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Ann and Johnny Baker are partners in Natural Maths, an educational publishing and consultancy business. The aim of Natural Maths is to bridge the gap between educational research and classroom practice in maths education. Their focus is on the effective implementation of mental strategies with real-life connections in the classroom. They have written more than 40 books together as well as many articles. They also present at Gifted and Talented days of excellence and maths camps, and are currently working together to create exciting learning and teaching environments involving interactive whiteboards. Ann is passionate about maths and pedagogy. Her mission is to engage students in worthwhile and realistic mathematics as well as to raise the intellectual quality of maths lessons. She believes it is vital that students have strategies for mental computation and develop the disposition to work through problem solving situations and investigations. Her frequent work in classrooms ensures that all of the strategies and activities she presents are tried and tested. Johnny was a founding member of the Centre for Mathematics Education at the Open University. He has lectured in universities and taught in schools. His current focus is on the role of technology in maths teaching. He also runs accelerated learning programs for gifted students that uses technology for maths in exciting ways. He is coeditor of the spreadsheet journal Spreadsheets in Education and runs an on-line maths competition for talented maths students. Ann and Johnny live in the Gold Coast Hinterland with their two sons and a small menagerie of pets and delight in the local flora and fauna that share their home and garden.

Content

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Introductio n ........ .......... Learning H . . .3 orizons . . . . .......... N3.1: Num . .4 ber Sense . . . . . . . . . . . .1 N3.3: Multip 6 lication . . . . . . . . . . . . .18 N3.3: Divisio n ....... .......... N3.4: Mone .20 y ........ .......... M3.1: Meas .22 urement . . . . . . . . . . . . .24 S3.1: Shape .......... . . . . . . . . . .2 CD3.2: Cha 6 nce . . . . . . . . . . . . . . . . .28 PA3.1: Patt erns . . . . . .......... Student Pro . .30 gress Chart . . . . . . . . . .3 2

natural maths s tr work sa mples fo ategies r book 3

© Blake Education ISBN 978 1 92114 344 1

Natural Maths Strategies - Assessment Guide for Book 3

Introduction

Introduction As teachers we want to make our planning decisions on the basis of a student's demonstration of learning. Techniques such as observations, interviews and work samples give windows into what students actually do know and can do. This book combines all three techniques by showing how work samples can be annotated with information gathered from observation and interview. The annotations summarise our own interpretation of a student's developmental level and plans that we might make for the future needs of the student. The work samples presented in this book have been chosen to demonstrate just how much diagnostic information can be obtained from student work samples. Diagnostic in this sense does not only mean looking for weaknesses or areas needing more support. It also means identifying strengths and planning to build on them. Quite often there are some surprises too. A student may know more than anticipated and have interesting strategies that can be built on. Sometimes a student knows less than anticipated. They may appear to be working at a particular level but when questioned reveal that their knowledge or conceptual understanding is wobbly. When working with mathematics outcomes, standards or levels it is important to be able to identify the developmental features from work samples and interviews and to be able to compare and moderate the samples with others. Being able to interpret the developmental levels of the students and plan effectively to meet the needs of individuals or small groups is at the core of good teaching. Generally it is possible to annotate three or four pieces of work in a lesson. By keeping a checklist it is possible to keep tabs on who has and who has not been observed yet.

Intensive Interviews Discussing a work sample with the individual student is like having an intensive interview. The student might explain what they are doing and thinking. Clarification can be sought and questions that will stretch the student to think further or differently can be asked. For example, if students use a repeated doubling strategy, they can be asked if there is a quicker way to find the total. The teacher might ask whether strategies such as chunking, compensating or the open number line could be used in the same context. Listening to the strategies the students feel comfortable with, and perhaps nudging them to try alternatives, provides further information on which informed planning can be based.

Annotations As the interviews take place it is essential to jot down what the students did and said, as well as their responses to your prompts and questions. Anecdotal information about attitude, disposition and perseverance can also be made. The intention is that the annotations should be written as positively as possible and in such a way that parents, other teachers and support staff can read, understand and ask questions about them. The work sample itself serves as a very tangible window into the current developmental level of the student and into the range of strategies and thinking that they are bringing to the tasks. It also provides evidence to support teacher evaluation and reporting.

Informed Planning Through intensive interviews and annotation of work samples it is possible to make informed decisions about what to do next either for the whole class, a specific group or an individual student. For example, we might ask: "Is the specific mathematical big idea involved being developed?" "Are there any misconceptions that need to be worked on?" "Is the intellectual level challenging enough or too challenging?" Answers to questions such as these point the way for future lessons.

Tracking Student Progress The dated and annotated work samples clearly show the development and mathematical progress of students. They are the evidence of the learning that is taking place presented in a way that you will find easy to use as the basis of a discussion about the individual student. The Student Progress Chart (at the back of this book) can be completed when the annotation of the work sample is complete and would make a cover page for your Work Sample records. It is useful to make reference to the dated work sample so that the student entry and the evidence to support the comments can be presented to parents and others.

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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Learning Horizons

Learning horizons The learning horizons that follow give the maths language, strategies, representations and consequent understandings associated with a big idea. They are based on classroom observations and research findings and describe the typical developmental stages that students pass through on their way to a deep understanding of that big idea. The purpose of each section is as follows: Language

identifies the informal words and terms that students may use in the early developmental stages of the big idea as well as the more formal language in which students need to be immersed. As we listen to students we can note the vocabulary that they use and make decisions about when and how to further enrich the mathematical vocabulary being used in class.

