Mental routines Problematised situations Games ... · unit containing mental routines, problematised situations, ... activities has been the research that suggests ... certain conditions

Based on world’s best practice! This series provides the core knowledge and understanding of the“big ideas” or concepts students require to become confident and enthusiastic maths users. Thisbook is organised into twelve units of work based on the current research into the developmentalsequence in which students generally acquire those concepts. Each unit is divided into fivesections:

Mental routines – 10-minute lesson starters with suggested closed and openquestions designed to engage students and arouse their enthusiasm

Problematised situations – challenges that encourage students to workmathematically with open-ended “real-life” situations and construct their own ideas.These lessons include a reflection session where mathematical language is used todescribe successful strategies and more formal methods are introduced anddemonstrated.

Games – fun activities designed to reinforce the strategies developed in each unit.

Investigations – open-ended investigations to encourage students to test and extendtheir skills.

Assessment activities – consolidation activities that students should readilyaccomplish at the end of each unit.

The series encourages the use of readily available concrete materials and is supported by over 50 photocopiable activity sheets and task cards. The CD-ROM included with this book is designedto help teachers to plan and personalise their maths program and to record individual student’sprogress.

The Natural Maths Strategies series is a complete school program, which also encourages the useof supplemental resources to ensure a variety of maths teaching and learning experiences.

CUTNMS Book 1 Cover.qxd 6/05/2006 5:09 PM Page 1

Natural Maths Strategies – Book 1Written by Ann Baker BPhil, DipRdg and Johnny Baker BScHons, PhD

Copyright © 2006 Ann & Johnny Baker and Blake EducationFirst published 2006

Blake Education Pty LtdABN 50 074 266 023108 Main RdClayton South VIC 3168Ph: (03) 9558 4433Fax: (03) 9558 5433www.blake.com.au

Publisher: Lynn DickinsonSeries editor: Sante D’EttorreDesigner: Domani DesignIllustrations: Shiloh Gordon and Oscar BrownCover design: Domani DesignTypesetter: Post Pre-press GroupPrinted by Thumbprints Utd, Malaysia.

This publication is © copyrighted. No part of this book may be reproduced by any means with-out written permission from the publisher.

CCOOPPYYIINNGG OOFF TTHHIISS BBOOOOKK BBYY EEDDUUCCAATTIIOONNAALL IINNSSTTIITTUUTTIIOONNSSA purchasing educational institution may only photocopy pages within this book in accordancewith The Australian Copyright Act 1968 (the Act) and provided the educational institution (orbody 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 LimitedLevel 19, 157 Liverpool StSydney, NSW. 2000

CCOOPPYYIINNGG BBYY IINNDDIIVVIIDDUUAALLSS OORR NNOONN--EEDDUUCCAATTIIOONNAALL IINNSSTTIITTUUTTIIOONNSSExcept 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 sys-tem, or transmitted in any form by any means, without the prior written approval of thepublisher. All enquiries should be made to the publisher.

National Library of AustraliaISBN: 1-921143-38-XISBN: 978-1-921143-38-0

1.Mathematics – Study and teaching (Primary). I. Title

372.7

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iii© 2006 Blake Education – Natural Maths Strategies – Book 1

CONTENTS

Contents and Natural Maths Planner

Contents

Introduction vi

Unit 1: Early Number Sense 2

Unit 2: Early Number Strategies 14

Unit 3: Extending Number Strategies 26

Unit 4: Patterns in Number, Space and Measurement 38

Unit 5: Beginning Chance and Data 50

Unit 6: Number in Context 62

Unit 7: Shape and Position 74

Unit 8: Exploring Number Situations 86

Unit 9: Informal Measurement 98

Unit 10: Shape and the Environment 110

Unit 11: Using Patterns 122

Unit 12: Putting it all Together 134

Activity Sheets 147

Task Cards 174

Maths PlannerThe Book 1 Maths Planner is the software package provided at the front of this book whichsupports the maintenance of planned units of work to cover the syllabus. The Maths Plannerhas facilities for:

2 incorporating the activities from this book and other schemes into a completemathematics progression

2 maintaining assessment records in conjunction with annotated work samples2 summarising class and individual progress.Each unit contains two examples of the type of assessment activity that will allow students toshow their understanding of a particular topic. Examples of student responses to these and tothe problematised situations are given in the associated publications Natural Maths StrategiesAssessment Guide for Book 1 with work samples. This book shows how work samples can beannotated to create a portfolio of the student’s achievement. The book also gives a summaryof key vocabulary, strategies, representations and understandings that students at this levelmight be expected to demonstrate.

