 4 and 2 are not equal. 1 4 1 4

1 4

1 2

1 3 2 4 These fractions do not have common denominators.

Lesson 55

285

1 2

1 2

2 4

1 2

 54

Fractions that do not have common denominators can be renamed to form 2 fractions that do have42 common denominators. Since 4 equals 21, we can 3 3 3 2 3 1 2 rename as 24. The fractions and 24 have common denominators. 4 4 4 2 4 4 4

2 2

2 2

 24  54

1 2 is renamed as . 2 4 3 4

 24  5422

Fractions that have common denominators can be added by counting the 1 number of parts, that is, by the numerators. 1 adding  22  24  22  24 2 2 1

 34

 34 5 4

1 4 1 4

1 4

2 3 4 4 The common denominator is 4.

2 2

 22  24

3 4

3 4

1 4 1 4

2 4

1 4

2 2 4 4

 1 14

1

24 3 4  4 4 5  1 14 2 34 2 5 44 + 4 = 4

1 4 1 4

 34

 34 1 15 2 24

1 2

 1 14

1 2

To add or subtract fractions that3 do 2not 5have common denominators, 2  form  4fractions that do have common 2 4 we rename one or more of them4 to 5 2 2or subtract. denominators. Then we 33add Recall that to rename a fraction, we 2   424   3454 2 5 4 2 2 4 2 2 1  we multiply it by a fraction equal to 41.Here 4 4 rename 2 by multiplying it by 2. 4 3 1 2 2 This forms the equivalent fraction , which can be added to 43. 2 4 4 4

1 2

3 4

1 2  24 2 2 2 2 32  24 2  342 2 2 414 2  2Then  4add. 23 2 5 1    3434 1 43 4

Rename .  3 4

Language 3 Math  2422 54 4 Review The least common multiple is the smallest whole 1  2  24 2 6 2that number is a 6 9 two 9 multiple of 3 3  4 numbers. 4 given 5 4



2 4

1 1 2 2

 54 2 2

 

3   434

4

5 5 4 4

 11

4 4 Simplify your answer, if possible.

1 2

 22  24 2 6 2 3find a common 3 9 To find the common denominator of two fractions, we 3 3 6  4denominators. 4 multiple 9of the The least common multiple of the 5 common denominators is the least denominator of the fractions.  11 4

 1 14 Example

4

1 6

Subtract: 1  1 2 6

1 6

1 1  2 6

Solution



4 1  1  1 414 5

3 6

The denominators are 2 and 6. The least common multiple of 2 and 6 is 6. 1  33  36 denominator of the two fractions. We do not need So 6 is the least2 common 3 1 to rename 61 . We1 change1 halves 32to sixths by multiplying  33 and  36 then we69 6 2 by 6 6 subtract. 6 2 6 2 23 1  16  16 99  6 2 3 6 3

1 6

3

1 2

 33  36

 16

 16 2 3

2 1 6 6 3 9

286

Saxon Math Course 1

2 3

1 2 3  8

6 9

2 6

 13

9

6 9

3 18 1 2 4 3 

 

1 1 1 2 23 66 6  1 6 1  161  1 2 1 2 1 2 12 6 6    2 6 3 6 6 6 3 3 1 3 11  1 3 11 3 36 2 3 1 36 3 13 3 3    2 3 2 3 6 1 26 6 23 8 2 23 16 1 1 1 81 1 3 3 8 1 3 1 1 1 3 2 2 8 8 2 2 Rename .   1      6 3 8 8 22 1 2 12 134 1 3 8 12 2 3 6 1 1 8 2 2 33 13 1 1 3 3   2 2      6 633 8 6   6 1 3 6 33 2 8 4 1 1 1 1 8 8 4 1 4 1 1 1   1 1   6  6   6 2 4 1 2 2 1 3 6 1 4   6  1 13  6 6 2 81 6  3 6 6  Subtract 3 from  36. 8 8 2 1 2 3 6 6 1 1 5 1 3  3 3 1    1 6 6 3 36 6 6 41 2  4 11 2 6  2 1 1 21 12 3 3 3 216 33 6 3 1 1 6 3 2 1  26 32  6 3 2 23 8 8 8 8 1 38 3 6 Reduce. 13 1 3 1 1 3 1 1 1 1 1 2 1      8  3  6 2  6 8 68 8 86 4 8 8 4 4 3 1   8 4 Practice Set Find each sum or difference: 5 1 3 5 3 1 3 11 35 2 8 4 8 1 2 8 2 4 2 8 8 3 a. 1 b. 3 c. 3 3 1 1 2 3 1 3 1 1 2 3 1 3 3 1 8 1 4   8   8    4 4 3 8 2 1 8 18 4 3 4 8 84 3 4 4 1 1 �1      3 2 1 3 8 1 8 8 8 4  8 4  8 2 3 4 8 1 1 3 3   3 8 4 1 2 1  d. 18 5 e. 5 5f.8 3 1 3  8 4 3 2 1 1 8 2 8 2 8 4 54 1    8 1 1 14 1 1 3 18 3     2   8 4 4 4 4 4 4 8 8 1 1   4 4 1 1 5 5 1 1 1  Evaluate 3 1How can we use1mental math to check 1our g. answers to a–f?   8 2 8 6 2 2 2 5 38 8 21 88 2 5 1 31 91 5 1 1 1 2 2 8 8 4   Strengthening Concepts Written Practice 2 8 5 4 1 4 4 4 1 1 3 1 1      5 2 8 1 1 1 4 4  8  4 4 4 2 8 4 1 1 * 1. Explain In the decimals chart, the memory cue by a whole 5 1 3 for dividing  1 (53) 2 2 2 1 4 1 4 1 2 8 1 is1 “up.” What does 4   33number 2 1 1that mean? 3 33 3 2 4 1 31 4   1 1 8 8  8 1 1  8 2 8 2 8 2  1 1 1 1 3 1  4 4 8 fit  * 2.6 Connect  How many 8 -inch-thick CD cases will on a 12-inch-long   8 6 6 8 2 (50) 8 shelf? (To answer the question, write and solve a fraction division problem using the method shown in Lesson 50.) 1 1 3 1 1 3   8 8 2 2 2 8 2 1 1 1 8 2 1 6 pounds. 1The 1prize-winning pumpkin 3 3. 3The average pumpkin 1 weighs   (15) 8 8 2 3 32 3 2 2 8 1prize-winning 8 3 2 weighs as much 2as 1  1 weighs 324 pounds. The pumpkin 8 1 1 1 1 3 8 2 1 1 1 1 3 2      how 8 pumpkins? 6 average 6 6 8 2 1 1 1 8 8 8 many 3 1    8 8 2 6 8 1 1 3 2 * 4. * 5. * 6.  1 2 1 8 2 (55) 8 (55) (55) 3 3 2 22 1 2 1 2 1 1 1 1 3 1 2 3 2     3 2 6 2 1 8 6 8 1 1 1 3 5 2 1 1      2 3 1 3 2 1 1 6 8 6 8 6 3 4 4 3 6 1 2 8 1 1 7. 6.282 + 4 + 0.13  8. 81 ÷ 0.9  (38) (49) 6 1 8 1 3 2   6 8 1÷ 10 1 9. 0.2 10. (0.17)(100)   (52) (46) 6 8 3 5 2 5 1 3 5 2 5 ? 1 3 1 13.  ? 11.  3 5  2 12.  3  3 4 4 6 (42) 3 8 24 4 4 3 8 24 4 4 6 3 (26) (29) 6

 16

 16

 16

 16

 16

 16

14. Mt. McKinley is the tallest mountain in North America. Its height in feet (32) is shown in expanded notation below. Write the height of Mt. McKinley in standard form. 2 (2 × 10,000) + 2 (3 × 2100)  + (2 × 10) 3 3 2

2 3

2 2  3 2

2 3

2 2  3 2

2 3

2 22  3 52

2 5

2 3

2 5

1 2

1

21

( 2  55 Lesson 5 )2

2 5

2 12 287 ( 2  3

2

3

3



2

3

5

3

5

2

(2  5)

2

15. Multiply 0.14 by 0.8 and round the product to the nearest hundredth.

(38, 51)

2 3

16. Compare: 2 3 (29)

3 1 3 4 4 3 5  21 5 2 3  4 6 34 6 3 3 1 3 4 4

2 2  3 2

2 2  3 2

* 17. The Copernicus impact crater on the moon has a diameter of about (47) 93 km. Assume the crater is round, and use 3.14 for π. What is the distance around the crater rounded to the nearest ten kilometers? 3 5 2 1 3  4 4 6 3 * 18. Model A 20-foot rope was used to make a square. How many square (38) 5 2 5 ?  the rope? Draw a diagram to help you feet of area were enclosed 6 by 3 8 24 solve the problem. ? 19. What fraction is six dimes? 5 5of 5 2 a dollar ? (29) 8  24 5  ?  6 3 8 24 8 24 5 multiple 2 20. What is the least common (LCM) of 3 and 4?  (30) 6 3 21. a. List the factors of 23.

5 ?  8 24

(19)

2 2 2is the name for 2 b. What number5 that has exactly two factors? 5 2 5 a whole ?    3 3 2 6 3 8 24 2 2 2 22. By what fraction should 2 be multiplied to form the product 1? 1  5 2 3 3 2 (30) 23. Compare: 32 + 42 52

2 3

2 5

5 ?  8 24

1 2 1 2 (2  5)

(9, 38)

2 2 2 5 2 2 1 15 2 ? 1 1 2  * 24. How many s2are in 2 ? ( 28524 )(  2 ) 3  2  5 3 6 3 (54) 5 2 3 2 22 2 22 2 2  25. 5  How many 12s are in 1212? 3 2 33 3 2 (15)

1 2

22 5

222 22 2  33 2 35 2 2 3 2

2 52  3 2

2 2  3 2

1 2

1 2 (2  5)

26. The window was 48 inches wide and 36 inches tall. What was the ratio (23) 1 of the window? of the height2 to the 2 width 1 2  2 (2  5) 3 2 2 2 of 27. What fraction of this group  circles is (29) 3 2 shaded? 1 2

1

2

(2  5) 32537 2 * 28. Analyze Reduce: 223555 (54) 232537 29. The performance  5  5at 7:45 2  2  3began 5 p.m. and concluded at 10:25 p.m. How (32) long was the performance in hours and minutes? 30.2 Conclude has three acute angles. 3  2  5 Triangle 3  7 ABC 2 ? 6 (28)    2Which  3  5 triangle  5  5 has 2 a. 3 one 9 angle? ? right 2 ? 6   b. Which triangle 3 ? has 9 one obtuse angle? D

A

C

Saxon Math Course 1

3 3

3 3

G

3 3

2 ? 6   3 ? 9

288

1 2 (2  5)

B

F

E

2 3 I

H

4

3

3 2 35

S4 S O N L6 E

56

Common Denominators, Part 2 Building Power

Power Up facts 4 6

Power Up G

3mental 4 math

a. Number Sense: 8 × 250 3 2 3 −5350 b. Number Sense: 462 c. Number Sense: 150 + 49 d. Money: $3.75 + $4.50 e. Fractional Parts:

1 2

1 2

of 15

 13

f. Number Sense: 30 × 40 g. Measurement: How many centimeters are in 10 meters? h. Calculation: 10 ×18, + 1, ÷ 9, × 3, + 1, ÷ 4, × 6, − 2, ÷ 52

9 12

2

problem solving

4

Terry folded a square piece of paper in half diagonally to form a triangle. Then he folded the triangle in half as shown, making a smaller triangle. With scissors Terry cut off the upper corner of the triangle. What will the paper look like when it is unfolded?

New Concept



Increasing Knowledge

In Lesson 55 we added and subtracted fractions that did not have common denominators. We renamed one of the fractions in order to add or subtract. In this lesson we will rename both fractions before we add or subtract. To see why this is sometimes necessary, consider the problem below. Tony and Catherine ordered a pineapple pizza. Tony ate half and Catherine ate a third. What fraction of the pizza did Tony and Catherine eat? 1 1 2 2

1 1 2 2

11 33

11 1 1 1 1   We cannot add the fractions 2 and 2 3 3 by simply counting the number of2 2 parts, because the parts are not the same size (that is, the denominators are different). 2 21 4 42

1 2

1 3

1 1 3 3

1 3

These fractions do not have common denominators.

Lesson 56

289

1 2

1 2

1  131342 and 13, the parts are Renaming 12 as 4221 does42 not help us either. In the fractions 2 still of different sizes.

1 2

1 2

1 4 1 4

1 3

2 4

1 3

1 3

2 4

These fractions do not have common denominators.

We must rename both fractions in order to get parts that are the same size. The least common multiple of the denominators can be used as the common denominator of the renamed fractions. The least common multiple of 2 and 3 1 1 1 1 (the denominators of 2 and 3 ) is 6. 3 3

1 2

2

1 6 1 1 6

6

1 3

 26

1 6 1 6

6

1 3 1 2 1 = =3 6 2 6 3 The common denominator is 6.

1 1  2 3

1 2

1 1  2 3

Example 1 1 2

 13

 33  36 Add: 1  1 3 1 2 2 2 2 26 Solution 5 6

1

2 Thinking Skill



2

1 2



1 3

3 3 1 1 21 3  6  2 13 2 2 326

 13

1 2

8 4 23   23 4 12

3 4

1

2 32

1 2



3 3



1 12

3 4 2 3 2 3 33 6 3

3 6

Then add. 1 5 4  122 6 3 3 3 2 6 6 6 9 3 1  33  12  15 4 41 2 2  6 8 2 4 32 3  13  22  26  3  4  12

5 6

3 3

1 2

3 4

290

3 4

Saxon Math Course 1

2 3

3 4

2 3

3 4 2 3 3 5 4 64 6 6  23

3 6

2 3

1  26 3 3 1  2 6



5 4 6 6 6

3 To find the fraction that remains, we subtract 23 from 43. The least common 4 multiple of 4 and 3 is 12. We rename both fractions so that their denominators are 12 and then subtract. 3 4

 36

3  33  4 1 21  4436 3  32



1 5 12 3 6   3 2 4 3 32 3 3 3 3 4 1 2 2   9a whole Risa saw that she had of a carton of 12 eggs. She used 9 3 33 36 1 3 3 4 33 of12 4  3  2 3 6 1  3  3   12   4 43 4 2to make 3 6 a batch2 of French toast 2for the 3 6 carton family. What fraction of a 8 2 4 3 2 5  3  4  12 2 24 8  22  26 1 2 2  34  12 6 1 2 2 carton  3of eggs  6was not used?  3  2 3 2 6 1 3 3 2 5 1 5 5 12 3 4 6 12 Solution 6 6

Example 2 1 2

9 34 33  12

1

5

5 6



1 3

 13  22  26

3 4

1 3

12 16 1 least common multiple of 2 and 3 is 6. The denominators are 2 and 3. The 2 3 3 1 3 1 9 3 1  33  12  33  fraction We rename so 2that 6 is the 4 add. 4 common denominator. Then 3we 2 each 6 1 2 1 1 4 3 8 2 2 11  11  13 and 1.   23 12   1 3    Rename 4 6 12 2 3 3 2 6 2 3 3 3 2 2

Explain

How do you rename 3a fraction?4 1

3 4 2 3

1 2 3  36 3

1 2

5 4 6 2 6 6 3

5 4 6 6 6

2 3

3

1 1  2 3

1 3 23 3

 36

 13  22  26 

5 6

3 6

 26

1 4 4   23 4 8 

5 6

3 4

3 4

5 6

 33  36

4 2 3

21 32

3 4

9 8 3 6 8 12 12 2 3

3 8 1 2

3 8

3 2 4 3 1  2

1 2 23 2 3 1 1 2 1 4 4 4 3   2 4 8 8 1 4 4 3   1 1 2 4 8 82  1 2 3 3 42 3 8 2 3 8  5 2 1 1 4 84   2 4 38 2 2 3 1 3 3 4 8 3 3 3 4 1 1 4 44   39 4 28 4 2 6 183 81 12 12 3 28 22 9 8 3 324 6 13 12 12 24 4 2 6 3 2 3 3 2 1 4 4 93 2 323 3 4  8 8 6 3 12 12 84 8 1 2 1 2  4 3 2 3 11 2 1  4 9 45 2 8 6 2 2 2 3 2 2 12 2 312 33  5 3 3 4 4 5 1 12  1 1 84  3 3 3 4 12 4 21 6 1 1 4 1 2 2  34 34 5 1 9 8 2 2 3 12 2 1 61 12 12 4  3 3 1 23 3 44 2 1 1 2 42 3   1 

3

 33  36 4 32 83  13  2  2 1 2

 13 8 22  26 1 2

2

3 4 2 3

2

6 5 6

1 2

6

3

3 3 4 4 2 3 2 2 3 Rename and  . 3 3 4 3 4



3 3



9 12

6 9  33  12 1 8  44 212 1

Then subtract. 12

3 4

9  33  12 1

8  23 2 44  12

1 12

12 12 3

8  23  44  12 3 4 1 4 4 3 3 4   6 1 8 8 2 4 3 8 3 89 8 8 12  3  12 4 We can check for reasonableness by thinking about the number of eggs. If 8  2  44  12 the carton holds 123 eggs, then there were nine eggs in the carton because 1 5 4 4 2 3 2 62 4 6 used 3 of 12 12 eggs, which 2 3 3 6 8 from 6  8 6 3 is 8 eggs. Subtracting 4 of 12 is 9. Risa 1 1 9 leaves 1 egg 3 in the carton, 3 4 12 42 which is 12 of a carton. 4



5 4 Renaming one or more fractions can also help us compare 2 6 fractions. To 6 3 compare fractions that have common denominators,6we simply compare the 2 3 2 3 3 1 4 3 2 numerators. 8 6 9 4 6 3 3   4 8 8 8 8 2 12 12 4 3 5 4 6 6 6

To compare fractions that do not have common denominators, we can rename one or both fractions 14 that8they do have 3 2 9 1 denominators. 3 3 common 2 so     4 3 3 44 12 4 3 12 4 Example 3 2 2 1 1     5 3 3 21 3 1 51 1 Compare: 1 8 2 2 2 8 2 12 1 1 1 13 Solution 2 1 1 3 2 2 2 2 1 24 1 12 12  1 3 3 3 4 3  1 denominator We rename 21 so1that the is 8. 1 10 1 1 12 2  1   124 3 3 4 4 4 1 4 4 4 4 1 3 3 4 1   6 3   1 1 8 8 8 8 8 2 4 8 8 2 4 8 8 2 2 123 3 34 1 3 1 4 4 3 4   3 62 We see that less . 3 8 6 38 3 4 1 1than 84 1 2 8 8 8 is 8 8 2 8 2  2 3  2 12 3 3 4 1 44 35 4 64 6 316 1 4 3  4 38 86 1 2 3 1 8 88 1 2 1 8 8 82 8 1 3 2 3 3 2 4 2 3 33 3 1 9 2 2 3 8 2 48 12  3 4 6 1 3 4 12 48  3 21 Therefore, 83is less4 than 21. 4 3 4 8 8 8 1 3 8 9 3 3 812 92 4 3 3 2 4     3 12   4   3 4 4 4 3 12 4 3 123 4 12 6 3 4 12  88 8 8 2 4 9 21 3 38  13  44  412 2  44 38 12 10 3  3  93 3 3 than 4 4 3 8 4 1is reasonable The answer2 because the numerator 4 is 3, is less 3, which 4 3 of12 8 29 6 32 1 3 4 128 3 2 8 4 1 2 9 8 8 8 8 88 8 2 4 8 3 3 1 4 3 3 9 3 2 4 1 half of the denominator 8, 4 indicating that 3is 6 3 3  less 4 than 2. 3    66 the fraction 112 12 2128 21 8 1 812 8 3 42 12 4 3 412 12 412 2 3 3 12 2 12 3 3 2 8 2 2 4 2 4 443 2  333 93  4 3 3 4 3  1 4 3 3 334 12 Example 4 3 345 12 5 3 2 6 4 2 3 64  3 2 24 3 4  4 3 3 8 9 3 3 2 2 4 3 4 3  4 Compare: 3 2 41 3 3 461 3 24 43 3 3 9334 23 12 4 1 32 4 41 113 4 8 3 2 3 3 8 3  4 4  4 483 4 4  12 6 36 4  3  12 4 8 48 3 4 8 8 8 8 3 8 2 8 8 8 2 Solution 1 3 2 1 2 13 3  1 22    4 4 5 3 5 3 5 24 4 2 The denominators are 3 and both fractions with a common 34 4. We rename 2 3 1 31 2  9 8   1 4 denominator8of6512. 29 3 32 3 2 3 3 4 12 26 12 2 3 1 31  4 3 2 4 21 33 33 3 9 9 1 3  3 84 1 3  121 12 23 24 4 8 2 8 1 1    1 3 443    3 43 43 4 12 3 4 3 33 244 43 3 12 12 3 34 4 12 12 2 10 1 11 1 1 1 1  3 1 92   82 9  3  4 11 1  We see that1254is3 less4than . 4 4   3 12 4 212 5 3 10 10 2 1 1 1 2 1 1 1 3 1 2   1 4 4 32 3  1 3 2 13 4 14 433332 10 2 2 42  33 4 291 4 3 3 10 3 4 3 Lesson 56 2 3   2   2 1   44 5 1 331 26 2 441 4 6 2 43 3  2 1 63 3 34 4 4 5 3      3 3 4 12 2 2 10 5 2  5 3 2 1    

212 8 8 124 12 123 3 8 6 8 4 33 8448 2 4 212 8 3 38 833 8 448 4 4 8 124 12 12 8 4 48 8 4 112 1 3 3 3 8 1 28 3 24 8 4 1 4 4 1 8 1 28 2 42 2 a341 �   2� 3 � 1 3 9 3 3 1 1 8 3 9 3 2 4 5 6 a 1�12 b1 6 2 1 2 1  2 b 1            2 4 8 2 8 8 8 8 8 8 2 3 8  8 2 8 2     3 2 3 3 5 5 4 3 123 4 5 512 43 32 12 4 2 2 2 5 5 3 42 12 2 � 3 � 5 2� 7 3 3 8 9 3 3 2 4     1 1 1 1 3 3 4 12 4 3 12 3 3 2 2 3 3 3 3 24 2 2 24 12 8 9 8  2   1 1 3 3 1 1 1 1 3 3 2 2 2 2 2 1 2 1 2 1 4 23 2 4 8 2 2 4 3 8 3 3 4 1 13  33  992 3 12 4 2 6 212  443 3 3 4 3 3 4 4 3 4 44  1    3 8 3 1 1 12 12 4 4 4 4 4 4 4 4 3 3 4 4 3 3 3 3 3 3 3 1 3 4 2 34 3 44 4  3  123 2 3 4 12 4  3 11 1 12 4 3 33    1 3 33 43 12 43 3 11122 10 2 11 3 11822 8 3 3 9 4 2 3 4 2 551 4 10  2 3 2 2 2 2 1 1 1 1 10 10 1 1 1 1    3  1  3 1 1  2 31 2 33  44  12  2 is3less  3  3 22 4 3 1 12 12 3  64 1Therefore,  3 4. 44    than   4 3   4  5 5 5 5 5 5 3 3 2 2 3 3 4 4 4 4 4 3 4 4 3 4 44 4 3 1 1 1 3 2 3  4 1 1 11 2 2 4 2 2 3 3 1 9 2 8 91 8 2 223 33 1 2 2 2 2 2 3 1 2 3 22 2 2 1 1 2 2 1 1 2 2 1 1 3 6 4 3 6  4 3 3 3 3 3 4 2 1 2 4 2 1 2 4 2 1 3 9 21 14 8 9 2 2 1 12 8 1 2 43  3  3 32 1 12 33 4 34 3 3 22 3 44      63312 3 Generalize 3to Example  34 63312 1 4 order  435 3 2 Use the answer 4 to arrange these fractions in 1 1 1       10 33 3  3 3 3 3 3 3 5 2 6 4 3 3 2 6 4 3 3 2 6 3 1 12 10 12 3 3 2 12 3  64 43  5 3 3 3 26 3 41 2 10 6 4 4 33 54 1 2 21 12 9 4 4 3 3 2 2 1 313 10 10 10 10 from least , , 121  4 3 2 18 611 11 11 to greatest. 1 1 1 1 3   4  4 2 3        3 12 5 12 4 3 434 44 44 44 44 4 4412 424 4 1 3 2 1 2 1 1 Practice Find each sum  4 Set  4  or difference:   1 1 2 323 3 3 3 32 5 2 43 2 5 2 3 2 4  3232 3 33 2 116  4 44 1 22 33 1414 333  5 b. 44 1 3 5 22 33 1 3 11 4 3 a. 1 252 2 22 2 1 6 3 c. 4 4 3 3   1  22 44 55 2626  43433 55 22  44  331  55 3 44 1 3 4 3 3 6 6 33 33 3 6 6 3 3 4 4 3 4 5 3 4 2  1 31 4 4 1 4 1 12 33 6 3 21 1 1     2 2 2 2 1 2 1 84 18 8 1 1 8 2   41 3  5 8 11 2 2  2 45  4 32 41 4 3 1 5 3 5   5 2 3 2 3 3  2 5 3 2 2 1 1 11 1 10   10  1  1 1 1 1 1 5 3 2   5 3 3 2 1 2 1f. 1 2 1 4 2 4 4 d. e. 1 3 2 4 24 32 32 3 3 10 2 1 3 3 3 1 2 21 1 1 2 4 2 1 4 2 2 1 4 2 3 4 3 2 1 2 1 4 2 2 3 3 3 3 113 1 2  3 1 6 5 4 10 16 43 43 5 5 15 3 3 3 3 3 2 23 4 3 6 4 3 3 2 6 4 3 3 2 6 3 8 3 9 3 2 4 6 4 3 2 3 2 6 4 3  2 1 1 2 3 1 11 1  21  3    103 10 1 1 1 1 31 44  2 3 4 2 1 3 4 12 4 3 12 10 1 1 4 1  4  34 3 34 2  4 35  4 4 64 6  44 3  5 4  10 10 4 1 31 2  31  4   pair2 of fractions 4 4 Before comparing, write each 6 4 3 3 4 3 3 3 4 54 3 4 2 3 3 2 23 1 4with3common denominators: 2 2 2 1  3 3 3232 33   2 116 4 4 2 2 1 1 22 3 3 33 2 2 1 1 4 4 22 3 2 4 2 1 3 344i.3 3 4 2 2 4 2 1 5 6 4 3 5 h.6  4 533 4 3 3 5 6 4 3 3 2 3 g.33 5 2 3 4 2    6566 44343 55 3 22 2 333 222 3 666 4442 333 555 6 3433 22626 44 3 333 555 3 5 6 4 3 2 4 3 23 1 2 4 43 3 3 4 35 6  34  2 3  65  4 3 Strengthening Concepts Written Practice

12 123 3 1 31 3 2  8 3 2 24 3 4 1 22 2 2 4 33 3 3 2 3  1 1 1 9 5 12 8   4  3 6 2 22 12 4 4 2

12 2

8 8 12 12 1 1 3 41

1 4

1 4 43 6 24 4 5 3 6 2 4 5 2 88 6 312 12 3 1 2317 4 3  13 8 348 6 1  4

8 8 12 12

* 1. (56)

* 2. (56)

1 1 1 1 common denominator. Analyze 9as9the 8 Add 9 9 8 9 49 and 3. Use 122 3 12 12 12 12 12 12 3 12 12 32 2 1 1 1 1 1 Subtract from . Use 6 as the common denominator. 4 3 2 4 33 2 23 43 6 2 2

1 2 1

42

2 3

4 4 12 3 6 4

1 6

2 3

1 6

5 3 5 6 4 3 1 3 52  3. Of the 88 keys on a 2piano, are3white. (29) 9 3 4 5 3 9 3 2a. 1 99 fraction 4 99 3 2 112 1 5 72 2 3 1 99 7 keys1are white? 5 of a3piano’s 8 88 What 12  13  3 3 12 2 1 2 3 3 3 3 4 2 4 1 4 5 5 3 2 6 4 3 2 6 4 3 8 88 3 6 81212 8 2 31212 1212 31212 4 2 5 2 65 keys? 4 3 5 6 34b. What 2 6black4 keys 2to3white 3 is the ratio of 57 2 5 2 1 5 6 4 3 3 1 2 5 13  3  1 2 43  8 3 8 8 8 3 6 5 5 2 6 4 3 3 2 1 1 10 apple pie, how many apples would 4. If 4712 apples are needed to make an 3 6 2 (29) 1 1 be needed to make two apple pies? 3 4 8 1 9 1 1 1  b  a5  b1 (8  10)  a6  1 b  a5  11 b (82  10)  12 1 a6 12 * 5. 10 100 10 100 Subtract from . Use 12 as the common denominator. 442 3 3 3 4 2 63 (56)2   1 1 9 81 9 5 6 4 3 b(8  10)  b a6 1 1 b  a5 1 2 1 1 1 1 1 1 1 1 answer. 2 1 12 21 00 10 5 24 6. 100 32 3112 and 4 2 4 1 3 12 2 6 6 42 6. Reduce your 2 2 2 2 Add 3 3 4 3 3 4 2 (55)  9 8 3 1 7 5 112 2 1 1 3 1 * 7. 8 Analyze Subtract 12 from your 42 9answer. 8 9 6. Reduce 3 3 6 8 8 1 8 9 9(55) 12 8 9 12 12 12 4 9 5 1 7 2 12 12 12 12 12 5  + $1.75 9. (0.625)(0.4) 3 8  1 12 8. $3 + 65¢ 8 9 9 18 1 3 6 (1)8 (39) 12 5 2 12 12 3 4 5 2 1 7 5 2 1 7 11 7 5 5  5 2   10. 62 ÷ 30.08 11. 3  1 12. 83 8 81 8 8 83 3 8 8 8 3 6 8 3 6 (49) (48) (29) 5 2 (8  10)  a6  1 b 1 a5  1 1b 1 7 1 2 1 1  students in the World 3 1 13. Forty percent of the Club 4 2 5speak 10 Language 100 8 8 8 100 3 3 3 6 4 2 (22, 33) 1 6 languages. 1more than two 2 1 1 3 3 1 b  a5  11 b 4  10) 2 a6  4 (8 42 13 7 6 3 2 5 a reduced a. Write 40% as fraction. 1 100 5 10 6 4 3 3 1 1 1 1 8 8 6 1 1 1 1 1 1 42 a6     (8 3 4 b b b  a5  2 b a5 10)10) a6a6   b 10) b a5 (8 1010 100 students in the World Language Club speak more than 10 How many100 100 b. 1 2 1 1 1 11 two languages? 42 5 22 1 3 7 5 4 b 3 6 1   1 3 (8  10)  a6  b  a5 8 8 8 3 6 100 10 5 5 2 1 7  3 1 1 1 6 1 1 8 8 3 292 Saxon Math Course 81  10)  a6  b  32 4 a5 b (8 5 1 7 1 1 2 1 1 1 1 1 10 3 12 5  42 8  4 213 8 100 3 3 3 6 6 2 2 8 3 6 2

8

6

8

14. Write the following as a decimal number: (46)

15. (16)

16. (47)

Estimate

(8  10)  a6 

1 1 b  a5  b 10 100

What is the sum of 3627 and 4187 to the nearest hundred?

Justify Molly measured the diameter of her bike tire and found that it was 2 feet across. She estimated that for every turn of the tire, the bike traveled about 6 feet. Was Molly’s estimate reasonable? Why or why not?

* 17.

Analyze

(Inv. 5)

What is the mean of 1.2, 1.3, 1.4, and 1.5?

18. The perimeter of a square is 36 inches. What is its area? (38)

19. Here we show 24 written as a product of prime numbers: (19)

2∙2∙2∙3 Show how 30 can be written as a product of prime numbers.

2 w�0 3

Find each unknown number: 2 2 2 21. m � 1 20. w � 0 m � 1 3 3 3 (43) (30)

2 w�0 3

2 2 �1 �n�0 m22. 3 (43) 3

2 �n�0 3

Refer to this bar graph to answer problems 23–26. Before you answer the questions, be sure you understand what the scale on the graph represents.

5 2 6 3 6

3 2

2�3�5 2�3�5�7

2 � 3 �2 5 2 � 3 � 53 � 7

5 2 6 3 6

5 2 6 3 6

3 2

3 2

2 3

Money Saved from $100 Earned

2�3�5 2�3�5�7

$100 $90

Savings 2 3

2 3

1 2

2 3

2 3

1 4

$80

1 2 a 2 � 3 b1 2

1 a2 �

13 44

1 7

$70 $60 $50 $40 $30 $20

1 7

$10 $0 1 2

1 2

Lisa

Naeem

Donato

1 23. How much did Naeem spend?

(Inv. 5)

2

24. How much more did Lisa save than Donato? 2 w�0 3 25. What fraction of his earnings did Donato save?

(Inv. 4)

2 m�1 3

(29)

* 26. (40)

* 27. (54)

Formulate Write a percent question that relates to the bar graph, and then answer the question. Analyze

Reduce:

2�3�5 2�3�5�7

5 2 6 3 6

2 3

Lesson 56

293

2 � 3 � 52 3 � 5 �37

2 2� 3

2 3

5 6

1 1 2 �2 a2 � 3 b 1 * 28. How many 32 s1 are in 12? a( 12 � 23 b) a 2 3 b 2 (54) 22��33��55 2 22 �77 2 2 3��55�rectangles 33 1 cm wide. On 29. Model Draw��3 three that are 2 cm long and (7) each rectangle show a different way to divide the rectangle in half. Then shade half of each rectangle.

* 30. Compare: (56)

Early Finishers

Real-World Application

294

22 33

55 66

1 3

1 3

1 3

Dalia and her three sisters make a pizza and decide to split it into four equal pieces (one for each person). Dalia finds out that her best friend is coming over, so she cuts her piece in half to share with her friend. What fraction of the whole pizza does Dalia have now? Show your work.

Saxon Math Course 1

2 5

2 5

LESSON

57

Adding and Subtracting Fractions: Three Steps Building Power

Power Up facts

Power Up H

mental math

a. Calculation: 8 × 425 b. Number Sense: 465 + 250 c. Number Sense: 150 − 49 d. Money: $9.75 − $3.50 600 30

1

e. Number Sense: Double 4 2. 1

f. Number Sense:

42

600 30

g. Measurement: Which is greater, a century or 5 decades?

1 2  2 3

h. Calculation: 2 × 2, × 2, × 2, × 2, × 2, ÷ 8, ÷ 8

problem solving

It takes ten hens one week to lay fifty eggs. How many eggs will sixty hens lay in four weeks?

Increasing Knowledge

New Concept

We follow three steps to solve fraction problems: Step 1: Put the problem into the correct shape or form if it is not already. (When adding or subtracting fractions, the correct form is with common denominators.) Step 2: Perform the operation indicated. (Add, subtract, multiply, or divide.) Step 3: Simplify the answer if possible. (Reduce the fraction or write an improper fraction as a mixed number.)

Example 1 600 30

Add:

1 2  2 3

Solution We follow the steps described above. Step 1: Shape: write the fractions with common denominators. Step 2: Operate: add the renamed fractions. Step 3: Simplify: convert the improper fraction to a mixed number.

Lesson 57

295

1 2 5 6

 33  36

 33  36

224  56 3 2 6 7 7   11 6  13 6 22 6 26

224  563 2 65 6

 13  22  26 3 6

1 6

1 2

 12

3 6

 12

5 5 5 1 3 5 3  56  56  6 6 6 2 3  6 6 1 2 2 1 2 2 2 2 4  13  22  26     5 3 2  5 6 31  3 23  2 63 6 1. Shape  5 6 6 3 3 1 2 3 6  56 5 3 3 7 1 211 3 3 633 1 16 5 1    1 2 2 1 6 2 2 64  1  3  2  6 6 2 262 2 3 466 2 3  2  6 6 21 6 2 2 6  3 1 2 2 6 2  3 2 2 2 6 4 2. Operate 3 1    7 1    2 3 2 36 1  16 3 2 76 6 6   1 63 21 67 161 2  16 6 6 1 2

 33  36

3. Simplify

5 6

 56

5 1 224  13  22  26 3 2 6  6 3 3 1 31 3 7 1 5 1   3 5 5 5 5 5 6  16 5  336 3 3 3 3 1 1 5 3 13 1 6 2 2 62 3 6          5 5 6 6 6 6 5 63 1 53 3 1 6 2 3 6 2 3 6 1 2 3 6Subtract:  2 562 4 2 4  3  36 6 3 2 13  6 6 62 2 265  13 1 2 2 2 1      6 2 2 4 4  6 2 2 4  1  22  22 3 3 2 3 6 2     326    326  2 1 2 2 3 3 2 6 1 32 2 246 2 6 3 4 32 26 6 2    26 7   5 5 3  5 17 1 35 5 3 3 5 3 3 3 Solution 3 1 1 1 2 3 6 2 2353 62 53 6 12   3 3 3 1 71 1 1 1 3 3 3 1 1      3 31 3 6  15 66  7 7 1 1 1 2 11 1 1 1 2 6  6  6 6  6 563  2 36  6  1 2 3 6  1  2 3 6  1 6       6 6 6 2 3 6 2 3 6 3 3 1 1 6 6 6 7 1 7 1 2 2 2 6 5 10 2 466 3 61 4 2 21  3 1 41 2 2 3 465 10 2 6 6 6 2 22 466 2 2 2 6 Step write 6 66 with 1 2 1 2 the 2fractions 6common  131:  2Shape:  226  2  22  denominators. 4 2 4     23  22  46 2     3 6 2 3 2 6    26   3 1 2 136 3 2 6 1 3 2 6 6 2 3 3  1 33  2  6 2  3 2 33 2  12 3 1  7 Step1 2: Operate:  1 1the renamed 3 1 2 3 6 71subtract 3 1  fractions.3 31 4 7 1 5 2 3 3 1 1 5 5 1 3 3  7 1 7 3   5 5 1011  5 10  156 10 6  21 6  62  156 2  564 361 6  1 164 6 2 2661 2 32 43 6 6 6 6 6 2  36  6 6 6  Step 3: Simplify: reduce 6 the 3 2fraction. 6 3 4 1 2 2 1 2 2 2 2 4 5 5 5 5       1 1 7 1 7 15 1 1 7 1 1   3 2 6 3 3 2  1. Shape  2 6 6      6 2 6 32 101 2 10 2 10 212 6 12 6 6 32 1 7 1 5 5 5 15 1  1  6 52 6 6  25 1 6 1 7 33 1 5 7 1  5 6 1 7 1 1 6 2 1 3 55 6 11 5  1      2. Operate 10 62 3 6 37   5 1 5 1 5 1 1 1 3  10 2 6 3 10 2 2 2126 4 63 6 2 121 61 5 1 1 212 2 6 2 3107 21 6 32 2  6 1 2 6 3 6 3 326   1 1   2 66 2 3 4 10 2 2 6 3 2 1 3 5 5 7 1 11 1 1 3 3 5 3 5 1 1 5 5 1 1    1   2 4 6 2 4 6 2 4 6  6 6 3 62 3 6 3 6 36 3 1 1 1 1 3. Simplify 3  35 3  1 1 121 3 2 313 1 1 1 3 2 3 5 3 5 1   2  1 561 1 1  4  22 36 22 3  6  4  6 36  1 4 6 5 10 5 10 6 2 63 4  3 45 10  2 6  2 13 4 413 5 1 74 2 difference: 652  16 3 43  4  Practice Set Find each sum or 2 6 2 10 2 6 3 3 3 31 1 3 1 1 1 12 1 3 2 2 3 2 3 1 31  3 1 51 2  2 43 1  3 1 11  1      a. b. c.        2 63 2 4 3 3 4 3 4 6 2 6 3 45 2 10 6 5 105 10 3 4 5 10 2 62 6 5 15 1 5 2 7 5 5 5 1 5 11 1 5 17  5 1 1 1 1 1 7 1 1           d. e. f. 6 26 2 6 3 10 12 2 2  115  1 10 2 2 67  317  112 6 12 6 6 2 6 23 6 6 210 5 1 1 2 3 2 6 3 216 32 1 10 3 210 2      2 5 10 4 2 6 3 4 3 4 52 10 3 5   4 (1 0.2) 3 5 1 52 Concepts 5 5 1 Practice 5 Strengthening 5 17 1 1 1 1 1 7 1 5 1 7 1 5Written 5 7 1 7 1 1 8   2      5 2 10 2 512 1  16 2 1 12 26   128 6  12 6 6 2 10 2 10 3 3  2     6 2 10 2 62 26 10 12 6 3 2 56 6 32 6 43 1 1 3 3 1 1 1. 1Analyze 2between 1 What is the difference the sum the 3 3 5 of 2 and 2 and 5 5 3 1  29) 2  3  3 b4  3 3 1 1 4 6 2 4 6 5 10 6 2 3(12, 6 3 a4 5 4 product of and ? 1 2 1 1 4 5 1 7 54 1 4 72 1 52 1 1        2 3 2 6 6 2 10 2 10 212 6 12 6 3 5 3 5 3 5 1 1 2. Formulate Thomas Jefferson was born in 1743. How old was he 5 2 3 1 2 3 3 1 1 1 3 3 3 3 3 5 1 2 ( 4 21 5 ) 2  ( 4  5(13) ) 2 4 6 4 1( 4 15 ) 2 6 5 3 2 5 4  6  2 4 4 6    2 3 6 of the United States 6 an3 5 was president 1800? Write 3 0.2) 32 3 5 3 in 4 6elected (1 a  2 1 2 when3 he23 1 2 1 1 1 1 2   2 3 6 3 3 b b a 5  4 4 85 8 5 2    11 1 equation problem. 52  4 4 3 and   3 solve the 2 5 3 16 3 72  2 6 56  13 5 1 1  233  ( )  6 6 3 5 3 3 5 4 0.2) 6 52  2 12 6 8 2 in 4 in ( 4 3 5 )3 10 4 23 3 5 3 5 1 8* 3. Subtract 8 from . Use 12 as the common denominator.  ( ) 2 4 6 4 6 3 3 5 4 5 1  6 3

5 1  6 3 2 Example

(56)

1 2 1 2  3  2 3 2 3 2 3 3 4  (1  5 0.2) 255 5 8 8

5 1 2 1 1 1 1 5 2 1 1     * 5. * 6. 5 32 5 6 3 (57) 36 653 33 5 1 33 2 2 1 (57) 6 (57) 2   a  5b 3 5 in in 4 4 3 3 8 8 5 4 2 4 4 34 3 3 23 3 8 8 3 * 7. Represent How many 5 s are in4 4 ? aQ 4  5 Rb

13 25

4 (54)

8. $32.50 ÷ 10

(52)

3 (4



3 5)

6

4

3 5 5 2 8 8 335 3 5 13 3 5 3  3 5  35 b a24 in 12. a 4  5 b 4a14 in 3 5 b 31 8in5 4 33 3 83 3 32 4 (29) 5 5 1 ( 41  )( 4  )  25 4  3 8 5 5 5 in in 8 2 4 8 8

9. 3 − (12  0.2) 4 14

(5, 38)

2 3 5 0.2) 63 ÷ 0.12 3 33 3 4  (1  10. 11. 53  2 5 (12  0.2)3  8 82 33 5 54 2 5 (49) 5 (48) 5 4 4 2 8 8 4  (1 3 0.2) 0.2) 3 3 3 2 3 4  (1 5  ( 4  5 ) 3 5 3  2( 45 5 ) 84 3 8 3 3 3 3 8 8 ( 4  5 )( 4  5 ) 3 5 296 Saxon Math Course 1 5 2 3 3 3 3 3 8 8 3 5 3 3 3 3  5) ( 4  5 ) 14 2 2(  )( 4  5 ) 2 5 3 5 3 5(  ) 3 5 3 4  (1  0.2)3 4  (1  0.2) 3 4  (1 4 0.2) 2 5  5  2 4 5 5  24 3 8 8 8 8 8 8

3  53 4 3 3 0.2)  5) 4

4 5 2 53 23   1 a64  3 51b 6 3 in in 3 2 45 3 3 a4  5 b

* 4.