Strategies

identifies strategies that students may develop naturally as they work on problematised situations. Watching out for and recording strategies used allows planning for the extension of and introduction of other strategies. Using clear and specific names (and meta-language) for strategies allows discussion and comparison of them on reflection. It also means that students can begin to take responsibility in identifying which strategy to use in particular situations and to say why.

Representations

identifies the many approaches that students may use to represent their thinking and to record their ideas. You will notice that these move from concrete through to the more abstract methods as students become more confident with the big ideas addressed and with representing them.

Understandings

identifies the underlying new understandings that students need to demonstrate in relation to the big idea concerned. They may be identified in student work samples and inform the questions that need to be asked during intensive interviews with students. They are fundamental to a deep understanding of the big idea and the mathematical connections within and between them.

The intention is not that teachers will set out to teach each of the items outlined, rather that they will use them as a guide in their observations and annotations of students at work and their work samples. The learning horizons make visible the types of experiences that a student may still need to have in order to develop the concept fully. In each section the level of complexity increases as you approach the horizon. Then, beyond the horizon, there are glimmers of what is to come. This allows teachers to plan for and track a broad range of learners.

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© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Applying landmark numbers to mental strategies for working with large numbers Confidently moving between different representations of numbers, particularly representing fractions in ratio, decimal and percentage formats

Working with decimal numbers and fractions to more places and effectively using the language of place value

Number sense

Relating fractions, percentages and decimals to everyday contexts and problems

Representing, comparing and ordering numbers to 9 999 using effective counting strategies

Language

Understandings

Real world situations need real world answers (0.15 is not a useful fraction when sharing pizzas) Density of the number line (1.5 lies between 1 and 2, 1.25 lies between 1.2 and 1.3) Decimal fractions can be used to describe tenths and hundredths Numbers can be decomposed in many Strategies Representations ways (by additive and Working systematically Representing counting multiplicative means) with materials to find patterns on open Internal zeros act as ways of representing number lines and number place value holders whole and decimal numbers grids and indicate that Using knowledge of the Creating, naming and there are no 10s, multiplicative nature of 3-digit comparing representations of 100s, … in that numbers (300 is also 3 × 100 or decimal fractions position 30 × 10 or 300 × 1) Using an open number line to Applying the additive nature of show counting on and back whole numbers to decimal fractions, patterns with large numbers (6 tenths is also 60 hundredths) Writing 3-and 4-digit numbers in Using knowledge of the additive numerals and words nature of 3-digit numbers (365 is also Representing whole numbers with 300 + 60 + 5) materials, on number expanders and with Number splitting calculators Counting on and back in 10s, 100s, 50s and 25s

Decimal fraction Tenths, hundredths Quarter, half Divisible, division Factor 50s, 100s, 150s, 250s Sum, total Increase 1000s, 100s, 10s, 1s Position, column 2-, 3-, 4-digit number

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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Confidently estimating and checking as a natural part of achieving mathematical goals Having access to a range of representations and choosing the most appropriate one for the task on hand

Extending the language to large numbers

Addition and subtraction

Identifying and solving addition and subtraction problems using a range of strategies and methods of recording

Language

Difference Landmark numbers (25, 50, 100) Ball park estimates Shortcuts Rounding and adjusting Repeated doubling Mental computation

Strategies

Rounding and adjusting Rounding and estimating Number splitting Build on 10s Repeated doubles Doubles Near doubles Rainbow facts

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Applying addition and subtraction strategies in a range of everyday contexts, particularly ones involving money

Understandings

Place value knowledge can be applied to addition of round numbers (345 + 500 only involves adding 3 and 5 in the 100s position) Selecting the best strategy from a range of known strategies (mental, pencil and paper, calculator) to match particular addition or subtraction situations makes computation Representations efficient Open number lines Recognising Chunking for addition addition and subtraction Chunking for subtraction situations Traditional methods presented in Calculator realistic problem situations

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Extending strategies to larger numbers and to fractional quantities Confidently choosing the appropriate representation for a given situation or problem

Using the language of multiplication and division and extending it to ratio, proportion and percentages

Multiplication and division

Applying concepts of multiplication and division in real-life situations

Identifying and solving multiplication and division problems using a range of strategies and methods of recording

Language

Remainder, left over Decimals Fractions (quarters, sixths, fifths) Fair shares, equal shares Sharing, dividing, multiplying Repeated doubling, halving Repeated addition, subtraction Halving, doubling Strategies Skip counting Rounding and patterns estimating Key strokes Using the rule for multiplying or dividing by 10 and 100 Applying known multiplication facts Doubling, halving Repeated doubling, halving Number splitting

Understandings

Number splitting (decomposition) is an effective strategy for multiplication Knowing when to use a decimal and when to use a remainder in division results (5 people into 2 teams leaves one person out, not 2.5 people per team) Numbers grow very quickly (exponentially) when doubling Using knowledge of the 10s rule when multiplying is Representations efficient Traditional methods Chunking for multiplication Invented methods Making jumps on a 100 square Calculator as a multiplication or division function machine Calculator key sequences for multiplication and addition

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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Confidently estimating and checking as part of the shopping process Developing the ability to express income and expenditure as part of an analysis

Developing an appropriate language to describe a range of financial transactions, particularly best buys, lay-bys and bank statements

Money

Becoming a discerning consumer as a result of confidence with financial transactions of different types