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Big Ideas in Maths


iv

Big Ideas in Maths

© 2006 Blake Education – Natural Maths Strategies – Book 1

Big

ideas

in

Maths

Big Ideas in Maths

The mathematical content of this series is organised around 12 big ideas that are relevant toteaching maths at this level. This chart gives an overview of the big ideas and their links to the 12units of work.

A full conversion chart that relates these big ideas to State and National Guidelines is suppliedseparately.

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6

Early NumberSense

Early NumberStrategies

ExtendingNumber

Strategies

Patterns inNumber, Space

and Measurement

Beginning Chanceand Data

Number inContext

N1.1 Number sense

N1.2 Addition and subtraction

N1.3 Multiplication anddivision

N1.4 Money

M1.1 Measurement

M1.2 Time `

S1.1 Shape

S1.2 Position in space

CD1.1 Chance

CD1.2 Data

PA1.1 Pattern

PA1.2 Equivalence

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v© 2006 Blake Education – Natural Maths Strategies – Book 1

Big Ideas in Maths


Big

ideas

in

Maths

Unit 7 Unit 8 Unit 9 Unit 10 Unit 11 Unit 12

Shape andPosition

Exploring NumberSituations

InformalMeasurement

Shape and theEnvironment

Using PatternsPutting it all

Together

Representing, comparingand ordering numbers to

20 using effective countingstrategies.

Using mental strategies andinformal representations ofaddition and subtraction.

Using informal strategiesand representations of

multiplication and division.

Recognising coins and theiruses in transactions.

Measuring by direct andindirect comparison using arange of non-standard units.

Using the language of timeand sequencing the days of

the week.

Naming, describing andconstructing with 2-D and

3-D shapes.

Using positional languageand informal spatial plans.

Using the language ofchance to describe everyday situations.

Collecting, interpreting andrepresenting data.

Creating and describingpatterns based on simple

rules.

Representing number in avariety of ways and

developing the concept of equivalence.

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Introduction

vi

Introduction


introduction

Introduction Introduction

vi

Introduction


introduction

The activities in this book provide starting points for three-part lessons that focus on the bigideas for teaching maths to 5 to 7-year-olds. It is intended that the activities also be used inconjunction with the Book 1 Maths Planner CD-ROM, which enables the teacher to maintaincomplete class records of progress. Accordingly, this book is organised into 12 units, with eachunit containing mental routines, problematised situations, an investigation, games and assess-ment activities to match the big ideas in mathematics. The units are intended as starting pointsfor teachers to build on to suit the range of learners that they are working with.

The maths curricula are divided into five strands:

2 Number (including Money)2 Space2 Measurement2 Chance and Data2 Patterns and AlgebraWithin each of the strands there are a number of big ideas, or concepts, which focus on thesyllabus. This book is organised into units of work based on the current research into the devel-opmental sequence in which students generally acquire those concepts. Guiding the choice ofactivities has been the research that suggests that “children are capable of grasping key math-ematical concepts at an earlier age than previously thought”. Researchers also “urge teachersto help children think mathematically rather than merely memorise algorithms and hone theircomputational skills” (see Checkley, K. 1999).

Three-part lessonsBoth the Maths Planner and this book are much more than a facility to help teachers meet theirsyllabus requirements. The activities provide the type of resource needed to implement athree-part lesson process. In outline, a three-part lesson includes:

2 a mmeennttaall rroouuttiinnee to develop the student’s self-confidence and repertoire inmathematical thinking

2 a pprroobblleemmaattiisseedd ssiittuuaattiioonn where the student applies their own thinking to a situa-tion that they can engage with

2 a time for rreefflleeccttiioonn in which strategies and solutions are shared, compared andformalised, through which:1. we begin from where the students are2. we build on their understanding through the sharing of ideas3. formal mathematical methods are subordinate to methods that the students

invent and which work effectively for them4. students learn to value each other’s ideas, working as a community of learners

rather than as individuals.This approach to the teaching and learning of mathematics has its roots in research findings,and brings these findings to life through activities that have been found to fully engage stu-dents in mathematical discovery, discussion and understanding.