5 8

31 5  42 in3

1 5 8 4 in

31 5  4 2 in 3

13. Fifty percent of this rectangle is shaded. Write 50% as a reduced fraction. What is the area of the shaded part of the rectangle?

(13, 33)

10 mm 20 mm

14. What is the place value of the 7 in 3.567? (34)

15.

(45, 51)

Estimate

Divide 0.5 by 4 and round the quotient to the nearest

tenth.

16. Arrange these numbers in order from least to greatest: (44)

0.3, 3.0, 0.03 17. (10)

1 1  2 6

Predict In this sequence the first term is 2, the second term is 4, and the third term is 6. What is the twentieth term of the sequence?

2, 4, 6, 8, . . . 5 2  6 3 18. In a deck of 52 cards there are four aces. What is the ratio of aces to all (23) cards in a deck of cards? * 19. a.

(31, 57)

3 4

3 5 5 2 8 8

3 5  20. What number 4 3 is (29) 21. (19)

Analyze

5 8

of 80?12 in

1 4

24. (10)

1  13

1 in. 4 1 in. 2

in

List the factors of 29.

22. What is the least common multiple of 12 and 18? (30) 3 1 1 1 2 55 174 24 7 * 23. Compare: 6 6 88 10 10 (56)

2 3

4 52

Calculate the perimeter of the 3 3 shown. rectangle a4  5 b b. Multiply the length of the rectangle by its width to find the area of the rectangle in square inches. Analyze

22  22  33  33  55  77 22  22  55  55  77  77

Connect What temperature is shown on this thermometer? −

1 1 2 1 2 in 99 8 in. 25. If the temperature shown on this 35 35 (14) thermometer rose to 12°F, then how many degrees would the temperature have risen?

10°F 11 33

0°F

–10 ° F 33 88

99 88

88 88

33

11

 22   33   11   44   22   66   11 

33 44

Lesson 57

297

5 7 8 5 510 7 7 8 8 10510 7 8 10

223357 2 2 222523533735757 7 2 22 225525537737 75  7 (54)  5  7 Reduce: 2  2  5  5 3 7  7 2  2* 26. 3  3Analyze 1 5 2 72  5  5  7  7 2  82  3  3  5 2 7 What fraction of the group of2 circles  2  5 is 577 8 10 27. (29) shaded? 5 8

5 3  3  5  78 5577

7 10

(50)

4 5

3 8

4 5 4 4 5 5 1 2

1 2 1 1 2 2

 7CDs on the shelf. Each 3CD is in a 1 2  3 stack 3  5 of Justify Ling has 2 a 9-inch 24  2  5  5  7  7 8 2 3 1 are in the 9-inch stack? Write 8 -inch-thick 5 case. How many CDs 2 plastic 3 and solve a fraction division problem to 4answer the question. Explain why your answer is reasonable.

3  33 3 55  77 5  55 4 773  377

* 29. 8383Subtract 2121 from 4545. Use 10 as the common denominator. (55)

4 4

3 4 3 4

30. The diameter of a regulation basketball hoop is 18 inches. What is the (47) circumference of a regulation basketball hoop? (Use 3.14 for π.)

Early Finishers 222Misty sister. These are the ingredients: 2223 3is 3335 555to 7 777bake a fruit cake for her 22 33going 3 3333 4444 1111 Real-World 2222 32 2225 55 5557 7777 777 55 cup butter Application 4

8888

2222

5555

1 cup sugar

3 4

2 cups flour 3 cups golden raisins 3 8

1 2

4 cup cherries 5

1 cup crushed pineapple (with juice) How many total cups of ingredients are needed for this recipe?

298

Saxon Math Course 1

4 5

3 8

223357 225577

7 10

* 28.

3 8 3 3 8 8

1 2

3 8 4 5

4 5

1 2

LESSON

58

Probability and Chance Building Power

Power Up facts

Power Up I

mental math

a. Calculation: 2 × 75 b. Number Sense: 315 − 150 c. Number Sense: 250 + 199 d. Money: $7.50 + $12.50 1 1 1

e. Fractional 6 , 3 , 2 , Parts:

1 2

1 4

of 25

f. Number Sense: 20 × 50

1

1

a 2 of 2 b

g. Patterns: Find the next number in the pattern: 4, 7, 10, 1 1 1 10 × 10, − 1, ÷ 11, × 8, + 3, ÷ 3, ÷ 5 h. Calculation:   1 1 1 1 2 4 4 , , , 6 3 2

problem solving

When Erinn was 12 years old, she walked six dogs in her neighborhood every day to earn spending money. She preferred to walk two dogs at a time. How many60 different of dogs60 could Erinn take for a walk? 3 6 combinations 3    0.60  0.6 60%  60%  100 10 5 100 5

New Concept

Increasing Knowledge

We live in a world full of uncertainties. What is the chance of rain on Saturday? What are the odds of winning the big game? What is the probability that I will roll the number needed to land on the winning space? The study of probability helps us assign numbers to uncertain events and compare the likelihood that various events will occur. Events that are certain to occur have a probability of one. Events that are certain not to occur have a probability of zero. If I roll a number cube whose faces are numbered 1 through 6, the probability of rolling a 6 or less is one. The probability of rolling a number greater than 6 is zero. Events that are uncertain have probabilities that fall anywhere between zero and one. The closer to zero the probability is, the less likely the event is to occur; the closer to one the probability is, the more likely the event is to occur. We typically express probabilities as fractions or as decimals.

Lesson 58

299

1

Range of Probability 1 2

0

1

Certain Unlikely not to occur

Likely Certain to occur

In this lesson we will practice assigning probabilities to specific events. The set of possible outcomes for an event is called the sample space. If we flip a coin once, the possible outcomes are heads and tails, and the outcomes are equally likely. Sample space = {heads, tails} Using abbreviations, we might identify the sample space as {H, T}. If we flip a coin twice and list the results of each flip in order, then there are four equally likely possible outcomes. Sample space = {HH, HT, TH, TT} Imagine you spin the spinner below once. The spinner could land in sector A, in sector B, or in sector C. Since sector B and sector C have the same area, landing in either one is equally likely. Since sector A has the largest area, we can expect the spinner to land in sector A more often than in either sector B or sector C. B

C A

Discuss Conclude

What is the sample space for the experiment? Are the three outcomes equally likely? Why or why not?

The probability of a particular outcome is the fraction of spins we expect to result in that outcome, if we spin the spinner 1 1 many times. Since sector A 1 1 1 1 1 1 a 2the of 2spinner b 1 1 takes up of the area, the probability that will in A is 12, or 0.5. , , , , 3,land 6 3 2 6 2, 2 4

1 4

In terms of sector Aa(12 of 12).b Sector B takes 1 1 of1 area, sector 1 11 B 1 is half the size a of b 1 6, 3, 2, 2 4 2 spinner. 2 up 4 of the area of the Therefore, 1 1 the probability that the spinner will a of 1 1 2 2b land in sector B is 4, or 0.25. 1 1 1 1 1 1 1 1 12 1 1 a1 2 of 2b , 3, 2,  This      1 1 sectors 6 2 4 is also the probability that the spinner will land in sector C, since 2 4 4 2 4 4 1 equal 1 B and1Care   in 1 size. We know that the spinner is certain to land in one 1 1 1 4 4so the probabilities of the three outcomes must add up to 1.   1 of the2sectors, 2 4 4 1 1 1   1 1 1 1 2 4 4 or 0.5 + 0.25 + 0.25 = 160 32  4  4  1 36 6 6 60 4 2 60 6 3     1     0.60  0.6 60% 60%  60%  60%  2 100 10 5 101 510 10 15 100 10 51 1 1 100 5 If we spin the spinner a large number of times, we would about 2 of 1 61,expect 3, 2, 4 3 6 6 60 4 60 3 1 1  160% b 0.6 1  A, about 160%    1 0.60 2 of 3 6 6 2 60 4 2B, the60% spins to 60 land in sector toa land in sector 3 4 of the spins 100 10 10 51 100 1 5 and2 6, 3,  2, 2101 0.6       60% 0.60 1 1 1100 a of b 1 5 60 1 10 10 10 5 100spins to land 5 C. 2 260 3 the 6 of 6 5 4 2 about 3 2 6, 3, 2, 2  4      in sector 60% 1 60% 60 3  0.6 6 0.60 60 100 10 5 10 10 5 100 5 5 3    0.60  0.6 60%  1 60%  100 10 5 100 5 1 1 1   1 2 4 4 1 1 1   1 2 4 4 1 1 1   1 2 4 4 1 1 1 6, 3, 2,

1 2 1 1 1 6 , 3 , 2 ,1

300

60% 

Saxon Math Course 1

60



6



3

3 6 60   60%  100 10 5 60  0.60  0.6 60% 

3 6 60   60%  100 10 5 60 3  0.60  0.6 60%  100 6 4 5 2 3 2   1

1

60%  6 4  10 10

Example 1 Meredith spins the spinner shown on the previous page 28 times. About how many times can she expect the spinner to land in sector A? In sector B? In sector C?

Solution 1 1 1 1 1 The spinner should land in sector 1 of 6 , 3 , 2 , A about 2 of 28 times. Instead 41 1 1 1 a of 2 b 1 1 2 multiplying 6by the fraction , we can simply divide by 2. , 3, 2, 2 4

28 times ÷ 2 = 14 times

1

1

1

a 2 of 2 b

1

a of 2 b the total 1 1 The land in sector 1 B 1about 4 of 28 times. We2 divide 1 2   1 number of spins 2 4 4 1 1 1by 4.   1 2 4 4 28 times ÷ 4 = 7 times 1 1 spinner16should , 3, 2,

As we noted1before, 1 the 1 probability that the spinner will land in sector B is    1 equal to the2probability 4 4 that it will land in sector C. So Meredith can expect the 3 60 6 60  in   0.60  60%  60% spinner to land in sector A about 14100 times, sector B about 7 times, and in  0.6 10 5 100 3 60 6 60 3    0.60  0.6 60%7 60%  sector C about times. 100 10 5 100 5 3 60 for 6the spinner 60in sector A in 28 spins. never to  land It would be 60% very unlikely 3     0.60  0.6 60% 100 be10 100 to always land 5 In 28 spins it also would very 5unlikely for the spinner in sector A. It would not be unusual, however, if the spinner were to land 12 times in sector A, 10 times in sector B, and 6 times in sector C. It is important to remember that probability indicates expectation; actual results may vary. In the language of percent, we expect the spinner to land in sector A roughly 50% of the time, and we expect it to land in each of the other sectors roughly 25% of the time. When we express a probability as a percent, it is called a chance. 1 1 1 1 1 a 2 of 2 b 1 1 6, 3, 2, 2 4 Example 2 1 1 1 1 1 aa2 60% of 2 b chance of rain tomorrow. 1 1 A weather forecaster says that there is , , , 6 3 2 2 4 1 1 Find1 the probability 1 1 1 a tomorrow. of b 1 that it will rain 1 1 12 2 6, 3, 2, 2 4   1 2 4 4 Solution 1 1 1  Tofindthe 1 probability that it will rain, we convert the chance, 60%, to a 2 4 4 1 1 1 fraction and a decimal.   1 2 4 4 3 60 6 60    0.60  0.6 60%  60%  100 10 5 100 3 60 6 60 3    0.60  0.6 60%  100 10 5 100 5 3 60 6 60 3  probability   0.60 as be expressed 60%  0.6 60% of rain can either 5 or 0.6. 100The10 5 100 60% 

6 4  10 10 6 4 2   1 10 10 5 1

The complement of an event is the opposite of the event. The complement of event A is “not A.” Consider the probability of rain for example. We are certain that it will either rain or not rain. The probabilities of these two possible outcomes must total 1. Subtracting the probability of rain from 1 gives us the probability that it will not rain. Lesson 58

301

1 1 4 3  5



60% 

3 6 60    100 10 5

60  0.60  0.6 100

3 5

1

6 4 2   10 10 5

2 5

1 – 0.6 = 0.4 6 4 2 60 3 2    0.60  0.6  the probability 1 60% So that it will 10 10 as5either 5 or 0.4. 100 5 not rain can be expressed The probability of an event and the probability of its complement total 1. In some experiments or games, all the outcomes have the same probability. This is true if we flip a coin. The probability of the coin landing heads up is 12, and the probability of the coin landing tails up is also 12. Similarly, if we16 roll a number cube, each number has a12 probability of 16 of appearing on the cube’s 1 1 upturned side. 2 6

1 6

To find the probability of an event, we simply add the probabilities of the number of outcome outcomes that make up the event. If all outcomes of an experiment or game of possibl number of outcomes innumber the event have the same probability, then the probability of an event is: of possible outcomes number of outcomes innumber the event number ofinpossible number of outcomes the eventoutcomes number of possible outcomes

number of outcome of possibl number of outcomes innumber the event 2 1   1 1 6 number of possible outcomes 3 number of outcomes in the event 1 2 6 probability that the upturned  A number cube is2 rolled. Find the number Math Language  6 number of possible outcomes 3 number of outcomes in the event 1 2 is greater than 4. Then find the probability that the number A number cube is   is not greater 6 3 number of possible outcomes a six-sided cube. than 4. number of outcome Its sides are of possibl Solution marked 1–6 5 number of outcomes innumber the event  number of outcomes in the event with numbers of possible outcomes 12 5 number of outcomes innumber the event  number of possible outcomes There are six possible, equally likely outcomes. or with dots. number ofinpossible 12 5 number of outcomes the eventoutcomes  number of possible outcomes 12 Sample space = {1, 2, 3, 4, 5, 6} 3 9 54 Two of the outcomes are greater than 4.   12 12 4 3 9 54 number of outcomes 1 12  4 3  212 9 event 5  4 in the    6 3 number of possible outcomes 12 4 9 123 54   1 2 2 4 12 12 44is 3. The complement Thus the probability of greater than of greater than 4 6 3 3 is not “less than 4.” Rather, the complement of greater than 4 is “not greater than 4.” Four of the six outcomes are not greater than 4. number of1 outcomes1 in the 5 2 2 2 4 event  4 5 2 Probability not greater =3 number3of of possible 3 outcomes 6 than 4 = 612 3 3 3 12 3 7 The calculation is reasonable because the probabilities 10 of an event and12 its 6 12 5 7 5 1 5 1 2 2 7 4 1 2 2 4 1, and 3 + 3 = 1. 3 6 complement total 12 12 12 6 6 3 3 3 12 12

Example 3

Example 4 3 6

3 9 54  36  12 12 4

3 6

10 12

4 yellow10marbles, 2 10A bag contains 7 5 red marbles, 3 121 orange marble. 6 looking, Brendan 12 12 Without draws and notes its color. What is the probability of drawing red? What is the complement of this event? What is the probability of the complement?

302

Saxon Math Course 1

10 12

7 12

7 12

green marbles, and 7 12 a marble from the bag

number of possible outcomes

Solution number of outcomes in the event 2 1   There are 12 possiblenumber outcomes in Brendan’s experiment 6 (each of possible outcomes 3 marble represents one outcome). The event we are considering is drawing a red marble. Since 5 of the possible outcomes are red, we see that the probability that the drawn marble is red is 1 1 2

6

5 number of outcomes in the event  number of possible outcomes 12

2 3 1 3

4 6

2 3 2 3

4 6

7 12

10 12

2 3

7 12 3 6

10 12

5 5 1 2 2 4 3 6 of drawing red 3 red.” Its probability The complement can be 12 3 is drawing “not 12 5 7 5 7 1 found by subtracting 12 from 1. 12 6 12 12of outcomes number in the event 5 5 7 7 5 5 2 7 outcomes 7 2 1 number of possible 5 4 129 1 3− = 3 12 12 6 12 3   12 12 12 12 12 12 4 3 10 7 So the probability of not red is 12. 6 12

Example 5

1 6

number of outcomes in the event 2 1   6 3 number of possible outcomes

7 12

In the experiment in example 4, what is the probability that the marble Brendan draws is a primary color?

Solution

5 number of outcomes in the event  number of possible outcomes 12 Red and yellow are primary colors. Green and orange are not. Since 5 possible outcomes are red and 4 are yellow, the probability that the drawn marble is a primary color is: 3 9 54   12 12 4

Practice Set

Juan is waiting for the roller coaster at an amusement park. He has been told there is a 40% chance that he will have to wait more than 15 minutes. a. Find the probability that Juan’s wait will be more than 15 minutes. Write the probability both as a decimal and as a reduced fraction. b. Find the probability that Juan’s wait will not be more than 15 minutes. Express your answer as a decimal and as a fraction. c. What word names the relationship between the events in a and 1 b?

1 2

3

A number cube is rolled. The faces of the cube are numbered 1 through 6. 1 3

d. What is the sample space?

1 2

1 6

e. What is the probability that the number rolled will be odd? Explain. f. What is the probability that the number rolled will be less than 6? g. State the complement to the event in f and find its probability. 1 2

1 6

1 22 1 3

1 2

1 31 6

4 6

1 22

2 3

1 2

2 3

1 6

5 12

1 3

1

22

5 12

4 6

Lesson 58

303

7 12

Refer to the spinner at right for problems h∙k.

1 3

h. What is the probability that the spinner will land in the blue sector? In the black sector? In either of the white sectors? (Note that 13 of 1 1 1 1 1 22 22 2 is 6.) 6

1 2

1 3

i.

Predict

1 2

1 6

1

22

What is the probability that the spinner will not land on

white? 1 6

1

22

j. State the complement of the event in i and find its probability. k. If you spin this spinner 30 times, roughly how many times would you expect it to land in each sector?

1 22

l.

Roll a number cube 24 times and make a frequency table 1 outcomes occurred 2 you 4 for the 6 possible outcomes. Which more than 3 6 3 expected? Represent

2 3

Written Practice

1 2

1 3

Strengthening 5 5 1 Concepts 1 5 1 8 2 8 2 8 2 3 1 10 1 and 1 and the 7 1 1. Analyze What is the difference between the sum of 3 2 6 3 (12,155) 1 1 1 1 1 6 1 12 112 product of 2 and 3? 22 22 3 6 2 6 5 1 1 8 2 held 2 1 dozen eggs. How many eggs are in 2. The flat of eggs 6 3 21 3 1 23 31 2 3 2 53 75 8 7 5 8 (29) 1 �ƒ � �3 � ƒ �3 �� ƒ � � nm�� �� n � �3n � 2 2 dozen? m � 8 �m2� 8 � 86 7 8 6 7 2 38 42 3 4 3 64

1 2

1 6

1 6

5 8

(38)

Analyze

Compare: 62 + 82

9. $32.50 × 10

3 3 3 3 a 4 � 5 b a 4 � 5 b* 16. (57)

17. (10)

Analyze Predict

7 8

10. (6.2)(0.48) (39)

(38)

(49)

1

8 Find the total price including 7% tax on a $9.79 purchase.

What number is next in this sequence?

1 8

304

1 8

perimeter of square is 1 7Analyze 7The 1 this 77 17 11 8 8 cm. What 8 8 area of 8 8 is 8 the 8 the 8 8 square? 4

1 8

Saxon Math Course 1

1

7 27

11. 1.0 ÷ 0.8

. . ., 0.6, 0.7, 0.8, 0.9, . . . 18.

1

72

5 535 3351 3 � � 6 646� �47462 4

3 1 3 12. 5 13. 7 58 12 3 2 3 7 8 5 314. 5 3 1 1 1 � � � � � 120 ÷ 0.5 m � �m � n�� 3�ƒ�3� n� ƒ �� 12 72 8 2 8 (49) 6 (30) 8 67 23 4 3 4 8 7 6 4(29) 6 4 7 2 17 1 1 5 3 533 3333 17 3 2 3 7 58 7 8 3 5 3 3 3 37 3 1 1 1 �aConnect �b abof �8 5 b � � � n � 3�� ƒ � 3 � ƒ �� 3 3both numbers 3 8 a864� 782 by 1 28, Julie5`doubled 182 and 5� 4b7dividing 4 6 5 b4 aInstead 4a 4 7 5 5 4 3 4 6 8 67 15. 8 4 2 (43) 4 5` 4 then divided mentally. What is the division problem Julie performed mentally, and what is the quotient?

7 8

5 3 � 6 4

102

1 3 3number: Find each unknown a � b 2 3 3 1 2 223 33 3 14 35 331 211 2 53 7 5855 7 5787 8587 3 8 � �ƒ ��3 � � � �m�� � * 6. m � �m m * 7.� � *� 8.� 3�� n�m 3 n� n� ƒ� 3� −ƒf � = � �� �� � � � � � � n n 3 3 ƒƒ 8 2 3 4 8 2 3 4 6 8 8 2 3 4 (57) (43) 6766 8 687 88 (57) 22 33 44 8 7678 4 7 (46)

7 8

57 38 �� 68 47

(18)

5.

1 2

1 6

22

3. In three nights Rumpelstiltskin spun $44,400 worth of gold thread. What was the average 3 value 3 he spun per 5night? 7 8 5 3 1 of the 2thread � � m� � n� � 3�ƒ� 7 8 2 3 4 6 8 6 4 5 1 5 55 1 151 1 3 3 3 3 3 3 4. Compare: 82 5 b a24 � 5 b a 4 � 5 b 2 8 88a24 2� (56) 8

1

22

5 8

1

1 2

3 1 5 `1 2

33 4 5`

8 8 77 8 ��7 8 8 7

2

3

3 55 3 ��4 6 6 4 3 3 a4 � 5 b

4

8 7

6

1

6 4

2

5`

2

3

1

1 1 * 19. How1 1 many 53`s5` are in3 34? a( 34 � 35 )b 77 22 22 4

(54)

Find each unknown number: 20. 0.32w = 32

21. x + 3.4 = 5

(43)

7 8

1 8

(43)

* 22. On one roll of a 1–6 number cube, what is the probability that the (58) upturned face will show an even number of dots? 23. Arrange these measurements in order from shortest to longest: (7)

2  5  2  3  3  77 2  2  2  5  5  87

1 10

2 3 � � 3 4

7 8

1 in., 3 cm, 201 mm 8

24. Larry3 correctly answered 45% of the questions. 252337 4 (29, 33)  2 or  5  than 2 less 5  7 half the a. Explain Did Larry correctly answer more2than questions? How do you know? 252337 3 1 2  2  2  5  5 b.7 Write 45% as1 2a reduced fraction. 4 1 1182

25. (52)

1 10

Justify

1 Describe how to mentally calculate 10 of $12.50.

     3 1 26. Reduce: 2 5 2 3 3 7 12 (54) 222557 4  3 1 35 7 1 252 3 5 1 * 27. Analyze 2 What is the sum of the decimal by points 1 2numbers represented 22 4 (50) 8  5 2 58 7 2 x and y on this number line?

252337 222557 28. (7)

1 10

1

12

x

1 10

y

3 1 12 3 4 2 1 3 3 3 5 1 2 2 1 75 8 7 25 32 m 33�733 � 2  5210 � m7 � �n � 1 �n1� �3 � 3 ƒ� 33 � ƒ � � 8 is21 23inches 6 7Then 8 6 8 Model rectangle inch wide. 1 24 3long4 and  758  7 2 that 2  22 Draw 22 52a55 4 4

1

8 7

draw a segment that divides the rectangle into two triangles.

29. What is the perimeter of the rectangle drawn in problem 28? (8)

1 1 3 3and3w = 0.75, what does A equal? * 30. If 10 A =10 lw, and ifa 3l = � 1.5 ba � 5 b 4

(47)

Early Finishers

Real-World Application

7 8 � 8 7

4

 2  3 of3stock  7 are bought and 2 of5 shares Millions sold each3business day, and 1 12      7 prices for every 2 2 2 5 5 4 Here are the closing records are kept for stock trading day. prices, rounded to the nearest eighth, for one share of a corporation’s stock during a week in 1978. 1

5 3�ƒ� 6

5

510 3 � 6 4

Mon. Tu. Wed.     3 1137 2 5 2 3 1 1 1 77 1 71 11 1 $15 $14 , , , 2 2    $15 2 2 2 5 5 8457` 82 8 6 3 2 8

Th. $14 34

Fri. 1

1

1 1a22 of 2 b $14

3 4

Which display—a line graph or a bar graph—is the most appropriate way to display this data if you want to emphasize the changes in the daily closing 1 1 and prices? Draw1 the  display   1 1 justify your choice. 2 4 4 10

60% 

3 6 60   100 10 5

60% 

60  0.60  0.6 100

Lesson 58

3 5

305

5 3 � 6 4

LESSON

59

Adding Mixed Numbers Building Power

Power Up facts

Power Up G

mental math

a. Calculation: 4 × 75 b. Number Sense: 279 + 350 c. Number Sense: 250 − 199 d. Money: $15.00 − $7.75 e. Money: Double $1.50. f. Number Sense:

800 40

g. Patterns: Find the next number in the pattern: 12, 24, 36,

1 1 2 1 2 6

Astronomers use the astronomical unit (AU) to measure distances in the solar system. One AU is equal to the average distance between Earth and the Sun, about 93,000,000 miles. On average, how many miles farther from the Sun is Saturn than Mars?

New Concept

Planet

AU from the Planet to the Sun

Mercury

0.39

Venus

0.72

Earth

1.00

Mars

1.52

Jupiter

5.20

Saturn

9.52

We have been practicing adding mixed numbers since Lesson 26. In this lesson we will rename the fraction parts of the mixed numbers so that the fractions have common denominators. Then we will add.

1 1 Add: 2  1 2 6

Solution

3 3 1 22  3  26

 1 16

 1 16 4 2 36  33

Step 1: Shape: write the fractions with common denominators. Step 2: Operate: add the renamed fractions and add the whole numbers.

306

Saxon Math Course 1

 1 16

4

Increasing Knowledge

Example 1

 1 16

36

h. Calculation: 4 × 12, ÷ 6, × 8, − 4, ÷ 6, × 3, ÷ 2

problem solving

3 3 1 22  3  26

12  3  16 1 2 1 2 2 3 Thinking 1 2 Skill

1 2 2 3 Discuss

12 1 1 1 12  1 23 3 2



2 23

2 the fraction. 1Step 23: Simplify: 1reduce 1 2 1 2 2 3 2 3 1. Shape

1  1 21 1 2  3  1367  3  3 4  6 6 2 2 2 2 4   2 3 11 21226  2 3  2  2 3 7 1 1 7 1 3 6  3  1 6  4 63 6  3  1 6  4 4 26

3 1 1 2 33 2 32 1 1 11 1 2 11  33  1 12  3  1 1 1 151  2 1 31  1   2 3 2 1 2 3 2 26 6 3 2 3 2 3 3 6 4 3 1 1 2. Operate 2 1 1 1 16 22  1 1    2 23  22  1 1 1 66 1 66 2 6 1 1 11 2 1 1 2 1 1 1 4 2 2 41 3 2 1 1 1 11  1 1 1 1 1 1   3 66  2 3 23 3 2 2 3 2 3 2 3 3 3 33 1 2 1 1  231 3 Simplify 3 1 1 1 1 1 3 313. 11 1 11 3  1 36 1 5 2 3  1 5  5 2 32 3 1 1 2  3  3 51  32 3 6 3 6 4 3 46 3 23 26 2 3 4 4 2 2 2 4    2 2 3 2 6  23  2  26 Example 31 3 3 7 1 1 1 1 11 1 1 1 21 7 1 1 3 6 5 3 5 2 5 32  1 3 1 51  3 5 2 3 3 1 3 6 46 3 23 6 3 3 6 4 6 33  1 6  4 46 2 1 2 1 1 1 2 1 2 Add:  2 1  111 1 1  111 1 2 3 13 3 1 2 12 1 1 1 2 3 3 1 7  3 2 21 3 1 1 1  3 5 1 3 3 1 1  3 512313 3 2 23 6 2 3 3 4 3 26 6 2 3 2 4 26 26 Solution 2 2 4 2 2 4  23  2  26  23  2  26 3 5 3 3 1 11 1 11 1 1 1 1 1 5the  1 1: Shape: 3write  3 denominators. 3 1 3 1 fractions 51 773  4 1 3 Step 7  with5common 1 7 1 4 3 23 6 2 8 4 2 6 4 3 2 6 2       3 6 3 1 1 6 2 4 63 6 3 1 6 4 6 1 1 1 Thinking Skill 11 1add 2 the 3 1 renamed fractions 1   1 1 1 1 Step 2: Operate: and add the whole numbers.     1 2 11 12 3 31 1 1 2 1 23 511 1 22 12 1 3 21 2 6 1 2 1 2 2 3 1 2 33 1 6 1 1  3  1 3 1  2 14  23 2 3 2 2Step 2 3 Verify 6 2 3 5 2 6 2 2 3 3 and 9 3: Simplify: convert the 2 improper 3 6 fraction to a mixed number, 2 2 4 Why did we 11 combine the mixed the whole number. 3333  2 number   2 with 1133331212  11  1122  6   1 3366 1 1 11 3 2 2 1 12 2 1 1 1 2 rename both  2 1 3 22 3222 2 1 1 2 11  2 1. Shape 7 1  2 1 1 123 1 33 2 2 1 4 4 22 122 2 6 26 3 2 3 12  2 6   322    22  fractions?   6  2 4 6 31 126 3 23  136 13 1 31 31 1 1 12 1 111  33  1633 114 1 32 23333 432233 2266   5 2 3 1 2 1 1     1 1 1 1 2 1 512  21 1 2 3 1 3 5 3 1 5 3  4 3  223 2226 1 6 3 3 1 46 11 43 3 3 1 221 331 1 66 2 6 2 77 2 2  1236 2 1 66 1 1 1 12 3 1 6 2 6 133 3366 2 3 2 2 22  3 44 3 2. Operate 23 3 2 2 3 5 22  2 5 9 3 9 22 1 24 1 74 7 2 2 11 1122  22 2233  22  2266 2 2     2 3 11  1122   22 6 33 231 6 26 4 63 66  3  2 3 2 3 2 3 2 3 7 1 1 1 1 6 31 1  2 2 1 6 31 1  2 2 1 7 2 1 2 1 23  33  61116  1 7 7 1 6 3366  442  1316 3 6 336  6 3 9 6 2 3 2 3 5 5 5 2 3 9 2 3 2 3 9 1 2 1  232 1 1 1 132 12 1 1 11 11 3. Simplify 1112  5 7131 14115 2 2 1 132142 231  1 3 1 1 1 1 1 321133  11 532 3 2 2 2 3 2 3     1 1 6 3 5 2 3 1 5 3 51 4 2 3 31 2 1 26 5 26 3 28 2 2 3 6 4 3 2 6 33 6 43 3 2 6 11 1111 11 11 1122 22 Practice Set Add: 111  111  1  11 11  11  22 1 2233 1 33 22 1 21 233 33 12 1 1 1 2 1 1 1 1 1 a. 1  1 b. 1  1 1 c.51  2 33 6 2 3 2 3 21 32 2 1 1 1 1 1 2 1 1 12 2 2   1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 3 1  1 112  21313  11 1 11213222 1 1111211213 5 6 3  21 3 5 1 1 1 1 11111 2d. 3 2351 2 5 23f.73 3 34 3 236 23 3 5 e. 5  3 3 1 7 4 5 2 32 23 1 6 9 4 3 2 6 2 8 3 4 2 46 3 23 6 26 8 33 3311 11 11 1111 11 522  311  2  11 55  33  33 5 3366 2 66 Concepts 44 3 41 433 33 31 3 3 1 Practice 1 1 1 1 1 Strengthening 1 Written 5 2 3 1 5 3 5 2 3 31  1 4 3 3 6 4 3 2 6 3 6 4 3 11  1211 3 11331 213111 6 23111 36111 3 1 1 5 3 5  21 3 What  1356 533  2116 35122  5 1.532Connect 244  5 of four tenths and 1 1 1 2 the3decimal 1 3 51 4 731 2 3613151 63 292 is the product 43numbers (40) 2 1 1 2 3 3 21 4936 2 6 15 2 6 2 3 23 6 26 3 4 2 hundredths? 2 8 four 33 3311 11 11 1111 11 3311  5533   11  33 33  55  4 3 22 for 4 3 6 class 4 3 26his 4 3 21 66 trip 2. Larry looked at the clock.1 It was 9:45 p.m. 1The bus 11 3 5 3 1 1 1 1 (32)      2 1 3 1 5 3 7 4 3 1 5 3 2 6 4 3 2 6 2 8 4 3 2 6 leaves at 8:30 a.m. How many hours and minutes are there until the bus 33  111 111 3111 111 4155 3 3 1 1 1 3 5 7    314 1113 3 5112 21326 6 leaves? 3 3 5 57322  34 88 7 3 6 1 2 4 3 2 6 4 3 2 6 2 4 3 2 6 2 1 1 2 5 2 2 6 5 2 3 3 9 2 3 9 3. Pluto orbits the 11 1 11 sun 11 five 22 billion, 11 at an average distance of about 22 22111  112121   11  22 22  11  (12) 2 6 2 3 2 6 2 3 2 6 2 2 6 2 33 nine hundred million kilometers. Use digits to write that distance. 1 1 1 2 1 1 1 1 2 3 2 1 1 2 2 1 1 2 2 6 2 3 21 62 2 2 3 5 1 1  111  2212 1 11111 1 12 2 21  2    2 * 4. 2 * 5. 1 2122 16 33 12  2 1 22 3 66 2 2 26 6 23 3 (59) (59) 5 2 3 9 66 66 11 1133 33 6. Compare: 7. Compare: 22 22 5 5 2 3 99 99 2 3 5 (56) (56) 5 2 3 2 3 3 6 3 6 1 2 1 2 5 56 2 3 9 22 3 9 33 11 36 6 1 2 123 26 5 2 3 9 Lesson 59 307 5 2 59 2 3 235 39 9 800 800 40 40

When should you reduce a fraction?

1 1 1 1 2 3



2 2

3 5  4 2

3 5 9. 3  5  24 2 (29) 4

14 4 1 8 8.83  3 5 5 5 5 (48)

1 2

* 10. How many 12 s are in 25 ? ( 25  12 )

2 5

11. (0.875)(40)

12. 0.07 ÷ 4

(39)

2

45

(50)

15. (16)

Estimate

(45)

What number is halfway

13. 30 ÷ d = 0.6 (49)

1 2 12 ( 5  (25)  2 ) between 0.1 and

(0.8) 2(0.8) 2

0.24?

2 (0.8) 2 4 5 1 78 Round 36,428,591 to the nearest million.

7 4 5

18

16. What temperature is 23° less than 8°F? 2 7 (14) 2 7 45 18 18 17. Estimate Miguela wound a garden hose around a circular reel. If the (47) 4 1 1 1 diameter of5the reel was 100 3 10 inches, how many inches of hose 3was wound on the first full turn of the reel? Round the answer to the nearest whole inch. (Use 3.14 for π.)

45 7

1 100

18

1 100

1 100

1 3

1 3

* 18. How many square inches are needed to cover a square foot? (38)

1 3

19. One centimeter is what fraction of one meter? (52)

20.

(46, 52)

Mentally calculate each answer. Describe how you performed each mental calculation. Justify

a. 6.25 × 10

3 5  4 2

b. 6.25 ÷ 10

1 2 one

* 21. a. With toss of a single number cube, what is the probability of (58) rolling a number less than three? b. Write the complement of the event in a and find its probability.

2 1 (5  2 )

22. Compare: (0.8) 2

(39, 44)

0.8

Refer to the line graph below to answer problems 23–25. Noontime Temperature During Week 80 4 5

7

Temperature (°F)

18

1 3

70

60 Sun.

Math Language Remember that range is the difference between the greatest and least numbers in a data set.

308

Mon.

Tues.

Wed.

Thu.

Fri.

Sat.

23. What was the range of the noontime temperatures during the week?

(18, Inv. 5)

24. What was the Saturday noontime temperature? (18)

* 25. Write a question that relates to this line graph and answer the (18) question. * 26. (50)

Nana can pack a bag of groceries in six tenths of a minute. At that rate, how many bags of groceries can she pack in 15 minutes? Write and solve a fraction division problem to answer the question. Connect

Saxon Math Course 1

1 2

2 1 2 (5  (0.8) 2)

(54)

2 2 514. 5Analyze 2 1 (5  2 )

3 15  4 22

1 3 54 8  3 5 4 25

1 4 8 3 5 5

4 5

4 5

27. One eighth is equivalent to 12 12 %. To what percent is three eighths equivalent?

(Inv. 2)

1

* 28. (46)

* 29. (56)

1

12 2 %

30. (47)

Early Finishers

Real-World Application

3

3

12 2 % 12 % 4 Mentally calculate the2 total cost of 10 gallons of gas priced 4 at $2.299 per gallon. Describe the process you used. Justify

Analyze 3 4

1

12 2 %

1

3 4

Evaluate

Arrange these three numbers in order from least to greatest: 3 , 4

3 4

the reciprocal of 34, 1

If P = 2l + 2w, and if l = 4 and w = 3, what does P

equal?

The Estevez family has three children born in different years. a. List all the possible birth orders by gender of the children. For example, boy, boy, boy is one possible order and boy, girl, boy is another. b. Use the answer to a to find the probability that the Estevez family has two boys and one girl in any order.

Lesson 59

309

LESSON

60

Polygons Building Power

Power Up facts

Power Up I

mental math

a. Number Sense: 2 × 750 b. Number Sense: 429 − 250 c. Number Sense: 750 + 199 d. Money: $9.50 + $1.75 e. Money:

1 2

of $5

f. Number Sense: 40 × 50 g. Primes and Composites: Name the prime numbers between 1 and 10. h. Calculation: 12 × 3, + 4, × 2, + 20, ÷ 10, × 5, ÷ 2

problem solving

Melina glued 27 small blocks together to make this cube. Then she painted each of the 6 faces of the cube a different color. Later the cube broke apart into the 27 small blocks it was made up of. How many of the smaller blocks had 3 painted faces? 2 painted faces? 1 painted face?

Increasing Knowledge

New Concept

Polygons are closed, flat shapes with straight sides.

Example 1 Which of the following is a polygon? A

B

C

D

Solution Only choice C is a polygon because it is the only closed, flat shape with straight sides.

310

Saxon Math Course 1

Polygons are named by the number of sides they have. The chart below names some common polygons. Polygons

Math Language The term polymeans “many,” and -gon means “angles.” The prefix of a polygon’s name tells how many angles and sides the polygon has: tri- means 3, quad- means 4, and so on.

Shape

Number of Sides

Name of Polygon

3

triangle

4

quadrilateral

5

pentagon

6

hexagon

8

octagon

Two sides of a polygon meet, or intersect, at a vertex (plural: vertices). A polygon has the same number of vertices as it has sides.

Example 2 What is the name of a polygon that has four sides?

Solution The answer is not “square” or “rectangle.” Squares and rectangles do have four sides, but not all four-sided polygons are squares or rectangles. The correct answer is quadrilateral. A rectangle is one kind of quadrilateral. A square is a rectangle with sides of equal length. If all the sides of a polygon have the same length and if all the angles have the same measure, then the polygon is called a regular polygon. A square is a regular quadrilateral, but a rectangle that is longer than it is wide is not a regular quadrilateral. We often use the word congruent when describing geometric figures. We say that polygons are congruent to each other if they have the same shape and size. We may also refer to segments or angles as congruent. Congruent segments have the same length; congruent angles have the same measure. In a regular polygon all the sides are congruent and all the angles are congruent.

Example 3 A regular octagon has a perimeter of 96 inches. How long is each side?

Solution An octagon has eight sides. The sides of a regular octagon are the same length. Dividing the perimeter of 96 inches by 8, we find that each side is 12 inches. Many of the red stop signs on our roads are regular octagons with sides 12 inches long.