Applying numerical operations to transactions involving money

Language

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Understandings

Mental computation methods can be applied to money situations Knowledge of equivalence (how many of each type of coin are needed to make a dollar) helps when dealing with multiple coins When dealing with money sometimes Strategies Representations rounding and Rounding and estimating Formal, abstract estimating are useful Rounding to the nearest methods strategies 10 cents or dollar amount Chunking as a method of calculating with money Applying knowledge of equivalence (twenty 5 cent Invented methods coins make a dollar) Using actual coins and notes Using the 10s multiplication rule Calculator methods including in money situations use of decimal point to separate Applying mental computation dollars and cents strategies to money

Denominations Equivalent to Total Amount Rounding amounts Nearest 10 cents, dollar Estimating

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Applying estimation and checking strategies to measurement situations and problems Extending the range and accuracy of measurement representations, particularly to maps with scales

Using the language of formal and informal measurement in a range of contexts

Measurement Using standard units when estimating, measuring and comparing the size of objects

Language

Fractions, half, quarter, third applied to measurement situations Metric measurement units: metre (m, cm and mm), kilogram (kg, g) and litre (L and mL) Accurate, approximate measurements Distance, length, height, capacity, Strategies mass, area Making estimations Temperature, based on experiences thermometer Using the relationships Estimate between measurement units (10 mm in a cm etc.) Tiling, covering, grid Developing benchmarks for measuring (having visual or Curved, physical memory of materials to straight base estimates on when standard lines tools are not available) Selecting appropriate units for measuring

Developing an intuitive sense of measurement units (about 15 metres, less than 2 km)

Understandings

Conservation of length (curved and straight lines) area, capacity and mass When measuring it is essential to select the appropriate measuring tool and unit for the purpose When covering an area there must be no gaps or overlaps in the tiles Area can be found by counting tiles but it is Representations more effective to use a Reading and using multiplicative strategy scales on measuring tools Reading and using abbreviations for standard SI units Recording measurements Using materials or grids to show and measure areas Using materials and direct comparisons when measuring

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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Knowing what is the appropriate resource for planning events and activities Choosing from a range of representations which is the most appropriate in a given situation or problem

Using the language of time to make and interpret a range of timetables and guides

Time

Having a reliable sense of the duration of events

Reading and recording time in digital and analogue formats; interpreting calendars and timetables in everyday contexts

Language

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Understandings

When adding time spans base 60 needs to be used or conversions from minutes to hours need to be made prior to or during calculation There is a base 60 relationship between seconds, minutes and hours Timetables help us to organise our daily lives and to make plans Strategies Representations based on them Converting minutes into Reading and telling time Mental computation hours and minutes and (analogue and digital) strategies and vice versa Calculating passage of landmark Applying mental calculation time using: numbers can be strategies to time applied to time • open number lines calculations Using quarter and half hour • chunking, formal methods landmarks to work out passages • hands-on methods with clocks of time

Duration, passage of time, time spans Cycles First, second, third, fourth quarter Analogue, digital clocks TV guides Timetables Hours and minutes

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Working with 2D and 3D shapes and justifying results in geometric terms Making scale models and designs, using 2D shapes to build 3D objects

Applying the language of shape to a broader range of situations, such as tessellations

Shape

Extending these understandings to more complex structures

Describing 2D and 3D shapes in formal terms; relating 2D and 3D shapes through nets

Language

Understandings

How workers in the real world work with 2D, 3D shapes and plans Properties of shapes effect their usefulness in particular situations Creating visual images in the head is the first step in the design process as well as in the construction process Strength and rigidity are major factors Strategies Representations when working with 3D Identifying important, Designing and making shapes relevant properties of 2D 2D shape puzzles and 3D shapes Visualising and creating Visualising the changes that nets for 3D shapes will be made by joining, Accurately measuring length cutting, folding shapes and angle when constructing Considering the strength and shapes and nets other properties of shapes when Drawing and designing plans and making design decisions elevations Fitting and joining pieces together Making models from plans and elevations Constructing models

Names of 2D and 3D shapes Angle, right angle, degrees Vertices Vertical, horizontal, diagonal, oblique, parallel vertices, apex, base, faces, Curved, straight, Nets, plans, elevations Properties, congruent

© 2006 Blake Education Natural Maths Book 1: Work Samples © Strategies 2006 Blake–Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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Applying these strategies to more complex plans and maps Selecting from a range of representations the one that is appropriate to a particular situation or problem and justifying their choice

Using the language of position to describe a locality and to interpret maps

Position in space

Creating maps and plans to describe location; using the major compass points

Language

Direction, 360, 180 and 90 degree turns, left, right Clockwise, anticlockwise, backtrack Compass points Rotations, slides and turns Alphanumeric plans, routes, pathways Key Shade, shadow

Strategies

Interpreting, following and giving directions physically and on a map or plan Using alphanumeric grids and plans Using compass points Using turns and degrees when giving instructions Using clockwise, anti-clock-wise, right and left when giving instructions

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The use of maps and plans is central to the design of a location, allowing ideas to be tested before implementation

Understandings

There are many ways of giving directions and we need to select from them to suit the purpose Maps and plans make it easy to navigate and identify locations Orientation and movement in space can be precisely described

Representations

Reading and constructing alphanumeric plans and maps Creating bird’s-eye view plans Marking routes and backtracking pathways Creating keys for maps and plans

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Quantifying likelihood in percentage and numerical terms in order to compare outcomes Extending the range of chance-data collection techniques, such as the use of a preference table

Developing the language needed to describe probabilities in numerical terms Being able to differentiate between subjective and objective descriptions of chance

Chance Identifying possible outcomes in a chance situation and determining the likelihood of outcomes

Understandings

Language

Numbers can be used to describe likelihood and chance Experiments and investigations can be used to model chance situations Data collected from experiments and investigations can be used as the basis of informed predictions

