individual responses â student math journal examples from grade 3 class, Quilchena .... use larger numbers and decimal
High-Yield Routines: Grades K-8
by Ann McCoy, Joann Barnett and Emily Combs published by the National Council of Teachers of Mathematics 2013
High-‐yield routines are structured activities that develop mathematical concepts and big ideas over time. There is “low” investment of time and planning for a “high” value in returns in terms of mathematical thinking and understanding. Mathematical routines are quickly and easily implemented, taking five to ten minutes per day. This book shares seven routines that can be used at a variety of grade levels and with a variety of mathematical content. Routines provide coherence across a program. The book refers to the mathematical practices which are part of Common Core in the USA. Curricular competencies or mathematical processes (current IRP) are referenced for our purposes here. For our Richmond and BC context, these routines weave mathematical concepts and content together with both core and curricular competencies. They also create opportunities for significant oral language development with content area vocabulary, essential for all of our students but particularly important for our English Language Learners and French Immersion students.
The Seven Routines 1. Today’s Number 2. Mystery Number 3. Alike and Different 4. Number Lines 5. Quick Images 6. Guess My Rule 7. How Do You Know?
Today’s Number The Routine: An intentionally selected “number of the day” is presented to students. Students create a variety of representations for the number. Students share and discuss their representations. Extensions: Instead of being given a number, students think of their own number individually, in pairs, in small groups, or as a larger group. Students work with their own number or trade numbers with another person or group and create, share and discuss representations of it. Add enabling constraints depending on focus at the time. Mathematical Content and Competencies: • demonstration of number sense, what numbers mean • flexibility and fluency with numbers • composition and decomposition of numbers • parts-‐whole relationships • place value concepts • operational skills and concepts • equivalent expressions • representing quantity • use of models • apply mental math strategies • communicate mathematical ideas • connect mathematical concepts to each other and to the world
individual responses – student math journal examples from grade 3 class, Quilchena Elementary, September 2014
math graffiti -‐ sharing and discussion example from grade 3 class, Quilchena Elementary, September 2014
Mystery Number The Routine: A set of clues is presented to students for a strategically chosen “mystery number.” Students discuss what each clue tells them about the mystery number, what some possible numbers may be, and what numbers it cannot be. Students use the clues to solve the mystery. Extensions: Students can think of their own mystery number and create a set of clues for it. The teacher can set requirements for the clues. For example, the teacher can specify the type of mystery number, the number of clues, and the terms or operations that need to be used in the clues. Students solve each others’ mystery numbers. Mathematical Content and Competencies: • demonstration of number sense, what numbers mean • flexibility and fluency with numbers • composition and decomposition of numbers • parts-‐whole relationships • place value concepts • operational skills and concepts • apply mental math strategies • communicate mathematical ideas • connect mathematical concepts to each other and to the world • reasoning, critical thinking and problem-‐solving skills • constructing logical arguments Grade 2&3 students at Diefenbaker Elementary brainstormed mathematical terms to use in their clues
Example of student work as they practiced writing their own clues for a mystery number for the first time at Diefenbaker Elementary in grade 2&3
Alike and Different The Routine: A set of two or more objects (numbers, shapes, etc.) is presented to students. Students consider, discuss and share the similarities and differences between the objects. Extensions: Place restrictions on how students need to represent the similarities and differences (a Venn diagram or other graphic organizer, using images, etc.), the number of similarities or differences that must be found, or the terms that must be used to describe the ways the objects are alike and different. Mathematical Content and Competencies: • comparative thinking • demonstration of number sense, what numbers mean • flexibility and fluency with numbers • finding relationships among mathematical objects and concepts • use of models • communicate mathematical ideas • connect mathematical concepts to each other and to the world • reasoning, critical thinking and problem-‐solving skills • constructing logical arguments Grade 8 students at Steveston-‐London Secondary School were asked to record their thinking, comparing a variety of values. April 2015
Then students were placed in groups to discuss their ideas. The students chose many ways to show their comparison of the different numbers. Students in grade 5 at Westwind demonstrated their understanding of area and perimeter through the Alike and Different routine. May 2015
Number Lines The Routine: A number line that is strategically marked with values is presented to students. Students are asked to locate a value on the number line or determine the value of a marked location on the number line. Students explain or show their thinking (how they know that is the location of the value or that it is the value of the marked position). Students can explain themselves in words or by annotating the number line. Extensions: The number of values to be determined can be changed depending on the level of the students. The number line can have fewer marked values, have different ranges of values (positive, negative, fractional, etc.). This routine can also be used for rounding. Mathematical Content and Competencies: • demonstration of number sense, what numbers mean • flexibility and fluency with numbers • composition and decomposition of numbers • parts-‐whole relationships • place value concepts • representing quantity and magnitude • use of models • apply mental math strategies • develop algebraic thinking • communicate mathematical ideas • reasoning, critical thinking and problem-‐solving skills • constructing logical arguments The following are examples of student work at Woodward Elementary. The grade 3&4 students were asked to locate 3 values (0.1, 0.4, and 0.7) on the number line and explain their thinking.