Lesson 60

311

Practice Set

a. What is the name of this six-sided shape?

b. How many sides does a pentagon have? c. Can a polygon have 19 sides? d. What is the name for a corner of a polygon? e. What is the name for a polygon with four vertices? 9 10

Written Practice 9

3  4

Strengthening Concepts 3 5 1 1  1 3 3 5 1 1 4 8 2 6 3 5 1 1 3  5 1 1 9 9  1 3  33 1 1  4 8 2 6 010 4 8 2 6 4 8What is the cost per ounce of a 42-ounce 2 6 of oatmeal 1. Formulate box (15) priced at $1.26? Write an equation and solve the problem. 5 4 3 3 1 a  b  5 4 3 5 3 * 2. Estimate Ling needs to purchase some gas so that she can mow her 5 (39, 51)4 3 3 1 with 1.1 gallons of gas. The a  b the station she  1 3 9 1fills a container 4 lawn. 3 At 3 3 5  a 4  3 b5 5 5 4 4  5 3 31 1 3   a a  b b5 13  5 3 4 3   station charges $2.47 per gallon. 1 1 5 10 3 55 44 33 3 4 8 5 53 3 3 3 a. How much will Ling spend on gas? Round 3 your answer to the 3 (5  5? nearest cent. 3 5 (3  1 ) 3 51 15 9 9 3 ?  is 1 3 8 3 3 5 5 5 b. Explain how to use estimation to check whether your answer 1 3 10 10 3 6 3 3 1 1 5 5 4 82 ( 5 4 3 ) 8 5? 8 ?5? ( 5 ( 5  3)3) 8 8 5 reasonable. 4 13 3 1 3 5 1 a  b   5 4 1 23 3 3 1 11 5 3 4 8 9 9 6  35 5 3 3 1  10 3. The 4 is 4100. 8  8What is the largest three2 12 6 10 smallest three-digit whole number (12) 3 5 1 1  3 digit whole 1number? 4 8 2 65 5 4 3 4 3 3 1  a  b3 53 5 3a 1 b 11 1 1 1 5 5 4 3 * 4. 4   3 5 3 3* 5. 1 13  3 5 3 33 1 2 6 4(57) 84 8 2 6 ? ( (59) 5 5 3) 3 1  1 3 13 1 5 3 56. 5a 4a 43 b 3 b 7. 3  1 5 4 3 1 4 3 5 35  3 (29) 5 (29) 3 3 1 3  3 3 1 3 3 5 1 1   * 8. 5 3 ? ) How many s are in ( ? ( ) 5 5 3 5 5 8 3 3 (54) 3 3 13 1 b b   Model 1 1 3 1 9. 3 How much money is 58 of $24? Draw a diagram to illustrate the 35 3 ( 5  3 )5(22) 3 3 31 3 5 3 3) 1 5  ? ( problem. 5 5 ( 53  3 ) 8 5? 8 3 5 1 (5  3 ) 8 10. (0.65)(0.14) 11. 65 ÷ 0.05 (40) (49) 5 3 5 1 13 ( 5  3( 5)  3 ) 8 8 12. A quadrilateral has how many sides? 9 10

10

(60)

* 13. (51)

Estimate Round the product of 0.24 and 0.26 to the nearest hundredth.

14. What is the average of 1.3, 2, and 0.81? (18)

15. (10)

What is the sum of the first seven numbers in this sequence? Predict

1, 3, 5, 7, . . . 16. How many square feet are needed to cover a square yard? (38)

17. Ten centimeters is what fraction of one meter? (52)

312

Saxon Math Course 1

Find each unknown number: 5 1 1.2 3 + 1.2 + 1.2 618.3xw= 8 8 (43)

3 1 5 5

3 3 20. m m 1 15  5 5 5 (49) 22. (19)

* 23.

(30)

4 y1 3

1 5 1 5 * 21. 6 6w  3 3 w 8 8 8 8 (58)

4 y 3

What are the prime numbers between 40 and 50? 9

Pedro cut a lime into thin 10 slices. The largest slice was about 4 cm 3 m1 5 5 in diameter. Then he removed the outer peel from the slice. About how long 3 5 1 was m theouter Round the answer 1 peel? 5 6  w  35 5 8 8 a 4  3 b to the nearest centimeter. (Use 3.14 5 4 3 4 for π.) y1 3 24. Connect a. To what decimal number is the arrow pointing? (47)

5 1 6 w3 8 8

List

19.

Analyze

(50, 51)

33? 5  45 8

9 10

5

6

7

b. This decimal number rounds to what whole number? * 25. (58)

5 4 3 a  b The face 5 of 4this 3spinner is divided into 8 congruent sectors. What is the sample space? The spinner is spun once, what ratio expresses the probability that it will stop on a 3? 3?

3 1  53 3 1

Explain

5

2

4 2

4 3 1 1 3

(5

3)

Mary found that the elm tree added about 38 inch to the diameter of its trunk every year. If the diameter of the tree is about 12 inches, then the tree is about how many years old? Write and solve a fraction division problem1to answer 5the question. 3 4 m1 5 6 w3 y1 5 8 8 3 27. Duncan’s favorite T V show starts at 8 p.m. and ends 5 at 9 p.m. Duncan (29) 8 timed the commercials and found that 12 minutes of commercials aired between 8 p.m. and 9 p.m. Commercials were shown for what fraction of the hour? * 26. (50)

* 28. (43)

Estimate

Connect Instead of dividing 400 by 16, Fede thought of an equivalent division problem that was easier to perform. Write an equivalent division 1 3 3 problem that has a one-digit divisor and8 find3the 8 2 quotient.

29. What is the total price of a $6.89 item plus 6% sales tax? (41)

3 8

1 3 2

5 8

3 4

* 30. Compare: (56) 51 6 1  a. 3 82 2 2

b.

5 8

3 1 6  4 2 2

3 4 Lesson 60

313

INVESTIGATION 6

Focus on Attributes of Geometric Solids Polygons are two-dimensional shapes. They have length and width, but they do not have height (depth). The objects we encounter in the world around us are three-dimensional. These objects, called geometric solids, have length, width, and height; in other words, they take up space. The table below illustrates some three-dimensional shapes. You should learn to recognize, name, and draw each of these shapes. Notice that if every face of a solid is a polygon, then the solid is called a polyhedron. Polyhedrons do not have any curved surfaces. So rectangular prisms and pyramids are polyhedrons, but spheres and cylinders are not. Geometric Solids Shape

Thinking Skill

Name

Description

Triangular Prism

Polyhedron

Rectangular Prism

Polyhedron

Cube

Polyhedron

Pyramid

Polyhedron

Cylinder

Not Polyhedron

Cone

Not Polyhedron

Sphere

Not Polyhedron

Verify

Why is a cube also a rectangular prism?

Name each shape using terms from the table above. Then name an object from the real world that has the same shape.

314

1.

2.

3.

4.

5.

6.

Saxon Math Course 1

Solids can have faces, edges, and vertices. The illustration below points out a face, an edge, and a vertex of a cube. Face: a flat surface of a polyhedron Edge: a line where two faces meet Vertex: a point where three or more edges meet

7. A cube has how many faces? 8. A cube has how many edges? 9. A cube has how many vertices?

Activity

Comparing Geometric Solids Materials: Relational GeoSolids Using the solids, try identifying each shape by touch rather than by sight. Discuss the following questions. • How can you tell if a solid is a polyhedron or not? • How are a cone and a pyramid similar and different? • How are a cone and a cylinder similar and different? • How are cylinders and right prisms similar and different? A pyramid with a square base is shown at right. One face is a square; the others are triangles. 10. How many faces does this pyramid have? 11. How many edges does this pyramid have? 12. How many vertices does this pyramid have? When solids are drawn, the edges that are hidden from view can be indicated with dashes. To draw a cube, for example, we first draw two squares that overlap as shown. Represent

Then we connect the corresponding vertices of the two squares. In both steps we use dashes to represent the edges that are hidden from view. Practice drawing a cube. 13. Draw a rectangular prism. Begin by drawing two rectangles as shown at right.

Investigation 6

315

14. Draw a triangular prism. Begin by drawing two triangles as shown at right.

15. Draw a cylinder. Begin by drawing a “flattened circle” as shown at right. This will be the “top” of the cylinder. One way to measure a solid is to find the area of its surfaces. We can find how much surface a polyhedron has by adding the area of its faces. The sum of these areas is called the surface area of the solid. Each edge of the cube at right is 5 inches long. So each face of the cube is a square with sides 5 inches long. Use this information to answer problems 16 and 17.

5 in.

5 in.

5 in.

16. What is the area of each face of the cube? 17.

Analyze

What is the total surface area of the cube?

A cereal box has six faces, but not all the faces have the same area. The front and back faces have the same area; the top and bottom faces have the same area; and the left and right faces have the same area. Here we show a cereal box that is 10 inches tall, 7 inches wide, and 2 inches deep. 18. What is the area of the front of the box? 10 in.

19. What is the area of the top of the box? 20. What is the area of the right panel of the box?

7 in.

2 in.

21. Combine the areas of all six faces to find the total surface area of the box.

316

Saxon Math Course 1

A container such as a cereal box is constructed out of a flat sheet of cardboard that is printed, cut, folded, and glued to create a colorful three-dimensional container. By cutting apart a cereal box, you can see the six faces of the box at one time. Here we show one way to cut apart a box, but many arrangements are possible. top

l e f t

front

r i g h t

bottom

back

22.

Here we show three ways to cut apart a box shaped like a cube. We have also shown an arrangement of six squares that does not fold into a cube. Which pattern below does not form a cube? Conclude

A

B

C

D

Investigation 6

317

In addition to measuring the surface area of a solid, we can also measure its volume. The volume of a solid is the amount of space it occupies. To measure volume, we use units that occupy space, such as cubic centimeters, cubic inches, or cubic feet. Here we show two-dimensional images of a cubic inch and a cubic centimeter: 1 in.

1 in.

1 cm

T

1 cm 1 in.

one cubic inch

1 cm one cubic centimeter

In problems 23–25 below, we will practice counting the number of cubes to determine the volume of a solid. In a later lesson we will expand our discussion of volume. 23. How many cubes are used to form this rectangular prism?

24. How many small cubes are used to form the larger cube at right?

25. How many cubes are used to build this solid?

Use the figures in problems 24 and 25 above to answer extension a.

extensions

a.

Draw the front view, top view, side view, and bottom view of each figure. Explain how they are the same and how they are different.

b.

Bring an empty cereal container from home. Open the glue joints and unfold the box until it is flat. On the unprinted side of the box, label the front, back, side, top, and bottom faces. Identify the glue tabs or any overlapping areas.

Represent

Estimate

Estimate the area of each of the six faces of the unfolded box. Do not include any glue tabs or overlapping areas in your estimate. Then estimate the amount of cardboard that was used to make the box. Did you find the volume or the surface area of the cereal box? Explain your thinking.

318

Saxon Math Course 1

Math Language A two-dimensional pattern that folds to make a threedimensional solid is called a net. Sometimes it is called a map.

c.

In problem 22, we show three nets that will form a cube. Investigation Activities 12 and 13 show nets for a triangular prism and a square pyramid. Cut out and fold these nets to form solids. Use tape to hold the shapes together. Describe how the solids are alike and different.

d.

How many blocks are in the tenth term of this pattern? Explain how you will represent the pattern to find the answer.

e.

Model

Represent

Represent

Sketch the front, top, and bottom of each 3-dimensional

figure.

f.

Figures A, B, and C were sorted into a group based on one common attribute. Classify

Figure A

Figure B

Figure C

Figures D and E do not belong in the group above.

Figure D

Figure E

What attribute is common to figures A, B, and C but not figures D and E? Sketch a figure that would belong in the group with figures A, B, and C.

Investigation 6

319

LESSON

61

Adding Three or More Fractions Building Power

Power Up facts

Power Up H

mental math

a. Number Sense: 4 × 750 b. Number Sense: 283 + 250 c. Number Sense: 750 − 199

1 4

d. Calculation: $8.25 − $2.50 __ _ __ 5  900 1 __ 12 2 Sense: 30 f. Number 4 5 5 _ _ _ 1_ g. Probability: What is the probability 4 5of rolling a 3 on a number cube? _5 1 _× 5, ÷ 2, × 4 h. Calculation: 6 × 10, ÷ 3, × 2, ÷ 4, 0 _5

1 4 1 4

1

1 4

problem solving

1 1 1   2 4 8

1

e. Number Sense: Double 12 2. 4



Increasing Knowledge

900 1 1 1 __ 12 2 Pedro ate half Pedro, shared a pizza. 4 Leticia, and Quan 4 30 the pizza, Leticia _ 5 6 6 7 1 7  __ ate 14 of the 18pizza, and ate 18 of 6 6the 8pizza. Together6the 8 Quan 4 6 three friends ate 45 what fraction of the pizza? 1_ To add three or more fractions, we find a common denominator for all_ 5 the fractions being added. Once we determine a common denominator, we 0 can rename the fractions and add. We usually use the least common denominator, which is the least common multiple of all the denominators.

Example 1 Add

1 1 1   and draw a diagram illustrating the addition. 2 4 8

Solution To add, we first we find a common denominator. The LCM of 2, 4, and 8 is 8, so we rename all fractions as eighths. Then we add and simplify if possible.

320

Saxon Math Course 1

7 8

900 30

0 A large piece of cardstock is 1 mm thick. If we fold it in half, and then fold it in half again, we have a stack of 4 layers of cardstock that is 4 mm high. If it were possible to continue folding the cardstock in half, how thick would 1 1 the1stack   of layers be after 10 folds? Is that closest in height to a book, a 4 a8man, or a bus? 2 table,

New Concept

1 8

1 2 1 4 1 8

 44  48  22  28

 11  18

7 8

We illustrate the addition with fractions of a circle. 900 30

1

12 2

1 2 __ = 1 4 4 8 = 1 1 2 8 5 _ _ _ 4 8 45 1_ 1 8 _5 0 1 11 11 7 2 to3simplify Verify Why don’t1weneed 8? 2 43 68

1 1 1 1 1 3 2 3 1 2 1 2 3 2 6 3 6

Example 2

7 8



1 2 1 4 1 8

 44  48  

1 2

1 2

2 2 1 1

 

2 8 1 8 7 8

6

66

1 1 1 1 1 6 66 2 33 2 3 6 1 2

1 3

1 6 1 3

1 6

1 2 1 6

3 3 1 1 1 1 111  13 1 1 3 1 1 1 1 2 1 3  11161 2 3 6 1 1 113 1 1 2211 1 3312 1 Add: 111  2 6 1 3 2  3 2 23 362 6 6 1 3 2 22 332 663221  62 3 2 2 6 2 23  2  26 3 3 3 2 1 1 13 6 13 1 1 1  61 12 1  3 121 6  3 Solution 1 11 11 11 1 1 11 2  3 13 11 3 23  3 16 2 11  3 16 3 6 3 2  3 12 6 1 6 23 36 2 3 6 1 26 2 1 2 2 6 63 2 3  2  2 62 3  2 6 2 6 6 7 7 6 rename 3 Thinking Skill A common11denominator is 6. We all fractions. Then we add whole 6 6the 3331 1 1 1 136 3 11 13 3 1 6 3 1 2      1 1 1 1 1      33we  3 3 3 6 Predict numbers,1 2and add the fractions. We simplify the result if possible. 2 3 2 6 3 6 6 12 6 6 1 6 1 2 21 61 2 2 222  1 2 1 1 1 11 1 1 1 22 113  12 26 63  7 3 6 6    2 2 2 2  7 If we rename 2, 3, 1 13 23  33  176 2 3 6 26 63 6 2 1 2  3 63 6 1 1 1 16 63 6 12216 133 1 66 2and 1 as fractions 1 1 31 1 1 1 1   3116  1 2 66 1 1 6 4 8 82       3 6 1 3 63 3 3 3 3 6   6 2 2 1 6 6 1 1 2 2 2 3 6 2 6 6 with a common  2 326 2  2 6 7 6 1 6 666  1 7 1 7 6 7 6 6 denominator that 6 6  36  6 1  36 1 1 1 1  3 636 1  3 6 3 3 is not the LCM, 3 1 1 3 1 1 1 6166  7 1 1 1 3 1 1 1 1 1  21 1 6 1 26  3 61 6 will the sum be       1  1 1 1   7 6 7 6 7 6 2 4 8 2 3 6 2 3 4 6 2 4 8 2 3 6 2 32 2 2 2 1 6 1 1 7 2 6 2 3  2  2 6 the same? Justify What steps do you use to simplify 6 ? 1 4 3 18 1 13 8 11 1 1 11 6 1 1 1 2 5 1 1 1 1 1 1 1 1      1  1 1 1 1    1  36 3 6 36 2 4 8 2 24 38 6 2 32 6 3 2 4 3 4 2 3 6 2 6 6 6 1 7 Add:1 6 6  7Practice Set 66  7 66 1 3 8 11 3 8 11 311 11 1 22 155 2 1115 1 4 11 3 1 1 1 11 1 1111  1 11111     b.    11 1 1 1 1  1 1         a. 2   2  1  113 1c. 141 1 1 1 2 3 6 4 8 3 6 2 2 4 28 4 228 43 286 3 2 62 3 362 4 3 2 4 3 4 2 3 26 3 2226 1 3 1 1 1 1 1 1 1 31 111 1 1 11 1 111 1 1 1 1 1 1 1 2 15 2 115 23 2 1 1 11 2  11 3  1 4 157 3 1 71 3 171 311 6  1 1 1f.1 1 1 1      1  11 1 1 1 1 1  1d.       e.   4 18 12 43 286 3 2 62 3 362 4 3 2 4 3 4 2 3 26 3 226 34 268 4 2 84 4 884 2 8 4 2 8 2

g. Find the perimeter of the triangle. 1 20 2 1 11 1 1 20 1 1 5 2 1 11 11 1 1 1 2 65 1 1 2 3 55272 1 13 715 2 61113 1 5 3 12 22 2 11 2 2     8 in.    1 1  1 1  1  1 1 1  1 1  1  1 2 23 1 63 2 4 8 2 24 52 13 3 4 7 2 3 1 6 1 2 2 31 4 6 8 2 44 3 in.88       1 1 1 18 2 1 1 2 1 1 1 1 1 1 1 2 3 6 2 4 8 20 4 820 2 12 22 5  1 5  5 1 2 5  1 2 2  1 3 6 6 2 2 3 2 3 3 2 3 2 2 3 2 4

1 1 3 1 1 1   2 2 41 3 8 6 1 4

1

22

0 1 6 22

1 in. 2 5111  1221 2522 1 53 2 32

1 1 1 1 2 111  221 1  1111   11 1 1 1 1  1 555 3 12 52 2 51 2 2221333241  3 23 32 3 2 32 23 2 4 3 2 1 that2involves 1 1 h. Select one of the exercises a–f and write1 a word problem 20 5 2 21 1 11 1 1 1 1 1 16 1 3 15  1 3 2 11 1 11 13 11 21 12 3 3 2  252  1 5  1 55  1  2 1 3 2  3 3  3 3 2  3 3  3 2  12 5  2 5  22 the adding fractions. 1 3 32 3 23 2 4 3 2 4 3 4 4 3 3 23 2 32 34 2 32 34 20 20 6 6

1 1 2 25 2 1

1 5 2 3 2

22 33

2 23 4 2144 42 5 3

2144

1 1 20 26222

1 22

3 2 2 3 1 1 1 13 1 4 1 2144 11 1 5 1  2 2  35 1 2 3 3 3 3 2 23 3 1 4 2 3 4 4 3 3 2 334  3 32 2 2 4 21444 2144 5 5 3 3

1 1 2 1 2 1 5 5  1 2 3 1 1 2 31 3 2 1 2 3 2 3 4

3 2 2 2144 4 24144 52144

20 6

33 2 55 3

3 2 33 3 5 3 1 4 3

42

2144

3 2422 3 2144 2 2 4 24144 24144 2144 5 3 5 3 2 42 2144 5 3

Lesson 61

321

22

1 3 1   2 4 8

3322 6644 88 22 22 33 33 4466 22 1 1 3 11 11 2 1 5 1 1 3 7 11 1 1   1  1  1 20  1  1  1 2 2 4 38 42 3 6 22 3 3 6 4 26 4 8

1 1 1   2 3 6

44

88

22

22 4422 8 33 22 4433 66 11 2 15 1 1 3 27  1  1 1 1  1 5 1 22 4 4 38 222 2 3 86

Concepts 3 17 1 1 1 11 1 1 1 12 5 1 1 3 Written 1 1 Practice 11 13 1 1 Strengthening 1 31 72  5 1 1 1   1 1  1  1 1      1  1  1   1  1  1 2 3 2 43 6 4 2 4 8 2 32 64 8 2 2 3 3 46 2 42 83 6 4 2 8 4 28 1 2 1 1 20 1 1. Convert improper Remember 1 5 to 5 2 2 1 51 31fraction 1 1 11 1 11 1number. 52  21 the 73  76 1to a mixed 12 2 3 3 2 (25) 1  1 1  11 1     1 1 1  11  1       1 1 1 number. 6 2 23 34 4 62 part 48 82 2 2reduce 32 the 63 fraction 42 of 84 the8mixed 4

1 2 1 1 2. A fathom is 6 feet. How many feet deep 20 is that is 2 2 fathoms 5 1 5 6 water 1 2 1 1 1 1 2 3 3 1 2 1 1 1 1 1 20 1 1 11 1 11111 2252  11 23 1111 11111  2 11 20 20 11 20 20 5  2 112  22 125511  52  52 55 3 deep? 53522  525  2233 2 11 22    6 2222 1 1 1 1 22 33  3 3 3 22 221122333 2 3 4 66 66 2 2222 33 22 3333 2244 22

20 (29) 6 20 6

11 a 2 13 11 1 2 1 1 1 1 3 1is the average 11 3. In2031 hours through 2 3  12tollbooth.  3 52 What  23 252 passed 51  13 56  1 2769 cars 2 4224  2 144 2 3 3 2 3 4 2 3 3 2 2 3 4 3 5 3 number of cars that pass through the tollbooth per hour?

1

22

(18)

11 2 1 11 11 1 1 31 1 1 1 2  3  23 5  522  1 3 1* 5. 523  24 2 3 34 4 33 3 2 2 3 (57) 3 2 2 2144 31 12 2 1 11 1 1 11 11 1 14 3 1 3  12   23  3 * 7.533  3 5  512  1 3 5  523 *26. 1 3 23 34 4 43 2 3 3 2 (61)2 2 4 53 3 2 (59)  62  6 8 3 3 2 2 8. Compare: 5 3 2 2 4 23144 2 (38,256) 3 22 33 2 26 2263 22 33 82 3 322 5 3 8 4 2 144 4 2 144 2144 144 a. b. 44 2 2144 144 5 3 44 5 332 3 33 55 55 3 2 3 2 342 2 52144 3 3 2 2 2 2 42 2144 9. 53 625 10. 3 2  a 8  338b 5 3 3 (29) 8  3 8 3 8 3 66 (29, 38) 6 5 3 25 3 2 3 2 3 2 3 2 3 2 2 s are in 3? ( 3 3 2 ) a 8  3 ab8  3 b 3  62  622 2 23 311. How 3many 62 2 428 32144 8 a 8 8 3 b 6 8 36 4 82144 3 (54) 8 3 5 3 5 3 12. (4 − 0.4) ÷ 4 13. 4 − (0.4 ÷ 4) (53) (53) 2144 n 1 2  the 7 in 8.7? 14. Which digit in 49.63 has the same place value as 3 10 100 (34) n 1 2  of $642.23 3 15. Estimate Find the and $861.17 to the nearest hundred 10 sum 100 (16) dollars. Explain how you arrived at your answer. 20 6

1

22

20 6

n1 12 n2   103 100 10 1003

n  100

3 2  8 3

 62

n 1  0 100

2 3

n 1 2 n 1   to draw a 23circle with a radius of 4 cm. 16. Elizabeth used a3 compass 10 100 5 3 2 10 100 2 (47)  62  3 a. What was the diameter of the and 6 circle? Describe 8 3how the radius 2 3 diameter are related. 5 3 2 (Use 3.142 for π.) 3 b. What was the circumference  62 of the circle?  3 8 6 8 3 3 2 2 * 17. Predict What is the next number in5this 2  6sequence?  (10) 3 6 8 3 …, 100, 10, 1, . . . 3 3 2 2 a8  3 b 3 8 18. The perimeter of a square is 1 foot. How many square inches cover its (38) area? n 1 2  3 10 100 19. Connect What is the ratio of the value of a dime to the value of a (23) quarter? n 1 2  3 10 100 Find each unknown number: n 1 2 20. 15m = 3 ∙ 102 21.  3 100 (4, 38) (42) 10 5 3 2 3 3 2 22. By what  6fraction  for 1 must 23 be multiplied to  23 b name a 8 with 8 form a fraction 6 8 3 (42) a denominator of 15? 23. What time is 5 hours 15 minutes after 9:50 a.m.? (32)

322

3 3 3 4

12 1 * 4. 5  212 2 3 (57)

Saxon Math Course 1

2 3 3 2 a8  3 b 3 2 a8  3

3 8

3 8

3 a8 

24. (38)

The area of a square is 16 square inches. What is its perimeter? Analyze

* 25. This figure shows the shape of home plate (60) on a baseball field. What kind of a polygon is shown? 26. (41)

The sales-tax rate was 7%. Dexter bought two items, one for $4.95 and the other for $2.79. What was the total cost of the two items including sales tax? Describe how to use estimation to check whether your answer is reasonable. Explain

27. Ramla bought a sheet of 100 stamps from the post office for $39. What (52) was the price of each stamp? * 28. (Inv. 6)

Represent

Draw a rectangular prism. A rectangular prism has how

many a. faces?

b. edges?

c. vertices?

Refer to the cube shown below to answer problems 29 and 30. * 29. Each face of a cube is a square. What is the (Inv. 6) area of each face of this cube? * 30. Find the total surface area of the cube by (Inv. 6) adding up the area of all of the faces of the cube.

Early Finishers

Math and Architecture

3 cm

The Pentagon in Washington, D.C. is the world’s largest office building. Each of the five sides of the Pentagon is 921 feet long. What is the perimeter of the Pentagon in yards?

Lesson 61

323

LESSON

62

Writing Mixed Numbers as Improper Fractions Building Power

Power Up facts

Power Up G

mental math

a. Number Sense: 5 × 40 b. Number Sense: 475 + 1200 c. Calculation: 3 × 84 d. Calculation: $8.50 + $2.50 1 3 of $25 10

e. Fractional Parts: 1 f. Number Sense: 3

$36.00

$25 10 5 36

5

36

5

36

5

36

g. Measurement: Convert 240 seconds into minutes. h. Calculation: 6 × 8, − 4, ÷ 4, × 2, + 2, ÷ 6, ÷ 2

problem solving

There are approximately 520 nine-inch long noodles in a 1-pound package of spaghetti. Placed end-to-end, how many feet of noodles are in a pound of uncooked spaghetti?

Increasing Knowledge

New Concept

Thinking Skill Model

Use your fraction manipulatives to 5 represent 3 6.

Here is another word problem about pies. In this problem we will change a mixed number to an improper fraction. 5 23 5 5 5 There were 3 6 pies on the shelf. The restaurant the 3 6 manager asked 3 6 6 6 server to cut the whole pies into sixths. Altogether, how many slices of pie were there after the server cut the pies? 1

$25

5

Saxon Math Course 1

6 6

5

3 3 4 We illustrate this problem with circles. 3 3 10 6 There were 3 6 pies on the shelf. 2 6 6 6 5 23 18 4 18 4 23 5 5 � � � � 6 6 6 6 6 6 6 6 6

�2 17 � cut the whole pies into sixths. Each whole pie then had six slices. 5� server The 3 3

324

5 3 � 6 6

1

The three whole pies contain 18 slices (3 × 6 = 18). The 5 additional slices 5 23 5 5 5 3 6 total to 23 slices from the 6 of a pie bring the (23 sixths).3 This problem 3 6 6 66 5 23 5 5 5 5 23 5 5 5 6 3 � that 3 6 is 3 66 3 6equivalent3to 6 . 3 � 3 3�6 6illustrates 6 6 6 6 6 6 23 5 5 5 3 3 Now we describe the arithmetic for changing a mixed number such as to 6 6 6 6 an improper fraction. Recall that a mixed number has a whole-number part and a fraction part. 6 6 6 5 18 18 23 5 � � � 6 6 6 6 6 18 6 5 6 236 6 5 6 236 6 5 6 6 5 fraction 23 23 18 5 18 23 whole number � � � � � � � � � � 6 6 6 6 6 66 6 6 66 6 6 6 6 66 6 6

5

5 6

36

18 6

18 6 5 36

5 6

2 17 5� � 3 3 �

5

36

18 6

23 6

18 6

23 6

18 65 36

18 65

3

6

23 66 6

5 6

5 3 � 6 6

�2 17 5� � of the mixed number will also be the denominator of the The denominator 3 3 �2 17 improper 5� � fraction. 3 3 �2 17 5 5� � 5 5 6 6 36 5 3�6 3 18 5 23 3 3 �66� 6 � 6 6� 5 � 23 23 3 � � 6 6 6 6 6 6 6 6 6 6 6 6

5

5

523 66



5 6

5 6

17 3 18 6

6

3�6

36 5 56 55 55 23 5 5 5 6 23 5 5 6 5 6 5 23 55 6 � � 66 � � � 3 3 3 3 3 3 3 3 3 3 3 3 3 6 6 6 6 6 66 6 36 66 66 6 66 66 6 6 6 6 6 66 same denominator 18 5 23 6 6 6 5 23 18 23 5 �2 � � � � � �In this case the 17 6 The denominator indicates 6 6 6 6 the 6 size 6 of the 6 fraction “pieces.” 6 6 6 5� � 3 3 fraction pieces are sixths, so we change the whole number 3 into sixths. We 5 5 56 23 5 5 5 66 18 53 �wholes 23 know that 5 � 66, which is 618 � 5whole is is 35× we6 3 63 3 6,�so 3three 3 3 one 6 36 66 6 . Therefore, � 566 18 6 6 623 6 18 6 6 6 63 66� 5 666 2365 6623 23 5 1823 5 18323 18 3 6 23 18 18 23 518 6 6 5 366 18 5 6 � � � � �� � � � � � � �� �� � add 66 and 6 to get 6 . 6 6 6 6 66 66 6 6 6 6 6 66 6 66 66 6 6 6 6 6 6 6 6 6 5 5 23 5 6 � 66 36 5 3 � 3 23 5 5 6 6 6 6 5 6 35 3 6 6 3 3 3 � 3�6 6 6 6 6 6 6 6 6 �2 17 5 23 23 65 623 18 23 6 6 65 623 6 5 18 23 6 5 23 18 18 �5 � � � 5� � � � �23 � � � � � �6� 5� ��5 � 23 6 6 6 3 6 66 6 66 66 6 6 6 6 66 66 66 66 �2 17�62 17 � 26 17 6 6 6 3 6 5� � 5� � 5� � 3 3 3 3 3 3 18 5 23 6 6 6 5 23 23 5 � � � � 23 � � 18 5 23 5 6 18 6 18 6 6 6 6 6 6 6 �66 � 66 � 5 � 23 � � 6 6 6 6 6 6 6 6 6 6 6 6

Example 1

�2 17 5� � 3 3

3 3 Write 2 4 as an improper 2 fraction. 4 4 �2 17 5� � 3 3

4 4

4

24

8 4

8 4

Solution

The denominator of the fraction part of the mixed number is fourths, so the denominator of the improper will 3 8 3 fraction 2 11 2 also be fourths.   2 35 3  3 35 3 4 4 4 4 4 4 2 43 24  28  4 3 42  4 4 4 24 2  4 4 4 4 4 34 3 3 4 3 84 4  1 equals 4, the We change the whole fourths. 2 Since whole 2 44 2 4 number224 into 2  2 4 4 8 4 8433 4 3 4 3 33 3 3 11 3 88 4 4 48 884 11 443 44 11 118  22     2 equals 2  × , which 222 2 4 244  2 number 2 is 44.44We add 4 and4 44 to get 444. 4 4 4 4 44 4 4 4 4 4 44 24 1744 4 2 4 44 2 2 2 2  2 5 13 2 1 3 5 8 33 11 21 3 2 5 3 3   3 28 3 11 5 32 5  2 4  45 3 4 3 2 5  4 4 4 4 3 3 4 3 8 3 11 8 32 11 2 3 2 2 5 3    2 53 5  5  2 3 4 4 4 3 3 4 4 4 4 3 4 3 33 11 2 88 2 113 8 22 3 118 3 2 2211 2  555     2  2  53 5  5 5  55  3 4 3 11 3 3 3 4 3 33 34 33 2Draw 3 44 44 44 Represent 444 4 33 4 5 103 6to2 show 12 that 2 4  = 4. 5 4   a model    2 17 2 2 2 5 5 3 3 3 5  2  17 52 1 32 2 52 1 32  25 5 13 3 25 13 3 3 3  2 17  22  17 2 2 2  51 1 3 2 25 2 12 1 22 5 3 32  23322 3  222 17  22 22 17 2 5 23 23 5 22 17 17 2 22 2 22 2 2 2 22 2  5 122    2 2 5 122 1 2 1 3 211 2 5 122 1 3 211 11533 53 3 55 5 33 5 5 3 355 3 33 55 3 3 33 3 33 3 2 5 10 2 12  2  12   Lesson 62 325 3 2 5 10   5 5 3  3  5 3 5 5 3 3 3 5 2 52 1210 2 12 3 2 5 3 10

24

2  4 4

24

4

4

4

Example 2 3 11 8 35 2 as an improper 3 4 2 5 fraction. 2  44 2 44  4  Write 4 3 3 4 2 4 3 4

3 2 4

8 4

4

8 4 5

5 6

Solution

4

3 4

11 4

23 6

36

5

36

We see that the denominator of the improper fraction will be thirds.

Thinking Skill  Evaluate 2 17

2 2 31 2 8 2 2 3 11 22 2 2  1 2 5   5 31 3 5 2 5 5 3 3 3 2 4 3 of 4 54 3 3 4 Which method 3 3 3 8 85 3 8 3 4 4 8 418 18   2  44  24 2 2 4 a mixed 2 Some 2 changing 4 4 4 to find 46of 46 4 4 mechanical method 4 4 use a quick, 4 4 people 64 the numerator number to an the improper fraction. Looking at the mixed number, they multiply the improper fraction denominator by the whole number and then add the numerator. The result is you prefer, 3 32 3 2 4 2 4 3 do 10 2 5  2 2 1712 3 8 4 4 2 2 1  2 42improper the   the numerator 2of 2 fraction. 2  42 12 4  method in 13 4 25 48 4 4 the 4 38 4  3 5 43 5 33 3 3  455 84 3 8 3 4 5 11 4 11 3 4 42  43 211 2 3 1 or4 3 2 example 2 8 3 8 2 2 11 2 4 4 4 4 4 4 4 4 � 4 4 4  17    5 2 53 2 4 2? Why? 5  5 52 � example 3 3 4 4 4 4 48 4 84 4 4 834 3 33� 311 311 8 3 4 2  42  4 4 4 4 4 4 4 4 4 4

8 4

2

3 23 11 44 6

11 4

2 5

8 4

3 11 8 11 3 3 8 3 2 2 5 3 2   5 23 5 23   Example 310  2 212 2 5 5  3  4 4 4 8 3 811 3 3 11 2 4 4 4 3 3 2 4 5 5 5 5  3 32 5 217     2 52 2 2 17 5 3 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 45 4 42 354  2 5 the 1improper 1 2 25 1 3 Then 2 5 multiply 2 1 3Write 1 3 and 2 5 as improper 1fractions. 3   3 5 5 3 3 3 3 3 2 2 2 2 fractions. What is the product?   5 5 5 3 3 3 33 3 5 12 60 60 5 12 2 2 � � �4 13 � 25 5 3 Solution 5 3 15 15  2  17 2 17 2 2 2 2 2 2 2 2 2 25 25 1 5 5  1 3 1 3 25 25 13 13 7 2 3 2 2 2 3 2 2 2 2 2 3  233 217 2 2  and 2 5 as improper 25 1 2 13 2 2 255First2we 1fractions. 2 13 1 355 2 10 10 3 12 2 1 312 3 write  3 5 5 3 3 3 3         3 2 3 323 3 53 2 5 25 5 5 2 2 25 2 2 2 2 1 2 2 1 2 1 2 1 5 5 3 3 5 5 5 3 3 3 5 5 3 4 1 1 2 3 1 6 1 3 5 3 4 4 6 10 5 12 60 60 12 3 5 2 5 10 10 2 12 3 2 2 512 � � �4        5 3 5 15 15 2 105 2 1012 2 12 3 3 3 3 5 55 3 55 5       3 53 5 5 5 5 5 60 5 12 605 12 60 60 5 5 12 12 �� 4 � �142 � 2 2 1 2 � 2 25 � Next we 512 3 � 12 35by 5 . 512 3 multiply 515 5 60 3 602 5 2 32 55 1215356012 60 60 6012 15 15 2 5 2 2 2 � � � � � � 4� � 4 1 3 � 4 2 51 3 � 2 51 3 � 2 5 560 5 12 3 35 3 3 5 5 5 3 15 15 15 15 3 15 15 5 60 5 12 2 2 5 12 � 5 12 60 � � 4 60 � 41 3 � 2 5 1 2 � 2 2 5 3 3 5 315� 5 � 1515 5 5 3 5 60 60 5 60 60 12 5 60 60 12 12 3 3 5 42 2 � 2 1 15 1 12 12 125 2 2 2 � � � � � � � � � � � 4 1 2 4 1 2 4 1 2 2 3 1 6 5 3 3 5 53 5 which 3 5 15 3 15 15an 5improper 15 15 fraction, 15 3 5 5 The 55we3simplify. 3 4 4 result is3 3 5 3 5 3 4 1 1 4 1 1 602 60 3 60 60 560 2 12 2 60 5 2 2 5 12 26 2 2 12 6 1 23 3 1 5 1� 60 1 431560 1223 1241 12 2 2 � �60 � 41 3� ��2� 45 132 � 2 21 3��42 5 55 2 5 1 34 � 2 52 54512�3 3 � 3 52 651 10 4 1 15 3115 15 60 4160 3� 4 �6 �1 � �4 3 �1 4 2553 4� 13 6 21 5 5 3 15 15 3 15 15 2 2 3 1 6 1 3 2 3 1 6 1 5 5 5 5 3 3 3 3 15 5 54 4 6 31 43 15 14315 5 41 15 5415 65 10 63 333 5 3 4 151 3 5 12 60 60 211122 � 2 22equals 334. 1 6 1 3 � � � 3 1 6 1 3 So 4 2 5 3 4 4 6 10 5 3 56012 1560 60 1 602 6012 3 60 5 5 12 12 35 2 2 12 2 2 2 3 4 4 6 10 34 12 15 5 3 3 5 3 15 1 1 1 5 3 4 1 1 1 1 115 1 1 1 � 1 � � �641� �64 1 3 �12 4651 3 �12 51 3 �3215 2 35 3 35 � 513�3 15 3 2 3 12 12 3 32 5 23 32 12 315 5 415 5 3 6 10 32 6 2 4 15 4 15 6 3 6 35 43 5 4 15 64 10 310 62 Practice Set Write each mixed number as an improper fraction: 60 5 2 12 5 60 2 21 60 2 1 13 131 1 52 3 3 1235 11 15 1 1 1 1 � 1 �� 4 23 543 � 3 334 �2 1113 1221141 3 62415 1235 12116 31 3 21 31 361 15 213 53 12 12 5 15 313 5166 15 615 1 3 363142 2336 12 a. 3 c. 234 b.26 2 10 3 5 4 4 6 10 2 6 3 6 10 3 2 4 6 10 5 4 6 10 353 2 634 4 46 610 3 5 3 1 1 1 1 1 1 3 3 16 1 5 2 3 53 1 12 23 3 6 13 1 4 1 1 1 11 1 64 3 1 3 105 11 11 43 2 3 1 6 1 12 3 3 32 2 2 3 1 1 3 3 12 d. 6 e. 1 f. 2 3 1 6 312 12 5 3 4 6 10 32 6 3 4 4 6 10 3 6 3 4 4 6 10 32 62 6 1

5 1 3 46

1

1 3 6 1

11

12

5 13 6 10 4 1

32

312 2

32 1

12 1

12 3

1

32 326

1 1 1 531 1 1 1331 1 1 g. 3 2 1 2 1 12 1 212 1 3 2 h.1 12 3 213 2 12 i. 3 10 2 3 6 3 1 2 1 3 6 1 6 10 3 31 2 6 3 2 2 2 2 11 121 1 1 1 12 3 j.2 Write 1 and 3 12 as improper 1fractions. Then multiply the improper 3 2

3

2

2

2

fractions. What is the product? 1

3121 2

1

1

13 2

32

1

32

1

1

22

Saxon Math Course 1 1

22

1

1

1 22

1

22 13 1 1 2 1 23 22 1

1

1

13 12 1 1 1 32 13 1

1

13

12

1

12 1 12

1

12

2

4 5

Written Practice 1

12

1 32

3 1 1 3 4 Strengthening Concepts 3

5

3

3 1 12

6

1 4

5 6

1

15 3

3 10

3 4 1 2 3 1 5 3 4 1 whole notes, 1there are 1. 1 In1 music notes, and eighth 1 half notes, quarter 1 3 3 (54) 2 2 12 32 2 notes. a. How many quarter notes equal a whole note? b. How many eighth notes equal a quarter note?

1 1 * 2. Don is 5 feet 2 12 inches tall. How 1 3 many inches 1 2tall is that? 1 (62)1 12 32 3. Classify Which of these numbers is not a prime number? (19)

A 11

B 21

C1 31 1 2

1 3 2 D 41

1 1 1 1 1 11 1 1 1 fractions, 2 2 * 4. 2Analyze 1 3 Write 1 3 and 1 2 as 2improper 2 1 3 and multiply 1 2the (62) 2 2 improper fractions. What is the product?