True, false Use of number, 1 in 5 Possibilities Table, diagram, graph Fair, unfair Likelihood, out of, predict Chance, probability

Strategies

Interpreting events and their effects on probability Observing actions and events to predict likelihood and chance Making predictions based on investigations with materials or with direct observation of events Listing all the possibilities

Representations

Recording actions and events as tallies, graphs, maps or tables to make a record and to predict likelihood and chance Using materials to model chance situations Acting out situations

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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Summarising data in terms of its key statistics Extending the types of representations to other types of graph

Using the terms for measures of spread (range) and central tendency (mean, median and mode)

Data

Understanding how summary statistics can be used to describe and compare data sets

Collecting and representing data and describing features of the chosen representation

Language

Understandings

Graphical representations need to be matched to purpose Data needs to be accurate to be useful Ease of comparison of data is important when selecting representational format Comparison allows trends in data to Strategies Representations be identified Creating surveys and Representing data in a observation schedules bar graph to compare Drawing conclusions based frequencies on data Choosing appropriate Transferring data from one intervals for data and showing these on a bar graph form of representation to another Using tallies to record and Sorting and classifying data for represent information graphing Showing information in pictograph form Locating information on graphical representations Creating appropriate scales and labels for axes

Values, scales Data collection Survey, observation Categories, sort classify Chart, matrix, graph, title axis

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© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Looking for patterns in tables of numbers and expressing the pattern in words and symbols Expressing patterns in terms of a rule; representing number sentences as equations and solving simple equations

Describing a pattern in terms of an algebraic rule and relating arithmetical expressions to everyday contexts

Patterns and algebra

Applying patterns, rules and formulae to everyday contexts

Creating and comparing number patterns based on relationships; working with arithmetical expressions that involve all operations

Language

Rules, sequences, counting patterns Change in magnitude Calculator, equals machine Order of operation Equivalent, not equivalent Equation

Strategies

Applying the order of operation rules, e.g. interpreting 3 + 2 × 4 as 3 + 8 = 11 rather than 5 × 4 = 20 Identifying the effects and changes on results when the order of operations is varied Inverse operation and backtracking Identifying, creating and using counting patterns Bridging 10s, 100s and 1000s when using counting sequences

Understandings

Recognising the importance of estimating an answer by rounding to friendly numbers Changing the order of operation or the order of the numbers will usually result in changes of magnitude Number patterns and sequences can help make sense of number situations and can be Representations used to solve problems Open number lines for and investigations showing mental strategies Chunking for multiplication and division Interpreting and writing calculator key sequences Representing a pattern by means of a table of values Diagrams for processes such as repeated halving Demonstrating patterns on a number grid Making models with materials

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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N3.1: Number Sense

N3.1: Number Sense What do you think is special about the number 168?

Annotation 1. Tried different ways of decomposing numbers (number splitting). 2. Represented the number as hundreds, tens and ones with MAB pictures. 3. Used tallies counting in 10s initially but then moving to 20s to speed up the process. 4. Showed 168 as notes but included a $3 note (impossible). When asked, knew there was no such thing as a $3 note and laughed at the silly mistake. 5. Experimented with mathematical expressions involving balance through inverse operations. 6. Wrote the number correctly in words. 7. Knew the identity property of multiplication by 1. 8. Knew that 168 is an even number.

Work Sample : Samantha (girl, 10-5)

Assessment Summary Samantha has demonstrated a strong number sense with place value into the hundreds. She knows that efficiency is important and chose tallies 'because she likes them' but soon realised that for large numbers an efficient counting strategy is important. She deliberately chose multiplication by 1 because she knows that the number remains unchanged. When looking at the overlaps in the mathematical ideas presented it suggests that Samantha is not risk taking. When prompted she had more mathematical knowledge than was shown. When prompted about other things she could have included, she had thought of doubling 168 but would not write it on the page in case it was wrong.

Informed Planning Samantha needs to be encouraged to be more playful in her approach to number and to be prepared to try things out even though she is slightly uncertain about them. More opportunities to experiment with numbers and increase the range of mathematical ideas and language before moving on to repeating this activity with 4-digit numbers will benefit Samantha.

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© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

N1.1: Number Sense N3.1:

What do you think is special about the number 168?

Annotation 1. Place value is shown in different ways, but with errors in some instances. 2. Knew that numbers can be represented with materials and tallies but avoided the 'slow' process of showing the number with them. 3. 168 correctly written in words. 4. 168 shown correctly as a mix of coins and notes. 5. Odd and even is used to describe parts of the number and the whole number. 6. 168 shown as the result of single additions or subtractions. 7. Expressed 168 as rounded to 170. 8. Divided 168 by 1 showing that the number is unchanged (identity property). 9. Repeated use of the equals sign as a trigger to do something, leading to an incorrect expression 168 ÷ 168 = 1 × 168 = 168. 10. Used knowing 10 squared as the trigger for trying to express 168 as the sum of squares.

Work Sample : Tim (boy, 9-11)

Assessment Summary Tim has thought broadly, trying to think of interesting ways of describing the special features of 168. When asked about 168 tens, knew instantly that that was actually 1680 and likewise with 168 hundreds, knew that that was 16 800. Tim admitted that he often made mistakes on the 'easy and uninteresting stuff'. Prompted about special types of numbers such as odd, even, square, said "prime" and knew that 168 was actually a composite number.

Informed Planning Tim has an interest in number and could be encouraged to further explore larger numbers. He would benefit from investigating square and triangular numbers and even cube numbers. He could also be encouraged to create more complex equivalent expressions but needs to see the equals sign as a balance rather than as a trigger to do something.