These students marked the number line from 0 to 1 in tenths in order to help their thinking while other students used different strategies.
Quick Images The Routine: An image is presented to the students for a short period of time (two-‐three seconds). This image can include a display of dots, geometric figures, base-‐ten blocks, or other materials. Students recreate what they saw by building, drawing or describing it. Students explain their own view of the image or model, including how they saw it or what it reminded them of. Extensions: The teacher can change for how long the image is shown, how many times it is shown and the difficulty of the image. Students also enjoy creating images or constructions for their classmates to represent. Mathematical Content and Competencies: • composition and decomposition of numbers • parts-‐whole relationships • representing quantity • use of models • visual-‐spatial relationships • positionality and transformational geometry • apply spatial strategies such as subitizing • communicate mathematical ideas • connect mathematical concepts to each other and to the world Grade 3&4 students at Woodward Elementary were shown an image that they were asked to recreate using geometric tiles and then explain in words.
A student at work recreating an image at Woodward Elementary.
Students started working in pairs to recreate the more difficult images at Woodward Elementary, negotiating what they each remembered of the image.
Guess My Rule The Routine: A set of number pairs or an operational rule is presented to students. Students analyze the number pairs to determine the “rule” or relationship between the numbers, or students consider the rule and determine pairs of “in” and “out” numbers that demonstrate it. Extensions: Students can apply a rule to a given set of “input” numbers or undo the rule given “output” numbers to find the input numbers. Students can create their own rule or set of number pairs and have a classmate determine number pairs or the relationship between the numbers. The teacher can specify what types of operations and numbers can be used. Multistep rules could also be used as a challenge. Mathematical Content and Competencies: • demonstration of number sense, what numbers mean • relationships between numbers • looking for generalizations, functional relationships • algebraic thinking • flexibility and fluency with numbers • operational skills and concepts • representing quantity • use of models • apply mental math strategies • communicate mathematical ideas • connect mathematical concepts to each other and to the world A grade 4 student at Woodward Elementary created a “Guess My Rule” where the rule was given and his/her partner had to find various input and output values. April 2015
Example of how students at Woodward Elementary used familiar numbers and operations to create their own “Guess My Rule” pairs of numbers.
Grade 3&4 students at Woodward Elementary challenging themselves and their partners by attempting to use larger numbers and decimals to create a “Guess My Rule”.
How Do You Know? The Routine: An open-‐ended question is presented to students. Students think about the question and discuss their responses with a classmate or small group. Students must explain and defend their answers. Extensions: Teachers can have questions of a variety of levels to differentiate the activity for different students throughout the classroom. Teachers can specify what must be used in a response, for example, key terms, diagrams, operations, etc. Teachers can have students write an answer for a specific audience. Students can write their own questions for each other. Mathematical Content and Competencies: • communicate mathematical ideas • connect mathematical concepts to each other and to the world • reasoning, explaining and building a strong answer • introduction to conjecture and proof • demonstration of number sense, what numbers mean • flexibility and fluency with numbers • use of models • apply mental math strategies Grade 8 students at Steveston-‐London Secondary School were asked to answer one of three questions using a variety of methods. Students used words, number lines and fractional diagrams to explain their thinking. Other students used mathematical concepts and terminology as well as images in their explanation. April 2015
Students were challenged to use algebraic reasoning in the final question. All students were prompted to use testing, examples, generalizations, algebraic reasoning and big math ideas such as balance, equality, proportionality, etc. in their answers, but many went beyond these suggestions using what worked for them.
A grade 6 student at Westwind uses pictures and words to explain how she knows which rectangle has the greatest perimeter. May 2015
This anthology of High-‐Yield Routines from classrooms from the Richmond School District was compiled during the 2014-‐15 school year by Janice Novakowski with the support of Richmond’s Math Mentor Teachers – Michelle Hikida, Braunwyn Thompson, Terra Hooyberg and Weily Lin as well as the support of Kirsten Rodgers, Queen’s University alternative practicum student.