5. If the chance of rain is 20%, what is the chance that it will not rain?

(58)

6. The prices for three pairs of skates were $36.25, $41.50, and $43.75. 1 What was the1average price for 1 a pair of skates? Estimate to show that 22 1 1 your answer is3 reasonable. 2

(18)

7.

(43)

1 1 1 3 1numbers 1 Instead1 of dividing 15 by21 23 , Solomon doubled 1 3 3both 13 m 4  3n  2  m 2 24*  3 2 2* 5 8 4 8 4 10 and then divided mentally. What was Solomon’s mental division problem and its quotient?

Evaluate

1  2

3 1 3 3 3 unknown 1 3 3 5 3 51 13 5 15 4 1 1 1each Find 4 3 n     mnumber: n   m2  2* 4 8  3 4 5 8 4 10 2 4 8 5 10 2 4 8 6 2 2 31 3 33 3 1 13 31 53 55 15 1 6 15 2 1 1       43  3    9. n  2 2* 2 2* * 8. m  4m  n 5 81 384 5 43 53 1 1(43, 56) 2 3 3 3 105 3 (43,159) 1 10 4 5225 1142 84 1864 262 21            n m4  3 n 10 6d5 = 0.456 2 4 8 10 65 2 2 11. 2 5 =63 8 410. 4 0.04w 8 2 5 3 2 1.522 1

2 2* 31 1 m  42 2 3 8 * 4

(43, 45)

31 33 4 3 10 54 8

(43, 49)

3 1 3 3 53 11 3 51 4 5 2 1 1 1 3 53     * 13.  * 12.  n  m  4 n3 8 4 10 5(61) 2 4 810 65 22 4 82 (57)5 6 3 2 2

3 1  42 12* 3 8 4

1 3 5   2 4 8

5 31 3 1 1 34 52 15 1   14.    n   6 102 5 (29)2 2 45 83 26 2 16. 1 − (0.2 − 0.03) (40)

1 4  2 5

2 1  3 2

1 4 2 1  15.  2 5 2 (54) 3 17. (0.14)(0.16)

(39)

18. One centimeter equals 10 millimeters. How many millimeters does (49) 2.5 centimeters equal? n

3 3  10 5

1 3 5   2 4 8

1 all of the 1common 4 2 factors 1 19.5  List of 18 and 24. Then circle the greatest   (19)6 5 2 2 3 2 common factor. * 20. (58)

5

Analyze

3

3 6of the marbles 2 4are red. Ten marbles are in a bag. Four

4 4

a. 1If one marble is drawn 3 3 3 from 1that33it5 will5 1 the bag, what is the probability   6 m4 3 2 2* n 5 8 4 10 2 4 8 be red? b. Write the complement of the event in a and state its probability. c. Describe the relationship between the event and its complement.

21. (38)

Analyze

If the perimeter of a square is 40 mm, what is the area of the

square?

Lesson 62

327

5  6

22. At 6 a.m. the temperature was −6°F. At noon the temperature was 14°F. (14) From 6 a.m. to noon the temperature rose how many degrees? 23. Lisa used a compass to draw a circle with a radius of 1 12 inches. (47)

1

32

a. What was the diameter of the circle? b. What was the circumference of the circle? (Use 3.14 for π.)

The circle graph below shows the favorite sport of 100 people. Refer to the graph to answer problems 24–27. Favorite Sport of 100 People

1

Baseball 40 Other 16

3 1 m4 3 8 4

1

2 2*

n

3 3  10 5

Basketball 22

Football 1 3   2 22 4

5 8

1

22

5 1  6 2

13

1 4  2 5

2 1  3 2

24. How many more people favored baseball than favored football? (40)

25. What fraction of the people surveyed favored baseball? (40)

26.

Was any sport the favorite sport of the majority of the people surveyed? Write one or two sentences to explain your answer.

27.

Since baseball was the favorite sport of 40 out of 100 people, it was the favorite sport of 40% of the people surveyed. What percent of the people answered that football was their favorite sport?

(40)

(40)

Explain

Connect

28. What number is 40% of 200? (41)

29. Here we show 18 written as a product of prime numbers: (19)

2∙3∙3 Write 20 as a product of prime numbers.

* 30. (Inv. 5)

Analyze

Judges awarded Sandra these scores for her performance on

the vault: 9.1, 8.9, 9.0, 9.2, 9.2 What is the median score?

328

Saxon Math Course 1

1

12

8

72

8

12 ,

100

LESSON

63

Subtracting Mixed Numbers with Regrouping, Part 2 Building Power

Power Up facts

1

2 2 3   3 2 1 3 1   2 3

2

12

33

Power Up D

mental math

a. Number Sense: 5 × 140 b. Number Sense: 420 − 50 c. Calculation: 4 × 63 d. Calculation: $8.50 − $2.50 1 8

1 e. Number Sense: Double 7 12. 8

f. Number Sense:

$25 100

$25 100

1 1 1 12 , 6 , 4 ,

...

g. Measurement: How many inches are in 6 feet? h. Calculation: 5 × 10, − 20, + 2, ÷ 4, + 1, ÷ 3, − 3

problem solving

Rhett chooses a marble at random from each of the four boxes below. From 2 2 4 which box is he most likely to choose a blue marble?3 3  2  3 6 3 1 3 B C 1 A 2 1  1 D 12 $25 2 $25 3 61 1 1 1 1 1 1 33 1 1 1 1 1 72 72 , , , . .,. , , . . . 8 8 8 8 100 1 12 6 4 12 6 4 100 2 6

New Concept 1 8

$25 1 1 1 1 7 2 48 we have practiced subtracting ... 12 , 6 , 4 ,mixed Since Lesson numbers with 100 4 2 2 2 42 regrouping. In this lesson we will rename the fractions with3common  3 3  3 3 2 6 3 2 6 denominators before subtracting. 3 3 3 3 1 1 $25 1 2 1 2  1  1 1  1 To subtract first rewrite the fractions with common 1 2 from 3 3, we 3 12 2 3 6 2 3 6 100 3 denominators. Then we subtract the whole numbers and the fractions. 1If 1 2 2 6 6 possible, we simplify.

1 8

1

12

Increasing Knowledge

2

33

2 2 4 3  3 3 2 6 3 1 3 1  1 2 3 6 1 2 6 When subtracting, it is sometimes necessary to regroup. We rewrite the $25 $25 fractions 100common denominators before regrouping. 100 with

$25 100

Lesson 63

329

9

9

9

4 3 1 3 1 4 33 1 4 3 3 Example 1 21 2 21  5  5 5  4 95 5 5 5  15  1 5  1 6 1 32 33 2 6 3 2 6 3 22 32 3 32 1 Thinking Skill 5  5 Subtract: 5  1 2 32 26 2 4 2 2 4 2 4 2 3  1  1 1 1  1  1 Explain 6 2 23 24 3 6 2 3 6 2 1  1 Solution Why did we 5 3 29 6 9 5 9 5 3 39 3 regroup in this 6 1 3 1 4 3 3 154 3 3 69 4 3 69 1 21 21 2   9 4 3    5 5 5 5 5 5 4 4 3 3 3  1 5  1 5  1 5 We rewrite the fractions with common example but not 3 3 3 1 1 1 2 3 6 2 3 6 2 3 6 1 2 1 2 1 2 2 31 2 5 23 1 2 5 3 1 5  1  463  5  5 5 5 5 1 53 in the previous 22 35 3 2 2 653 2142 6 51 3 1 61 denominators. Before we 2 31 2 3 3 1 11 132 1 25 2321243 1can 9subtract, 12 11 2 1 5  4 2 9 9 211 6  131716 3761  31 6  1 64 5  3 54  32 4  2 46  11671 2 5 After 3 3 we    problem? we must regroup. subtract, 2 2 4 2 2 4 4 2 4 4 21 3 2 11 3 33 2411 3 33 4 1 33 423 343 2 22  463 36 12 4 236 2 6 36 1 169 1 62 6 11 4 2 21 1 21 2 9 1  11 2 331592 4163  5  35 1  3 5 4 5 25 6 5 1 7 4 1     5 15 15 1 3 2 6 2 4 65  we simplify if2possible. 32 3 24 6 3 32 6 3 2 6 4 45 42 533 6 2 3 6 363 323 1 32 2 32 3 1 331 31 3 136 21 1 21 2 3 5 5  5 56 5 56 5 5 5 5  1 5  12 5 2 12 4 2 2 4 2 66 3 2 3 2 6 3 25 6 3 4 3 2 3 2 1 9 32 1 9 1  1 9 31 1 6 6 6 3 3 2 4 33 6 2 4 33 6 2 6 2 2 2 4 2 264 2 4 4 3 3 3 1 3 1 1          21 2 1 1 1 1 1 1 35 5 2 3 6 52 2 31 1  1 15 23 3 1 3 65 1 1 61 1 1 1 1 11 5 1 155 1 515 1 5 15 1 5 5 2 3 35can 4use 2 6check 13 7 1 32 6 3 12 4 5 3 4 1 126to3 6the1 7 31 6 52 5 3 4 6 33 2 1we 6 33 32 3 Justify 31 1 52 3 12 1addition 23 67 11 26 11 1 24 3 1 11 2 11 46 answer? 2 Why3 4 2 4 3 6 6 2 3 2 4 3 2 3 6 2 4 3 2 4 3 6 6 6 6 6 5 1 3 5 13 5 4 1 3 2 4 12 4 6 3 2 1 6 21 6 7 5 1 3 7 513 7 6 51 3 1 6 11 6 4 2 21 2 3 42 2 42  44 2 42 7  3 63 6  6 26 4  32 31 4 32 6 31 2 346  4 3 4 6  123 13 124 4   1 52 1 13 2 4 3 6 2 3 6 6 6 3 2 3 62 3 62 6 32 51 3 5 32 51 11 11 11 11 1 11 11 1 11 11 1 2 1  34  3 5 Practice 4  2Subtract: 6 5 2 3Set 4 1 26  17 6 55 3 1 17 55 3 16 7 515 1 3 156 1  1 54 6 12  1 534 2  1 34 22  1 3 2 34 32 6 2 3 2 2 32 34 6 2 3 2 2 4 14 3 2 1 1 4433 2 4433 1 1 64636 3 1 1 6 6 3 3 5 3 5 3 5 1 1 2 2 1 1 1 42 2312 1 1 2 12 1 1 21 1 1 1 2 6 6 5 243 3 7 5 84 35 8 1 6 2 6 33 7 23c.  a. 5  3 55 616 35 24 1 6  16 4 36b. 2 14 36256 6 3 36 126 3 668 7 3 1 4  1 14  3 4 6 3 4 2 3 4 3 3 2 63 26 2 43 2 4 63 4 2 83 42 63 6 2 16 26 3 12 3  1 32  3 34  5 42 4 34 5 2 6 3 2 6 3 4 3 5 32 51 11 1 11 32 51 11 11 11 11 11 1 2 1  34 1 5 2 34  26 4 1 2 6 d.1 7 6 3 17  36 7 e. 36  14 6  1 4  f.1 4  1 34 4 6 2 23 34 32 43 66 23 23 2 4 32 2 43 3 66 6 2 9 4 3 1 3 1 2 5 51 32 51 32 5 3 25  21 5 1 5  11 5 1 2 1 2 1 1 1 g. 4  1 5 6 15h. 3 6 15 5 8 51 8 1 43 5 2 32 3 2 6 2 2 21 2 1 25 1 1 5 4 51 34 3 1 4 51 3 32 5 44 3 32 2 13 66 2i.5 48 2 6 352 2 4 3 6 12 4 3 4 6 4 13 1 3 6 2 6 3 6 5 1 3 6 23 6 5 8 2 5 8 8 3 3 4 3 2 5 4 3 4 1 4 2 32 1 22 1 2 6 3 6 61  2 62 8  3 4 3 43 142 3 33 65 6 3 2 6 41 22 4 1 15   6 Write 3 a word 2 problem 6 3 4 subtracting that involves j. Formulate 3 mixed 2 6 52 3 22 5 5 3 5 52 32 11 15 11 1 numbers. 1 1 1 12 1 1 1  1 4  16 225 22 9 1 15 2 2 22 4 3 16  38 6 5 38  54 8  5 43 4 23 2 52 1 25 32 63 6 36 6 6 32 2 54 6 3 1 5 4 33 1 5 4 3 51 32 51 33 2 32 1 2 1 1 2 1 2 1 45 6 21 8  58 15 4  14  14 6 1 3 6 5136 5 83  43  5 43 2 32 1 25 4 2 1 2 2 3 56 36 3 4 6 Strengthening 36 2 Concepts 62 2 62 3 4 3 3 4  1 5 3 6 Written 2 3 Practice 31 5 1 1 1 2 1 1 1 5 3 52 3 22 5 52 35 1 11 11 1 1 12 2 1 2 1 1 41 2  32  21  3 2 12 4 7 6 1 4   16 23 2 1 5  1 42  16 4 3 16  38 6 5 38  54 83  5 4 2 4 1 25 2 13  2 4 32 522between 6 6 2 3 6 4 32 63 6 32 2 6 3 * 1. 4 Connect 3 4 3 What3is23 the difference 62 the sum of 0.6 and 0.4 and (12, 53) 9 5 the product of 0.6 and 0.4? 4 3 3 1 3 6 5  5 2 3 6 * 2. Analyze Mt. Whitney, the highest point in California, has an elevation (14) 2 21 4 3 sea level. 5 1 1 1 above 2 1 1 1 1 Death 1  15 1   2of 14,4946feet 1  1can see 4  1Valley, 3 4 7  3From there 6 one 32 2 3 6 4 3 2 4 3 6 6 2 3 2 which contains the lowest point in California, 282 feet below sea 5 level. The floor of Death Valley is how many feet below the peak 3 6 of Mt. Whitney? 5 5 3 1 1 2 2 1 1 41  11 6 3 8 5 43 22 15 3 5 2 1 1 6 3 2 6 3 4 1 7 3 6  1 Conclude 4 It 1 * 3. was 39° outside at 1 p.m. By 7 p.m. the temperature had 4 3 6 6 (10) 2 3 2 dropped 11° and was below freezing. What was the temperature at 7 p.m.? On what scale is the temperature being measured? Explain

5 1 4 1 6 3

5 3 1 2 1  3Write the 8mixed  5number 4 23 as an improper 6 * 4. 2 2 fraction. 2(62) 6 3 4 * 5.

3 4

2

1

43

22

Explain

7. (30 × 15) ÷ (30 − 15)

5 8

330

25 38

1

22

Round $678.25 to the nearest ten dollars. Describe how you decided upon your answer. 5 6 3 2 1 2 1  w  3  1a.m.? How did you find 6. Explain1 15What time 8 is3 2 2 hours after 10:15 3 2 8 4your (32) answer?

(62)

5

1

15

(5)

2 2 21 1  31  1 9. w  3w  3 3(43) 32 2

6 36 3   10. 8 (55) 48 4

Saxon Math Course 1 3 4

55 22 22 11 ww33 11 88 33 33 22 3 23 2 5 3 23 2 1 15     6  65  5 4 4 54 5 48 8 4 54 5

8. Compare:

(56)

1

18

5 1 6 5 4 8

5 8

2 3

2 1 w3 1 3 2

52 2 1 3 1 83 3 2

2 6 13 w  3  1 3 8 24



3 4

6 3  8 4

5 3 23 2 3 2 1   6 5  12. 8 4 (29) 54 5 4 5 4

1 65 3 * 11. 6  5  4 88 4 (63)

3 2  4 5

5 1 6 5 4 8

13. (54)

3 2  4 5

5 1 6 5 4 8

3 2  4 5

3 2  4 5 3 2  4 5

14. (1 − 0.4)(1 + 0.4) (53)

15. How much money is 60% of $45? 13 1 184 18 16. 0.4 ÷ 8 17. 8 ÷ 0.4

3(41) 4 3 4 3 4

(49) 9

(45)

16 33 43 3 2 5 2 1 1   5in this sequence? 18. 3Predict 5 number  1 What is the next  6 5 w 28 34 6 (10) 3 2 4 8 4 5 0.2, 2 0.6, 2 0.4, 4 0.8, … 1 1  1 18 3 2 6 9 9 4 3 is the tenth prime number? 19. 3 3 What 1 3 14 5 5 5 (19)  5 5   3 6 2 3 2 63 6 * 20.2 What 2 2(8, 2  459)  4 is the perimeter of this rectangle? 1   1 1 1 6 1 3 51 1 1 3 2 1 3 6 12 2 1 1 1 1 5 3 4 2 6 1 7  3 1 8 in. 6  1 4 1 2 3 4 53 2 4 3 6 6 2 3 2 5 3 3 6 6 1

11 8 5 2 2 1 5  2 8 3 3

3

3 2  4 5

3 in. 4

1

32 11 11 1 3 1 5 2 1 51 1 1 1 1 1 1 2 4  2 6  1 6 41 7  3 7  386  1 6  1 4  1 4  1 32 43 66 23 2 3 4 4 2 6 3 2 6 2 3 * 21. A triangular prism has how many (Inv. 6)

5 1 4 1 6 3

a. faces? 5 3 1 2 6 3 8 5 2 6 b. edges? 3 4

2

1

43

1

22

1

22

15

c. vertices? 52 32 5 1 3 2 3 6  38  5 8  54 3 2 6 3 4 6 3 4

2

43

1 1 1 22.2 12Write 2 12 and 1 15 as improper 2 2Then multiply the improper 15 2fractions. 2 (16) fractions and simplify the product.

23. This rectangle is divided into two congruent (31) regions. What is the area of the shaded region?

30 cm 10 cm

* 24. A ton is 2000 pounds. How many pounds is 2 12 tons? 25. (50)

Connect

Which arrow could be pointing to 0.2 on this number line? A

–2

26. (47)

3 4

1 2

(15, 62)

B

–1

C

0

D

1

Evaluate The paper cup would not roll straight. One end was 7 cm in diameter, and the other end was 5 cm in diameter. In one roll of the cup,

a. how far would the larger end roll?

2

Use 3.14 for π.

b. how far would the smaller end roll? (Round each answer to the nearest centimeter. Use 3.14 for π.)

Lesson 63

331

22

2

4

4

27. Jefferson got a hit 30% of the 240 times he went to bat during the season. Write 30% as a reduced fraction. Then find the number of hits Jefferson got during the season.

(29, 33)

28. Jena has run 11.5 miles of a 26.2-mile race. Find the remaining distance (43) Jena has to run by solving this equation: 11.5 mi + d = 26.2 mi 1

1

3

1

29. The sales-tax rate2was 7%. The two each. What 2 2 CDs cost $15.49 4 4 was (41) the total cost of the two CDs including tax? 3 1 1 * 30. Rosa is mixing paint in ceramics class. 2 2 She mixes 2 teaspoon of yellow 4 5 3 1(57) 1 paint with teaspoon of red paint to make orange paint. How much 2 4 4 12 orange paint does Rosa make?

1 22

Early Finishers

1 Real-World 2 Application

1 2

3 4

1 2

332

1

22

The drama club had their first annual meeting this afternoon. The officers 5 the new members this year. They ordered 3 had decided to 1order pizza for all 4 4 12 one cheese, one mushroom, and one tomato pizza. Due to the rain, the turnout for the meeting was small and there was a lot of pizza left. They 5 5 51 31 1 1 1 13 had 4 of the pizza,4 2 of the mushroom pizza and 124 of the tomato 12 2 12 2 2cheese 44 pizza left. How much leftover pizza did the drama club have?

Saxon Math Course 1

5 12

1 4

LESSON

64

Classifying Quadrilaterals Building Power

Power Up facts

Power Up J

mental math

a. Number Sense: 5 × 240 b. Number Sense: 4500 + 450 c. Calculation: 7 × 34 d. Calculation: $7.50 + $7.50 $75

1 4 of $75 10

e. Fractional Parts: 10 f. Number Sense:

$20.00 1 4

g. Measurement: How many meters are in 200 centimeters? h. Calculation: 6 × 8, ÷ 2, + 1, ÷ 5, – 1, × 4, ÷ 2

problem solving

Emily has a blue folder, a green folder, and a red folder. She uses one folder each for her math, science, and history classes. She does not use her blue folder for math. Her green folder is not used for science. She does not use her red folder for history. If her red folder is not used for math, what folder does Emily use for each subject? Make a table to show your work.

New Concept Math Language The prefix quadrimeans four. A quadrilateral is a polygon with four sides.

Increasing Knowledge

We learned in Lesson 60 that quadrilaterals are polygons with four sides. We can classify (sort) quadrilaterals by the characteristics of their sides and angles. The following table describes the various classifications of quadrilaterals: Classifications of Quadrilaterals Shape

Characteristic

Name

No sides parallel

Trapezium

One pair of parallel sides

Trapezoid

Two pairs of parallel sides

Parallelogram

Parallelogram with equal sides

Rhombus

Parallelogram with right angles

Rectangle

Rectangle with equal sides

Square

Lesson 64

333

Notice that squares, rectangles, and rhombuses are all parallelograms. Also notice that a square is a special kind of rectangle, which is a special kind of parallelogram, which is a special kind of quadrilateral, which is a special kind of polygon. A square is also a special kind of rhombus.

Example 1 Is the following statement true or false?

Thinking Skill Justify

All parallelograms are rectangles.

How can we rewrite the statement in example 1 so that it is true?

Solution We are asked to decide whether every parallelogram is a rectangle. Since a rectangle is a special kind of parallelogram, some parallelograms are rectangles. However, some parallelograms are not rectangles. Since not all parallelograms are rectangles, the statement is false.

Example 2 Draw a pair of parallel lines. Then draw another pair of parallel lines. These lines should intersect the first pair but not be perpendicular to the first pair. What is the name for the quadrilateral that is formed by the intersecting lines?

Solution We draw the first pair of parallel lines. We draw the second pair of lines so that the lines are not perpendicular to the first pair. At right we have colored the segments that form the quadrilateral. The quadrilateral formed is a parallelogram.

Practice Set

a. What is a quadrilateral? b. Describe the difference between a parallelogram and a trapezoid. c.

Model

Draw a rhombus that is a square.

d.

Model

Draw a rhombus that is not a square.

e.

Verify

True or false: Some rectangles are squares.

f.

Verify

True or false: All squares are rectangles.

Written Practice

Strengthening Concepts

* 1. When the sum of 1.3 and 1.2 is divided by the difference of 1.3 and 1.2, what is the quotient?

(12, 53)

2. William Shakespeare was born in 1564 and died in 1616. How many years did he live?

(13)

334

Saxon Math Course 1



3. Duane kicked a 45-yard field goal. How many feet is 45 yards?

(15)

* 4. (60)

Explain

88 88

1 ?  4 100

Why is a square a regular quadrilateral?

8 82 8 23 1 each 219 1 1 3 1 2 2How 9 1 ? 81 88? 1 312long1is 1 A regular ? * 5.  has a perimeter   hexagon    5   3  5  53 of363 inches. (60) 83 8 34 2 310 4 100 84 8100 88 4 2 3 32 443 410 4 side? 100 2 42 6.

(42)

91 62 6  10 2

1 ? 1 13 22 13 2 9 11 12 9 1 1 19 1 8 ?8  8 28  83 2 ?8  * 8.  7.  5 3 5  5 3      6 3  4 100 4 100 8 44 3 1024 23 104 2 21 4 8(5)100 8  8 38  84 3 (59)24 33 2

?8  7 1 ? 1 13 22 13 2 9 1 12 9 1 1 19 71 1 1 8 ?8  8 28  83 2 7  9. 3  10.   6   2 6 2 5 3 5* 2 6 *11. 5 3    8  8 38  84 3(61) 24 33 244 3 1024 (57) 100 4 100 8 23 104 2 210 82 2 (63) 8 2 4 8 100 8 12. Compare: 2 × 0.4 (44)

2 + 0.4

13. 4.8 × 0.35 (39)

1 ?  4 100

1 2

3 1 4 2

1 2 15.

14. 1 ÷ 0.4 3 31 4 42 $0.12 pencils

(49)

3 can4 Mr.

many Velazquez buy for $4.80? 9 1 2 1 1 2 (15) 3 1 7    5  3 6 2 3 4 2 3 4 10 2 2 8 16. 1Estimate 1 Round of 0.33 and 0.38 to the nearest 3 1the product 3 3 (51) 2 4 2 4 2 4 hundredth.

88 88

3 4

1 2 How

3 4

1 2

3 4

17. Multiply the length by the width to find the (31) area of this rectangle.

1 in. 2 3 in. 4

1 2

3 4

* 18. (64)

19. (19)

Conclude Analyze

Is every rectangle a parallelogram? What is the twelfth prime number?

20. The area of a square is 9 cm2. (38)

a. How long is each side of the square? b. What is the perimeter of the square?

Refer to the box shown below to answer problems 21 and 22. * 21. This box has how many faces? Draw a net (Inv. 6) to show how the box would look if you cut it apart and flattened it. * 22. If this box is a cube and each edge is (Inv. 6) 10 inches long, then a. what is the area of each face? b. what is the total surface area of the cube?

1

1

12 22 23. There are 100 centimeters in a meter. How many centimeters equal (15) 2.5 meters? 2  2  22 32 32  3  3 1 1 1 1 * 24. Write the mixed numbers fractions. Then multiply 1 2 and 2 2 as improper 22 12 2  2  32 52 53  5  5 (62) the improper fractions and simplify the product.

Lesson 64

335

* 25. (19)

The numbers 2, 3, 5, 7, and 11 are prime numbers. The numbers 4, 6, 8, 9, 10, and 12 are not prime numbers, but they can be formed by multiplying prime numbers. Verify

4=2∙2 6=2∙3 8=2∙2∙2 Show how to form 9, 10, and 12 by multiplying prime numbers. 26. Write 75% as an unreduced fraction. Then write the fraction as a decimal number.

(33, 35)

1

12

1

2 227. Reduce: (54)

28. (43)

3 2

Analyze

22233 22355 Find the missing distance d in the equation below. 16.6 mi + d = 26.2 mi

6 75  52  15  3 34 Refer to the double-line graph25 below to answer problems 29 and 30. 4 100

16

Daily High and Low Temperatures high low

Temper ature ( °C)

12 8 4 0 –4 –8 Sun.

Mon.

Tues.

Wed.

Thu.

Fri.

Sat.

* 29. a. The difference between Tuesday’s high and low temperatures was (18) how many degrees? b. The difference between the lowest temperature of the week and the highest temperature of the week was how many degrees? * 30. (18)

336

Predict If the daily high temperature dropped 5 degrees the day after this graph was completed, what probably happened to the daily low temperature? Explain.

Saxon Math Course 1

LESSON

65 Power Up facts mental math

Prime Factorization Division by Primes Factor Trees Building Power Power Up H a. Number Sense: 5 × 60 b. Number Sense: 586 − 50 c. Calculation: 3 × 65 d. Calculation: $20.00 − $2.50 e. Number Sense: Double 75¢. f. Number Sense:

$75 100

1

1 10

g. Primes and Composites: Name the prime numbers between 10 and 20. h. Calculation: 9 × 9, − 1, ÷ 2, + 2, ÷ 6, + 3, ÷ 10

problem solving

Use the digits 6, 7, and 8 to complete this multiplication problem:

New Concepts prime factorization

Thinking Skill List

What are all the factors of 4, 6, 8, and 9?

23_ × _ 166_

Increasing Knowledge

Every whole number greater than 1 is either a prime number or a composite number. A prime number has only two factors (1 and itself ), while a composite number has more than two factors. As we studied in Lesson 19, the numbers 2, 3, 5, and 7 are prime numbers. The numbers 4, 6, 8, and 9 are composite numbers. All composite numbers can be formed by multiplying prime numbers together. 4=2∙2 6=2∙3 8=2∙2∙2 9=3∙3 When we write a composite number as a product of its prime factors, we have written the prime factorization of the number. The prime factorizations of 4, 6, 8, and 9 are shown above. Notice that if we had written 8 as 2 ∙ 4 instead of 2 ∙ 2 ∙ 2, we would not have completed the prime factorization of 8. Since the number 4 is not prime, we would complete prime factorization by “breaking” 4 into its prime factors of 2 and 2.

Lesson 65

337

In this lesson we will show two methods for factoring a composite number, division by primes and factor trees. We will use both methods to factor the number 60.

division by primes

To factor a number using division by primes, we write the number in a division box and begin dividing by the smallest prime number that is a factor. The smallest prime number is 2. Since 60 is divisible by 2, we divide 60 by 2 to get 30. 30 2  60

15 5 3  15 2  30 2  30 2  60 Since 30 is also divisible by 2, we divide 30 by 2. The quotient is 15. Notice 2  60 how we “stack” the divisions. 1 5 3  15 5 5 2  30 3  15 2  60 2  30 Although 15 is not divisible by 2, it is divisible by the next-smallest prime 2  60 number, which is 3. Fifteen divided by 3 produces the quotient 5. 2 2 3 15 1 30 5 3  15 5 5 2  30 2  60 2  30 3  15 2  60 2  60 2  30 2 60 Five is a prime number. The only prime number that divides 5 is 5. 30 2  60

2

30 2  60

23

15 2  30 2  60

15 2 2 2 330 2  60

5 3  15 2  30 2  60

1 5 5 3  15 2  30 2  60

By dividing by prime numbers, we have found the prime factorization of 60.

2

23

60 = 2 ∙ 2 ∙ 3 ∙ 5

Example 1 Use division by primes to find the prime factorization of 36.

Solution We begin by dividing 36 by its smallest prime-number factor, which is 2. We continue dividing by prime numbers until the quotient is 1. 1 1 3 3 3 9 2  18 2  36 36 = 2 ∙ 2 ∙ 3 ∙ 3

1

338

Some people prefer to divide only until the quotient is a prime number. When using that procedure, the final quotient is included in the prime factorization of the number.

Saxon Math Course 1

1 5 5 3  15 2  30 2  60

factor trees

To make a factor tree for 60, we simply think of any two whole numbers whose product is 60. Since 6 × 10 equals 60, we can use 6 and 10 as the first two “branches” of the factor tree. 60 6

10

The numbers 6 and 10 are not prime numbers, so we continue the process by factoring 6 into 2 ∙ 3 and by factoring 10 into 2 ∙ 5. 60 10

6 3

2

2

5

The circled numbers at the ends of the branches are all prime numbers. We have completed the factor tree. We will arrange the factors in order from least to greatest and write the prime factorization of 60. 60 = 2 ∙ 2 ∙ 3 ∙ 5

Example 2 Use a factor tree to find the prime factorization of 60. Use 4 and 15 as the first branches.

Solution Thinking Skill Connect

What other whole number pairs could we use as the first two branches of a factor tree for 60?

Some composite numbers can be divided into many different factor trees. However, when the factor tree is completed, the same prime numbers appear at the ends of the branches. 60 4 2

15 2

3

5

60 = 2 ∙ 2 ∙ 3 ∙ 5

Practice Set

a.

Classify

Which of these numbers are composite numbers? 19, 20, 21, 22, 23

b. Write the prime factorization of each composite number in problem a. c.

Represent

Use a factor tree to find the prime factorization of 36.

d. Use division by primes to find the prime factorization of 48. e. Write 125 as a product of prime factors.

1

62

1

62

2

23

3 8

Lesson 65

339

f.

Written

1 2. The African white rhinoceros can reach a height of about 5 6 2 feet. How 3 1 1 2 many inches is 5 6 2 feet? 62 23 8

(15)

3. Jenny shot 10 free throws and made 6. What fraction of her shots did 3 she make? What percent of her shots did she make? 5 3 8 3 3   1 5 5 3 8 * 4. Represent Make a factor tree for 40. Then write the prime factorization 1 30 15 5 (65) of 40. 2  60 3 15 5 2  30  5 2 30 3 15 2  60   * 5. Classify Which of these numbers is a composite 2  60 number? 2  30 (65) B 31 C 41 A 21 2  60

(29, 42)

1

23

7. Four of the ten marbles in the bag are red. If one marble is drawn from 1 that 1 the1 marble3will not be1 1 ratio1expresses 1 the bag, what the probability   8 1 2 15  m  2 2 3 6 12 6 2 4 8 red?

(58)

8 3



3 8

3 5

1

3 1 1 11 11 1 3 1  2m  2    15 15m  * 9. 4 8 12(61) 12 6 26 2 4 8

11 11 1 1  12  2 * 8. 8  81  2 3 6 2 3 6 (61) Find each unknown number: 3 3 4 1 1 11 11 1 3 1   10. 15 15m   2m  2 12 6 12 2(43) 6 2 4 4 8 8 1 1 1   12 6 2 5

13 8

5

13 8

1 8

12 1  23 2

4

37 4

37

1 5 5 5  25 15. 1 1 − (0.2 1 5  50 1 11 +1 20.48) 12 1 4 4 (38)       37 37 5 5 5 5 2 32 3 23 2 16. Explain What is the total cost of two dozen erasers 5  25are priced 5  25 that (15, 41) 5  50 5  50 at 50¢ 1 each1 if 8% sales tax is added? Describe a way to perform the 4 37 calculation mentally. 5 5 5 5 5  25 5  25 117. Connect The store manager put $20.00 worth of quarters in the 5  50 5  50 5  5(15) change drawer. How many quarters are in $20.00? 5  25 25 4 5  *5018. 7 25

14. Compare: (56)

2 1 1 1    2 3 3 2

a. faces? 25 7

b. edges?

340

4

37

A pyramid with a square base has how many

25 7

4 25

1

4n 8 n 3 13 1    y  y 11. 25(42) 100 25 100 8 38 3

5

(Inv. 6)

4 25

5

13 8 4

3 1 5 3 1 13 8 13 3 3 1 48 n 8 1  2 =4 0.0144  12. 13. 8   y m 412w 15  4 (45) 8 25 100 (29) 8 3 1 8

1 8

62

1

2

8 3

 38  1

3 5

* 6. Write 2 23 as an improper fraction. Then multiply the improper fraction (62) by 38. What is the product?

2

62

1 1 12  2 36 6

 

Write the prime factorization of 10, 100, 1000, and 10,000. What patterns do you see in the prime factorizations of these numbers? 1 3 3  Strengthening1Concepts Practice 2 6  3 3 2  12 2 6 1. The total land area of the world is about fifty-seven million, 2 five 24 hundred 2  12 (12) 2 48square six thousand square miles. Use digits to write that number of  2  24 2  48 miles. Generalize

c. vertices? Saxon Math Course 1

4 25

4 25

4 2

4n 4   25 25 100 1

* 19. Use division by primes to find the prime factorization of 50. (65) 2 1 1 1 4   37 2 3 3 2 * 20. Connect What is the name of a six-sided polygon? How many vertices (60) does it have? 1 1  2 3

2 1  3 2

* 21. Write 3 47 as an improper fraction. (62)

22. The area of a square is 36 square inches. (38)

a. What is the length of each side? b. What is the perimeter of the square?

23. Write 16% as a reduced fraction. (33)

24. How many millimeters long is the line segment below? (7)

cm

25. (7)

1

2

$75

3

4

5

6

7

1

A 100 meter is about 1 10 yards. About how many meters long is 3555 an automobile? 222555 Estimate

* 26. Write the prime factorization of 375 and of 1000. What method did (65) you use? 27. Reduce: (54)

28.

3555 222555

The radius of the carousel is 15 feet. If the carousel turns around once, a person riding on the outer edge will travel how far? Round the answer to the nearest foot. (Use 3.14 for π.) Describe how to 35 105 7 mentally check whether the answer210 is reasonable. 3  105 2  420 2  210 5  35 420 many2answers  210 3  105 29. Eighty percent of the 20 answers were correct.2 How were (29, 33) 420 2  2  210 correct? 2  420 30. Verify The prefix rectangle means is a1 35 “right.” A rectangle 210 “rect-” in105 7 (54) 3  105 “right-angle”2shape. every square also a rectangle? 7 7  420 Why2is 210 5  35 2  210 2  420 5  35 3  105 2  420 3  105 2  210 2  210 2  420 2  420 (47)

Estimate

Lesson 65

341

LESSON

66

Multiplying Mixed Numbers 2

2

Building Power 2 23

Power Up facts

23

23 2

2

23

10 3 yd

2

10 3 yd 2

23

2

23 2

10 3

2

10 3 1

22

Power Up J

mental math

a. Number Sense: 5 × 160 b. Number Sense: 376 + 99 c. Calculation: 8 × 23 d. Calculation: $1.75 + $1.75 1 3 of $30 10

e. Fractional Parts: 1

f. Number Sense: 3

$60.00

$30 10

g. Measurement: Which is greater, 5 years or a decade?

problem solving

h. Calculation: 8 × 8, − 4, ÷ 2, + 3, ÷23, + 1, ÷ 6, ÷ 2 1  12 10 2 3 2 1 1  10 2 a square and In this 2figure 3 a regular pentagon share a common side. The area of the square is 25 square centimeters. What is the perimeter of the pentagon?

New Concept

Increasing Knowledge

Recall from Lesson 57 the three steps to solving an arithmetic problem with fractions. Step 1: Put the problem into the correct shape (if it is not already).

Thinking Skill Verify

Step 2: Perform the operation indicated.

How do we write a mixed number as a fraction?

Step 3: Simplify the answer if possible. Remember that putting fractions into the correct shape for adding and subtracting means writing the fractions with common denominators. To multiply or divide fractions, we do not need to use common denominators. However, we must write the fractions in fraction form. This means we will write mixed numbers and whole numbers as improper fractions. We write a whole number as an improper fraction by making the whole number the numerator of a fraction with a denominator of 1.

Example 1 2 23

342

A length of fabric was cut into 4 equal sections. Each of 4 students 2 2 2 2 1 received 2 3 yd of fabric.10 How before it was cut? 2 3 was there 10 22 3 3 ydmuch fabric

Saxon Math Course 1

1

32

2

23

2

23

2

10 3 yd

2

23

Solution This is an equal groups problem. To find the original length of the fabric we 2 2 2 2 2 2 2 2 2 1 2 3we write 2 3 and 10 34 yd multiply First, in fraction form. 2 3 2 3 yd by 4.10 10 3 2 2 2 10 3 3 3 yd

2 23

2 2 4 3

2

23

8 4  3 1

2

23

8 4 32   3 1 3

1

32

32 2  10 3 3

Second, we multiply the numerators to find the numerator of the product, and we multiply the denominators to find the denominator of the product. 2 2 4 3

2

23



1 3

32 2  10 3 3

1 2 203 20 2 12 2 21 1 1 21 3 5 41 1  1 2 5 41 2  1   32  33 2 3 2 2  133 4 2  3 2 3 23 3 6 6 6 3 2 3 1 2 2 1 2 2 1  1  1 2 1  3 2 21 31 3 1 3 4Reading Math 23  31 Second, 32  3 we2 multiply22the of the fractions.  terms 33 3 3 4 2 Recall that 1 11 52 4 5 4 20 20 1 2 1  10    3 3 2 2 2 1 the terms of a 23 3 2 3 6 6 6 2 3 3 1 2 2 3 1 2 2 1 fraction are the   1 1 1 1 1 3 2 23 Third, we simplify the product. 2 3 3 4 2 3 numerator and 1 1 1 1 1 1 1 1 1 2 1 1      a b 7 3 1 2 1 22 2 1 2 2 2 2 2 2 2 3  1 the denominator. 2 1 2 5 2 4 3 2 5 4 2 202 20 1 3 3    3 3 2 4 1 2 3 2 3 6 6 6 2 3 3 1 2 2 3 1 2 2 1 2 2 1  1  1 1 1 3 2 2 31 2 3 3 4 2 3 3 2 3 3 3 1 2 1 Sketching 3a rectangle on a2 grid is a way to 1 2  3 the reasonableness of a 2 check 21 31 3 2 2 2 23 2 1 2 2 23 2 4 2 1 2 3 2 3 223 2 3 2  3213 10 10 3 2 2 of 2 2 and 3 2, we use 8342 a 8 4 product of 3mixed the yd 2 3numbers. 2 3 To illustrate 10 3 product 3 yd 10 2 2 2 32 23 1 2 2 1 2 2 that 1 by2 24 2so1 that 2 1 3 1 fits 2 8 3 the grid.2 at3least 243 10 3 yd1 1 10 12 3 grid 3 3yd 3 is10 1 2 3by 3 2 rectangle 2 2 on 1 2 1 310 1a 2 2 8 43 3 4 31 2 4 2 3 3 4 2 3 3 2 3 3

2 1 2 233

31

3 1 $30 d4

1 3

10

10

8 4 1032   3 1 3

8 Evaluate How can we2 use estimation2 to check whether 4 our answer8 is 2   2 21 1 23 2 4 23   12 10 10 3 1 3 3 reasonable? 2 2 3 3 5 4 5 4 320 20 2 1 2 1 2 2 1 2 2  1 3 2 3 1 3 2 2 31 2 1 23  3 2 1 33  46 6 1 2 6 1 3 3 3 2 3 3 8 4 8 4 32 32 2 2 Example2 2 2 4    1 23 23 3 1 3 1 3 3 3 5 4 5 4 20 20 1 1 2 1    3 3 Multiply: 2  1 6 1006 7516 2 3 1 12 32 22 3 231 12 5 4 20 20 2 1   1 2 2 2 1  2 1  32  2 2 3 Solution 1  32  3 2 1 2 2  5 75 2 3 3 100 2 3 2 3 3 12 3 16  1 6 6 8 2 2 4 2 3 8 3 4 1 3 32 32 2 2 3 21  3 2 3     10 23 3 2 3 4 4 23 2 3 1 3 1 3 3 3 3 1 First, we write the$30 numbers in fraction form. 3 10

5 4 1 2 12 3 2 3

2 3

1

$30

1 3

1 2

1 1 2 1 2 3

1

8 4  3 1

Third, we simplify the product by converting the improper fraction to a 8 4 32 8 4 2 2 2 mixed number.    23 23 $30 $30 1 2  4 1 3 1 3 1 3 3 32 810 43 8 410 32 2 1 32 1      23 1 10 1 35 14 35 124 3220 3 20  332  3 1 2 1 2 3 2 3 6 6 6 2 3 3 2 2 2 2 2 1 23 10 3 22 Before the fabric2was cut it was 10 3 yd long. 2 3 3

2

4

2

23

23

2

1 2

1 2 3  1 21 3 32

3 4

$30 10

1 3 2 2 3$30 10



1 2

10

1 2

2

2 10 1 3 3

3



1 2

2 10 3

$30 10

1

2 3

2 10 3

3 2 2 4

23

1 2

3 1 2 11 3 1 2 2 2  3 32 3 3 4 2 3 We sketch3the  1rectangle and estimate the area. There are 6 full squares, 4 a quarter square. Since 2 half squares equal a whole 5 half squares, and 3 1 1 square, the area is3 about 2 3 1 8 4 square units. 2 2 2 2 2 3 4 2 31

3 4

Lesson 66

343

2

32

32 2 3 32

3 2 2 3 32 6 32 26363 3 6466 33 664 2 3 6 3 23

3 3 15 4 5 4 5 4 520 4 2020 1 2202 1 1 22 1     3    3 3 132  31 2  3 1 1  1121  1 2 1 2  31 2  3 2 1  2221  2 2 3  1 23  1  6631 2  3 63 32 3 2 3 2 3 2 6 3 6 61 2  3 3 3 3 33 4 3 4 2 32 2 32 3 Practice Set Multiply: 1 1 2 1 12 1 2 21 3 1 3 3 2 1  23 1 2 1 21 2 33 2 2 1 22 2 2 21 2 2 3 3 1 3 3 4 1 41 1 2 c.1 1 2  2 3 22  23 a. 1  1 3 b. 1 13 2 1  31 13 2 2 1 2 3  1 2 3 2 3 2 3 3 4 3 4 3 2 43 2 3 2 32 32 3 2 33 33 3



3 21 1 21 2 2 2 3 2 3 1 22 2 2 2 1  1  1  1 1 d.1 1  3 1  3 2 e. 2 2  2 3  1 3  1f. 3  1 32 2 32 3 3 3 4 3 4 2 33 3 3 3 4 3 1 1 23 1 1 21 23 1 3 h. 1 2  2 2 13 2 22 3 2 g. 3  1 3  1 2 3 1i.2 2 23 3 2 1 1 3 34 4 3 33 2 34 23 3  1 3  12  2 2  2  3 34 4 2 3 3 3 2 j. Check the reasonableness of the products in e and h by sketching 3 2 3 1 1 21 1 rectangles 3  13 ona1grid. 2  22  2 2  3 2  3 3 4 4 2 3 33 2 k.