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Introduction

introduction

Part1

vii© 2006 Blake Education – Natural Maths Strategies – Book 1

Number sense, fluency with mathematical and strategic thinking and estimation skills are thefoundational building blocks of all later mathematics. Worksheets and mental arithmetic testsare anathema to risk-taking, reflective thinking and seeking out efficient strategies that willdevelop automaticity in number facts based on deep understanding.

Three years ago we began to question the relevance of paper and pencil, worksheets and “drilland kill” methods in the development of foundational basic number facts and understandings.We began by testing a few mental activities that involved the whole class simultaneously in funand relevant activities. As we did so, we observed that when students are engaged in mentalactivities, certain conditions need to be present for them to obtain maximum benefits. Theseare:

2 Concrete materials need to be provided for students to use as tools by students.2 Feedback is immediate, and involves sharing and discussing strategies as well as

showing equal respect for all students.2 Errors are seen as learning opportunities for all.2 Questions provide success for all as well as challenges for some.2 All students need to be engaged at their own level during the process.2 Students see themselves as a community of learners where everyone has a role to

play in the development of thinking and learning.It was with these criteria in mind that we began to explore the potential of “mental routines”as we have chosen to call them.

The purpose of mental routines is to develop useful strategies that will lead to mastery and asolid foundation in basic maths concepts. Mental strategies as far as possible should relate tothe methods that students develop intuitively and within their own culture. They should alsorelate as far as possible to the ways in which those strategies are applied in the real world.

This means that mathematics instruction must use contexts and pedagogies that allowstudents to use their own cultural, ethnic, and gender preferences and approaches.

Ladson-Billings, 1994

When we refer to the conditions that need to be present for the effective development of men-tal strategies, we see that this view is clearly reflected.

The mental routines make an excellent lesson starter as they arouse enthusiasm and encour-age the students to feel part of a learning community. They need last no more than tenminutes, but in that time every student has been engaged and challenged to take risks withtheir current understanding.

Part 1:Mental

Routines

Part 2:Problematised

Situations

Part 3:Reflection

Introduction

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introduction

Part1

Classroom managementFor each mental routine, provide a laminated mini-whiteboard of the task resource card foreach student and suitable writing materials. We call these “mini-whiteboards”, as felt high-lighter pens can be used to ring or mark ideas on the laminated cards.

The teacher begins by posing simple, closed questions that enable everyone to be successful.Soon, the questions change to a more open type, where more than one answer can be found.This enables students to begin to work at their own level. Finally, the process is flipped, and thestudents ask the questions, trying to determine a solution to the problem that the teacher hasposed.

Note: Each laminated mini-whiteboard can be reused for up to two weeks by changing thelevel of the question content as the students‘ vocabulary, skills and strategies improve.

Early Number Sense

5

YYOOGGHHUURRTT TTUUBBSS

UNIT


1

Target strategies2 Counting and counting on2 Subitisation2 Number recognition2 Estimation

Closed questionsFor this routine give each student the laminated margarinetub card and ask them to record their answers in their ownway onto the card. For example they might draw the coun-ters being dropped in or they might use numbers.

Listen as I drop the counters into the margarine tub. Howmany counters are in the tub?

There are four counters in the tub so write the number or draw four counters in the tub. If Idrop two more in, how many will there be then?

There are two counters in the tub. Listen and count as I drop some more into the tub. Howmany are in the tub now? When the students are good at this, start with a low even numberand drop the counters in by twos and see how they count or record then.

Open questionsListen as I shake the tub. Estimate from the sound when I rattle the tub, how many are in thetub?

There were three counters in the tub. I dropped a few more in. Listen as I rattle the tub.Estimate how many might be in the tub now?

I put counters into the tub two at a time. There are less than ten counters in my tub. How manycounters might I have in my tub?

Flip questionsI have six counters in my tub. Your job is to ask me questions to find out how many were in mytub to begin with.

Encourage the students to use counters or marks, and to find a way of keeping track of whatthey have found out from the questions. You may need to model this a few times. As the stu-dents get good at this, encourage them to ask trickier questions or increase the number ofcounters being used in the tub.