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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N3.3: Multiplication

N3.3: Multiplication The tuck shop ordered 15 packets of 24 bread rolls. The bread rolls are $6.45 a packet. How many bread rolls altogether and how much would it cost? The tuck shop complained to me that kids today would never know how to work that out without a calculator. They think your brains have rotted. Let's prove them wrong.

Annotation 1. Jola clearly understood the problem as evident in her drawing of the problem components. 2. Used a repeated doubling strategy represented by vertical chunking. 3. Found the total for the first 8 packets correctly. 4. Found the total for the remaining 7 packets correctly but never added the two subtotals to complete the answer. 5. Used repeated doubling for the $6.45s as well. 6. The final two subtotals were added as if they were two lots of $12.90 where in fact they were $12.90 + $6.45. 7. The four subtotals of $25.80 were correctly added together giving an answer for 16 packets not the 15 packets asked for.

Assessment Summary Jola effectively uses drawings or diagrams to identify what the problem is about. Jola has developed a repeated doubling strategy and is very efficient with it. In this instance Jola did not make the switch from working with combinations of doubles to the combination of a double and a single. In fact the strategy has been overgeneralised. When working with a large number such as 15, using a multiply by 10 strategy would be more effective.

Informed Planning While repeated doubling is working as a strategy, Jola should be encouraged to use a more efficient strategy such as number splitting followed by multiplying by 10 and halving to multiply by 5 or using a strategy such as chunking. Multiplication is avoided so consolidation of multiplication tables or strategies for deriving them needs to be addressed.

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© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Work Sample : Jola (girl, 9-6)

N3.3: Multiplication

The tuck shop ordered 15 packets of 24 bread rolls. The bread rolls are $6.45 a packet. How many bread rolls altogether and how much would it cost? The tuck shop complained to me that kids today would never know how to work that out without a calculator. They think your brains have rotted. Let's prove them wrong.

Annotation 1. Matt used number splitting to make 15 more manageable. 2. Found 10 × 24 then halved it to find 5 × 24. 3. Used chunking to add the two smaller products together. 4. Used chunking to work out $6.45 × 24. He wanted to chunk with money, even though he had never done that before. 5. Having looked back at his setting out for the earlier problem, Matt made the error of working out 24 packets and not 15. 6. Having multiplied all of the subparts they were chunked in two manageable groups and then added correctly to finish off.

Assessment Summary Matt is confidently number splitting as well as chunking for addition and multiplication. He is seeking efficient strategies as evident in his multiplication of 24 by 15. Matt's understanding of place value is well established as demonstrated by the ease with which he multiplies by 10 and 20, even with money.

Informed Planning Matt needs to be encouraged to check his answers and methods against the question to avoid 'silly' mistakes with his answers. Although he used chunking for money for the first time and very effectively too, he could be prompted to look for efficient strategies involving rounding 45 cents to 50 cents and compensating. He is also ready to move toward more traditional methods for showing his working.

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Work Sample : Matt (boy, 10-2)

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N3.3: Division

N3.3: Division Some people, trying to get fit, wear a pedometer to measure how many footsteps they take in a day. If you needed to take 10 000 footsteps in a day how many times would you need to walk around the classroom block?

Annotation 1. Jarrod drew a picture of the problem. 2. Identified the situation as a repeated addition problem rather than a division problem. 3. Used the addition sign between most of the numbers to signal that he was adding, prompted could not identify anything inappropriate about this or suggest an alternative method of separating the numbers. 4. Used a compensating method of addition (add 200 subtract 3) to explain the accuracy of his answers. 5. Used a calculator as a repeated addition machine (197 + = = = and so on) to check his working out. 6. The final addition resulted in a total greater than 10 000 but he did not see this as relevant in his total of the number of laps or recognise it as part of a lap. 7. Touched and counted the number of additions that he had made and wrote 45. Knew he wasn't quite right and crossed it off but did not try to fix it up.

Assessment Summary While Jarrod's strategy almost worked it was not an efficient one. However because of the compensation strategy, it was an effective one for Jarrod. A lot of effort went into the computations each time although Jarrod did use doubling for the first few additions. By the end of his list he was mentally tired and found it hard to finish off.

Informed Planning Jarrod needs experiences with situations involving smaller numbers where he can be encouraged to make links between repeated addition and multiplication and the inverse repeated subtraction and division. Jarrod also needs to review the role and meaning of symbols in equations as well as representations such as a table to organise strings of numbers.

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© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Work Sample : Jarrod (boy, 10-3)

N3.3: Division

Some people, trying to get fit, wear a pedometer to measure how many footsteps they take in a day. If you needed to take 10 000 footsteps in a day how many times would you need to walk around the classroom block?

Annotation 1. Luke was slow to start and eventually copied Jarrod's diagram idea. 2. Wrote 197 to remind him how many steps around the pod. 3. Recognised the problem as a division situation. 4. Used a calculator to carry out the division and was surprised (showing everybody the long string of numbers). 5. Wrote the answer exactly as it appeared on the calculator display. 6. When asked what the answer meant he did not know but when prompted knew that 0.5 was equivalent to ½ and could say that the decimal part of his answer was greater than a half. 7. After a pause was able to give a real world answer of 51 because he recognised that another half lap would be needed to get back to the starting point that he pointed to on his diagram.

Assessment Summary Luke was hesitant about how to get started and how to make sense of the problem but the diagram and detail he added to it really helped him. There was real excitement at the long string of numbers after the decimal point. Although Luke recognised the landmark decimal of 0.5 as ½ he did not recognise the 0.75 landmark as equivalent to ¾.