Write3and solve a word problem about multiplying a whole 3  123 3 3 1 1 2 1 3  21 1 1 1  1 3and a1mixed 3number 2 2 2  2 2 3 2 2  3 2  3 34  1number. 4 3 33 4 33 2 34 4 2 2 Formulate

3 1 23 1  1 2  2Written 3 23 2  2 2 Practice 2 34 4 2

1

Strengthening Concepts

3 3 3 1. 3 Fifty test are multiple choice. Find  60  1percent 1 questions on the 3  1of 3the 3 3 4 4 4 the number of multiple-choice questions 3 the 1 test. 3  1 on 4 4

(29, 33)

2.

(58)

3 of the 30 students in the class are boys. 3Twelve 3131 4 4 of boys to girls in the class? a. What is the ratio Analyze

3 3 3 3b.If1each 3 student’s 1 3  1name is placed in a hat and one name is drawn, 4 4 4 what is the probability that it will be the name of a girl? 3 4

2

2

2

23 10 3 yd 2 Some railroad rails weigh 155 pounds per yard. 2How much 100  7 1 23 2 1 2 2 2 would a 33-foot-long rail weigh? 2 3 3 100  7  100 100 75 1 2 2 1 2 2 1 7511 1   1 21 1 1 1 1 2 2 1 2 100 1 75 2 2 1  1 2 75 12  1 1 3 2 100 2     75 * 4. 1  2 * 5. 2 2  12  2 111 2 1  322 3 100 35  75 2  5 12 75 3 3 100 2 3 3 100   75 7523  62 5 12 2 2 3 3 2 3 100 3 2 4 3 26 2 (66) (66) 1 5 2100 2 3 3 100  75  75  75100  100 1 1 2 2 16. 2The 1 75 111 1 1 the 1 1 is1the 11 1 1 1 1 1 14 41 141 441 22 2 100 1 1average numbers is 200.   12 12    2 of  35   1235  2 1  2 2 1  3 3 1  1 2 2 1  2 2sum 1What 35 2 five 3  of  75  75100  5 75 5 2 2 5 3 326 612 312 6 4 12 2 2 3 3 (18) 4 2 2 45 52 252 552 2 3numbers? 33 3 100100 100  75 100 100 22 11 1 1 11 1 1 1 11 1 4 11 1 4 1 21 1  75 11 111 1 22 2  75 2 35  12 1  21  22 2 2  2 2  1  3 1  3 1  3  35 12 1  35  12  75100 75 11 75 4 1312 114 4 1 2 1 1100 175 1 1123 5 16 1231112 6 4 112 1 22 12 175 445 1 1 4 223 2 10032 115 1175 5 3 100 4 3 100 2 2 2 3 3 100 6 75   1 124 21 5  3  * 9.    7. 2 2  2 1  2 1  22 1  2 22  11  8. 12 32 1  3 2 35  35  12 1 12 35 100 1 1  5100 5 5 5 5 5 2 3 2 2 3 6 12 4 2 2 33 2 2 3 6 12 4 2 2 2 5 5 75 33 (5) 1003 75100 175 2 3 6 12 4 2 2 (59) (61) 2   5 2 2 1 3 5 2 3 3 100  75 2 3 6 12 1 1 100 0 1 1175 1 11 1 1 111 1 1 14 111 44 111 4 11 4 1 1 4 1 2  75 $30   12   12  3 1  3   *1  11. 35 12.  35 10. 3  1235  12  5 2 12(63) 4 2 4 65 2 2(29) 55 242 5 22 5 3 (54) 5 3275 6 3 5 6 212 75 0 3 100 3 12 2 5 2 10 2 1 1 1 1 1   a b 13. 0.25 ÷ 5 14. 5 ÷ 0.25 2 2 2 22 2 (45) (49)2 2 2 12 1 1 1 11 111 1 1 1 1 1 1 1 1 11 11 111 1 11 1   is the product   a b 2to  a b214? 1722  % 15. 2What 1222and 1a b 1a b 7 % 13 2 2 2 12 2 2 2of 2 the2answers 2 2 problems 2  222 2 a b2 2 2 2 (39) 2 2 2 2 2 2 2 2 1 11 1 116.11 Verify 1 1 to1 1 111? 1 11 1 1 1of the 1 1 is11 1 2 10 equal       Which a ba b following  7 % 7 % a b  7 % 2 3 2 22 2 2(54)22 22 2 2 2 2 2 22 22 22 2 2 222 2 2 1 1 1 1 1 1 1 1 11 1 1 11 1 1 1 1 1 1 1 1  A 2 C 7 2   a b2 B a b % 7 2% 7 2% a b 1 1 1 1 1 121 121 2 1 2112 12 12 2 1 21 21 1122 12 12 2 221 2 2 1 2 12     a b a b 1   7 2 %1 7 2 % 1 7 2 % a b 1 1 1 2 2 2 2 2 2 2 2 2 2 22 2 2 2  212 2 2 1  2 2  a b 7 2% 2 2 2 2 2 2 2 2 2 * 17. Represent Use a factor tree to find the prime factorization of 30. 11  1 1 1 1 1 11 1 1  b 7 2(65) % ab 7 2 % 7 2% 22 2 2 2 2 2 2 18. If three pencils cost a total of 75¢, how much would six pencils (15) cost? 3.

(15)

344

Analyze

Saxon Math Course 1

1 1  2 2

19. Seven and one half percent is equivalent to the decimal number 2 (41) 1 1 1 1  a 0.075. b If the sales-tax rate is 7 2 %, what is the sales tax on a $10.00 2 2 2 purchase? * 20. (60)

One side of a regular pentagon measures 0.8 meter. What is the perimeter of the regular pentagon? Analyze

21. Twenty minutes is what fraction of an hour? (29)

22. The temperature dropped from 12°C to −8°C. This was a drop of how (14) many degrees? The bar graph below shows the weights of different types of cereals packaged in the same size boxes. Refer to the graph to answer problems 23–25. Weight of Cereal

20 Weight (in ounces)

1 1  2 2

5557 222555

15 10 5 0

Fruit and Flakes Flakes

Puffed

Type of Cereal

23. What is the range of the weights?

(Inv. 5)

24. What is the mean weight of the three types of cereal?

(Inv. 5)

* 25. (13)

* 26. (65)

Write a comparison word problem that relates to the graph, and then answer the problem. Formulate

Connect

Use division by primes to find the prime factorization of 400.

27.

Analyze Simon covered the floor of a square room with 144 square floor tiles. How many floor tiles were along each wall of the room?

28.

The weight of a 1-kilogram object is about 2.2 pounds. A large man may weigh 100 kilograms. About how many pounds is that?

(38)

(46)

Estimate

29. Reduce: (54)

* 30. (60)

Classify

A

5557 222555 Which of these polygons is not a regular polygon? B

5557 222555

C

D

Lesson 66

345

LESSON

67

Using Prime Factorization to Reduce Fractions

facts

555 125  1000 2  2  2  5  5  5

Building Power

Power Up

125 1000

$30 100

Power Up J

mental math

1

a. Number Sense: 5 × 260 b. Number Sense: 341 − 50

1

1

555 1   2 22555 8

125 1000

375 1000

1 1

1

1

1

c. Calculation: 3 × 48 d. Calculation: $9.25 − 75¢ e. Number Sense: Double $1.25. f. Number Sense:

5 5

125 1000

$30 100

g. Measurement: Which is greater, 3 yards or 5 125 feet? 125 $30$30

1000 1000

h. Calculation: 6 × 6, − 1, ÷ 5,100 ×100 2, + 1, ÷ 3, ÷ 2

problem solving

555 125 Copy this problem and fill inthe 55 1000 2 missing 2  2  5digits:

_ _ _,_ _ _ × 7 5 55 55 5 125 125  999,999 1000 1000 2 22 22 25 55 55 5

125 $30 1 1 1000 100 125 Increasing Knowledge 555 $30 1 125 1 $30 125 1000 100 1000 8   2  5 125  5  5 1000 2 2 8 100 $30 1 1000 1 1 1 11 11 1 One way to reduce the terms and then 100 fractions with large terms  5to factor 5 5555is 1 1 125125  begin  $30 To reduce 125 , we could reduce the common factors. 1000 1000the 221000 255555 5 8 8 by writing 2 2 2 100 1

New Concept

1 115 11 1 125 1000. 5  5 prime factorizations of 125 and 375 1000  2 5 2  2  5  5  51 5 125 55 125 Explain  1000 55  5  5 1 5 5 5 1000 125 2  2  1000 255  25  2  5  5  5 2 What are two  1000 2  2  2  375 5 375 55 5 5 1 1 strategies for We see three pairs of 5 s 125 that can be reduced. 1000 5 5 reduces to 1. 1 1000 5  5  5 Each finding the prime  1  12  12  5  5  5 1000 2 factorization of a 555 1 125 1 1 1 1  number? 1000 8 5 8 2 112 12 15  5125 555 1 1 1 1 125 1 1 1  1 5  5  51 1000 8  2  2  2  5 55 5 5 5 1000 8  5  15  5 8 125 2 82$30 125 1 1 2 1factors We multiply the remaining and reduces to 18.  1 1find 1 that 1000 1000      15 1 8 2 2 2 5 15 100 1 51  51  5 1 125 1  Example 10001 8 2555 58 2  2375 1 1 1 5 1000 1 375 5 1 3 1 5  5  5 375 Reduce: 5 375  1000 375 5 1 2555 1000 5 1000 1 2  2 1 5 Thinking Skill 5 15  5 1000 5125  Solution Discuss 1000 52  2  2  5  5 1 5 375 1000 5 1 When is it helpful We write the prime factorization of both the numerator and the denominator. to use prime 1 1 1 factorization 3555 3 3555 375 375   = to reduce a 1000 1000 2  2  2  5  5 1 5 1 21  2  2  5  5  5 8 555 11 1 1 125 fraction?  1000      2 2 2 5 5 5 8 remaining Then we reduce the common factors and multiply the factors.

Thinking Skill

1

346

Saxon Math Course 1

1

1

1

1

1

3555 3 375 2  2  2  5  5  5 1000 8 1

1

875 1000 375

1 1 8 8

3 8 1 8

1

10005 5

48 400 3 8

125 500 1 1

36 81

1

48 400

1

1

1

1

1

1

1

1

1

1

1

48 48  5 125 125  5 875 3  5 875 5 3 3  5  5375 3  400 4001000  5003 1000 500 1000 2  2  2  5  5  5 28  2  2  5  5  5 8 8 1

1

1

1

1

1

1

1

1

36 36 81

375 81 1000

3 8

1

125  5  5375  5prime factorization 3  Practice 5  5  5 Set336 3Write 3 3 375of both the numerator 3 the and the denominator of   81 500 1000 8  2  1000 55 8 8 5 2  2  2  5  5  5 28  2each fraction. Then reduce each fraction. 1

875 a. 1000

125 48 b. 500 400

36 36 125 d. 500 81 81 7 1 w 3 n  10 2 Strengthening Concepts5 100

Written Practice

7 1 w 10 2 1 2

48 875 400 1000

48 875 125 48 c. 400 1000 500 400

875 1000

3 4

1

3 4

3 1 3 2 3 4

1 2

36 81

125 500

1 7 4 2 4 8

36 81

2 2  29 3

* 1. Allison is making a large collage of a beach scene. She needs 2 yards of 3 13 (66) 3 1 blue ribbon for the ocean, 12 yard of yellow24ribbon for the 4sun, and 4 yard 2 3 1 1 1 625 2 1  2 costs $21000 of green ribbon3 for a yard. How much money 3 the4grass. Ribbon 3 4 3 will Allison need for ribbon? 3 1 1 4 4 72 2 1 w 3 n 1 7 2  feet. A nautical  2feet. 2. Estimate 10 A mile2 is 5280 A 2 is about 4 mile 2 6080 9 5 100 4 8 3 (13) nautical mile is about how much longer than a mile? 3 n 1 7 2  Verify Instead 2 2 dividing 4 3of 2$1.50 9 3$0.05, Marcus formed an 3. by 1 5(43) 100 4 8 1 3 2 4 2 4 equivalent problem by 3 1 division 1 1 mentally 625 multiplying both the dividend   3 2 1 2 1000 and the divisor 4 by 100.3Then4he performed the equivalent division 33 4 problem. What is the equivalent division problem Marcus formed, and 3 1 1 1 625 12 quotient? 4 1 what 2 is the1000 3 4 Find each unknown number: 4. 6 cm + k = 11 cm

5. 8g = 9.6 (43) 7 1 w 3 n 1 7 2 7 1  w 6. 3 10 217. 7 n 2 4 2 2  29  49 8 3 2 4 (42) 52 100 2  2 (43) 10 5 100 4 8 3 * 8. The perimeter of a quadrilateral is 172 inches. What is the average 3 1 (60, 64) 1 2 4 length of each side? Can we know for certain what type of quadrilateral this is? Why or why not? 3 1 1 1 625 3 1 1 37 1  2 625 1 2 1 3 w 3  w 4 1000 37 n 4 2 1000 37  21 1  n2 3 3 1 7 2 1 4 9. 10 3 4−2($46.75 3 +10 4 2  42 2  29 2 4210. (2 2 (0.2 90.3) $100.00 $9.68) ×100 0.3) − × 5 5 4 8 3 100 4 8 3 (5) (53) 7 1   1 7 2 w 3 n 1 37 n 2 Analyze 42  2   10 * 11. 12. 2 *42 9 2Analyze 2  29 8 3 5 (63) 100 4 5 8 100 3 (38,466) (3)

1

12 7 1 w 10 2

3 1 3 2 3 4

3 625 3 11 1 1 1 1 13. 3  2 14. 1  2 1  32  2 *1000 34 4 (66) 3 4 3 4 3 (59) 3 1 1 1 1 1 625 625 16. $6.00 ÷ $0.15  31.44 2 ÷2 601000 1  2 115. 1000 (45) (49) 34 4 3 4 3

625 1000

17. Five dollars was divided 3 4 people. How much money did 1 evenly among (15) 1 3 1 2 4 each 1 2 receive? 4

1

12

3 4

12

1

3 4

1

3 4

12

1

12

3 4

Lesson 67

347

5 7 1 w 10 2

100

4

8

3

3 2 1 1 625 1 71 1 9 2 4  23 3  2 42  2 1000 3 4 3 Conclude 4 The 8 area of a regular quadrilateral is 100 square inches. What 3 1 1 1 625  2 3 is the 2 name of 1 the is its perimeter? What 1000 3 4 3 quadrilateral? 4

3 n  5 18. 100 (60)

* 19. Write the prime factorizations of 625 and of 1000. Then 3 1 1 (67) 1 625 3 2 1  2 reduce 10007. 1 3 4 3 4 w 3 n 1 7 2 7 1  w 10 2 4 2 2  29 3 n 1 7 2 5 9 100shown below?  * 20. What  the rectangle 4 8 3 10 2 2 area of2 the 4 is 2 5 100 4 8 3 (31, 66) 1

3

1

3 1 3 2 3 4

1 1 1 2 3 4

3 4

1

12

625 1000

(Inv. 6)

3 4

1

1

12

1 4 1 1 2 3 4

3 4

625 10003

4

in.

21. Thirty-six of the 88 piano keys are black. What fraction of the piano 1 (29) 12 keys are black? * 22.

12

1 123 3 2 3 4

1 in. 2

* 23.

(30, 62)

Draw a rectangular prism. Begin by drawing two congruent

Represent

rectangles. Analyze

1

12 ×

=1

3 4

24. There are 1000 meters in a kilometer. How many meters are in (15) 2.5 kilometers? 25. (50)

Connect

Which arrow could be pointing to 0.1 on the number line? A

–1

26. (47)

1

12 27.

(Inv. 6)

1

12

D

0

1

Estimate If the tip of the minute hand is 6 inches from the center of the clock, how far does the tip travel in one hour? Round the answer to the nearest inch. (Use 3.14 for π.) Connect

A basketball is an example of what geometric solid?

28. Write 51% as a fraction. Then write the fraction as a decimal number.

(33, 35)

* 29.

What is the probability of rolling a prime number with one toss of a number cube?

* 30.

This quadrilateral has one pair of parallel sides. What kind of quadrilateral is it?

(19, 58)

(64)

348

BC

Represent

Conclude

Saxon Math Course 1

LESSON

68

Dividing Mixed Numbers Building Power

Power Up facts

Power Up I

mental math

a. Number Sense: 5 × 80 b. Number Sense: 275 + 1500 c. Calculation: 7 × 42 d. Calculation: $5.75 + 50¢ 1 of 4 $120 10

e. Fractional Parts: 1 f. Number Sense: 4

$120 10

$48.00

g. Measurement: Which is greater, 1 meter or 100 millimeters?

problem solving

h. Calculation: 7 × 8, – 1, ÷ 5, × 2, – 1, ÷ 3, − 8 8 4 2  2 4 3 1 3 8 4 2  4 Megan2 has 1 white socks, and black socks in a drawer. In 3 many gray3socks, the dark she pulled out two socks that did not match. How many more socks does Megan need to pull from the drawer to be certain to have a matching pair?

New Concept

Increasing Knowledge

Recall the three steps to solving an arithmetic problem with fractions. Step 1: Put the problem into the correct shape (if it is not already). Step 2: Perform the operation indicated. Step 3: Simplify the answer if possible. In this lesson we will practice dividing mixed numbers. Recall from Lesson 66 that the correct shape for multiplying and dividing fractions is fraction form. So when dividing, we first write any mixed numbers or whole numbers as improper fractions.

Example 1 2

2

1

1

2 3 is pouring 2 3 cups of plant 2 4 amounts to feed Shawna 4 food into equal 4 plants. How much plant food is there for each plant?

Lesson 68

349

Solution

1 4

$120 10

2

2

2 2 4 3

8 4  3 1

1

1

2

2 4 divide 2 3 Shawna is dividing 2 3 cups of plant2 food into four 4equal groups. We 3 by 4. We write the numbers as improper fractions.

2

23

To divide, we find the number of 4s in 1. ( That is, we find the reciprocal of 4.) 4 1 8 1  Then we use the reciprocal of 4 to find the number of 4s in 83.  1 4 3 4 1 8 4 1 8 8 1 1  8 8 1 4 1   3 1 8 1 4 1 4 8 3 3 4 12 4 1 8  1 4 1 8 4 1 3 184 88 812 1 8 83 8 8 1 1 1  18  4 1  831  4 1  1  1  4 3  441312   3 1 4  8 3 4 12 1  8 3 4 12 3 4 12 3 1 4 4 1 8 8 2 2 8 8 1 1 4 1 4 1  3 4 1412 23  8 3 8 8   8 8 1  1 4 8 3 1 1 1 12 3  34 12   3 1 4 1 4 1 3 41  4 1 12 8 182 82 3 4 12 2 2 4 1 8 8 2 8  1 8 8 the 1 1   4 3 2 We simplify answer. 3 3 8 12 2 3 8 1 214 4 1 3 2412 3  3 1 4 12 22 2 223 8 12 3 4 12 8 14 1 2  2 3 23 12 4 3 8  24 2 3 112  33 42 12 3 2 4 4 41 8 212 3 1 88 2 8 31 8 1 2 1 1 2 1   3  3 1 4 4 1 2 4 1 4 1 4 1 8 8 2 4 4   82 3 8   1 2 1 2 8 1  18 1 1 3 1282 12 31 1 8 88 8 81 a 12 3  142 2341 88  is232 cup 14of plant 2 23 2 4 33 3Notice There plant. 4 34dividing     1 number 4 3 3each 1 4 1that 4food for 12 12 3 3 1 3 4 12 2 1 4 12 24 3 12 3 8 12 4 1 1 3 4 1 3 2 8 2 8 1 2 2 1 2 by84 is equivalent to finding the of 2 by 4, we 2 1 4 4 dividing 1  243 12 2 number.2 Instead 43 3 22  11 1  1 4 of 2 8 4 8 31833 21 8 8 3 3 13111 1 2 883 1  8 12 3   2 8 8 1 1 3 1 4 4 2 4 1 2 1    3 1 4 could8 have found of 223 .3 8 4 121128 22 11 28  8311 3328 24 12 2 1  48 3 1 directly 444   1 3 2 1  2 2  1 2 2  4 3 3 1 43 4 123 8 3 3 12 4 1 13 4 22 3 33 222 23  131 2 8  2 4 3 4 12 3 4 12 8 8 1 2  1 8 3 1 2 8 1 28  2 3 2 14 23 3 3 3 2  1 1 4 48  23 122 3 8 23 1   4 1 8 12 2 3 2 2  1 1 2 1 4 3 4 12 1 8 2 4 1 8 8 2 2 1 8 1 1 2 3 8 2 1 2 1 2 8 1 Example  4 1 28 1 8 83 1 2 341 2 4 3 2 22  2 2 8  23 3 2  2 2  2  1 4 3   4 4 3 3 1 4 3 1 4 4 1 123   12 3 2 12123 3332 312 3 3 3 32 2 3 3 2 3 8 3 4 12 3 8 2 1 1 2 3 8 2 1 3 8 16 2 2 2 3 8 8 2 1  1  3  4 2  2 3 3 82  2 2 3 1  211 4 8 2 32 Divide: 1 2  1 32 2 2 323 83 23 2223 3 16 3 2 9 2 16 2 3 32 3 2 42 12 23 3 38 2 8 22  12 8 4 3 8 8 1    1 12 4 3 3 8 16 3 2 32 83 12 3   2 23 23 12  3  238 9    2  16 12 3 2 84 2212 3 31 13 3 34 1  2822 3 3 32  23313 93131 2 3 8 891  2  3 3  Solution 2233 2 2 416 3 3 3 2 239  8 16 2 2 2 2  1 4 4 3 3 8 12 3 7 2 2 8 8 13 3  2162 2 3 1 22 8 1 1 3 2 1 1 8 2 32 116 2 31131 8 1 2 8 3 2 2222 8 2 2 16 3 1 8 32 2 22 3 2  2932932  11   11 43 4 2 3 834    2 3numbers as 2 4    1 4 3 the mixed 9 9  4 Skill 3 3 12 Thinking 3 We write improper fractions. 3319 2 3 3 2 12 3 3 2 3 2 3 22 2 2 2 22 3 3 31633 2 9 8 3216 16 3 3 3 7 3 8 3 8 16 2 2 1 16 3 Justify 8 7 162  1 1  89   8 3 1 2 91 91  29 2  13 16  212 8 2 33 13 2 1  22 16 4 2372 3 39 2 3 7  2  18 3 43  3 3 2 2 3 16 99 11 3 1 1921621 23 4 2 316 2 1 3 8 2  1 3 2 3 2 16 16 7 7 16 3 9 2213  1 Describe in your 9   27  1 2 33 49 2 9  16 2 4  1 1 16 1 9 9 329 2113 28 9 3 3 9 2 9 3 2 29 9 own words how 3 of2 3 .) 8 2 168 29 find  1the 4 number of 3 s2in 93 1. ( That is,8 we 9 To divide, we find the reciprocal  2  21   16 7 1 16 3 3 2 3 2 3 1 to divide1mixed  2 3 of 3 322 23 16893382 222 3 8 1 2 1 3 8 2 83 3 3 8 2 7 91 of 3 to8 find 16reciprocal 2 163 s in 781. 1  16 92 2 43 82 1 1 163 the number    9 2  1 4 Then we use the of      1 1 3  2 2  1 4   1 1 5 5 5 4 32 2 33  3 numbers. 2 3 93 33 3 3 333 2 9 3 39 2 9 3 3 2 2 9 2 162 3 9 2 2 2 3 9 3 3 1 2 7 16 16 7 3 16 2 4 16 3 182  28 23 16 of 1 83 11of 1 3 8 2 1 2 2 162 8 9 3 1  4 3 921 13 3  5 3 5  41  35 2 9 98 32 9 2 3 3 58 5 5 9 4     1 3 3 3 9 3 3 2 3 1 2 3 8 2 2 3 8 16 2 2 3 8 13  4 21  212  33 21 3 1 2 33 3 1623 of 13 9 2 14 4 3 3 9   of 1 5 5 5   33452 92 83 2of213 163 1 4 3 3 2 of 1 1 2 3 25 3 2 9 5 33 5 5 5 4 168153 244 3 5516113 4  3 16 2  of 1 16 5 3 4 127 3 5  3 3 12 3 5 3 99 1 2 9 2  3 16  9 3 8 16 16 2 2 9 7 1 4 2 of 1 9 3 8 16 7 3 8 16 16 16 2 2 7 7 16 3 16 3  162 5 3 3 8 51   1 1 3 1 2 of 2   5 4 1  915 2    49 9 1 1 1 3 4 2 3 of 13 2 3 3 2 2 2 3 995 31353  9 9 1 4 9of921 59 39 9 59 3 9 5 5 4 3 3 1 2 3 3 1 2 1 16 14 2 3of 2 5 7 2 316 of 1 1 4 31improper of 1 We16simplify of221 7fraction 16 2  75 16 59 as shown5 below. 4 2 16 1 5 5 4 9 16 51the 5 3 9 1 16 9 7 16 7 16 91 9 92 13 of 225  1 of 2 9 9  1 of 2 9 1 279 39 9 5 16 9 3 of 2 5 1659 3 of 2 3 1 3  15 3 1 2 9 4 9 of 1 1 2 1 3 2 3 1 4 of 1 5 4 591 5 5 3 3 of2 2 7 4 1 2 16 3 3 1 2 3 3 3 3 1 2 1 2 16 16  1 7 1 16 2   5 3 1 4 3 2 of 1 4of 1  4 of 2 51  2 1 3 of 152  3 2  3 of41 1  9 9 9 9 3 of92 5 1 9 55 5 5 3 5 5 4 4 55 5 4 5 54 1 2 1 Practice 2 3 3 1 2 Find each product or quotient: Set of 2 3 1 2 of 2 3 1 1of4 1 3 3 3 21 3 1 5of2 2 2 3 5 3 1 4 543 5 5 33 55  4 1 3 4 of 1 5 2 2 41 5 5 5     1 4 3 2 of 1 a. 1 4 2 b. 3 of 1 5 2 5 5 55 5 34 43 1 1 2 1 of 21 1 4 2 3 of 2 of 2 5 5 5 4 5 3 5 11 22 1 3 3 2 31 2 1 2 1 3 2 d. c.of2 1 3 3 ofof 22 1 4 of 2 1 4of4 1 5 2 3 of 2 5 5 5 33 5 3 5 5 5 5 5 4 3 1 2 2 2 1 1 2 1 2 e.ofGeneralize Why is dividing by2 of 4 the by 14? 24 2 2 same as2multiplying 3 3 of 2 5 3 1 2 1 2 5 3 5 3 of 2 of 2 5 5 3 1 23 of 2 5 3 1 2 1 2 of 2 Saxon of 2 Math Course 1 5 3 350 5 3 Math Language Reciprocals are two numbers whose product is 1.

2 24 200

3 m

2 3 12 5 8

2

3 1 16

4 1 4

1 2

1 4

1 2 1 2 2 3

1

82

2 1 2 1 2 11  2 3 1 2 2 1 21  1  31  2 71 1 2 1 1  1 13g. 21222311 21 1 1 2 3 2 f. 1  314 71 1 1 2 1 2 1 1 1 2113 3    w 1 1 1 1 1 1 3  3 2 22 n 2  31 4 213 1  3 2    1 7 2 1 1 1 7 1 2 3 2 2 4  2 1 1 1 7 1 8 1 2 24 5  21 4 3 2 3  5 10 2 3 1 2 3 8 2 2m 2 335 3 2121 3244 3 3  1 2 4 2 41002 2 2 200 4 2 2 1 2 1 16 31 1 1 2 2 1 38 1 2 113 2 1 3 2 21  13 1  1 7  113  13 2 1 h. i. 7  1 7 24  2 2 4 3 4 n 32 1 w 1 2002 1 22 13 m 1 542 1 12 1 1 3 8 108 1 5  1 3 1 316 1 1  24  2 3 2 3 2 18 2 2 100 1 2 1 divides 2 he 1 2 1 2 2 3 three1 projects. time 23 23 12his 2 2211 11 1 1 1 If2 2 11 1 1 2  4 j. Gabriel has1214 hours 1to finish 1111 1 1 1 1 1 1 1 1 1 3 2 3 22 21  2 1 1 3 2 2 1  7 1 1 1 1 1 1 4 1 2 1 3 2 4  8 1 2 8 1 2 3 22 s832212 2 3 341 2equally, 3 2 3 2 1 2 what fraction of an hour can he spend on each project? 24 24 1 1 22 3 1 1 2 4 2 2 4 2 4 4 2 4 2 3 m 2   2   4 1 1 5 4 1 1 3 4 2 m 5 3 7 2 1 1  81 71 2 200 200 81 3 n 33 w 1 1 1 1 2 3 1 2 2 116 1 3 5  4 10 1 1 1 1 1 1  12  1 5 31 212  8 1 2 8 1 4 100 8 2 3 Strengthening Concepts 2 4 2 4 2 2 4 2 4 2 2 1 2 Written3 Practice 3 2 32 7 n 3 w 1 13 1  22 1 21 1  12 1 11 1  11 3 2  5 10 5 1 3 1 3 1 2 4 100 8 21 1 3 1 2 2 1 13 3 1 2 2 1 1. What is the difference 1 between the12 sum of and and the product 3 3 of 12 12 24 2 4 2 4 4 2 4    3 3 3 3 1 1 4 1 1 m 5 3 3 (12, 55) 1 24 24 24 123 121 13 1 14 11 200 4 1  200      1 m 5 4 1 1 m 5 3 8 16 7 1 m 5 7 and ?  8 1 2  200 200 1 2 1 1 1 2 1 3 1 32 1 w  8n2 16 3 341 1 3 1 3 8 12 16 1 2 1 833  1 16  1w 1 13  3225  2 10 1n 4 2 4 1 51 3210 3 1 3 2 2 23 1 2 100 2 2 2 1 1 1 24411100 7  15 824 3 3 4 3 3 2 3 2 3 2 1mile in two 2 4 3 2 2  42. m 45 2a  2seconds 1  2 1   2 1 1 1 1 1 m 5 Bill ran half minutes fifty-five seconds. How many  200 200 3 1 2 1 1 1 2 3 2  1 1 3 3 83 16 3 (15) 8 16 11  11  2 17  1 1 2 21  2 1 2 1 1   2 31 3 2 2 1 1 2 2 4 4 1 is that? 2 3 2 42 71 32 2 121 2 1 1 2 41 1 1 33 1 1 3 1 24 24 24 1 1 3 2 33 7  1 4 4200 200 200 m  5m  m51  5 1  14  3 2 3 21 1 13 1 3 7 1 1 21 3. 3 2 1 3 2 22211  3 2221 2   3 3 7 7 3 w 1 1 3 2 n 3 3 3 8 16        The gauge of a railroad—the distance between the two tracks—is 3 3 1 1 1 7 1 3 1 2 1 1 1 7 1 1 1 1 7 1 24 2 1 1 3 1 1 1 1  1 18 3       3 3 w 1 1 1 1 1 1   3 2 n w 1 1 3 2 n 5 10 5 3 w 1 1 1 3 2 n 2 352(15)1 1251 2 4 10  2 3 2 51 32 3 4 5 1 1 221 4 s 1 2m  1 2 541 114 3 4 1 1 2   1 3 5 3 1 18 23 3 51 4 1 100 8 usually 1  110 How 3 200 2 2 10 1 16 1 4 feet 7 inches. inches3 is 4 8 2100 81many 34 that? 2 42 3 8 3 2 22 2 3 3 4 22 100 8100 4 2 2 4 2 3 2 4 2 3 3 7 3 1 1 7 1 1 1 w 11 11 1     3 w 1 1 1 1 1 3 3 2 n 1 1 1 1 3 2 n 1 1 1 3 15 1 24 2 1  1 3  341 10 5 * 4. 101 1 22 2 5211  38 24513 42 2 1 s 5 8  441 2 100 3*35. 1 82 m 4 2 1003 82 422 3 200 32 3 4 3 82 1 (68) (68) 3 27 1 7 2 1 13 232317 1 1 1 1 1 1 1 1 1 1 11 3 1 3 3w w 3 2 n 3 w n 12  n 8 a total 1510 2many 6. In 1 83 Yvonne scored of2 108 points. points 38 24 1 32six games 4 13 2 2 (18) 4   5 4 25 4  5 10 10 4 2 2 31 2How 3 2 7 41 2 2 3 1 1 1 1 1 1 1 242 3 4 53 4100 10 4 100 4 4 4 33 w3 per 1 did 1 she 1 1 8 2 11  41 1  200 3  2 24 n 1 2 m game average? 2 4 2 4     5 10 5 1 3 1 1 3 2 3 3 8 1 4   s 1 m 5 200 3 48 3 100 8 2 3 2 3 2 16 1 24 14 14 3 3 1 24 1 24 4 200m  5 4 3 3 1 4 reduce m3 200. 5 1 71 * 7. Write 1the of 24 and  3200. 200 5 1  s 16 3 3 prime factorizations w 3  28 3Then n (67)  5 8 16 10 4 100 3 3 3 3 3 1 3 3 3 24 24 3 Find 24 1 unknown 14 14 14  4200  4 5  m 1 5 each 1  4200 m 1 5  1 number: m 200 3 13 8 16 3 8 16 8 16 3 3 3 1 7 24 1 47 n 3 1 4 w s * 8. m  5  1 9. 3  2  3 16 1 10 2 n 3 200 w (59) 1 18 1 1 (59) 15 4 100  10 5 1 3 1 3 1 3 7 73 4 100 8 23 3 23 n 3 1 2 3 7 n 2 3 2 10 3 1  131 w 11. 134 w3 5  2 105 1n 5 51 10 w 1 1 1 1 10. 25d 1 = 0.375 1   s 1 100 8 2 3 5 1 3 1 (45) 3 (42) 4 42 10 100 8 2 3 2 3 2 7 3 7 7 3  n3  2 3 n 2 3 n 2 11 1 1 11 11 1 31 11 1 1 1 w 3  w 13 1w 1  1 5  13 5*13.   10 1 3 1 s14. 10 5 s * 12. 5  1 13  13 3   1 13 * s 3 5 7 10 23 23 2 8  n 4 100 4 (63)100 3 84 w100 1 8(66) 2 1 23 1 3 1 42 1 23 (68) 21 3 1 2 3 2  5 10 5 1 3 1 3 1 4 100 8 2 3 2 3 2 3 15. What 1is3the area of a rectangle that s is 4 inches s 1 4 inches 3 long and 14 (31, 66) 4 wide? 1 2

1 4

3

14

3 1 3 4 14

16. (3.2 + 1) − (0.6 × 7) (53)

* 18. (52)

19. (16)

s

20. (43)

Analyze

s

17. 12.5 ÷ 0.4 (49)

The product 3.2 × 10 equals which of the following?

A 32 ÷ 10 Estimate

thousand.

B 320 ÷ 10

C 0.32 ÷ 10

Find the sum of 6416, 5734, and 4912 to the nearest s

Instead of dividing 800 by 24, Arturo formed an equivalent s division problem by dividing both the dividend and the divisor by 8. Then he quickly found thesquotientsof the equivalent problem. What is s s the equivalent problem Arturo formed, and what is the quotient? Write the quotient as a mixed number. Verify

1 1 4 3 2

1 4 3 2 Lesson 68 351

21. The perimeter of a square is 2.4 meters. (38)

a. How long is each side of the square? b. What is the area of the square?

22. What is the tax on an $18,000 car if the tax rate is 8%? (41)

23. (58)

24. (60)

If the probability of an event occurring is 1 chance in a million, then what is the probability of the event not occurring? Analyze

Classify

Why is a circle not a polygon? 1 11 1 Compare:  4 4 3 32 2

1 1 4 3 3 4  2 2 1 1 1  4line segment 3 4 to 26. Estimate Use a ruler to find the length of3this 2 2 the (17) nearest eighth of an inch.

* 25.

(66, 68)

27. (28)

Analyze

Conclude

1 1 4 3 2

Which angle in this figure is an obtuse angle? 1 4 3 2

X

W

M

Y

28. Write 3% as a fraction. Then write the fraction as a decimal number.

(33, 35)

29.

(Inv. 6)

4

1 2

1 4 3 2

Early Finishers

Connect

A shoe box is an example of what geometric solid?

30. Sunrise occurred at 6:20 a.m., and sunset occurred at 5:45 p.m. (32) How many hours and minutes were there from sunrise to sunset?

Each 4 by 4 grid below is divided into 2 congruent sections.

Choose A Strategy

Find four ways to divide a 4 by 4 grid into 4 congruent sections.

352

Saxon Math Course 1

LESSON

69 Power Up facts mental math

Lengths of Segments Complementary and Supplementary Angles Building Power Power Up J a. Number Sense: 5 × 180 b. Number Sense: 530 − 50 c. Calculation: 6 × 44 d. Calculation: $6.00 − $1.75 e. Number Sense: Double $1.75. f. Number Sense:

$120 100

1

72

g. Measurement: Which is greater, 36 inches or 1 yard? h. Calculation: 6 × 5, + 2, ÷ 4, × 3, ÷ 4, − 2, ÷ 2, ÷ 2

problem solving

Nathan used a one-yard length of string to form a rectangle that was twice as long as it was wide. What was the area that was enclosed by the string?

New Concepts lengths of segments Reading Math We can use symbols to designate lines, rays, and segments:

¡

AB · ¡ ·

AB AB · AB

·

AB ¡ AB

Increasing Knowledge

$120 to designate 1points. Recall that we may use two points Letters are often used 72 100 to identify a line, a ray, or a segment. Below we show a line that passes through points A and B. This line may be referred to as line AB or line BA. We · ¡ · · may abbreviate line AB as AB. AB AB AB AB

A

The ray that begins at point A and passes through point B is ray AB, which may · ¡ · be . The portionAB of¡ line AB between ABabbreviated AB· AB·and including WX points A XY BA AB(or segment AB which can be ABabbreviated AB (or BA). WX and B is segment AB BA), · · AB ¡ WX · XY · = line¡ ¡ · · AB AB WY AB · AB AB AB AB In the AB AB AB segments: XY. The WY ABAB WX XYWX , XY , and WY WY figure below we canAB identify WX three ¡ · ¡ · · = ray AB AB AB AB WX of XY equals XYthe length WY . AB AB AB length AB XY WY AB of WX AB plus WX WX XY WY of WY the length AB = segment AB WX BA XY WY W

BA BA

BA

B

BA

BA

X

Y

BA

Lesson 69

353

Example 1 In this figure the length of LM is 4 cm, and the length of LN is 9 cm. What is theLM length of LMMN?LN LN MN LM LN MN MN L

M

N

Solution LMof MN equals LN MN LN MN The length the length of LN . With the MN LM LM of LM plus LNthe length information in the problem, we can write the equation shown below, where the letter l stands for the unknown length: 4 cm + l = 9 cm Since 4 cm plus 5 cm equals 9LM cm, we find that LNthe length of MN is 5 cm.

complementary and supplementary angles

Complementary angles are two angles whose measures total 90º. In the figure on the left below, ∠PQR and ∠RQS are complementary angles. In the figure on the right, ∠A and ∠B are complementary angles. S

A

R

Thinking Skill

60°

Verify

30°

Can an angle that is complementary be an obtuse angle? Explain.