Informed Planning Luke needs to be immersed in strategies for interpreting and drawing or diagramming problems. He is ready to be introduced to decimals to hundredths and to the associated landmark decimals. He clearly likes the appeal of 'big' numbers.

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Work Sample : Luke (boy, 9-10) 21

N3.4: Money

N3.4: Money Matt has been collecting 5-cent and 20-cent coins and finds that he has 44 coins and a total of $4. How many of each type of coin did he have? Emily has been collecting 10-cent and 50-cent coins and finds that she also has 44 coins but $8 in total. How many of each type of coin did she have?

Annotation 1. Mikayla clearly identifies what the problem is asking her to do. 2. Thinks about how to approach the problem and plans to use actual coins, thinks logically that she should start by making $2 with each denomination. 3. Uses a systematic trial and error approach swapping four 5-cent coins for a 20-cent coin until she has a solution that works. 4. Records what she has found out and double checks, finding her mistake and creating a fix-up strategy for it. 5. Repeats the earlier trial and error strategy for the second set of coins again changing the denominations to match the given criteria.

Work Sample : Mikayla (girl, 10-3)

Assessment Summary Mikayla is restating the problem in her own terms and thinking of an approach that will work for her. She uses actual coins and a trial and error approach followed by the systematic exchanging of coins which shows that her method is much better than random guess and check. While this strategy works for her it demonstrates little use of money-based strategies such as knowing how many coins of a particular denomination make $1.

Informed Planning Mikayla could be encouraged to use a trial and error strategy based on coin values rather than on the actual coins. Reviewing the number of each denomination needed to make a dollar and relating this to the problem might help her move on from concrete materials for money problems.

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© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

N3.4: Money

Matt has been collecting 5-cent and 20-cent coins and finds that he has 44 coins and a total of $4. How many of each type of coin did he have? Emily has been collecting 10-cent and 50-cent coins and finds that she also has 44 coins but $8 in total. How many of each type of coin did she have?

Annotation 1. Chloe used a trial and error strategy based on her knowledge of how many of each type of coin are needed to make a dollar. 2. She knew she needed to double the amount and almost double the number of coins but her inexperience with proportional reasoning led to halving and doubling the wrong coins. 3. Saw her error and made more informed estimate. 4. Used the earlier mistake to mentally switch denominations and number of coins to create a correct solution. 5. Applied her earlier trial and error and new insights to create a fairly close estimate. 6. Knew immediately how to fix up her answer. 7. Checks her answers each time and really enjoys the success.

Work Sample : Chloe (girl, 10-7)

Assessment Summary Although trial and error was used the initial estimate was based on her current understanding of the denominations. Chloe is expecting her thinking to make sense and really stops and thinks about it when it doesn't (1, 2 and 3). She is confidently multiplying money amounts by 10 showing solid place value understanding applied to money.

Informed Planning Chloe would benefit from situations where proportional reasoning is involved so that she begins to develop relational thinking. She could comfortably work with more complex money problems of the same type. In fact allowing her to pose her own problems might lead her to some situations where proportional reasoning is required. © 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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M3.1: Measurement

M3.1: Measurement We all know what a metre is and how it is different to a millimetre. We could measure the classroom in metres, but it would be best to measure a spider in millimetres. What if we used the wrong units to measure? There would be some very big or very small numbers to remember.

Annotation 1. Trad decided to measure the thickness of a book. 2. Correctly multiplied by 10 to convert cm to mm. 3. Demonstrated understanding of the relationship between units (10mm = 1cm, 100 cm = 1m, 1000 m = 1km). 4. Drew up a table to show the conversions from each unit but has missed out the tens column and used his own invented convention for showing the decimal place, giving it a column of its own. 5. Knew that in order for the table to make sense the units on every row needed to be named. 6. The arrows have been used effectively in the algorithms to show the effect of multiplying or dividing by powers of 10. 7. Clearly showed his range of measurements in a table next to his original measurement task.

Assessment Summary Trad has demonstrated deep understanding of place value in relation to measurement units and has applied multiplication and division by powers of 10 effectively.

Informed Planning Because Trad has demonstrated a good understanding of linear measurement units he can now be challenged to use inappropriate measurements to other measurement situations such as, mass, capacity, area, volume.

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© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

Work Sample : Trad (boy, 10-11)

M3.1: Measurement We all know what a metre is and how it is different to a millimetre. We could measure the classroom in metres, but it would be best to measure a spider in millimetres. What if we used the wrong units to measure? There would be some very big or very small numbers to remember.

Annotation 1. Kuldeep has measured the distance around the class block. 2. Decided to show his information on a table labelled from mm through to km with place value grid attached. 3. The coloured arrows clearly show his conversion process with one colour being used to show each multiplication by 10. 4. The process is reversed for division by 10. 5. The unit labels were added without hesitation after he was asked what each of the numbers on the grid represented. 6. The answer and a reasonable variation expressed as a fraction of a kilometre are given. 7. Referring back to the original question the inappropriate measurement was given. This amused Kuldeep who was surprised to see the magnitude of the number when inappropriate units were used.

Work Sample : Kuldeep (boy, 9-11)

Assessment Summary Relationships between linear units of measurement are clearly understood as is the rule for multiplying by 10 and multiplying repeatedly by 10 for 100 and 1000. Colour coding the place value to show the movement of the numbers was effective.

Informed Planning Kuldeep also needs to be challenged to apply his knowledge of measurement units to other measurement situations (mass, capacity, area, volume).

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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S3.1: Shape

S3.1: Shape You have a maximum of fifty straws and balls of dough to make a scaffold that will support this small heavy object 30 cm from the ground for at least 30 seconds. You will be charged for materials so you need to make the scaffold as cheaply as possible.

Annotation 1. Nicky intends all straws to be the same uncut length. 2. Nicky is unaware the diagonal straws would be too short to fit across the squares. 3. The sketch is of a 2D design with no attempt at 3D. 4. An attempt was made to create a costing for a 3D building by multiplying the number of straws and dough balls (used instead of paper clips) by 2. 5. Marks on the straws were made to keep count after several failed attempts at counting systematically. 6. Used number splitting to multiply by 6. 7. Saw the relationship between double 25 (the cost per straw) and her attempt at doubling the number of straws. 8. To multiply by 50, knew to multiply by 100 and then halve the result.

Assessment Summary Nicky shows limited ability to think, plan or draw in 3D. She has inappropriately applied a doubling strategy to her single facade to create a 3D estimate of straws and dough balls required. Her actual design with points on the top would not support an object. As can be seen Nicky's number sense is good whereas her spatial reasoning not so well developed.

Informed Planning Nicky needs hands-on experiences with creating skeleton shapes and exploring the relationship between the diagonals and vertical or horizontal lines comprising a square. She also needs to be encouraged to create images of 3D shapes in her head and to develop strategies for drawing them.

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Work Sample : Nicky (girl, 10-1)

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

S3.1: Shape

You have a maximum of fifty straws and balls of dough to make a scaffold that will support this small heavy object 30 cm from the ground for at least 30 seconds. You will be charged for materials so you need to make the scaffold as cheaply as possible.

Annotation 1. Earl began by deciding that the straws would be cut to 15 cm. 2. Drew a series of 3 squares to make a tower and then drew 3 more offset behind and joined them to create a 3D tower. 3. Dots show his first count which led to his decision to add another tower. 4. Purple 'dough balls' were enlarged to show which had been counted. 5. Yellow dashes show which straws have been counted. 6. Extra 'full length' straws added for strength and to bring the total to 50. 7. Costing for straws shows use of doubling and the rule for multiplying by 10. 8. Chunking used for costing dough balls. 9. Total by traditional method with dollar sign added as an afterthought.

Work Sample : Earl (boy, 10-5)

Assessment Summary Earl met the criteria of the problem in every way. He realised that he would need diagonals for strengthening the tower, although in reality more diagonals were needed. He also knew that the diagonals would be longer than the sides and allowed for this in his original cutting of the straws.

Informed Planning Earl has a good basic knowledge and ability to think, plan and draw in 3D. What he needs now is more experience with exploring how to design and make structures that are strong and rigid. Exploring how to make structures with a range of shapes and materials would be beneficial.

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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CD3.1: Chance

CD3.1: Chance In any hour of the day there is a one in four chance that news will be on TV. Agree or disagree? What are the chances that there will be news on at any time between 6 am and 9 pm?

Annotation 1. Luke drew up a table with hour by hour timeslots but soon realised he would not have enough space so reduced the size of his spaces. 2. Systematically wrote all of the hours between 6 am and 9 pm as 6 – 7, 7 – 8 and so on. 3. Began making tallies for each news or news related show. 4. At 6 – 7 pm he noticed that he had already marked the 6 o'clock news in the box above. 5. Double checked all other entries and found other similar errors. 6. Decided to fix up his error by going over the arrows in purple and writing '1 to' under them meaning one minute to the hour. 7. No 'Agree or disagree?' statement was made but he answered orally that his chart showed that the chance was about 50-50.

Work Sample : Luke (boy, 10-6)

Assessment Summary Luke's idea of creating a table, even though not well-planned, was an effective strategy for showing clearly at what times of the day the news related programs were on. He realised his error with the time overlaps and was able to cover his tracks – a good learning experience for him.

Informed Planning Luke would benefit from a focus on planning; although he clearly understood the problem and had an idea for solving it, he did not systematically consider its parameters. He could easily state that 6 am through to 9 pm would require 15 timeslots but could not easily see how this could have helped with his planning of the page. Similarly he rushed ahead without considering the timeslot overlaps.

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© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

CD3.1: Chance

In any hour of the day there is a one in four chance that news will be on TV. Agree or disagree? What are the chances that there will be news on at any time between 6 am and 9 pm?

Annotation 1. Skye thought that there definitely would be news every hour of the day. 2. Used the actual TV guide as part of her work. 3. Used a key to explain her decisions (pink shows the actual news and blue the shows that have some news in them). 4. Used digital representations of the times. 5. Listed all of the times when a news show was on. 6. Ticks used where there was more than one news show at a time to indicate how many were on. 7. Related the data collected back to the original question and statement of chance, realising that she was definitely wrong in her estimation of probability.

Work Sample : Skye (girl, 9-11)

Assessment Summary Skye has been systematic and logical in her approach however her representation of the data does not make it easy to 'see at a glance' whether every hour has a news slot. At reflection time, the representation made it hard for her to answer questions, such as: ‘During what time periods were there no news shows?’

Informed Planning Skye needs opportunities to explore different ways of representing data to make it easy for others to extract meaning visually. She needs help making connections between the purposes for representing data and the format that will suit the purpose. © 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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PA3.1: Patterns

PA3.1: Patterns Tilers often have to create interesting tiling designs that sometimes include borders. They may need to cut tiles in halves or in quarters to create the borders. How would you tile a bathroom floor using two colours and with a fancy border?