Q

P

B

C

We say that ∠A is the complement of ∠B and that ∠B is the complement of ∠A. Supplementary angles are two angles whose measures total 180º. Below, ∠1 and ∠2 are supplementary, and ∠ A and ∠B are supplementary. So ∠A is the supplement of ∠B, and ∠B is the supplement of ∠A. B

A

75°

105° 2

75°

1

105°

C

D

Example 2 In the figure at right, ∠RWT is a right angle.

R

a. Which angle is the supplement of ∠RWS? b. Which angle is the complement of ∠RWS?

Q

W

S T

Solution a. Supplementary angles total 180º. Angle QWS is 180º because it forms a line. So the angle that is the supplement of ∠RWS is ∠QWR (or ∠RWQ). b. Complementary angles total 90º. Angle RWT is 90º because it is a right angle. So the complement of ∠RWS is ∠SWT (or ∠TWS). 354

Saxon Math Course 1

Practice Set AC

BC ACthe length of AB BC is 26 mm. AB a. In this figure the length of AC is 60 mm and BCthe length of AB. Find A

1 8

B

C

1 11 that measures1how many b. The complement of a 60º angle is an angle 8 28 2 1 degrees? 2

c. The supplement of a 60º angle is an angle that measures how many degrees? d.

Conclude If two angles are supplementary, can they both be acute? Why or why not?

e. Name two angles in the figure at right that appear to be supplementary.

3 1

f. Name two angles that appear to be complementary.

Written Practice * 1.

(28, 64)

2.

(54)

2

Strengthening Concepts 45 AC Then draw AC Draw a pair of parallel lines. a second pair AB of parallel 72 BC lines that are perpendicular to the first pair. Trace the quadrilateral that is formed by the intersecting pairs of lines. What kind of quadrilateral did you trace? Model

Connect

RS What is the quotient if18the dividend is 12 and the divisor is 18?

3. The highest weather temperature recorded was 136ºF in Africa. The lowest was −129°F in Antarctica. How many degrees difference is there between these temperatures? 45 45 72 72 4. Estimate A dollar bill is about 6 inches long. Placed end to end, about (15) how many feet would 1000 dollar bills reach? (14)

* 5. Write the prime factorization of both the numerator and the denominator (67) of this fraction. RS reduce the fraction. RS45 Then 72 45 72 * 6. (64)

Conclude In quadrilateral QRST, which segment appears to be 12345 parallel to RS? 12345 Q R RS

T

S

2 w1 5

1 3 2 2

8 n  25 100

2 1 2 3

7. In 10 days Juana saved $27.50. On average, how much did she save per day?  42   53  4  5 1  2  31  3 1 5 3 5 3 1 3 2 1 m m1   42   53  4  5 3 2  2 4 21 8 4 8 1  2  31  4

(13)

Lesson 69

355

4 3 3 53 3 3 3 1 5 n 32.42 3 5 1 12 1 m28 1 n 2.4 23 2  2 4m1 1  f 8 482 w 8 1 f 2 2  1 24 4  8 1 1 2 w1 42 3 1 35 5 255 100 3 33 5 0.08 25 100 0.08 54  5 35 5 33 3 1  21324 43 3 3 1 13 5 1  2.4 1 2  33 4 32 5 3 m 1   21 5  1  15 3  5 5 1m3  *28. 1  2  3  9. 8 n 2 1   3 2 1 5 m 1           3 2 1 1 2 3 4 5 2 4 8 4 8 m1 1 2 3 4 5 2 4 8 4 8 (61)  1 (5) 2 2  1 w1 2 4 45 8 8 23 2 4 8 4 25 100 3 123 3 4 55 1 0.08 Find each unknown number:  34  855 1  5 3 3 3 1n2 2 1 32 3 124 352.435 23 1 1 3 2.4 n 22 33 1  33 2 1 m  3 m  1 3 5 1 232 1143 25 8 3 1  2  m 1 13  5 1 5    13 1w2   124 1 2 2210. 1m 131   2 4 8 f    f  1 3 5 2 4 8 4 8f  5 4 8 4 8   3 11. 1 2 1 5 5 52 3 0.08 100 3 4 4 3  f 4 4 3 8 25 100 3(63) 5 3 4 0.088 (56) 3 5 3 3 1 4 3 3 2 1 m  12  5 8n 8 n 32 21 1 2.4 2.4 2 f 2 1 2 2 2 4 8 2 2.4 2 1  2n   128 3 2n 2 2   121  1 2.4  14w  18 8  1 w 413.  12. 5 w 1 23  2 3 211  25 0.08 0.08 1 100 2 1 w 1 5 5 25 3 25 100 3 25 5 100 3 25 100 3 0.08 3 5 35 0.08 (30) 5 (42)

5 5

1  2  31 4  3 5 5 3 2 1 1  2  32 4  4 5 8

n 2 2 2 113  2 100 51 w 2 3 2 1 2 3

8 2n2  1 2w 11 2 5 1 *14. 25 2  100 13 (68)5 5 3 2 1 16. 2 1 5 (49) 3

2.4 n 2 8 21 2 1 2.4 2 12.4 0.08 2  2  1  2 * 15. 2  1 5 35 3 3 25 100 3 0.08 (66) 0.08 2.4 0.08

2.4 0.08

17. a. What is the perimeter of this square?

(38)

b. What is the area of this square?

* 18. (65)

2.5 m

How can you determine whether a counting number is a composite number? Explain

* 19.

Represent

(65)

Make a factor tree to find the prime factorization of 250.

20. A stop sign has the shape of an eight-sided polygon. What is the name (60) of an eight-sided polygon? 21. There were 15 boys and 12 girls in the class. (29)

a. What fraction of the class was made up of girls? b. What was the ratio of boys to girls in the class?

22. (43)

1

1

1

1

1

1

Instead of4dividing doubled before 4 2 by 1 2, Carla 1 22 2 2 both numbers 2 2 dividing mentally. What was Carla’s mental division problem and its 1 1 42 12 quotient? Verify

1

1

42 12 1 1 1 What is the 4 23. 1 2reciprocal of 2 2? 2 (30, 62) WX XY XY WY WY WX 24. There are 1000 grams in 1 kilogram. How many grams are in 2.25 (15) kilograms? WX 1

42

1 1 1 25. How manyXYmillimeters long is the 4WX 1 1 1 line 4 12 2below? 2 WX WY 1 2 (7) 4 1 1 2 1 22 2 221 1 4

cm

*1 26. WX

4

(69)

1

356

2

3

4

1

22

XY 1 2WY 2

1 4

1

WX WY XY length of XYWY Connect The length of WX is 53 mm. The is 35 mm. What WX XY WY 4 XYthe length of WY ? is W

1 4

4

1 1 1XY 22 2

1 4

Saxon Math Course 1

X 1 4

Y 1 4

1

22

WY

27.

(Inv. 6)

WX

Draw a cylinder.

Represent

XY

WY

28. Arrange these numbers in order from least to greatest: (50)

29. (33)

0.1, 1, –1, 0 Draw a circle and shade

Represent

shaded?

1 4

of it. What percent of the circle is

30. How many smaller cubes are in the large cube shown below?

(Inv. 6)

Early Finishers

Real-World Application

Taylor and her friends at school decided to make ribbons for their classmates to wear for spirit week. Taylor’s mother offered to buy two rolls of ribbon. If $120 1 each roll of ribbon is 25 yards in length and each ribbon is cut to 7 2 inches 100 long, how many ribbons can Taylor and her friends make to give away at school? Show your work. Hint: 1 yard = 36 inches.

$120 $120$120 100 100 100

1

1

1

72 72 72

Lesson 69

357





LESSON

70

Reducing Fractions Before Multiplying Building Power

Power Up facts

Power Up G

mental math

a. Number Sense: 5 × 280 b. Number Sense: 476 + 99 c. Calculation: 3 × 54 d. Calculation: $4.50 + $1.75 1 3 of $250 10

e. Fractional Parts:

$250 10

$90.00

$250 10 g. Geometry: A square has a perimeter of 24 cm. What is the length of the sides of the square?

1 3

f. Number Sense:

h. Calculation: 5 × 10, ÷ 2, + 5, ÷ 2, − 5, ÷ 10, − 1

6 15

3 2 6   5 3 15

6 15

6 New 15

5

2 Concept

1

3 2 2   5 3 5Skill Thinking 1

Connect

2 5

Sometimes we 3 can reduce the 3 numerators and 5 1 the denominators 5  5 6 5 of both fractions. Reduce the following: 4 3  9 8 1

5 1 1   6 5 6

1 1 1 1 5 9

Front

Bottom 9 in.

Back 12 in.

12 in.

9 in.

3 2 6   5 3 15

Before two or more fractions are multiplied, we might be able to reduce the fraction terms, even if the reducing involves different fractions. For example, 1 3 in the multiplication below we see that the number 3 appears as a numerator 2  2 5 3 5 and as a denominator in different fractions. 1 3 2 6 6 2 33 22 66  66reduces to 22 15 5   5 3 15 15 15 15 55 33 15 55 We may reduce the common terms (the 3s) before multiplying. We reduce to 11 by dividing both 3s by 3. Then we multiply the remaining terms.

3 3

1

3 33 22 22  2  2   5 3 5 55 33 55 1 11

11

By reducing before we multiply, we avoid the need to reduce after we multiply. Reducing before multiplying is also known as canceling. 33 33

10 2 1 Saxon Math Course 5

12 in.

2 in.

Increasing Knowledge

1

10 6  358 5 9

Top

Right Side

The Crunch-O’s cereal company makes two different cereal boxes. One is family size (12 in. high, 9 in. long, 2 in. wide) and the other is single-serving size (5 in. high, 3 in. long, 1 in. wide). Each of their boxes is made out of one piece of cardboard. To the right is a net of the family size box. Use this diagram to draw a net for the single-serving box. 2 5

Left Side

problem solving

1

6 9

3 3

2 3

4

3

4 3  9 8

6 15

Example 1 Simplify:

Solution

5 1  6 5

5 5

1 1

5 1  6 5

5 5

1 1

We reduce before we multiply. Since 5 appears as a numerator and as a 51 51 5 1 5 1 1 1 1 denominator,  we reduce 51to  1 both 5s by 5. Then we multiply the 1 by  1 dividing 5 65 551  1 6 5 9 remaining 6 terms. 5 6 1 1 1 1 5 1 1 1 1 5 1 5 1   5 9  5 1 5 1 6 5 5 1 6 6 5 1 5 1 5 6 1 1 1 1 1 1 6 10 6 2 2 5 1 15  110  1 1 1 1   5 5 5 9 9 59 9 3 1 6 5 66 5 6 1 Example1 2 10 6 5 5 1 1 5105 12 1 1 1 1 5 9 5  5 11 1 5 1Simplify: 5 1 1 1 1 5 6 6 5 1 1   5 9 5 1 1 1 1 515 65 1 5 5 1 51 5 1 65 10  1 5 1 5 9 1  1 5 6  1025 10621 5  52 66 1 6 10 2 2 2 51 6 4 1 1 5 5 5 6Solution 6 5 6 6 10 6  1 5 9 55  4 9 53 11 9 3 9 3 1 3 5 3 9 53 the 1 5 1 5 in fraction 2 2 1 First we write 41 1 1 51 form.  1 numbers  5 1 10 6 11 1 12 1 5 5 10 6 6 6 52 61 4 10 5 5 13 1 13 11 1 1   1 6 3 10 5 9109 53 5 9 5 9 5 1 2 515 9 6 36  6 6 1 1 1 1 5 9 5 1 1 1 2 1 2 2 2 1 1 1 1 1 1 1 5 11 5 1 4 1 5 5 4 1 5 1 15 1011 5 5  1 1  1  1 1 1 1 1 1 1  5 5  1 696with559. 154with 4 5 1 5 1 5   1    510 9326 96  pair 10 and Thinking   3 39 6352 5 6 6  5 mentally 5 66 9We 5 6 Skill 6 6 5 66 5 1 5 5 9 3 3 5 6 1 1 31 15 1 1 5 11 15 5 1 Discuss 1 3 1  1 1 5 5 15 51 55 5 1 5 1 51 5 610 1  1110 1 6 61 1  5 1 5 15 2 265  10 26 2 51  4  151 1 1 91  6 2 2 10  Why might you5 10 5 1 5 9 6 5 1 6  15 6 4 9 92 5 52 5 5 6 5 6 1 1 6 5 6 5 5 10 3 136 51 want to reduce 9  5  3 6 4 1 4  11 6510 56 10 5102 106 10 6 3105 110 36 2 2 106 1 29  5 6 32 2 1 2 before you 13 2 6 to 2 by We    5to 2 dividing 5 by 55 5both and 5.1 We 5 1 51 1 1 1 5  5 9 5 91 95 by 3 9 5 reduce 9 55 9  reduce 3 1 5 95 110 9 3 13 1 3 1 1 multiply? 6 22 6 2 6 2  6 dividing both 62and59 by 3.6 5 61 55 61  1 9 1 95  1 5 1 1 1 1 16 65121  5 1 10 5 2 16 2  1 610 1  61 10 12 222151 6 2510 1 1 2   1     5 5 9 9 4 1 1 1 1 1 4 1 5 9 5 9 3 5 5 156 1 5 9 5 9 1 5 6 2 10 6 5 5 5 5 6 6  6 1 5 1 15 11  6 4 4 110 1 11 1  1 5 5 3 53 31 3 5 1 61 11      96 25 9 1 5 1 2 5 9 995 5 3 3   6 5 626 5 62 2 2 6 21 6 2 3 63 1 16 1 1 124 62 5 516 24 10 42 2 12 62 4 510 2 10 6 10 1    1 1 1 1 10 6 10 4 4 1 6510 4 4      1  3 2    3 6 3 3 3terms. 9 3 5wesimplify 5 the10 53 Then 53 5  63 23 9remaining 2 product. 1 5 multiply 5 39 We 9 9 35 5 the 5 6 9 5 1 10 6 10 1 3 6 12 1 1 3 3 3210 22 10 2 24  12 4  1 10 6 5  5 9 5 9 3 9 1 5 1 1 4 5 5 5  21 5 6 10 6 10 10 5 5 52 5 66 63 4 3 6 2 10     1  5 102 2 592 5 10 3 103  2 32 66 6 5 5 9 9 5 5 2 223 25 2 2 55 1 63 5 1 19 93 95 5 3 695 6  9 1 1  1 3 5 5 6 5 5 55 5 5 5 5 5 55 5 5 25 22 5 5     62 42 4  141 6 22 6 2 2 6 6 22 6 2 6 2 106  10 6 6 34 3 11 Example 3 92 52 3  2 2 3 3 4 1 19 3 10 6 5 4 5  10 6 5 45 345 1 1 51 55 5 Simplify: 2 2 2 2 5 5 2 2  1 3   3   3 5 56 2 23 3  14 9 6  2 5 1 6 3 2 26 22 2 3 6 1  10 6 4 10 46  141 94  1 15  6 5 5 10 6 4 13 1 5 5 3 21 2    3  3 33   5 39 5 3 3 9 3 55 22 5 5 2 2 95 52 3 5 5 2 5 1 3 3 5 1 2 Solution             1 1 1 1 3 5 5 1  6 6 5 25 5 26 55 2 6 5 5 6 2 5 2 5 6 2 5 5  number Thinking Skill This is a division problem. We first find6the of s in 1. The answer is 2 6 2 5 5 5 5 5 5 2 5 5 5 5 2   5  2 2 the5reciprocal Justify the use of 2 to find the number . Wethen 65 reciprocal 2 5 1  of 6 5 5 5 55 5 52  6 5 5 6 66 25 2 255 1   of s in . 5 5 how 6 Explain 2 6 to 22 6 6 2 6 2 6 2 divide any two 5 2 5 2 1  5 2  5 2 fractions. 21 5  6 5  5 2 5 2 5 22 5 5 26 5   1  1  6 5 6 5 2 5 5 22 5 5 2 55 22       5 2 1 1 5 2  6 5 2 5 26 55 1  6 5 2 5 Lesson 70

359

6 9

6 9

6 6 9 9

2 3 6 9 6 9 2 3

6 9

1 2 11 1 1 1 1 11 21 121 2  4 2  4 12 241  2 31 223  2 3  2 4 3 32 4 3 4 4 24 32 433

33 42 3   44 53 4

8 2 9 38 9    9 310 49 10

8 9  1 1 2 1 1 9 10 2 4 1  2 We cancel 2 we multiply. 3  Now we have a multiplication problem. 4 2 3 3 before 4

82 221 8 1 2 98 2 19 1 9 1 1 1 1 1   1 12 1  112 1   1 22 22 212 5 102 52 9 3 510 95 335 9 2 3 110 2 2 1 1 1 6 1 5 1 1 3 1 1 1 1 1 1 1 32 11 2 1 1 11 2 1 1 21 1 5 1 2 55 22 515 2 2  5 1 1 5 1 4  2 3 1 2 3 2  2 11 1 2 2 3 1 2 1  651 2521 52 38 2 3 9 36 2 5 6 1 3 3 2 43 3 4 Note: 4 32 3 5 We 4 5may 1 1 cancel only when fractions  1 multiplying. 62 3 3 3 1 1 1 15 61 3the5terms 3 of 361 12 2 A division 2 2      5 5 3 9 3 10 1 1 1 3 3 3 51 3 as a multiplication problem before we 51 61be 51 61 3 must 61 problem 3 rewritten 1 5 3 4 1 53 4 1 55 4 3 9 58 9 5 3 5 4 5 95 552 93 5 2 5 2 2 1the 5 5 1 1        cancel the terms of may  cancel 6 5 the   5 6of 6 terms 6 5 2 10 10not 6 10 69 10 6 52 4 fractions. 6We62do 3 4 5 36 3 4in5addition 6fractions 633 54 or 3 3 4 2 3problems. 8 9 8 29 3 8 9 1 1 1 2 subtraction 3 3 3  9 8 9 8 9  3 3 3 3 4 2 4 2 4 23 3 98 2 19 1 19 2 2 82 2 1 4 15 1 14 5 3 431 5 8 4 3 4 9 10 9 10 9 10     1   1  1 12 12 14 3 3 9 8 3 3 9 8 4 2 3 3 9 8 4 2 2 2 12 2 5 5 4 9 10 4 3 4 9 10 4 3 4 9 10 5 5 5 9 5 4 5 5 10 5 3 93 3 10 9 10 3 2 2 2 2  2   25   5 5 4 multiplying: 4 before 3 5 4 3 4 9 10 9 10 Practice Set Reduce 6 54 6 3 2 4 10 96 10 3 4 3 4 3 3 2 3 2 8 39 8 9 18 92 2 4 1 1   b.  2 1  4  c.  2 a.  3 2 4 5 4 5 4 3 5 4 3 4 43 9 4 10 9 101 29 10 1 1 1 1 3 3 4 1 1 1 1 1 5 2 51 2 1 1 21 1 1 2 1 1 1 1 12 1 1 21 2 2 1511 5in1 fraction 2 5             1 Write form. Then reduce before multiplying. 1 2 4 2 4 1 2 1 2 3 2 3 2 4 1  12 1 42 32 2 31331 22 41 133 1 4 3 5 5 45 65 59 5635 59 53 9 2 5  424  4    23  23  23  2 1 1634 2 452121 3   1    61 2 1 14 54 2 21 1 1 4 5 63 2 3 3 4 13  f. 143 1 4 6 2 56 32 110 63 6 10 2 6 10d. 2 6 42 2 3  24 3 1 3 e.4 211 2  2 31 32  23 1 4 23  1 3 4 3 4 4 32 43 4 2 32 3 43 1 21 2 11 1 1 2 12 2 1 12 1 9 1 Rewrite  4 2division 42 4  a2 multiplication 2 each 1problem 2 1 as 3  2 3  8problem. 1 2 3  2 Then reduce 1  1 4 2 3 3 4 4 4 2 3 2 3 3 4 3 4 12 2 5 3 5 9 3 10 3 4 3 4 2 3 before 8 9 8 9 2 3 multiplying.     3 4  2 3 24 2 233 428 2 223 28 82 9 32 98 8 9 12 8 9 9 11 9 11   2 8 2 2 19  1 1 1 1 4 5 4 5 3 4 3 42  92 10 10   9  1112 95 1 19 91 1012 8 544 9 8 2 2 2 2 22 2 1 12 5 4 3 10 5 4 3 4 9 10 3 4 1 1 1 53 3 9 1310 9 i.1531210 1 112 5210  h.39  g.  5 3 2 252 2 222 2 2 2 2 12 9 12 9 1 8 9 2 1510 2 1 52 5 5 5 3 25  23 2 92 10 10 3 89  23 8 9 3 2 1 1 11  1  1 12  1 12 1 12 22 22 222 5 2 510 2 5 3 5 3 5 9 3 9 3 9 10 3 510 8 2Concepts 8 25 894 2 19 2 2 2 2 19 51 1 1 11 1 1 Written Practice 2  Strengthening     1  51  12  1 512 2 9 12 2 5 22 22 9 1 3 9 36 9 10 5 3 510 6 51022 10 262 1 1 1 2 1 5 2 31 5 1 31 5  2143  1 19 1 5 1 9 5 14 52 1 1 5 19 21 2  4 2  4 1  2 1  12 3 2 51 5 seven 42321 345  2 34 1 5244 2 from 21867 2 44purchased 4 4 2 3 1. 2 Alaska 3 45 4155 293 55 3  million, 2 was in for2 45 Russia    two 5 9 5 5 5 5 5 5 9 4 2 3 3 4 4 2 3 3 2 3 (12) 6 5 6  62 5 6 210 66 2 41035 6 410 6 6 5  5 9 amount. 5 56 Use 52 5 9write 56 that 4 5 5 9 45 5 5 56thousand 5 64 2 10 6 10 6 6 2 10 6 hundred dollars. digits to       6 5 6 5 6 52 6 2 6 10 26 10 6 10 6 2. 5Connect 5 45eighth 5 5 59 equal 55 9a half 5 note? 9 5 4 5 How 4 many notes        (54) 5 5 5 6 2 10 6 6 6 6 2 6 2 10 6 10 6 2 2 2 2 8 2 8 2 9 19 1 1 1 1 1     2Verify2 Instead 9 82 ,21 92 2 2 8 122 by 9 21 28of12dividing 1doubled 1 1 numbers 3. Shannon 12 1 1 1 1 1 5 2 21  2   112both 5 3 5 3 9 3 9 5 3 10 10 2 112 2 212 2 2 2 22 (43) 5 5 5 5 3 9 3 10 5 5 3 9 3 10 2 3 9 3 10 2 and then divided. Write formed, as2well 3 5 2 2 the5 division 1problem 1 Shannon  n1 7 2 m1 ƒ   102 3 3 4 6 2100 8 2 3 3 4 6 as its quotient. 2 2 2 2   5 3 5 3

Reduce before multiplying: 5 5 5 5 9 5 9 5   5 4  5 4  5 5 54 5 5 9 5 5 5 9 5 9 5  * 6.  6 2 *64. 2  10 6 10 6 * 5.  5 5 6 2 106 6 2 10 (70) 6 5 66 2 6 10 6 (70) 6 (70) 8 2 1 2 2 1 1 1 7. What 5 is halfway 3number 9 3between 2 and 1 on the number line? (17) 3 5 5 2 2 1 1 2 1 1 1 m 21   2 7 9. 22  5  n  2 3  n  1ƒ 4 m6 1 8. 2100  102 3 3 4 6 3 4 7 8 2 3 2 2100  10(59) 3 6 8 2 3 3 (38) 5 3 5 2 2 1 1 2 n1 10. 7  2 11.4.37 3 4 m 1 + 12.8 ƒ + 6 2100  102 3 6 (63) 8 2 3 3 (38) 4 6 5 2 3 4 2 12. 0.46 ÷ 5 13. 60 ÷ 0.8  10 2 100 3 6 (45) (49) 8 2 1 2 8 1 1 2 2 2 1 1 5 2 2 1 1 1 2  2 2 3 1 4 5 What 9is the13average 3 2 n1 14. Evaluate the by 7the 5 of2 3 9three 3numbers  10 100 3 2marked 6 8 2 3 (18) 8 2 1 2 number line? (First2estimate whether the 5 3 average 5 on this 2 1 decimal 1  2 2 3 2 1arrows 1 1 n1 4 7 2 m1 ƒ  5 2100 3 9  10 3 3 2will6be more 8than 52 or less3than 5.) 3 4 6 8 2 1 2 2 1 1 5 3 9 3 8 2 1 2 2 1 1 1 5 3 9 3 2 4 5 6 8 2 1 2 2 1 1 1 5 3 9 3 2 5 4  6 5

360

5 4  6 5

Saxon Math Course 1

15. (49)

The division problem 1.5 ÷ 0.06 is equivalent to which of the following? Verify

A 15 ÷ 6

B 150 ÷ 6

C 150 ÷ 60

16. There are 1000 milliliters in 1 liter. How many milliliters are in 3.8 liters? (39)

Find each unknown number: 5 2 51 2 1 51 21 1 2 22 21 2 2 35 3 5 3 5   2 7 n21  n m1  18. 3  24  ƒ m  3  47 3 2 47 17. n1  m 1 1 1ƒ  19. ƒf   2 100  2 10100 10 1006 3102 68 3 2 68(43) 32 8 32 3 3 (30) 3 32 3 4 6 4 4 6 (56) 6 20. A pyramid with a triangular base has how many

(60, Inv. 6)

a. faces?

5 2 1 5 2  1 21 1 2 2 3 4 7 2  32  4 n7 1 2 m 1 n  1ƒ  2 b.2100 edges?  10 31 2 261100 2 10 8 2 32 81 2 2 81 22 8 52 2 5 1 2 6 323 5 83 52 3 1 2 33 5 5 2 2 1 1 1  1 1  1 1 2 12 32 21 43 2 14 27 3 1   27 4 2 7n  ƒ  12 n  1 m 1n m11 ƒ  m ƒ1  5 3 2100 95 32 31095 10 3210  100 83 2 34 6 4 6 4 6 33229100 63  28 6 2 3 3 3 3 6 8 c.32 vertices? Write the numbers in fraction form. Then reduce before multiplying. 8 2 1 2  21 2  1 1 *1 22. 8  2 2 * 21. 1  1 1 5 3 9 33 5 2 (70) 9 3 228 1 2 (70) 8 2 2 12 18 2 1 2 12 1 2 1 1  11  1 1 339 5 3Refer 9 3 53 59 3 to 2the line graph 2 below to answer problems 23–25. 2 John’s Waking Pulse 70

5 2 3 4 6

2 1 1 1 5 3

Beats per Minute

2100  102 68 3

2 n1 3

1 1 7 2 8 2

2 m1 3

ƒ

66 64 628

9

2

0 Sun.

2 3

Mon.

1 2

Tues.

Wed.

Thu.

Fri.

Sat.

23. When John woke on Saturday, his pulse was how many beats per (18) minute more than it was on Tuesday? 24. On Monday John took his pulse for 3 minutes before marking the graph. (18) How many times did his heart beat in those 3 minutes? 25. (18)

* 26. (67)

Formulate

Write a question that relates to the graph and answer the

question. Write the prime factorization of both the numerator and the denominator of this fraction. Then reduce the fraction. Analyze

72 300

72 300

AB

AB

BC

DC

BC

DC

BD

BD Lesson 70

361

BC BC ABthe length ofDC In rectangle ABCD the length of AB is 2.5 cm, and is 1.5 cm. Use this information and the figure below to answer problems 27–30. 72 72 72 72 300A 300 300 B 300

DC BD

Analyze

D 72 BC AB AB AB 300 27. What is the perimeter of this rectangle?

C

(8)

28. What is the area of this rectangle? (31)

Real-World Application

1

1

3

2

4 3 a  b 9 8

362

3

2

1

1

3

2

4 3 a  b 9 8

* 30. If BD were drawn on the figure to divide the rectangle into two equal (31) parts, what would be the area of each part? 1

Early Finishers

1

BD

(64)

DC

1

1

1

1

1

1

1

1

4 3 4 3 44 33 44 33 4 3 4 3   a  bHe b b a  roller a  ab anew Roland went to the local Super 88 9 8 paint 9 8 9 8Store 9 yesterday. 8 99 88 99bought 3 2 and 2 2 2 3 and roller pan for $8.97, a gallon of milk for3 $2.89, a3 magazine for $1.59, two identical gallons of paint without marked prices. He paid a total of $47.83 before tax. Find the price for each gallon of paint. 1 1 4 3 4 3  a  b 9 8 9 8 3

Saxon Math Course 1

2

DC BD

DC BD

4 3 a  b 9 8

4 3  9 8

BC ABsegments perpendicular * 29. Name two to DC . BC

BC DC

BC DC

BC AB

BD

INVESTIGATION 7

Focus on The Coordinate Plane By drawing two number lines perpendicular to each other and by extending the unit marks, we can create a grid called a coordinate plane. y

(–3, 2)

6 5 4 3 2 1

–6 –5 –4 –3 –2 –1 –1 –2 (–3, –2) –3 –4 –5 –6

(3, 2)

1 2 3 4 5 6

x

(3, –2) origin (0, 0)

The point at which the number lines intersect is called the origin. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis. We graph a point by marking a dot at the location of the point. We can name the location of any point on this coordinate plane with two numbers. The numbers that tell the location of a point are called the coordinates of the point. Thinking Skill Explain

On the coordinate plane, where will a point whose ordered pair contains two negative numbers be located?

The coordinates of a point are written as an ordered pair of numbers in parentheses; for example, (3, −2). The first number is the x-coordinate. It shows the horizontal �( )� direction and distance from the origin. The second number, the y-coordinate, shows the vertical �( )� direction and distance from the origin. The sign of the coordinate shows the direction. Positive coordinates are to the right or up, and negative coordinates are to the left or down. Look at the coordinate plane above. To graph (3, −2), we begin at the origin and move three units to the right along the x-axis. From there we move down two units and mark a dot. We may label the point we graphed (3, −2). On the coordinate plane, we also have graphed three other points and identified their coordinates. Notice that each pair of coordinates is different and designates a unique point: (3, −2) (3, 2) (−3, 2) (−3, −2)

Investigation 7

363

Refer to the coordinate plane below to answer problems 1–6. y

Thinking Skill Conclude

If you connected the points in alphabetical order (start with AB and end with HA), what type of polygon would you make?

C D

6 5 4 3 2 1

–6 –5 –4 –3 –2 –1 0 –1 E –2

F

–4 –5 –6

B A 1 2 3 4 5 6

x

H G

1. What are the coordinates of point A? 2. Which point has the coordinates (−1, 3)? 3. What are the coordinates of point E? 4. Which point has the coordinates (1, −3)? 5. What are the coordinates of point D? 6. Which point has the coordinates (3, −1)?

Visit www. SaxonPublishers. com/ActivitiesC1 for a graphing calculator activity.

The coordinate plane is useful in many fields of mathematics, including algebra and geometry. In the next section of this investigation we will designate points on the plane as vertices of rectangles. Then we will calculate the perimeter and area of each rectangle. Suppose we are told that the vertices of a rectangle are located at (3, 2), (−1, 2), (−1, −1), and (3, −1). We graph the points and then draw segments between the points to draw the rectangle. y

(–1, 2)

6 5 4 3

(3, 2)

1 –6 –5 –4 –3 –2

(–1, –1)

364

Saxon Math Course 1

0 –2 –3 –4 –5 –6

1 2

4 5 6

(3, –1)

x

We see that the rectangle is four units long and three units wide. Adding the lengths of the four sides, we find that the perimeter is 14 units. To find the area, we can count the unit squares within the rectangle. There are three rows of four squares, so the area of the rectangle is 3 × 4, which is 12 square units. Use graph paper or Investigation Activity 15 to create a coordinate plane. Use the coordinate plane for the exercises that follow. 7.

The vertices of a rectangle are located at (−2, −1), (2, −1), (2, 3), and (−2, 3). Represent

a. Graph the rectangle. What do we call this special type of rectangle? b. What is the perimeter of the rectangle? c. What is the area of the rectangle? 8.

The vertices of a rectangle are located at (−4, 2), (0, 2), (0, 0), and (−4, 0). Represent

a. Graph the rectangle. Notice that one vertex is located at (0, 0). What is the name for this point on the coordinate plane? b. What is the perimeter of the rectangle? c. What is the area of the rectangle? 9. Three vertices of a rectangle are located at (3, 1), (−2, 1), and (−2, −3). a. Graph the rectangle. What are the coordinates of the fourth vertex? b. What is the perimeter of the rectangle? c. What is the area of the rectangle? As the following activity illustrates, we can use coordinates to give directions for making a drawing.

Activity

Drawing on the Coordinate Plane Materials needed: • 4 copies of Investigation Activity 15 10.

Verify Christy made a drawing on a coordinate plane as shown on the next page. Then she wrote directions for making the drawing. Follow Christy’s directions to make a similar drawing on your coordinate plane. The coordinates of the vertices are listed in order, as in a “dot-to-dot” drawing.

Investigation 7

365

y

x

Christy’s Directions On your coordinate plane, draw segments to connect the following points in order: a. (−1, −2)

b. (−1, −3)

c. (−112, −5)

d. (−112, −6)

e. (−1, −8)

f. (−1, −8 12)

g. (−2, −9 12)

h. (−2, −10)

i. (2, −10)

j. (2, −9 12)

k. (1, −8 12)

l. (1, −8)

m. (112, −6)

n. (112, −5)

o. (1, −3)

p. (1, −2)

Lift your pencil and restart:

11.

a. (−2 12, 4)

b. (2 12, 4)

d. (−5, −2)

e. (−2 12, 4)

Carlos wrote the following directions for a drawing. Follow his directions to make the drawing on your own paper. Draw segments to connect the following points in order: Conclude

a. (−9, 0)

b. (6, −1)

c. (8, 0)

d. (7, 1)

e. (6, 12)

f. (6, −1)

g. (9, −2 12)

h. (10, −2)

i. (7, 1)

j. (6, 112)

k. (−10 12, 3)

l. (−11, 2)

n. (−10, −112)

o. (9, −2 12)

q. (−7, −8)

r. (−10, −8)

m. (−10 12, 0) p. (−3, −3 12) s. (−9, −112)

366

c. (5, −2)

Saxon Math Course 1

Lift your pencil and restart:

12.

extensions

a.

a. (−10 12, 0)

b. (−11, −12)

c. (−12, 12)

d. (−1112, 1)

e. (−12, 112)

f. (−1112, 2)

g. (−12, 2 12)

h. (−11, 3 12)

i. (−10 12, 3)

j. (−1112, 8)

k. (−9 12, 8)

l. (−7, 3)

m. (−6, 2 12)

n. (−7, 3)

p. (−4, 5)

q. (−1, 2)

o. (−6, 5)

On a coordinate plane, make a straight-segment drawing. Model Then write directions for making the drawing by listing the coordinates of the vertices in “dot-to-dot” order. Trade directions with another student, and try to make each other’s drawings. Use whole numbers, fractions, and mixed numbers to write the coordinates for each point. Represent

y

C

4

D E

3

B

2

1

A

F

0

b.

Use the given coordinates to identify the point at each location. Express the coordinates as whole numbers or decimal numbers. Represent

1

2

3

4

y 6

x D

C

5

A

4

E

3 2

B

1 0

c.

Generalize

F 1

2

3

4

5

6

x

Graph these points on a coordinate graph. Then connect the

points. (4, 2), (6, 2), (2, 4), (8, 4), (4, 6), (6, 6) • What polygon did you form? • Name a set of points that when connected would form a hexagon inside the hexagon you drew.

Investigation 7

367

LESSON

71

Parallelograms Building Power

Power Up facts

Power Up D

mental math

120  1.2 f

64 224

a. Number Sense: 5 × 480 b. Number Sense: 367 − 99 c. Calculation: 8 × 43 d. Calculation: $10.00 − $8.75 e. Number Sense: Double $2.25. f. Number Sense:

$250 100

g. Geometry: A square has an area of 25 in.2. What is the length of the sides of the square? h. Calculation: 8 × 9, + 3, ÷ 3, × 2, − 10, ÷ 5, + 3, ÷ 11

problem solving

Griffin used 14 blocks to build this three-layer pyramid. How many blocks would he need to build a six-layer pyramid? How many blocks would he need for the bottom layer of a nine-layer pyramid?

Increasing Knowledge

New Concept

In this lesson we will learn about various properties of parallelograms. The following example describes some angle properties of parallelograms.

Example 1 In parallelogram ABCD, the measure of angle A is 60∙. Reading Math Give two other ways to name ∠C.

a. What is the measure of C?

B

A

b. What is the measure of B? D

C

Solution a. Angles A and C are opposite angles in that they are opposite to each other in the parallelogram. The opposite angles of a parallelogram have equal measures. So the measure of angle C equals the measure of angle A. Thus the measure of ∠C is 60∙.

368

Saxon Math Course 1

A flexible model of a parallelogram is useful for illustrating some properties of a parallelogram. A model can be constructed of brads and stiff tagboard or cardboard. Model

Lay two 8-in. strips of tagboard or cardboard over two parallel 10-in. strips as shown. Punch a hole at the center of the overlapping ends. Then fasten the corners with brads to hold the strips together. 10 in.

8 in.

If we move the sides of the parallelogram back and forth, we see that opposite sides always remain parallel and equal in length. Though the angles change size, opposite angles remain equal and adjacent angles remain supplementary. With this model we also can observe how the area of a parallelogram changes as the angles change. We hold the model with two hands and slide opposite sides in opposite directions. The maximum area occurs when the angles are 90°. The area reduces to zero as opposite sides come together.

Discuss The area of a parallelogram changes as the angles change. Does the perimeter change?

The flexible model shows that parallelograms may have sides that are equal in length but areas that are different. To find the area of a parallelogram, we multiply two perpendicular measurements. We multiply the base by the height of the parallelogram. height

Math Language Remember that supplementary angles have a sum of 180°.

b. Angles A and B are adjacent angles in that they share a side. (Side AB is a side of ∠A and a side of ∠B.) The adjacent angles of a parallelogram are supplementary. So ∠A and ∠B are supplementary, which means their measures total 180°. Since ∠A measures 60°, ∠B must measure 120∙ for their sum to be 180°.

base

Lesson 71

369

The base of a parallelogram is the length of one of the sides. The height of a parallelogram is the perpendicular distance from the base to the opposite side. The following activity will illustrate why the area of a parallelogram equals the base times the height.

Activity

Area of a Parallelogram Materials needed: • graph paper • ruler • pencil • scissors Represent Tracing over the lines on the graph paper, draw two parallel segments the same number of units long but shifted slightly as shown.

Then draw segments between the endpoints of the pair of parallel segments to complete the parallelogram. 4 units high

5 units long

The base of the parallelogram we drew has a length of 5 units. The height of the parallelogram is 4 units. Your parallelogram might be different. How many units long and high is your parallelogram? Can you easily count the number of square units in the area of your parallelogram? Model

Use scissors to cut out your parallelogram.

Then select a line on the graph paper that is perpendicular to the first pair of parallel sides that you drew. Cut the parallelogram into two pieces along this line.

We will cut here.

370

Saxon Math Course 1

Rearrange the two pieces of the parallelogram to form a rectangle. What is the length and width of the rectangle? How many square units is the area of the rectangle?

Our rectangle is 5 units long and 4 units wide. The area of the rectangle is 20 square units. So the area of the parallelogram is also 20 square units. By making a perpendicular cut across the parallelogram and rearranging the pieces, we formed a rectangle having the same area as the parallelogram. The length and width of the rectangle equaled the base and height of the parallelogram. Therefore, by multiplying the base and height of a parallelogram, we can find its area.

Example 2 Find the area of this parallelogram: 5.2 cm

5 cm

6 cm

Solution We multiply two perpendicular measurements, the base and the height. The height is often shown as a dashed line segment. The base is 6 cm. The height is 5 cm. 6 cm × 5 cm = 30 sq. cm The area of the parallelogram is 30 sq. cm.

Practice Set

Conclude

Refer to parallelogram QRST to answer problems a–d.

a. Which angle is opposite ∠Q?

T

Q

b. Which angle is opposite ∠T? c. Name two angles that are supplements of ∠T.

S

R

d. If the measure of ∠R is 100°, what is the measure of ∠Q? Calculate the perimeter and area of each parallelogram: e.e.

f.f. 10 m

12 m

6 in. 8m

5 in.

8 in.

Lesson 71

371

g.

A formula for finding the area of a parallelogram is A = bh. This formula means Analyze

Area = base × height The base is the length of one side. The height is the perpendicular distance to the opposite side. Here we show the same parallelogram in two different positions, so the area of the parallelogram is the same in both drawings. What is the height in the figure on the right?

9 cm

6 cm 12 cm

Written Practice

h

12 cm

9 cm

Strengthening Concepts

1. What is the least common multiple of 6 and 10?

(30)

2.

(14)

3 3 Everest, 3 2 3 is Mt. 1 whose 2 1 peak3 is 3 1 on 2 land 3 11  The highest point  1 2 4 6  63  1 2 4 2 6 3 8 4 3 4 8 3 8 4 3 4 8 2 4 22 29,035 feet above sea level. The lowest point on land is the Dead Sea, which sea level. 3 to 1371 feet 3 3What is the1 2 dips 1 below 2 3 1   1 2 4 6 63 2 difference in elevation 3 8 between these 4 3two points? 4 8 2 4 2 Analyze2

3 31 2 3 1 3 2 1 1 32 3 3 1 1 3 1 3. 6 2 1 movie  p.m., The If the movie at 37 1:15 2  minutes. 4  3  2 2lasted 105 4  6 started 26  3at what 166 1 7 3 8 4 45 3 321 4 4 3 8 8 23 1 25  1 4 4 2 2 5  14 5  1 815  $75.004 8 4 8 15  $75.00 782 time did it end?

3 3  4 8

12 3 4  6 23 8

15  $75.00 2 3  3 8

1 1 7  3 reduce the 1 5 4–7, 5 fractions, In problems if possible, before 1 8 4 8 72 15  $75.00 multiplying. 3 3 3 3 3 3 1 2 1 2 2 1 2 1 1  22 3 3 3 3 138 1 1 4 2 213 11 4 3 2 343 8 4 381 4 1 6 4  66  2 35311  1 7 5 1  1 17 1 5  3 * 4.5  3 8 2 26  3    * 5.        2 26 6 34 41 41528 $75.00 8 (70) 3 8842 38 72 7 281 $75.00 3 4 3 3 7 24 2 461 8 1 8 2 154 (70) 32 1 3 2 4 3 4   1 2 4 6 63 2 3 8 4 3 4 8 2 4 2 2 3 1 3 3 1 1 22 3 31 1 3 3 1 7. 34 26 63 2 1  2  * 6.  1  2 4  6 6*  3 4 2 4 33 8 (70) 4 8 4 2 4 8 42 2 (70)

3 3  4 8

1 4 6 2

(32)

1

72

Analyze

53

1 8

15  $75.00

3 13 7 1 1 2 3 17 1 3 2  31 123 1 5  3115  6 1 5  41  6 8. 6  3 1  2 1 3  21 710. 1 9.7154  28  $75.00 4 7 22 15 15  $75.00 51  3 38 5 48418(63) 81 4 5  3 4 3 82 5 (63) 1 72 1 18 8 7 15  $75.00 4 8 7 2 15  $75.00 72 8 4 53 5 1 1 8 4 8 12. 15  $75.00 11. (3.5)2 72 (2) 1 1 7 1 7 (39) 5 1 5  51  3 1 8 15  $75.00 4 8 7 12 15  $75.00 72 4 8 13. (1 + 0.6) ÷ (1 − 0.6) (53) 1 1 7 53 5 1 1 1 14. Quan ordered8a $4.50 bowl 4 of soup. 8 15tax The rate was 7 2 % (which  $75.00 72 (41) equals 0.075). He paid for the soup with a $20 bill. (59)

a. What was the tax on the bowl of soup? b. What was the total price including tax? c. How much money should Quan get back from his payment? * 15. What is the name for the point on the coordinate plane that has the (Inv. 7) coordinates (0, 0)?

372

Saxon Math Course 1

* 16. (Inv. 7)

Refer to the coordinate plane below to locate the points

Represent

indicated. y

D E

6 5 4 3 2 1

–6 –5 – 4 –3 –2 –1 0 –1 F –2 –3 –4 –5 –6

C

B A

1 2 3 4 5 6

G

x

H

Name the points that have the following coordinates: b. (0, −3)

a. (−3, 3)

Identify the coordinates of the following points: d. E

c. H Find each unknown number: 17. 1.2f = 120 (49)

120  1.2 f 120 64  1.2 18. 224 f (49)

64 224

* 19. Write the prime factorization of both the numerator and the denominator (67) of this fraction. Then reduce the fraction. 120  1.2 f

120  1.2 f

64 224

64 224

20. The perimeter of a square is 6.4 meters. What is its area? (38)

21.

(Inv. 2)

22. (47)

$250 100

23. (7)

Analyze

What fraction of this circle is not

$250 100

$250 100

shaded?

If the radius of this circle is 1 cm, what is the circumference of the circle? (Use 3.14 for π.) How did you find your answer? Explain

Estimate

$250 100

A centimeter is about as long as this segment:

About how many centimeters long is your little finger?

Lesson 71

373

24. (10)

Water freezes at 32° Fahrenheit. The temperature shown on the thermometer is how many degrees Fahrenheit above the freezing point of water? Connect

50° F

40° F

30° F

64 224 25. Ray watched TV for one hour. He determined that commercials were shown 20% of that hour. Write 20% as a reduced fraction. Then find the number of minutes that commercials were shown during the hour.

(29, 33)

26. Name the geometric solid shown at right.

(Inv. 6)

* 27. (60)

Analyze This square and regular triangle share a common side. The perimeter of the square is 24 cm. What is the perimeter of the triangle?

28. Choose the appropriate unit for the area of your state. (31)

A square inches

B square yards

C square miles

* 29. a. What is the perimeter of this (71) parallelogram? b. What is the area of this parallelogram? * 30. (69)

In this figure ∠BMD is a right angle. Name two angles that are

7 cm

8 cm

10 cm

Conclude

B A

M

a. supplementary. b. complementary.

374

Saxon Math Course 1

C D

LESSON

72 Power Up facts mental math

Fractions Chart Multiplying Three Fractions Building Power

72 120

2 3

8 5

3 4

3 4

Power Up H a. Number Sense: 3 × 125 b. Number Sense: 275 + 50 c. Number Sense: 3 × $0.99 d. Calculation: $20.00 − $9.99 e. Fractional Parts:

1 3

of $6.60

f. Decimals: $2.50 × 10 g. Statistics: Find the average 45, 33, and 60. h. Calculation: 2 × 2, × 2, × 2, × 2, − 2, ÷ 2

problem solving

Kioko was thinking of two numbers whose average was 24. If one of the numbers was half of 24, what was the other number?

New Concepts fractions chart

Increasing Knowledge

We have learned three steps to take when performing pencil-and-paper arithmetic with fractions and mixed numbers: Step 1: Write the problem in the correct shape. Step 2: Perform the operation. Step 3: Simplify the answer. The letters S.O.S. can help us remember the steps as “shape,” “operate,” and “simplify.” We summarize the S.O.S. rules we have learned in the following fractions chart. Fractions Chart +− 1. Shape

2. Operate

Write fractions with common denominators. Add or subtract the numerators.

×÷ Write numbers in fraction form. × Cancel.

÷ Find reciprocal of divisor, then cancel.

Multiply numerators. Multiply denominators. 3. Simplify

Reduce fractions. Convert improper fractions.

Lesson 72

375

• Below the + and − symbols we list the steps for adding or subtracting fractions. • Below the × and ÷ symbols, we list the steps for multiplying or dividing fractions. The “shape” step for addition and subtraction is the same; we write the fractions with common denominators. Likewise, the “shape” step for multiplication and division is the same; we write both numbers in fraction form. Math Language Recall that canceling means reducing before multiplying.

At the “operate” step, however, we separate multiplication and division. When multiplying fractions, we may reduce (cancel) before we multiply. Then we multiply the numerators to find the numerator of the product, and we multiply the denominators to find the denominator of the product. When dividing fractions, we first replace the divisor of the division problem with its reciprocal and change the division problem to a multiplication problem. We cancel terms, if possible, and then multiply. The “simplify” step is the same for all four operations. We reduce answers when possible and convert answers that are improper fractions to mixed numbers.

multiplying three fractions

To multiply three or more fractions, we follow the same steps we take when multiplying two fractions: Step 1: We write the numbers in fraction form. Step 2: We cancel terms by reducing numerator-denominator pairs that have common factors. Then we multiply the remaining terms. Step 3: We simplify if possible.

Example Multiply:

3 3 2 1  5 4 3

3

15

8 5

3 3 2 3 8 1  15 5 4 3 5 Solution 3 3 2 33 3 2 3 8 8 1   1  improper 1 5 5 fraction 5. Then we reduce where possible 4 3 125 5as1 the 3 First5we write 4 before multiplying. the remaining terms, we find the product. 4 2 8 3Multiplying    2 1 3 5 4 5 1 2 1 8 13 4 2 14 3      2 3 2 5 1 104  45 2 1 23 5 1 8 1 1 2 8 3 24 8 3 4       3 5 4 35 5 4 5 1 1 1 the fractions chart from this lesson. Practice Set 1 a. Draw 5 3 fractions. 1 1 adding 2 1 three steps for b. Describe the   31 2 2 6 5 2 3 4

3 2 4

1 2 1 1 2 3

c. Describe the steps for dividing fractions. Multiply: 2 4 3   d. 3 5 8

376

1 4

Saxon Math Course 1

1 2 14 3 2  1  4 5 8 2 3 10

1 51 3   2 64 5

1 1 4 e. 2  1 2 10

11 5 32 3  1   2 22 6 53

3 1 3 2 1 2  23  1  2 1  1 2 2 4 3 2 3 4

1 1 1 2

Written Practice

Strengthening Concepts

2 4 3   3 5 8

1 1 4 2 1 2 10

1. What is the average of 4.2, 2.61, and 3.6? 2(18) 4 3 1 12 4 3 1 1     4 4 1 5 3 2 1 2 1 1 2 3 5 8 2 103 5 8 10 12 tablespoons  31 2 2. Connect Four tablespoons equals 4 cup. How many 5 2 6 2 3 (54) would equal one full cup? 3. The temperature on the moon ranges from a high of about 130°C 5 3 1 1 1 25 3 of how many 1 1 2 23 to a low of 1 about   −110°C.  1is adifference 2 3This 2   2 3  1 1   21 5 5 2 6 2 2 3 6 2 2 3 34 4 4 3 4 2 1 1 degrees?   4 2 1 3 5 8 2 10 4. Four of the 12 marbles in the bag are blue. If one marble is taken from (58) the bag, what is the that the 3 is 1 3 7 1 1 1 probability 7 1marble 1 n  w  2 13w  w2 3 n  1w 5 2 blue? 2 3 2 5 12not 2 3 12 a. blue? b. 5 3 1 3 1 2 1 2 1 1 3 7 1 relationship 1  the  2 theevents 1 1 b? 3  1between 2 c.4What in a and n  1 w word names w2 2 5 3 6  2 3 4 2 3 2 5 12 2 3 5. The diameter of a circle is 1 meter. The circumference is how many (7, 47) 1 1 1 1 centimeters?(6(Use 3.14   10)  for .) b (6a3 b 10)  ba4  b  a3  a4 π 10 100 10 100 6. Connect What fraction of a dollar is a nickel? 1 1 b (6  10) (29) a4  b  a3  10 100 Find each unknown number: 17 3 1 1 1 3 1 1 7   21 w3 7 7. n   w 12 13 1  wn  3 w 8. 1 2 2 3 3 2 5 (56)  5 (43)1 2w  3 12 w  2  n  12 2 5 12 2 3 7 9. 1 1 10. 1 − w = 0.23 1w w2 3 12(59) 2 3 (43) (14)

1 4

n

1 3  2 5

11. Write the standard decimal number for the following: (46) 1 1 1 1   a4b 1 b  a3 1 b (6  10)  a4  b(6a310) 10 100 10 100 b (6  10)  a4  b  a3  10 100 1 1 b (6  10)  a4  b  a3  10 Estimate Which of these numbers is closest to 1? 12. 100 (50)

A −1

B 0.1

C 10

13. What is the largest prime number that is less than 100? (19)

* 14

(64)

15. (38)

Classify

Which of these figures is not a parallelogram?

A

C

B

D

Connect

A loop of string two feet around is formed to make a square.

a. How many inches long is each side of the square? b. What is the area of the square in square inches?

Lesson 72

377

* 16. (69)

Conclude

Figure ABCD is a rectangle.

M

A

B

a. Name an angle complementary to ∠DCM. b. Name an angle supplementary to ∠AMC. D

C

Refer to this menu and the information that follows to answer problems 17–19. Menu Grilled Chicken Sandwich Green Salad Pasta Salad

$3.49 $3.29 $2.89

Juice: Small Medium Large

$0.89 $1.09 $1.29

From this menu the Johnsons ordered two grilled chicken sandwiches, one green salad, one small juice, and two medium juices. 17. What was the total price of the Johnsons’ order? (1)

18. If 7% tax is added to the bill, and if the Johnsons pay for the food with a (41) $20 bill, how much money should they get back? 19. (1)

Make up an order from the menu. Then calculate the bill, not including tax. Formulate

20. If A = lw, and if l equals 2.5 and w equals 0.4, what does A equal? (47)

* 21. Write the prime factorization of both the numerator and the denominator (66) of this fraction. Then reduce the fraction. 72 120

2 3

Refer to the coordinate plane below to answer problems 22 and 23. y 72 2 120 3

J

6 5 4 3 2 1

K

I

L

B

C

1

–6 –5 – 4 –3 –2 –1 0 3 –1 –2 –3 H –4 G –5 –6

A

1 2 3 4 5 6

F

E1

D

3

* 22. Identify the coordinates of the following points: (Inv. 7)

a. K

b. F

* 23. Name the points that have the following coordinates: (Inv. 7)

378

a. (3, −4)

Saxon Math Course 1

b. (−3, 0)

8 5

3 4

x

3 4

  4 5 8

4 2 1 2 10

1 1 21 4 3 1 2 41 31 1 5 3 3  4 1 2 of  1 lines.  4 Then draw a second pair of parallel  2 * 24. 2 parallel 2  1 Draw    2Model 3  13  2 5 182 apair 10 2 3 10 2 6 5 2 5 8 3 (64) 3 2 4 4 3 1 4 3 1 522 331 3 4 perpendicular 1 11 21 lines 1 to the 2 first pair of lines and about the same distance 2 24 3  2    3  1 1 21  1        1 1 2 1 2 1 2 1 Trace the quadrilateral that is formed by the intersecting lines. Is 5 8 2 2 2 10 3 apart. 2 3 2 22 633 510 34 45 8 the quadrilateral a rectangle? 1 1 5 32 3 1 21 3 2 1 51 3   2 3  1  21  1 2 3  1  2 * 26. 2 2 6 53 4(72) 2 32 4 3 2 64 5 11 5 32 3 1 21 3 2 1 2  23  1  21  1 2 * 28. 1  1 3  1   27. 2 53 4 22 6 (54) 2 32 4 3 2 3 (68)

1 51 3   2 64 5

(72)

29. (0.12)(0.24) (39)

Early Finishers

30. 0.6 ÷ 0.25 (49)

LaDonna had errands to run and decided to park her car in front of a parking meter rather than drive from store to store. She calculated that she would spend about 20 minutes in the post office and 10 minutes at the hardware store. Then she would spend 5 minutes picking up her clothes from the cleaner’s and another 30 minutes eating lunch.

15

30

45 60

Real-World Application

1 2 1 1 2 3

* 25.

0

1 4

The sign on the meter read $0.25 = 15 minutes $0.10 = 6 minutes $0.05 = 3 minutes a. How much time will LaDonna spend to finish doing her errands? b. If the meter has ten minutes left, how much money will she need to put into the meter?

Lesson 72

379

LESSON

73 Power Up facts mental math

Exponents Writing Decimal Numbers as Fractions, Part 2 Building Power Power Up J a. Calculation: 4 × 112 b. Number Sense: 475 − 150 c. Calculation: 4 × $0.99 d. Calculation: $2.99 + $1.99 e. Number Sense: Double $3.50. f. Decimals: $3.50 ÷ 10 g. Statistics: Find the median of the set of numbers: 30, 61, 22, 46, 13 h. Calculation: 3 × 3, × 3, + 3, ÷ 3, − 3, × 3

problem solving

How many different ways can Gunther spin a total of 6 if he spins each spinner once? 1 1

2

exponents Math Language Recall that an exponent is written as a small number on the upper right-hand side of the base number.

380

2

4

3

2 3

New Concepts

1

Increasing Knowledge

Since Lesson 38 we have used the exponent 2 to indicate that a number is multiplied by itself. 52 means 5 ∙ 5 Exponents indicate repeated multiplication, so 53 means 5 ∙ 5 ∙ 5 54 means 5 ∙ 5 ∙ 5 ∙ 5 The exponent indicates how many times the base is used as a factor. We read numbers with exponents as powers. Note that when the exponent is 2, we usually say “squared,” and when the exponent is 3, we usually say “cubed.” The following examples show how we read expressions with exponents:

Saxon Math Course 1

52

Thinking Skill Explain

3

10

“five to the second power” or “five squared” “ten to the third power” or “ten cubed”

How does the 34 “three to the fourth power” quantity two to 5 2 “two to the fifth power” the third power differ from the quantity Example 1 three to the Compare: 34 43 second power?

Solution We find the value of each expression. 34 means 3 ∙ 3 ∙ 3 ∙ 3, which equals 81. 43 means 4 ∙ 4 ∙ 4, which equals 64. Since 81 is greater than 64, we find that 34 is greater than 43. 34  43

Example 2 Write the prime factorization of 1000, using exponents to group factors.

Solution Using a factor tree or division by primes, we find the prime factorization of 1000. 1000 = 2 ∙ 2 ∙ 2 ∙ 5 ∙ 5 ∙ 5 We group the three 2 s and the three 5 s with exponents. 1000 = 23 ∙ 53

Example 3 Simplify: 100 ∙ 10 2

Solution We perform operations with exponents before we add, subtract, multiply, or divide. Ten squared is 100. So when we subtract 102 from 100, the difference is zero. 100 − 102 100 − 100 = 0

writing decimal numbers as fractions, part 2

We will review changing a decimal number to a fraction or mixed number. Recall from Lesson 35 that the number of places after the decimal point indicates the denominator of the decimal fraction (10 or 100 or 1000, etc.). The digits to the right of the decimal point make up the numerator of the fraction.

Lesson 73

381

Example 4 Write 0.5 as a common fraction.

Thinking Skill Connect

What is the denominator of each of these decimal fractions: 0.2, 0.43, 0.658?

Solution We read 0.5 as “five tenths,” which also names the fraction the fraction.

5 10.

We reduce

5 1  10 2

5 1  10 2

5 10

Example 5 Write 3.75 as a mixed number.

Solution Thinking Skill Verify

How do you to 34?

75 75 reduce 100 100

The whole-number part of 3.75 is 3, and the fraction part is 0.75. Since 0.75 has two decimal places, the denominator is 100. 3.75  3

3 4

75 100

3

3 75 3 100 4

We reduce the fraction. 3.75  3

75 100

3

3 75 3 100 4

Practice Set

Find the value of each expression: 3 375 375 c. 223∙ 52 75 b. 275 a. 104 3.75  33.75 + 2433375  33.75  33  3 100 100 100 4100 4100 4 3 75 75 3 75 3 d. Write 75 75 the prime factorization 1of 72 using exponents. 333 3 3 33 3 3.75  3.75 7 2% 100 100 100 100 100 4 4 4 Write each decimal number as a fraction or mixed number: e. 12.5

1

7 2%

h. 0.05

Written Practice 1

1

7 2%

7 2%

g. 0.125

i. 0.24

j. 10.2

Strengthening Concepts 1 7 2% Formulate

1

1 7 2% 7 2% 1. Tomas’s temperature was 102°F. Normal body temperature (38) is 98.6°F. How many degrees above normal was Tomas’s temperature? Write an equation and solve the problem.

2.

Jill has read 42 pages of a 180-page book. How many pages are left for her to read? Write an equation and solve the problem.

3.

If Jill wants to finish the book in the next three days, then she should read an average of how many pages per day? Write an equation and solve the problem.

(11)

3 75 3 100 4

f. 1.25

(18)

Formulate

Formulate

* 4. Write 2.5 as a reduced mixed number. (73)

* 5. Write 0.35 as a reduced fraction. (73)

382

Saxon Math Course 1

3 3 3 tax 1 when 13 1 1 (0.075) 11 1 1 1 6. What is the total3 cost of 1 a 1$12.60 �2� �22 � � 11 7 25% � 23 � sales 3 �item 3� 1 5� � 1 3 4 3 4 3 3 4 6 3 2 4 4 6 2 2 is added? 3 31 1 1 31 1 3 1 1 �3 � 1 3 � 2 1 � 1 3 5*18.� 3Analyze 3�2 �1 5* 7. � 1− 102) ÷ 52 (100 3 4 6(72) 4 2� 2 � 143 2 3 4 6 2 4 2 (73) 3 3 1 1 1 1 � � � � 2 1 3 2 1 5 �3 3 3 11 31 1 1 1 3 3 13 1 1 3 3 �4 1 1 6 2 � 2 � 1 9. 3 � 2 � � 21 � 15 � 33 � 2 4 � 110. � 15 � 3 4 3(61) 4 36 2 3 4 (63) 43 4 26 2 4 2

3 � 4

(41)

3 1 �2�1 4 3

�2�1

3 1 3�2 �1 3 4

1 3

3 1 1 1 5 � 3 * 11. � 1 6 2 (68) 4 2

(49)

* 13. Compare:

1

(44, 73)

1

1

0

1

1

1

1

1 3�2 � 3

1

1, , , , 0 AB 1, 2, b. , 4, 0 0.125 AC AB AC BC 253 a. 521 3 3 1 1 2 101 4 1 10 0.3 �2�1 �1 3�2 �1 5 �3 4 3 3 4 6 2 4 2 14. The diameter of a quarter is about 2.4 cm. 1 1 1 BC AB AC (47) 1, , AB AC BC 10 , 4 , 0 a. 2What is the circumference of a quarter? (Use 3.14 for π.) 1 1 1 1, 2, 10, 4, 0 AB AC 1 1 1 1 1 1 b. What is the ratio of the radius 1, 2, 10, 4, 0 AB 1, 2, 10, 4, AC BCof the 0 AB ACquarter to the BCdiameter of the quarter?

1

1, 2, 10, 4, 0

1 1 1 2 , 10 , 4 ,

3 1 �2�1 4 3

12. 7 ÷ 0.4

3 1 �1 4 2

AB

AC

BC Find each unknown number:

1

1

1

BC

BC

AB

1, 2, 10, 4, 0

15. 25m = 0.175

16. 1.2 + y + 4.25 = 7 (43) 1 1 1 1 1 1 1, , , , 0 1, , , , 0 1 1 1 17. 2 10 4 10 4in 123,456.78? Which digit is in the ten-thousands 1, 2, 10, 4, 0 (34) AB AC BC 2place (45)

1 1 1 1, , , , 0 2 10 4 3 1 �2�1 4 3

1 1 1 these 18. Arrange in order from least to greatest: , 3, 0 numbers (56) 1, , 1 3 11 3 11 13 13 11 2 10 4 �2�1 �3 2��211 �1 1 13 �5 2 ��3 1 5 ��31 4 3 3 4 6 2 4 4 3 3 4 6 22 3 3 1 1 1 1 1, , , , 0 3� 5 �31 1 1 �1 1 21 �1 1 2 10 4 6 1,2 , , 4, 0 2 1, , 3 , , 04 2 10 4 2 10 4 * 19. Write the prime factorization of 200 using exponents.

3 1 �1 4 2

(73)

1 1 1 , , ,0 2 10 4

1 1 1 , , ,0 20. The store offered a 20% discount on all tools. The regular1,price 2 10of4a (41) hammer was $18.00. a. How much money is 20% of $18.00?

1 1 1 1, , , , 0 2 10 4 1

1

1

1, 2, 10, 4, 0

b. What was the price of the hammer after the discount?

* 21. AB

(69)

1

1

1

1

1

1

of 2AB The is 50 mm. What 1,The , 4, 0 1, AClength of ACBC , 10is , 416 , 0 mm.AB BC 2 , 10length is ACthe length of BC? Connect

A

B

22. One half of the area of this square is shaded. (31) What is the area of the shaded region?

1 1 1 1, , , , 0 2 10 4

23. (64)

* 24. (73)

Verify Analyze

C

6 in.

1 1 1 1 1 1 , ,square , 0 1, a,rectangle? , ,0 Is1,every 2 10 4 2 10 4 22  23 2

Lesson 73

383

* 25. (72)

Explain The fractions chart from Lesson 72 says that the proper “shape” for multiplying fractions is “fraction form.” What does that mean?

Refer to this coordinate plane to answer problems 26 and 27. y

J

K

I

6 5 4 3 2 1

–6 –5 – 4 –3 –2 –1 0 –1 –2 –3 H –4 G –5 –6

A

L

B

C 1 2 3 4 5 6

F

E

x

D

* 26. Identify the coordinates of the following points: (Inv. 7)

b. L

a. H

* 27. Name the points that have the following coordinates: (Inv. 7)

b. (3, 0)

a. (−4, 3)

* 28. If s equals 9, what does s2 equal? (73)

29. Name an every day object that has the same shape as each of these geometric solids:

(Inv. 6)

* 30. (71)

a. cylinder

c. sphere

b. rectangular prism

d. cube

The measure of ∠W in parallelogram WXYZ is 75°.

W

Z

Conclude

a. What is the measure of ∠X? b. What is the measure of ∠Y?

Early Finishers

Real-World Application

Y

X

A teacher asked 23 students to close their eyes, then raise the hand up when they thought 60 seconds had elapsed. The results, in seconds, are shown below. 61 55

65 67

73 63

80 66

35 83

56 70

57 51

71 54

52 66

86 64

39 41

58

Which type of display—a stem-and-leaf plot or a line graph—is the most appropriate way to display this data? Draw your display and justify your choice.

384

Saxon Math Course 1

LESSON

74

Writing Fractions as Decimal Numbers Writing Ratios as Decimal Numbers

Power Up facts mental math

1

2 Power Building

0.5 2  1.0 10 0

1 2

2 1

Power Up I a. Number Sense: 3 × 230 b. Number Sense: 430 + 270 c. Calculation: 5 × $0.99 d. Calculation: $5.00 − $1.98 1

e. Fractional Parts: 4

1 4

of $2.40

4 1

f. Decimals: $1.25 × 10 g. Statistics: Find the median of the set of numbers: 101, 26, 125, 84, 152 h. Calculation: 5 × 5, − 5, × 5, ÷ 2, + 5, ÷ 5

problem solving

A 60 in.-by-104 in. rectangular tablecloth was draped over a rectangular table. Eight inches of the 104-inch length of cloth hung over the left edge of the table, 3 inches over the back, 4 inches over the right edge, and 7 inches over the front. In which directions (left, back, right, and/or forward) and by how many inches should the tablecloth be shifted so that equal amounts of cloth hang over opposite edges of the table? What are the dimensions of the table?

New Concepts writing fractions as decimal numbers 1 2

1 4

Increasing Knowledge 1

We learned earlier that a fraction bar indicates division. So the fraction 2 also 1 1 means “1 divided by 2,” which we a decimal 2 can write as 2  1. By attaching 2 point and zero, we can perform the division and write the quotient as a decimal number. 0.5 0.5 2  1.0 2  1.0 10 10 0 0 0.5 1 1 We find that 2 equals the decimal 2  1 number 0.5. To convert a fraction to a 2 2  1.0 1 decimal number, we divide the numerator by the denominator.1 0 4 1 1 40 1 4 4 1 2

1 4

2 1

2 1

1 4

1 4

1 4

1 2

1 2

4 1

0.25 4  1.00 4 8 20 20 0

4 1 1 4

4 1

Lesson 74

0.5 2  1.0 10 0

385

10 0

2  1.0 10 0

2

Example 1 1 Convert 14 to a decimal number. 4

8 20 20 0

4  1.00 8 20 20 0

4 1

Solution 1 2

1 0.5 1 1 4  1 by 4,” which is 2  14 The fraction 4 means “1 divided 2 1.0 2  point and zeros, we 4  1. By attaching a decimal 10 can complete the division. 0

1 4

0.25 4  1.00 8 20 20 0

Example 2 15

1 4

1 4

Solution

2

15 16

15

Use a calculator to convert 16 to a decimal16number.

2

75

75

4 1

Begin by clearing the calculator. Then enter the fraction with these keystrokes. 0.4 5 � 2.0 15 After pressing the equal sign, the display shows the decimal equivalent of 15 16: 16

15 16

0.9375 15 answer 16

The is close to 1.

2 reasonable15 16because 5

2 3 2 both 15 than but7 25 4 15 7 5 less 7 50.9375 are 16 and 4

0.4 11 2 2 Write 7 5 as a decimal 7 5number. 16 5 � 2.0

15 16

Solution

2 5

31 32

15 16

2 5

24

15 16

3 64 2 5

15 16

2

2 75

2 2 2 2 The whole number part of 7 5 is 7, which we of the decimal 75 5 write to the left 5 2 15 2 2 3 1 2 1 2convert 2 to a decimal point. We by dividing 2 by 5. 7 7 7 1 75 3 5 45 1 5 5 16 5 5 4 8 4 5 4 8 20 0.4 2 15 2 2 2 31 11 2.0 5 7 7 � 5 5 16 5 5 16 32 2 15 2 2 15 2 2 2 2 2 15 Since 5 16 equals 0.4, the 7mixed number 7 5 equals 7.4. 5 716 716 5 5 5 5 531 11 3 3 1 7 3 24 31 2 1 7 24 11 Model4 Use a calculator to check the answer. 75 3 16 32 64 3 5 4 8 20 10 25 16 32 64 3 2 2 7 5 5 4 3 writing ratios Converting ratios to decimal numbers is similar to converting fractions to 4 1 5 43 3 1 7 2 1 7 0.4 decimal numbers. 7 5as decimal 4 45 3 10 8 20 25 3 3 1 7 1 7 5 � 2.0 numbers 4 5 24 3 10 3 3 125 7 1 4 31 8 20 4 3 3 32 64 5 4 8 20 10 31 11 24 Example 4 3 64 3 3 1 7 1 7 16 32 4 3 31 11 5 4 8 20 10 25 A number cube is rolled once. Express the probability of16rolling an even 32 24 3 1 7 7 21 31 31 1 2 73 7 3 64 number 3 10 number. 455 32 8 20 25 875 204 as a decimal34 105 25 24 3 64 1 3 31 1 7 11 7 3 24 4 3 32 16 5 8 20 10 25 3 64 1 1 24 4 3 64 5 4 8 15 16

15 16

11 16

1 8 31 32

1 4 3865

24 1 3 64 7 8 20 7 Saxon Math Course31 3 24 20 3 10 64 31 32

1 8

0.4 5 � 2.0

Example 3 15 16

2 5

7 25

3

3 10

7 25 11

31

24

75

3 4

7 20

3

3 10

24

3 64

1

45

1 8

7 25 3

7 20

24 64

0.4 0.4 0.4 0.4 0.4 0.4 2.0 5 2.0 5 2.0 5�� 2.0 5�� 2.0 5�� 2.0 5 Solution

0.4 0.4 0.4 2.0 5 � 2.0 5 � 2.0

2

2 5

22 75 5

75

7 45

11 16

31 11 16 32

31 32

3124 3 64 32

32

1 Probabilities are often expressed as 36decimal numbers between 0 and 1. 2 3 1even, the 2 3 2 2 on a 3number 1 probability 11 1of Since22three of22 the six22numbers 3 1 2 1 31 1 7 52 11 34cube are 4 7 7 25 4 4 15 33 7 55 45 5 5 4 8 8 7 4 4 4 55 5 47 5 8 5 number 5 5 4 8 5 5 is 66, which 4 rolling5 an even equals . 22

3 3 6 4

1

1 73 31 11 17 1 by 2.7 73 73 7 We convert by1 dividing 24to a decimal 3 10 45 4 3 10 3 5 8 20 2510 4 85 8 20 20 25 25 0.5 2 1.0  3124 11 24 31 31 11 11 24 31 11 31 11 3124 11 24 3 3 3 24 16 3264 16 16 32 32 64 0.5 0.5 3 3 3 64 Thus 16 the probability of number 32 16 64with one 16 rolling32an even 3264 64 roll of a number cube 22 1.0 1.0 is 0.5. 0.5 24 24 23 64 1.0 3 64

Practice Set 3 4

11 16

31 4 45

31 11 16 32

4

1 5

31 32

17 17 8 20 20 8

Convert each fraction or mixed number to a decimal number: 3 3 3 3 17 11 1 7 1 1 1 3 31 1 7 1 a. 3 b. 4 c. 17 4 4 4 11 4 45 5 5 8 4 4 8 8 20 20 6 2 4 85 20 5 4 8 20 45 8 3 73 e. 3 3 10 2010

73 3 2510

3 33 3 10 10

11 4 85

1 8

3124 3 3264

You may use a calculator to convert these numbers to decimal numbers: 31 31 31 11 24 11 11 24 24 11 24 31 31 11 11 24 3 3 24 i. 3 0.531 3 3 g. 16 h. 32 3 64 16 32 16 32 64 64 16 64 16 2  1.032 32 64 16 32 64 24 j. In24 3 3 a bag are three red marbles and two blue marbles. If Chad pulls 64 64 one marble from the bag, what is the probability that the marble will be blue? Express the probability ratio as a fraction and as a decimal number.

d.

Written Practice * 1. (73)

7 20

73 37 3 3 20 10 10 20

17 8 20

7 25

f.

73 7 3 3 3 10 25 2510

7 25

Strengthening Concepts What is the difference when five squared is subtracted from four cubed? Analyze

* 2. On LeAnne’s map, 1 inch represents a distance of 10 miles. If Dallas, TX (15) and Fort Worth, TX are 3 inches apart on the map, approximately how are they? 3 many miles apart 1 6

* 3. Convert (74)

3 24

2

4

to a decimal5number.

* 4. Tito spins the spinner once.

(58, 74)

a. What is the sample space of the 0.5 2  1.0experiment?

1

4 3

24 a b. What is the probability that he spins number greater than 1? Express the probability ratio as a fraction and as a decimal. 5 3 4 1 2 1 1 7  1   3fraction. * 5. Write 0.24 as3a  reduced 5 5 2 6 4 8 8 (73)

1 2 1   2 3 6

4 5

3

2 5

3 1 1 4 2

1 3 3 3

1 7 3 1 4 8

5 3 4   8 5 5

Lesson 74

387



24

5

4 5

3

3

3

4 5

24

24

1 2 1   2 3 6 2 1  3 6

1 7 3 1 4 8

3

4 5

24

4

24 5 6. Formulate Steve hit the baseball 400 feet. Lesley hit the golf ball (13) 300 yards. How many feet farther did the golf ball travel than the 4 5 baseball? After converting yards to feet, write an equation and solve the problem. 3 1 2 1 1 1 72 1 5 3 14 7 1 5 3 4 11    h  8, then  3 what  1 does  31  3 1 equals 3 A 3  7. If A = bh, and if b3 equals 123and 5 5 5 5 2 6 4 2 8 3 6 8 4 8 3 8 4 32 (2) equal? 5 3 4 3 40 1 7 1 1   1 3 1 3 3 5 5 4 8 8 3 4 2 96 * 8. Compare: 32 3+3 (73) 5 3 14 2 1 1 32 15 1 3 41 40 7 1 5 3 4 3 40 3 11 1     3 3 3 1  11 7  3  1 3  3    13  3 4   1 210. 1 5 25 3 6 3 8 9. 4 2 8 3 68(63)5 54 96 8 3 8 5 5 4 23 96 4 2 (61)

1 2 1   2 3 6

7 1 5 3 4 3 1 1 7 2 15 3 41 11   3  1 3  3    13  3 12. 3  1  11. 8 3 8 5 5 4(66) 23 4 2 8 3(72) 68 5 54

5 3 4   8 5 5

1 3 3 3 3 24

13. (68)

4 15. 5 (Inv. 6)

16. (15)

3 1 1 4 2

40 96

Represent

40 96

3 1 1 4 2

40 96

14. (4 + 3.2) − 0.01 (38)

Draw a triangular prism.

LaFonda bought a dozen golf balls for $10.44. What was the cost of each golf ball? Write an equation and solve the problem. Formulate

17. Estimate the product of 81 and 38.

4 5

(16)

18. In four days Jamar read 42 pages, 46 pages, 35 pages, and 57 pages. (18) What was the average number of pages he read per day? 5 3 4 3 1 7 1 1  1 is the least  common   12? 3 What 3  3 of 6, 8, and 1 19. multiple 5 5 4 8 8 3 4 2 (30)

1 2 1   2 3 6

40 96

20. 24 + c + 96 = 150 (3)

21. Write the prime factorization of both the numerator and the denominator (67) of this fraction. Then reduce the fraction. 1 7 3 1 4 8

5 3 4   8 5 5

1 3 3 3

* 22. (47)

3 1 1 4 2

40 96

PQ

PQ

If the perimeter of this square is 40 centimeters, then Analyze

a. what is the diameter of the circle? b. what is the circumference of the circle? (Use 3.14 for π.) 23. Twenty-four of the three dozen cyclists rode mountain bikes. What (29) fraction of the cyclists rode mountain bikes? * 24. (64)

Classify

Why are some rectangles not squares?

3 4

388

Saxon Math Course 1

AB

Q

25. (17)

Connect

Which arrow could be pointing to 34? A

B

0

26. (64)

C

1

2

3

In quadrilateral PQRS, which segment appears to be

P

PQ

Q

AB

b. perpendicular to PQ? 27.

4

Conclude

a. parallel to PQ?

(Inv. 6)

D

DC SAB

PQ

The figure at right shows a cube with edges 3 feet long. SR RS a. What is the area of each face of the SR RS cube?

R

DC

Analyze

PS

SP PS

DA

SP 3 ft

3 ft

b. What is the total surface area of the cube?

3 ft

2 3

Refer to this coordinate plane to answer problems 28 and 29. 2 3

y

3 4 3 4

D

6 5 4 3

A

1 –6 –5 – 4

C

–2 –1 –1 –3 –4 –5 –6

1 2

x

4 5 6

B

* 28. Identify the coordinates of the following points: (Inv. 7)

29. PQ

AB

(64)

b. origin

a. C Connect

PQ PQ ABCD is AB and One pair of parallel segments in rectangle

DC. Name a second pair of parallel segments.

30. Farmer Ruiz planted corn on 60% of his 300 acres. Find the number of (41) acres planted with corn. SR RS PS RS

PS

DC

SP

DA

AD

CB

BC

2 3

3 4

Lesson 74

389

SP

LESSON

75

Writing Fractions and Decimals as Percents, Part 1 Building Power

Power Up facts

Power Up K

mental math

a. Calculation: 504 × 6 b. Number Sense: 625 − 250 c. Calculation: 3 × $1.99 d. Calculation: $2.50 + $1.99 e. Number Sense: Double $1.60. f. Decimals: $12.50 ÷ 10 g. Statistics: Find the median of the set of numbers: 28, 32, 44, 17, 15, 26 h. Calculation: 6 × 6, − 6, ÷ 6, − 5, × 2, + 1

problem solving

The sum of the digits of a five-digit number is 25. What is the five digit number if the last digit is two less than the fourth, the fourth digit is two less than the third, the third is two less than the second, and the second digit is two less than the first digit?

Increasing Knowledge

New Concept

A percent is actually a fraction with a denominator of 100. Instead of writing 1 1 3 25 � 2 (%). So 100 the denominator 100, we can use a percent equals 25%.100 3 sign 3 2 1 1 3 �2 3 2

Example 1 1 � 1 25 2 3 100 3 2 3 3 25 Write 100 as a percent. 100 100

3 100

3 100

3 100

Solution A percent is a fraction with a denominator of 100. Instead of writing the 1 1 3 3 25 denominator, we 100 write a percent100 sign. We write 100 as 3%. 3 �2 3 2 3 ? 3 � 10 10 100 Example 2 Write

3 ? � 10 100

3 10

3 10 3 10

3 10

390

Saxon Math Course 1

3 ? as a percent. � 10 100 10

3 10

3 10 30 � � 10 10 100

10

3 ? � 10 100

3 10

10 10

10 10

3 10

10 10

3 10 30 � � 10 10 100 30 100

3 10 30 � � 10 10 100

3 1

1 1 3 �2 3 2

3 100

25 100

3 100

3 100

0

Solution First we will write an equivalent fraction that has a denominator of 100.

Thinking Skill Explain

How do we write 3 ? 3 an equivalent 3? 3 � � 10 We multiply 10 10 100 10 100 fraction?

3 10

3 10 3 10

by

10 10 .

10 10

3 ? 3 � 10 10 100 3 10 330 10 30 � � � � 10 10 10 100 10 100 3 10 30 � � 10 10 100

30 100

3 ? 10 3 � 10 10 10 100 3 10 30 30 � �We write the fraction 100 as 30%. 10 10 100

3 10 10 10

3 1 � 10 1

10 10 30 100

30 100

Example 3 Of the 30 students who took the test, 15 earned an A. What percent of the students earned an A? 15 1  30 2

Solution

1 2

50 1 50   2 50 10

50 50

1 2

Fifteen of the 30 students earned an A. We write this as a fraction and reduce.

15 1 50 1 1 8 80 12 2 2 50  0.08  30 2 0.8  0.80 100 100 15 1 15 11 1 100 50 1 50 50 5 151 1 1 1 1 1 50  To write 2 as a fraction with    by .  a denominator  we multiply 2 2 50 of 100, 30 2 30 22 2 50 2100 50 100 302 2 2 50 1 50 50 5 1 1   2 2 50 2 50 100 100 8 80 12 50 0.8  0.80  0.12  0.08  The fraction 100 equals 50%. 100 100 100 8 80 8 80 8 12 12 12   0.800.08   0.8 0.80 0.08  0.8  0 0.12  0.08  0.12 0.8 0.12 100 100 100 100 100 100 100 100 8 80 Example 4 0.8  0.80  0.08  100 100 15 50 1 50 1 1 Write 0.12 as a percent.  2 2 50 30100 2 50 Solution 50 50 0.12 

15 1  30 2

0.12 

12 100

Example

4

100

100

100

The decimal number 0.12 is twelve hundredths.

50 100

0.12 

12 100

0.08 

8 100

0.8  0.80 

80 100

Twelve hundredths is equivalent to 12%.

Example 5

50 100 1 2

Example 5

15 1 Write 0.08 as a percent.  30 2 Solution

1 50  2 50

50 50

1 2

The decimal 0.08 is eight hundredths. 0.12 

12 100

0.08 

8 100

0.8  0.80 

80 100

Eight hundredths is equivalent to 8%. 50 100

Lesson 75

391

Example 6 Example

6

Write 0.8 as a percent. 15 1 1  2 30 2 Solution

50 1 50   2 50 100

50 50

1 2

5 100

The decimal number 0.8 is eight tenths. If we place a zero in the hundredths place, the decimal is eighty hundredths. 31 31 1 8 100 0.08 100

31 12 0.12 100 100 1 31 100

1 100

11 10 100

1 10

1 100 0.8

100

1 11 80 100  10 0.80  10 100

3 7 37 Eighty13 hundredths equals 80%. 50 10

50

25

25 50

27 5 25

1 3 10 50

2 5

3 50

3 7 50 25

7 25

27 25 5

2 5

27 25 5

2 5

2 5

Notice that when a decimal number is converted to a percent, the decimal point is shifted two places to the right. In fact, shifting the decimal point two 50 100 places to the right is a quick and useful way to write decimal numbers as percents.