Annotation 1. Storm did not complete the tiled border in the time given. 2. No thought or planning given to creating a repeating pattern and more than two colours used. 3. Tiles were drawn in a random arrangement of halves and quarters and then coloured randomly too. 4. Began afresh after the reflection and discussion of patterns and symmetry. 5. Drew all tiles in quarters and applied the rule of no like colours touching along an edge. 6. The inner row was partially completed before the outer row was begun. 7. The outer row worked and fitted the rule by accident rather than by design.

Assessment Summary Spatial thinking and visualisation proved challenging for Storm. Her first attempt demonstrated no preplanning or thought of a repeating design. In effect there appears to be a gap in her understanding of repeating patterns and symmetry. The pattern eventually arrived at resulted from colouring one tile and repeating that tile for the inner border.

Informed Planning

Work Sample : Storm (girl, 10-3)

Storm needs opportunities to work with concrete materials to manipulate, rotate and flip tiles to create and describe repeating spatial patterns. She will benefit from other spatial visualisation activities requiring mental manipulations of simple shapes.

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© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

PA3.1: Patterns

Tilers often have to create interesting tiling designs that sometimes include borders. They may need to cut tiles in halves or in quarters to create the borders. How would you tile a bathroom floor using two colours and with a fancy border?

Annotation 1. Jai completed the border. 2. The inner row was completed first using only half tiles. 3. The simple rule of no like colours touching along an edge was used. 4. The outer edge was planned to use quarters and apply the same colour rule. 5. The result of the rigid rule was that there was no symmetry in the design. 6. The mistake in the bottom row led to the creation of one odd corner requiring two colours to complete. A fix-up was considered as can be seen with the overlap of green but discarded. 7. The key makes it clear how many of each tile would be needed to complete the design. The half tiles were counted by touching in 2s. The quarter tiles were counted slowly in 4s with no relationship between them and the halves being noticed.

Work Sample : Jai (boy, 10-6 )

Assessment Summary Jai set out with a pattern rule in mind but at no point stopped to consider or plan to make it symmetrical. Jai did not consider using a mix of half and quarter tiles in one square and had not noticed the equivalence of two quarters in a half until it was drawn to his attention.

Informed Planning Jai would benefit from tessellation activities with tiles of different shapes and of different fractional sizes. His understanding of the area model of fractions needs to be developed further through such experiences. © 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

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Student Progress Chart

Student Progress Chart Student:

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Topic

Big Ideas

Number sense

N3.1 Representing, comparing and ordering numbers to 9 999 using effective counting strategies

Addition and subtraction

N3.2 Identifying and solving addition and subtraction problems using a range of strategies and methods of recording

Multiplication and division

N3.3 Identifying and solving multiplication and division problems using a range of strategies and methods of recording

Money

N3.4 Applying numerical operations to transactions involving money

Measurement

M3.1 Using standard units when estimating, measuring and comparing the size of objects

Time

M3.2 Reading and recording time in digital and analogue formats; interpreting calendars and timetables in everyday contexts

Shape

S3.1 Describing 2D and 3D shapes in formal terms; relating 2D and 3D shapes through nets

Position in space

S3.2 Creating maps and plans to describe location; using the major compass points

Chance

CD3.1 Identifying possible outcomes in a chance situation and determining the likelihood of outcomes

Data

CD3.2 Collecting and representing data and describing features of the chosen representation

Pattern

PA3.1 Creating and comparing number patterns based on relationships

Equivalence

PA3.2 Working with arithmetical expressions that involve all operations

Comment & Date

© 2006 Blake Education Natural Maths Strategies – Book 3: Work Samples © Blake Education ISBN 978 1 92114 344 1 Natural Maths Strategies - Assessment Guide for Book 3

WS Ref

Natural Maths Strategies Assessment Guide – Book 3 Written by Ann Baker BPhil, DipRdg and Johnny Baker BSc. Hons, PhD Copyright © 2006 Ann & Johnny Baker and Blake Education First published 2006 Blake Education Pty Ltd ABN 50 074 266 023 108 Main Rd Clayton South VIC 3168 Ph: (03) 9558 4433 Fax: (03) 9558 5433 www.blake.com.au Publisher: Lynn Dickinson Series editor: Shelley Barons Designer & Typesetter: Domani Design Printed by Thumbprints Utd, Malaysia. This publication is © copyright. No part of this book may be reproduced by any means without written permission from the publisher. COPYING OF THIS BOOK BY EDUCATIONAL INSTITUTIONS A purchasing educational institution may only photocopy pages within this book in accordance with The Australian Copyright Act 1968 (the Act) and provided the educational institution (or body that administers it) has given a remuneration notice to the Copyright Agency Limited (CAL) under the Act. For details of the CAL licence for educational institutions, contact: Copyright Agency Limited Level 19, 157 Liverpool St Sydney, NSW, 2000 COPYING BY INDIVIDUALS OR NON-EDUCATIONAL INSTITUTIONS Except as permitted under the Act (for example for fair dealing for the purposes of study, research, criticism or review) no part of this book may be reproduced, stored in a retrieval system, or transmitted in any form by any means, without the prior written approval of the publisher. All enquiries should be made to the publisher.

National Library of Australia ISBN: 1-921143-44-4 978-1-921143-44-1

This is a companion book for Natural Maths Strategies Book 3. This book provides the tools needed to make an informed and accurate assessment of individual student's level of mathematical competence. Annotated work samples, from students working with the Natural Maths Strategies program, illustrate how to interpret students' work for assessment purposes. The authors have also provided learning horizons, that identify the developmental stages that underpin each of the "big ideas" in the Natural Maths Strategies program. These can be used as the basis of informed planning and direct teaching.