Practice Set

31 1 100

1 100

11 10 100

Write each fraction as a percent: 31 31 1 31 1 a. b. 100 100 100 100 1 13 d. 10 50 10

3 50

37 25 50

e.

7 25

11 100 10

1 1 3 c. 10 10 50

27 5 25

2 5

f.

3 50

3 7 50 25

7 25

2 5

g. Twelve of the 30 students earned a B on the test. What percent of the students earned a B? h. Jorge correctly answered 18 of the 20 questions on the test. What percent of the questions did he answer correctly? Write each decimal number as a percent: i. 0.25 l. 1.0

Written Practice 1.

(30, 62)

j. 0.3

k. 0.05

m. 0.7

n. 0.15

31 100

1 100

1

Strengthening Concepts Connect

What is the reciprocal of two and three fifths?

2. What time is one hour thirty-five minutes after 2:30 p.m.?

(32)

3. A 1-pound box of candy cost $4.00. What was the cost per ounce (1 pound = 16 ounces)? 31 1 100 100 4. Estimate Freda bought a sandwich for $4.00 and a drink for 94¢. Her (41) grandson ordered a meal for $6.35. What was the total price of all three items when 8% sales tax was added? Explain how to use estimation to check whether your answer is reasonable.

(15)

5. If the chance of rain is 50%, then what is the chance it will not rain?

(58)

392

Saxon Math Course 1

1

3 4

6.

(Inv. 6)

3 4

3 4

20 4

3

3 Draw a cube. How many 4edges does a cube have? 20 4

Represent

* 7. 33a. Write

(74, 75)

3

3 20

3

33 3 number. 3 as a decimal 20 20 20

4

3 20

20

3 20

3 3 20 20

3 20

b. Write the answer to part a as a percent.

3 4

3 20

3 3 20 4

3 * 8. 4 (75) 3 3 20 20

3 3 4 4

3 20 3 20

a. Write b. Write

as a

3 3 20 20

3 fraction 20 with

as a percent.

3 3 20 20

a denominator of 100. 3 4

3 20

3 20

7 n 1  5m3 * 9. Write 12% as a reduced fraction. Then write the fraction 10 as 100a 8 (33, 74) decimal number. 7 n 1 2  3 5 7  m n 3 m5  1m   10 100 8 3 3 3 3 3 10 100 Find each unknown number: 4 20 20 2 7 n 11 1 1 1 1 a 111 1 b1 n 371  mn  172  11nm  3 1 2m 1  5 1a311   3m  1 2 3a 3 1 57m  3 b 1 1       10. 11. 5 m 5 m 3 3 m 1 10 100 8 3 6 10 100 (42) 10 100 8 3 6 (43) 3 63 2 6 3 b3 26 2 33 2 10 8100 6 28 3

7w 7=  n n 1 1 1 1b  1 1 1 1 m 21 2 13 1 5  0.95  51 mm 3 3213. 3 6 b13 16 m 712. 1n−10 1 1 a 21a 1 13  10 5100 100 8 3 2 (59) 8 6 3 6         m 3 3 a b 1  m 1 3 10 100 8 37 6n 21 3 62 2 1 3 1  31 2 1 1 11 111 1 1 1  1 1 1 15 1m 1 21 m1  1 1  3 a m  1b  *10 15. 3 100 6 3 m 5 114. 3 b a 1 b 1 m  13 3a1  3 8  1  13 3 2 6 83 6 2 2 3 3 2 18 623 3 2 (72) 6 2 3 2 6 32 (57) 30 (38, 43)

7 1n 2 7 1 n1 3  m 3 m  15 m3 10 8 100 8 3 10 6 100

16. (0.43)(2.6)

17. 0.26 ÷ 5

(39)

18 30

18 30

18 30

18

30 test. What * 18. Nathan correctly answered 17 of the 20 questions on the (75) 7 n 18 1 1 1 1 2 1  3 correctly? 5  m18 did 3 Nathan a  b m  1answer percent of the questions 18  100 10 8 3 6 2 3 6 30

30

19. (47)

Estimate 18 18 tire 30 rolled 30

20.

Connect

18 30 18 30

(45)

(34)

30

The diameter of the big tractor tire was about 5 feet. As the one full turn, the tire rolled about how many feet? Round the answer to the nearest foot. (Use 183.14 for π.) 30

Which digit in 4.87 has the same place value as the 9 in

0.195?

21. Write the prime factorization of both the numerator and denominator of (67) 18 30 . Then reduce the fraction. 22. What is the greatest common factor of 18 and 30? (20)

23. If the product of two numbers is 1, then the two numbers are which of (30) the following? A equal 24. (64)

Verify

B reciprocals

C opposites

Why is every rectangle a quadrilateral?

25. If b equals 8 and h equals 6, what does (47)

* 26.

(65, 73)

D prime

bh 2

equal?

Find the prime factorization of 400 using a factor tree. Then write the prime factorization of 400 using exponents. Represent

Lesson 75

393

3

1 2

* 27. (Inv. 7)

Represent Draw a coordinate plane on graph paper. Then draw a rectangle with vertices located at (3, 1), (3, –1), (–1, 1), and (–1, –1).

28. Refer to the rectangle drawn in problem 27 to answer parts a and b below.

(8, 31)

a. What is the perimeter of the rectangle? b. What is the area of the rectangle? * 29. a. What is the perimeter of this (71) parallelogram? b. What is the area of this parallelogram? 30. (64)

Early Finishers

Math and Science

12 cm

15 cm

20 cm

Model Draw two parallel segments of different lengths. Then form a quadrilateral by drawing two segments that connect the endpoints of the parallel segments. Is the quadrilateral a rectangle?

The Moon is Earth’s only natural satellite. The average distance from Earth to the Moon is approximately 6202 kilometers. This distance is about 30 times the diameter of Earth. a. Simplify 6202 kilometers. b. Find the diameter of Earth. Round your answer to the nearest kilometer.

394

Saxon Math Course 1

LESSON

76

Comparing Fractions by Converting to Decimal Form

Power Up facts mental math

Building Power

AC

bh 2

Power Up G a. Calculation: 4 × 208 b. Calculation: 380 + 155 c. Calculation: 4 × $1.99 d. Calculation: $10.00 − $4.99 e. Fractional Parts:

1 5

of $4.50

f. Decimals: $0.95 × 100 g. Probability: How many different four digit numbers can be made with the digits 6, 4, 2, 9 using each digit exactly once? h. Calculation: 8 × 8, − 4, ÷ 2, + 2, ÷ 4, × 3, + 1, ÷ 5

problem solving

New

In the 4 × 200 m relay, Sarang ran first, then Gemmie, then Joyce, and finally Karla. Each girl ran her 200 meters 2 seconds faster than the previous runner. The team finished the race in exactly 1 minute and 50 seconds. How fast did each runner run her 200 meters?

1 1 7 7   Increasing Knowledge 1 1  Concept 5  5  7 7   3   5 drawing   3  pictures of fractions and by writing 5 We have compared fractions by 2Another 2   3 3   fractions with common denominators. way to compare fractions is to 2  2    2 form.   2 convert the fractions to decimal 2 2   1 1 Example 1 7 7    5 5    fractions. First convert each fraction to decimal form. Compare these 3 3   3 35 5 3 3 5 5 2 2   5 5 5 8 5 8 3 35 5 3 3 5 5 8 8 2 2   5 5 8 8 5 58 8 Solution

We convert each fraction to a decimal number by dividing the numerator by 0.625 0.60.6 0.625 the denominator. 8 5 3.0 8 5 3.0 5.000     0.60.6 5 0.625 0.6255.000 3 35 5 3 3 5 8 8 5 5 3.0 5.000  3.08 8  5.000 5 5 5 58 8

0.60.6 5 5 3.0  3.0

0.625 0.625 8 8 5.000  5.000 Lesson 76

395

We write both numbers with the same number of decimal places. Then we compare the two numbers. 0.600 < 0.625

Thinking Skill Explain

How do we compare decimal numbers?

33 << 55  55 88

55 88

33 55

Example 2 3 < 5 0.753  0.7 0.75 Compare: 0.7 5 8 443.00 3.004

5 8

3 3< <5 5 55 88

3 3 Since 0.6 is less than 0.625, we know that is less than 58. 58 5 5

3 4

3 5

33  0.7 0.7  44

0.75 0.75 3.00 4 43.00  3 > 0.7 4

5 8

3 4

33 44

33 0  0.7 44

33 >> 0.7 0.7 44

3 < 5 5 8

3  0.7 4

Solution

5 30 < 5 5 8

1 8

35 58

3 4 5 8

3 8

3 5

5 8

75 0.75 00 4  3.00

15 25

3 5

1 3 8 8 20 3 8

2 5

1515 3 3 33 11 33 22 4 0.7 0.7 First we write the fraction as a decimal. 2020 8 8 5 88 55 2525 5 5 15 0.75 15 33 33 22 33 33 11 44 22 0.7 0.5 0.325 0.7 0.5 0.325 5 5 5 5 5 5 5 5 20 8 8 25 83 8 8 25 83 3 3 3 20 33 5 33 << 55 33 << 55 33 3 333 55 33 33 5 3 33 53 33 5 45 3530.7 > 3.00  0.7 < < > > 4  0.7  0.7  0.7 >> 0.7  0.7 0.744 0.744 0.7 55 4 55 88 4 8 585 44 4 85 8 44 5 4 54 44 5 88 44 88 55 88 4 Then we compare the decimal numbers. 3 3 53 33 < 5 3 < 5 33 < 50.73  0.7334  0.734 > 0.73 > 0.7 3 > 0.73 44 85 4 54 8 4 4 44 4 5 4 815 8 0.75 > 0.70 3 3 4 2 0.7 0.5 0.325 15 3 3 1 2 4 5 3 3 3 3 5 0.75 is greater 5  0.7 255 Since 8 3 is greater than 0.7. < 5 than 0.7, 0.7 > 0.7 know that 48 8 0.75 5 0.75 0.75 0.75 4 we 0.75 54 5 20 8 25 45 8 0.75 3 < 5 344 3.00 3 > 3 3.00 44 3.00 3.00 3.00344 3.00  0.7   0.7  0.7 4 4 5 4 4 8

Practice 0.75 Set 0.7 4  3.00

4 5

Change the fractions to decimals to compare these numbers: 3 11333 221133 22 15 2 33 15 3434 315 3 223315 15 3315 0.5 11 33 0.325 b. c. 0.7 0.7 0.7 0.7 5a. 20 20 8820 8 20 825 20 558888 55 25 8 555525 25 555 20 8 888 25 5525

4422 4422 0.5 0.7 0.7 0.5 5555 555

3 23 2 0.5 0.5 8 58 5

33 0.325 0.325 0.5 0.5 88

15 34 3 23 3 2 315 2 315 d.3 0.7 4 2 e.42 0.5 f. 0.325 0.7 0.5 0.5 0.325 0.7 0.325 0.325 0.7 0.5 55 55 58 8 5 825 5 525 5 25 55 8 15 3 3 3 3 1 2 4 2 0.7 0.5 0.325 Strengthening Concepts 5 5 5 5 20 8 Practice 8 25 8 Written 15 3 3 2 4 2 0.7 0.5 0.325 5 5 5 5 25 * 1. Connect What is the product8of ten squared and two cubed? 21 3 58 8

(73)

2.

(50)

Connect

What number is halfway between 4.5 and 6.7?

3.

Formulate It is said that one year of a dog’s life is the same as 7 years of a human’s life. Using that thinking, a dog that is 13 years old is how many “human” years old? Write an equation and solve the problem. 1 2 1 22 4 5 * 4. Compare. First convert each fraction to decimal form. (15)

(76)

22 11 44 55

11

2222

* 5. a. What fraction of this circle is shaded?

(74, 75)

b. Convert the answer from part a to a decimal number. c. What percent of this circle is shaded? 4 ?  5 100

1 1 1 6 3 2 3 4 2 396

Saxon Math Course 1

11 11 11 66 33 22 33 44 22

44 ??  55 100 100

11 11 a2 a2 bba3 a3 bba1 a 22 33

1 2

6. Choose the appropriate units for measuring the2 circumference of a 1 1 22  juice glass. 5 4 1 2 1 meters A centimeters B C kilometers 22 5 4 1 12 2 12 1 to1a decimal 1 * 7.2a.Convert number. 2 (73, 5 74) 54 45  4 2 2 2 2 b. Write 3.75 as a reduced mixed number. (7, 27)

2 21 1 5 54 4

1

1

22 22 1 2 5 4

1 * 8. a. Write 0.04 as a reduced fraction. 2 2 (73, 75) b. Write 0.04 as a percent.

1 half1of each 1 number Instead of dividing 200 by 18, Sam found 6 3 2 3 4 2 quotient and then divided. Show Sam’s division problem and write the 4 ? 1 1 1 1 1 1 as a mixed  6 number. 3 2 a2 b a3 b a1 b 5 100 5 3 4 2 2 3 4 4? 4 1 1 1 11 1 11 1 1 1 11 1 11 ? ? 1 1    6 6 3 63 2 b a3b a3 b a1bba1 b bb a1 b  10. 11.  a2 a2 a2 a3 5 5100 3 3 4 34  2 3242  2 2 5 100 2 2 3 32 5 53 5 (61) (42) 100 9.

(43)

Verify

4 4? ? 1 11 11 1 1 11 11 1 1 1 * 12.  Analyze * 13. 5  52  2 6 6 3 3 2  2 a2 ba2 a3 bba3 a1 bba1 b 5 5100 3 34 42 2 2 23 35 5 2 2 (72) 100 (68) 4 ? 1 1 1 1 1 1 1    6 3 2 a2 b a3 b a1 b 52 Find each unknown 5 100 number: 2 5 3 4 2 3 2 4 ? 1 1 14. 16.7 + 0.48 + n = 8 1 15. 12 − d = 4.75  a2 b a3 b(43) a1 b 52 (43) 5 100 5 2 3 2 17. 4.3 ÷ 102

16. 0.35 × 0.45 (39)

(38, 52)

18. Find the median of these numbers:

(Inv. 5)

0.3, 0.25, 0.313, 0.2, 0.27 19. (16)

20. (19)

Estimate

Find the sum of 3926 and 5184 to the nearest

thousand. List

Name all the prime numbers between 40 and 50.

* 21. Twelve of the 25 students in the class earned A s on the test. What (75) percent of the students earned A s? Refer to the triangle to answer problems 22 and 23. 22. What is the perimeter of this triangle?

R

(8)

* 23. (69)

24. (7)

20 mm

Angles T and R are complementary. If the measure of ∠R is 53°, then what is the measure of ∠T ? Analyze

Estimate

T

16 mm

12 mm S

About how many millimeters long is this line segment? cm

1

2

3

4

5

Lesson 76

397

4  5

1 5 52 2

* 25. This parallelogram is divided into two congruent triangles. (71)

12 cm

10 cm 20 cm

a. What is the area of the parallelogram? b. What is the area of one of the triangles? 26. How many small cubes were used to form this rectangular prism?

(Inv. 6)

* 27. (Inv. 7)

28. (28)

bh Sketch a coordinate plane Graphbh point A ACon graph ACpaper. 2 2 (1, 2), point B (−3, −2), and point C (1, −2). Then draw segments to connect the three points. What type of polygon is figure ABC?

Represent

In the figure drawn in problem 27, a. which segment is perpendicular to AC? Conclude

bh 2

b. which angle is a right angle? 29. If b equals 12 and h equalsAC 9, what does (47)

* 30. (64)

1

equal? 1

Draw a pair of parallel lines.5Draw a third line perpendicular to 5 the parallel lines. Complete a quadrilateral by drawing a fourth line that intersects but is not perpendicular to the pair of parallel lines. Trace the quadrilateral that is formed. Is the quadrilateral a rectangle? Model

1 5

1 5

398

bh 2

Saxon Math Course 1

LESSON

77

Finding Unstated Information in Fraction Problems Building Power

Power Up facts

Power Up K

mental math

a. Calculation: 311 × 5 b. Number Sense: 565 − 250 c. Calculation: 5 × $1.99 d. Calculation: $7.50 + $1.99 e. Number Sense: Double 80¢. f. Decimals: 6.5 ÷ 100 g. Statistics: Find the median of the set of numbers: 134, 147, 125, 149, 158, 185. h. Calculation: 10 × 10, × 10, − 1, ÷ 9, − 11, ÷ 10

problem solving

Kathleen read an average of 45 pages per day for four days. If she read a total of 123 pages during the first three days, how many pages did she read on the fourth day?

Increasing Knowledge

New Concept

Often fractional-parts statements contain more information than what is directly stated. Consider this fractional-parts statement: Three fourths of the 28 students in the class are boys. This sentence directly states information about the number of boys in the class. It also indirectly states information about the number of girls in the class. In this lesson we will practice finding several pieces of information from fractional-parts statements.

Example 1 Diagram this statement. Then answer the questions that follow. Three fourths of the 28 students in the class are boys. a. Into how many parts is the class divided? b. How many students are in each part? c. How many parts are boys? d. How many boys are in the class?

Lesson 77

399

e. How many parts are girls? f. How many girls are in the class?

Solution We draw a rectangle to represent the whole class. Since the statement uses fourths to describe a part of the class, we divide the rectangle into four parts. Dividing the total number of students by four, we find there are seven students in each part. We identify three of the four parts as boys and one of the four parts as girls. Now we answer the questions. 28 students 1 are girls. 4

7 students 7 students

3 are boys. 4

7 students 7 students

a. The denominator of the fraction indicates that the class is divided into four parts for the purpose of this statement. It is important to distinguish between the number of parts (as indicated by the denominator) and the number of categories. There are two categories of students implied by the statement—boys and girls. b. In each of the four parts there are seven students. c. The numerator of the fraction indicates that three parts are boys. d. Since three parts are boys and since there are seven students in each part, we find that there are 21 boys in the class. e. Three of the four parts are boys, so only one part is girls. f. There are seven students in each part. One part is girls, so there are seven girls.

Example 2 There are thirty marbles in a bag. If one marble is drawn from the bag, 2 2 2 the probability of drawing red is 5 . 5 5  30  12 a. How many marbles are red?

2  30  12 5

b. How many marbles are not red? c. The complement of drawing a red marble is drawing a not red 3 2 1 1   1 marble. What is the probability of 5 5 drawing 4a not red marble? 4 d. What is the sum of the probabilities of drawing red and drawing not red?

Solution 2 2 2 2 18 2 2 2  30  30 red.   3025  3518 12  30  12  12 12 a. The probability of5drawing red is 5, so 5 of the 305marbles are 30 30 5 5 5 2 2 3 18 18 2 2  30  12  1  35  35 5 5  30  12 30 30 5 5 5

2 5

There are 12 red marbles. 2 5

400

Saxon 1 3 Math Course 2 1 5

51

4

1 4

2 3  1  35 25135 51 14 5 14

1 4

1 4

b. Twelve of the 30 marbles are red, so 18 marbles are not red.  35. c. The probability of drawing a not red marble is 18 30 d. The sum of the probabilities of an event and its complement is 1. 2 3  1 5 5

Practice Set

Model

Diagram this statement. Then answer the questions that follow.

Three eighths of the 40 little engines could climb the hill. a. Into how many parts was the group divided? b. How many engines were in each part? c. How many parts could climb the hill? d. How many engines could climb the hill? e. How many parts could not climb the hill? 2 2 2  30  12 5 5  30  12 f. How many engines could not climb the hill?5

2 5

2 18 5 30

 35

2 18 5 30

Read the statement and then answer the questions that follow. The face of a spinner is divided into 12 equal sectors. The probability of 3 3 1 1 2 spinning red4on one spin is 4. 5 1 5 5 1

2 5

1 4

g. How many sectors are red? h. How many sectors are not red? i. What is the probability of spinning not red in one spin? j. What is the sum of the probabilities of spinning red and not red? How are the events related? 2 3 Strengthening Concepts Written Practice  1

2 3  1 5 5 5 5 1. The weight of an object on the Moon is about 16 of its weight on Earth. (29) A person weighing 114 pounds on Earth would weigh about how much on the Moon?

* 2. (22)

Estimate the weight of an object in your classroom such as 1 a table or desk. Use the information in problem 1 to calculate what the 6 approximate weight of the object would be on the moon. Round your answer to the nearest pound. Estimate

* 3. Mekhi was at bat 24 times and got 6 hits.

(29, 75)

a. What fraction of the times at bat did Mekhi 2get x a 1hit? 5 b. What percent of the times at bat did Mekhi get a hit?

2 x1 5 Lesson 77

401

 35

* 4. (77)

Model

Diagram this statement. Then answer the questions that follow.

There are 30 students in the class. Three fifths of them are boys. a. Into how many parts is the class divided?

1 6

b. How many students are in each part? c. How many boys are in the class? d. How many girls are in the class? * 5. a. In the figure below, what fraction of 1 the group is shaded?

(74, 75)

6

b. Convert the fraction in part a to a decimal number. c. What percent of the group is shaded? 2 x1 5 * 6. Write the decimal number 3.6 as a mixed number. (73)

Find each unknown number: 7. 3.6 + a = 4.15

(43)

9.

1

(58)

8. 2 x  1 5

(30)

Explain If the chance of rain is 60%, is it more likely to rain or not to rain? Why?

* 10. Three fifths of a circle is what percent of a circle? (75)

11. A temperature of −3°F is how many degrees below the freezing temperature of water? 31 � 2 1 3 71 � 3 7 1 2 17 1 1�3�7 � � �3 13 � 1 0.35 3 � 1 3� 0.35 5 20 5 52 3 2 4 85 20 2 3 3 2 4 8 * 12. Compare: 1 3 7 (73, 76) 1 1 1 3 1 3 7 1 1 � � 3 �a.10.35 37� 1 1 � 2 1 � 3 3b.1 3 �2 1 23 � � 3�1 3�1 2 4 8 3 0.35 3 3 5 5 20 2 3 2 4 8 3 3 7 1 1 1 1 3 1 31 73 7 7 1 � 21 2 13 1 1 1 � � � � � � � � � � � � � 3 1 3 1 3 1 3 1 0.35 0.35 3 1 3 1 3 1 3 1 3 7213. 321 32 1 843 1 87 3 3 1 1 11 3 73 � 31 1 1 20 20 � 0.35 5 7 355 � 115� 22 42 14. � 3� � 1 3 � 13 � � 3 13 � 13 13 � 33 � 1 0.35 20 (57) 2 3 2(63) 4 5 8 5 3 3 20 5 2 5 3 23 4 8 3 3 (10, 14)

3 1 3 �1 5 5

1 2 1 3 1 7 3 1 7 1 1 3 2 11 3 7 1 � 0.35 � 3� � 1 3 � 1 � � 316. � 13 � 1 3 � 1� 15. 2 3 20 5 2 5 3(61) 2 4 5 8 5 23 4 8 (66) 3 3 11 3 7 1 1 1 1 3 1� 7 3 1 1 � 3 � 13 � 1 � 3� 1 3 � 1 � 17. 1 � 33 � 1 18. 1 � 3 23 4 (68) 8 3 3 3 2 4 58 5 3 3 (68) 19. What is the perimeter of this rectangle? (8)

1 1 1 �33� 1 3 3

1 1 �3 3

1.5 cm

20. What is the area of this rectangle? (31)

0.9 cm

* 21. Write the prime factorization of 1000 using exponents. (65)

22. Coats were on sale for 40% off. One coat was regularly priced at $80. (41) a. How much money would be taken off the regular price of the coat during the sale?

b. What would be the sale price of the coat?

402

Saxon Math Course 1

23. Patricia bought a coat that cost $38.80. The sales-tax rate was 7%. (41)

a. What was the tax on the purchase? b. What was the total purchase price including tax?

24.

Classify

(64)

Is every quadrilateral a polygon?

25. What time is one hour fourteen minutes before noon? (32)

26.

(Inv. 2)

What percent of this rectangle WX XY XY appears to be WX shaded?WX Estimate

A 20% XY B 40%

WX

WX

XY YZ

WX YZ

WX XY YZ ZW ZW

X ZW

C 60% YZ

ZW

D 80%

WX WX XY WX XY * 27. Represent Sketch on graph paper. Graph point WX a coordinate XYplane XY WX (Inv. 7) W (2, 3), point X (1, 0), point Y (−3, 0), and point Z (−2, 3). Then draw YZ WX XYZW ZW XYWX XY WXYZ , XY , YZ, and . ZW YZ ZW WX XY * 28. a. Conclude Which segment in problem 27 is parallel to WX? XY

X

(71)

WXis parallel to XY ? b. Which segment in problem 27 WX

WX

YZ

ZW

WX XY XYWX XY 29. Write the prime XY factorization of both the numerator and the denominator (67) of this fraction. Then reduce the fraction. 210 210 WX XY 350 350 210 210 210 WX XY 350 350 350 30. a.

210 350

(47, Inv. 6)

Connect

The moon has the shape of what geometric solid?

1 1 The diameter of the moon is about 2160 miles. Calculate 8 8 1 the approximate circumference of the moon using 3.14 for π. Round 1 1 8 8 the answer8to the nearest ten miles. 210 210 210 210 350 350 350 3501 8 210 Early Finishers Simplify each prime factorization below. Then identify which prime 350 Choose A Strategy factorization does not belong in the group. 210 Explain your reasoning. 32 × 23 22 × 33 × 5 22 × 350 32 × 52 35 24 × 72 1 8

1 8

1 8

b.

Estimate

1 8

1 8 1 8

Lesson 77

403

LESSON

78

Capacity

facts

1

22

Building Power

Power Up

1 2

Power Up J

mental math

a. Calculation: 4 × 325 b. Number Sense: 1500 + 275 c. Calculation: 3 × $2.99 d. Calculation: $20.00 − $2.99 e. Fractional Parts:

1 3

of $2.40

f. Decimals: 1.75 × 100 g. Statistics: Find the median of the set of numbers: 384, 127, 388, 484, 488, 120. h. Calculation: 9 × 11, + 1, ÷ 2, − 1, ÷ 7, − 2, × 5

problem solving

Raul’s PE class built a training circuit on a circular path behind their school. There are six light poles spaced evenly around the circuit, and it takes Raul 64 seconds to mow the path from the first pole to the third pole. At this rate, how long will it take Raul to mow once completely around the path?

New Concept

Increasing Knowledge

To measure quantities of liquid in the U.S. Customary System, we use the units gallons (gal), quarts (qt), pints (pt), cups (c), and ounces (oz). In the metric system we use liters (L) and milliliters (mL). The relationships between units within each system are shown in the following table: Thinking Skill Discuss

Name some real world situations where we use the word quarter.

Equivalence Table for Units of Liquid Measure U.S. Customary System 1 gallon = 4 quarts 1 quart = 2 pints 1 pint = 2 cups 1 cup = 8 ounces

404

Saxon Math Course 1

Metric System

1 liter = 1000 milliliters

Math Language Note that we use ounces to measure capacity and weight. However, these are two different units of measurement. When we measure capacity we often refer to fluid ounces.

Common container sizes based on the U.S. Customary System are illustrated below. These containers are named by their capacity, that is, by the amount of liquid they can contain. Notice that each container size has half the capacity of the next-largest container. Also, notice that a quart is one “quarter” of a gallon.

Milk Half & Half

1 gallon

1 gallon 2

1 quart

1 pint

1 cup

Food and beverage containers often have both U.S. Customary and metric capacities printed on the containers. Using the information found on 2-liter 1 1 seltzer bottles and 12-gallon milk cartons, we find that one liter is a12little more 2 2 than one quart. So the capacity of a 2-liter bottle is a little more than the 1 1 1 1 capacity of a 12-gallon container. 2 2 2 2

2-liter bottle

1 -gallon 2 container

Example 1 A half gallon of milk is how many pints of milk?

Solution Two pints equals a quart, and two quarts equals a half gallon. So a half gallon of milk is 4 pints.

Example 2 Which has the greater capacity, a 12-ounce can or a 1-pint container?

Solution

1 2

1 2

1 2

A pint equals 16 ounces. So a 1-pint container has more capacity than a 12-ounce can.

Practice Set

a. What fraction of a gallon is a quart? b. A 2-liter bottle has a capacity of how many milliliters? c. A half gallon of orange juice will fill how many 8-ounce cups? d.

Explain The entire contents of a full 2-liter bottle are poured into an empty half-gallon carton. Will the half-gallon container overflow? Why or why not?

Lesson 78

405

1 2

1 2

Written Practice

1 2

1 2

1 2

1 2

1 2

Strengthening Concepts

1 1 1. What is the difference when the product of 12 and 12 is subtracted from 2 2 1 1 1 1 1 1 1 1 the sum of 2 and 2 ? 2 2 2 2 2 2

(12, 55)

1 2

1 2

1 2

1 2

1 2

2. The claws of a Siberian tiger are 10 centimeters long. How many millimeters is that?

(7)

3.

(25)

* 4. (77)

Sue was thinking of a number between 40 and 50 that is a multiple of 3 and 4. Of what number was she thinking? Analyze

Model

Diagram this statement. Then answer the questions that follow.

Four fifths of the 60 lights were on. a. Into how many parts have the 60 lights been divided? b. How many lights are in each part? c. How many lights were on?

4 m1 5

d. How many lights were off? 4 4 4 4 4 4 4  w  1m  1  x  1  w  1  1 x  1 m1 y 5 5 5 5 5 5 5. Classify Which counting number is neither a prime number nor a5 (65) 4 4 composite number? 4 4 w1 x1 m1 y 1 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 1 4 Find each  11 w 1  1  1 m y w x w 11 x  x 1 1 1 m unknown 1 m  number: y y  12  1 5 5 5 5 5 5 5 5 4 4 45 4 5 45 4 5 44 2 w  m 1 1 y 1 6. m  1 7. x 1 w  1y  1x  1 5 5 5 5 (43) 5 55 (30) 5 4 4 n 4 4 4 4 4  x  1 4 1 y3 1  1  w  1 8.  x  51m  1y  51 w  1 9. 3  5 5 5 5 5 (43) (42, 43) 4 100 4 3

y

4 1 5

1 2

4 4 4 4 4 4 4 * 10. a. What fraction of4the rectangle below is  shaded? 4 44 4w  1wy4 1  1 144x 1x11x  1 y 4y1 y  1w51  mwm 1    1 3 1 15 m x m3 1 5 5 5 5 5 5 5 3 ? 5 1 1 5 2100 514  5 23 5 55 141 23 5   0.355 0.35 2100 b. Write the 5 answer to part a as a5decimal number. 6 2 4 100 6 2 2 4 4 4 4 4  4  4 4  4 4  4    1 y 15 11is shaded? mw 11m3 1  x   w 1 1 w yc.1What x4 1 1 3x  of y ? 1 2 5 percent the4 rectangle 4 43 5 5 51 5 4 m5 2 5 5 5 5 5 0.35 5    100 1 2    w 1 x6 1 y  41100 5 2 2 3 6 5 5 5 5 3 5 3 1 5 1 3 5 ?1 3 3 3 ?1 3 2 ?15    23 14       23 2 0.35 2 14100 0.35 2100 0.35 14100 2 5 5 5 6 2 4 100 26 6 2 4 6 100 2 2 5 3 54 1 3 3 5 ? 3 ? 1 23100 14 3 4      23 2100 1  2 0.35 2100 10.35 5 5 6 2 6 2 4 100 2 34 610 n 1.15 1 mixed 1 number. 5 1 * 11. Convert 3the decimal to ?a 3 ? 1 5 3 3 1number    0.35 100 3 144 23 3 3 1  2  5  4 2 0.35 2100 14(73) 23 100 4 5 6 2 4 100 2 5 6 2 4 100 2 3 6 * 12. Compare: 3 (73, 76) 3 3 ? 4 3 3 35 51 3 ? 15  113 23  32100 151424 2 14100   2  100  0.35 b. 22100 +1223312 2 a. 0.35 0.35 2 1000.35 14 20.35 21 5 6 2 5 6 4 1 2 1 2 1 1 2 1 1 5 1  2 1  22  1 16 22 61 2 5 2 34 1 2 24 100 23 2 2 3 2 3 2 33 2 2 3 2 3 4 5 2 5 3 5 3 14  33 n3 5 ?1 5 11 11 3 2 ?3 5 ? 1 2 1 3  100 2 14. 4 1 3    15.  3  ?  1 0.35 2100 213. 141   2 12 2100 0.35 2 5 1 1 2 5 4 100 4 3 51 1 1 2 2 2 3 6 2 4 100 2 5 6 2 4 100 2 6 4 3 6 2 4 100 2 3 6 (57) (63) 1  2 21002 1 1  (51)  3 6   0.35 2 2 3 5 2 3 3 2 2 6 2 4 100 2 3 6 21 1 2 21 22 1 1 22 1 2 1 11 11 2 1  1 2 1 2 21  22 1  1 22 18. 16. 1  2 1  21 117. 32 2 3 1 2 2 1 2(68) 13 2 32 2 33 2 2 12 1 33 2(68)212 3 2 22 1 2 1 (66) 2 1 1 2 1  12  2 2  11  2 2 2 23 3 3 22 3 3 2 2 3 2 19. a. What is the perimeter of this square? (38) 1 2 2 1 1 1 2 1 2 2 1 1  212 2 1 square? 2 1  1 1is2 the 1 2 1 2 2b. What 1 2 area of this 3 2 3 3 2 2 in. 2 3 2 3 3 2 2 2 12 221 112 12 1 22 2 1 21 11 1 1 1 2 11 122 12 22  12 2 12 3122  22132 112 2 121 1 2 11 2 2 2 23 3 3 223 23 2 33 3 2 32 2 3 2 (74, 75)

1 1 2 21  22 1 1 21  12 2  12  1 1 22 1 1 1 2 2 1 21 12 1 1  2 21 33 2 12 3 2 2 2 23 3 1 1 2 3 2 2 22 2 2 3 32 2 3 2 3 406 Saxon Math Course 1

1

22 1 2 1 3 2

1 2

3 5

0.35

20. (64)

5 1 3 ? 1 2 5     6 2 4 100 2 3 6 “The opposite sides of a rectangle are parallel.” True or

14  23

2100

Verify

false?

* 21. What is the average of 33 and 52? (73)

* 22. The diameter of the small wheel was 7 inches. The circumference was about 22 inches. Write the ratio of the circumference to the diameter of the circle as a decimal number rounded to the nearest hundredth.

(51, 74)

1 2 1 2 2 3

1 23.2 2 1 is 2 12 feet? 1  2 How many 1 2 inches 2 (66) 3 3 2 24.

A

4 m1 5 25. (64)

26.

(65, 73)

Which arrow below could be pointing to 0.1?

Connect

(50)

4  w –1 1 5 Represent

B

4  x  10 5

C

D

y

4 11 5

4 m1 5

Draw a quadrilateral that is not a rectangle.

Represent Find the prime factorization of 900 by using a factor tree. Then write the prime factorization using exponents.

27. Three vertices of a rectangle have the coordinates (5, 3), (5, −1), and (−1, −1). What are the coordinates of the fourth vertex of the rectangle?

(Inv. 7)

Refer to this table to4 answer a and b. 5 1 3* 28. (78)0.35  2100 1  23 5 6 2

3 teaspoons = 1 tablespoon 16 tablespoons = 1 cup 2 cups = 1 pint 2 pints = 1 quart 4 quarts = 1 gallon

3 ?  4 100

1 2 5 3 236 0.35 5

a. A teaspoon of soup is what fraction of a tablespoon of soup? b. How many cups of milk is a gallon of milk? * 29. Estimate A liter is closest in size to which of the following? 1 (78) 2 1 2 2 1 C 12 gallon D gallon 1  2A pint 1  2 B quart 2 1 2 3 2 3 3 2

1 2 1 2 2 3 * 30. In 1881 Clara Barton founded the American Red Cross, an organization (78) that helps people during emergencies. The Red Cross organizes “blood drives” in which people can donate a pint of blood to help hospital patients who will undergo surgery. How many ounces is a pint?

Lesson 78

407

LESSON

79

Area of a Triangle Building Power

Power Up facts

Power Up K

mental math

a. Calculation: 307 × 6 b. Number Sense: 1000 − 420 c. Calculation: 4 × $2.99 d. Calculation: $5.75 + $2.99 e. Number Sense: Double $24. f. Decimals: 0.125 × 100 g. Measurement: How many liters are in a kiloliter? h. Calculation: 2 × 2, × 2, × 2, − 1, × 2, + 2, ÷ 2, ÷ 2

problem solving

The restaurant serves four different soups and three different salads. How many different soup-and-salad combinations can diners order? Draw a diagram to support your answer.

New Concept

Increasing Knowledge

In this lesson we will demonstrate that the area of a triangle is half the area of a parallelogram with the same base and height.

Activity

Area of a Triangle Materials needed: • pencil and paper • ruler • scissors Model

Math Language Recall that congruent polygons have the same size and shape.

408

Fold the paper in half, and draw a triangle on the folded paper.

While the paper is folded, use your scissors to cut out the triangle so that you have two congruent triangles.

Saxon Math Course 1

1 2

Arrange the two triangles to form a parallelogram. What fraction of the area of the parallelogram is the area of one of the triangles?

We find that whatever the shape of a triangle, its area is half the area of the parallelogram with the same base and height.

Recall that the area of a parallelogram can be found by multiplying its base by its height (A = bh). So the area of a triangle can be determined by finding half of the product of its base and height. h

h b

b

parallelogram triangle bh 1 1 1 1 A = 2 bh Area of a triangle  bh A 2 2 A = bh 2 2 bh 1 1 1 1   Since multiplying by and dividing by 2 are equivalent operations, the Thinking Skill Area of a triangle bh A 2 2 2 2 2 formula may also be written as Analyze Why are bh 1 1 1 1 bh Area1 of a triangle 1 bh A 2 2 2  by 21 Amultiplying 2 2 2 2 A  (8 cm) (4 cm) and dividing by2 In the following examples we will use both formulas stated above. From our 1 2 equivalentA  (8 cm) (4 cm) 2 calculations of the areas of rectangles and parallelograms, we remember that operations? the base and the height are perpendicular measurements. 1  (8 cm) (4 cm) A 2 Example 1

Find the area of the triangle at right. 4 cm

5 cm

8 cm

Solution Reading Math The area of a figure is expressed in square units. Read the abbreviation “cm2” as square centimeters.

bh 1 1 Area of a triangle  bh A 2 2 2 The area of the triangle is half the product of the base and height. The height must be perpendicular to the base. The height in this case is 4 cm. Half the product of 8 cm × 4 cm is 16 cm2. A

1 (8 cm) (4 cm) 2

A = 16 cm2

Lesson 79

409

Example 2 Find the area of this right triangle:

5m

3m

4m

Solution We find the area by multiplying the base by the height and then dividing by 2. All right triangles have two sides that are perpendicular, so we use the perpendicular sides as the base and height. A

(4 m) (3 m) 2

A = 6 m2

Practice Set

Find the area of each triangle: a. a.a.

b. b.b. 6 ft 66 ftft

in. 6 in. 66 in.

7 ft 77 ftft

10 in. 1010 in.in.

in. 8 in. 88 in. 10 1010 ft ftft

c. 3939 c.c. 39 mm mm mm 15 mm 1515 mm mm

9 1

25 mm 2525 mm mm

56 mm 5656 mm mm

e.

1.

11 cm 1111 cmcm

cm 6 cm 66 cm

cm 5 cm 55 cm

Predict If the height of the triangle in c is doubled to 30 mm, would the area double? Calculate to check your prediction.

Written Practice (8)

d. d.d.

Strengthening Concepts Explain If you know both the perimeter and the length of a rectangle, how can you determine the width of the rectangle?

* 2. A 2-liter bottle contained 2 qt 3.6 oz of beverage. Use this information (78) to compare a liter and a quart: 1 liter

1 quart

3. Mr. Johnson was 38 years old when he started his job. He worked for 33 years. How old was he when he retired?

(11)

4.

(64)

Verify

Answer “true” or “false” for each statement:

a. “Every rectangle is a square.” b. “Every rectangle is a parallelogram.”

410

Saxon Math Course 1

9 1

5. Ninety percent of 30 trees are birch trees.

(23, 41)

a. How many trees are birch trees? b. What is the ratio of birch trees to all other trees?

* 6. Eighteen of the twenty-four runners finished the race.

(75, 77)

a. What fraction of the runners finished the race? b. What fraction of the runners did not finish the race? c. What percent of the runners did not finish the race?

* 7. (79)

This parallelogram is divided1into1two3congruent triangles. 3 1 1 3   2 3 n2 3 3 5 4 2 4 3 What is the area of each triangle? 3 1 1 1 3 1 1 3   2 3 n2 3 w  3 5 4 2 4 3 4 Analyze

15 mm

3

2

* 8. 10 ÷ 10 (73)

10. 1.2(0.12) (39)

12 mm

1 3 1 1 3 20 mm 11 1 3 1 3   23 3  n2  2 3 3 2 4 3 5 4 23 5 4 3 9. 6.42 + 12.7 + 8

13 5 1  2  3 wn  44 6 3

w

(38)

11. 64 ÷ 0.08 (49)

31 35 1 1 1 31 5351 1 1 1 15 5 1 1 1 1 3 3 1 1 1 311 11 3 313. 1212(9 1 1 3   3   3w wb 10) 1) 32 3 wn(6 33 3 3 n2 2 3a3     2 3  3  2n 322  3   12. 33 4 nw n 2 w 4 4(68) 2 44 6463 31004 46 6 3 5 4(72) 32 3 5 5 4 4 2 3 2 435 53 442 46 3 3 1 Find each unknown number: (9  10)  (6  1)  a3  100 b 14. 10 – q = 9.87

1 1 3 3   3 5 4

(43)

15. 24m = 0.288 (45)

1 5 1 1 3 1 31 1 53 1 1  2  317. w  1  2  33  16. n  22