Task Card 3 (4 cards)

Mental

routines

178 © 2006 Blake Education – Natural Maths Strategies – Book 1

task card 3

Target strategiesgive a focus to thelesson.

Card masters areprovided for themini-whiteboards.

Closed questionsenable the teacherto see who has “got it” and whichstrategies are being used.

Open questionsshow the studentsthat there is oftenmore than onemethod and morethan one rightanswer to a question.

Flip questionsgive the studentsthe opportunity topractise thelanguage of maths.

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ix© 2006 Blake Education – Natural Maths Strategies – Book 1

Introduction

introduction

Part2

We use the term problematised situation to describe the type of activities that will allow stu-dents to engage with realistic (to them) situations as described in the research from theFreudenthal Institute. The situations provide the kinds of challenges that encourage studentsto construct their own ideas, strategies and mathematical understandings as they grapple withthem. The students, as described earlier, are developing their own mathematical tools, whichcan be formalised by the teachers when appropriate.

The problematised situations provided have multiple entry points and many methods of solu-tion. If the numbers are too hard, they can be reduced; if they are too easy, they can beincreased. Some students will draw pictures or act out the solution with objects, whereas others may use a more symbolic approach using numbers or tallies. Some will present solu-tions in an organised fashion, whereas others will be more muddled. It is the sharing andreflecting on the range of strategies that will broaden the possibilities for the students andallow them to enter into mathematical thinking from their very first experiences. The focus inthe primary classroom is shifting towards an emphasis on mathematical reasoning and prob-lem solving in a true sense. This new focus helps students learn to describe, compare anddiscuss their multiple approaches to solving real problems. In the classrooms where we havebeen working, we have noted that all the students engage well with the problems and haveshown increased interest in maths along with a really firm conceptual understanding. It is notdifficult to teach the students algorithms and procedures when they are ready for them andhave firm foundations in place.

The reflection, as described further on and included in each of the presented problematisedsituations, is central to this approach. Part of the preparation for the reflection is the processof observing the strategies that the students use and of listening to their explanations. Fromthe information gathered, it is possible to extend, consolidate and formalise learning duringthe reflection process. When the students are working, it is possible to gather informationabout what they do know and what they can do. For instance when a student is touch count-ing all the things drawn or set out, a simple question such as “Do you have to count them allor is there something else you can do?” may act as a prompt from which the students candemonstrate that they can count on or count by 2s. Annotating the work samples makes it pos-sible to record this information so that decisions about future planning can be made. A rangeof work samples will eventually give a clear picture of a student’s progress towards understand-ing the big ideas.

Part 1:Mental

Routines


Situations

Part 3:Reflection

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introduction

Part2

Classroom managementThe body of a three-part lesson is often taken up with a problematised situation in which theactivity is introduced with as little scaffolding as possible. The activity can be structured toenable the students to work independently, in small groups or collaboratively in larger groups,either as they wish or to suit the teacher’s assessment purposes.

The problematised situations require the students to work mathematically, to draw on theirown experiences and often to invent their own methods of recording and finding a solution.

As the students are working on the problem, the teacher has opportunities to observe meth-ods of recording, strategies used, problems encountered and fix-up strategies used. This isimportant preparation for the final reflection stage.

8

HHOOWW MMAANNYY TTHHIINNGGSS DDIIDD TTHHEE CCAATTEERRPPIILLLLAARREEAATT AALLTTOOGGEETTHHEERR??


PROBLEMATISED

SITUATIONS

ResourcesThe students‘ pages for The Very Hungry Caterpillar, counting materials, Kid Pix.

Activity guideUse five of the students’ pages (Monday to Friday) and ask them to find a way of working outhow many things the caterpillar ate altogether. Tell the students that they can use countingobjects if they want to but that you want them to draw their method of working it out so thatyou can follow their thinking. Kid Pix can also be used for this activity.

You may want to change this activity to suit the different entry levels of your students, so forsome students Monday to Wednesday will be enough, whereas for others the whole week canbe used. For some students showing their mental thinking is still important whichever combi-nation is chosen.

ReflectionUse three or four of the students’ methods as the basis ofthe reflection, so that the students see a range of differentstrategies. Do not be afraid to use an example with an errorbecause often the students learn from their mistakes and fixit up as they explain their methods. After hearing about thedifferent methods, ask the students:

“Which method of recording made it clearest to you howthe answer was found?”

“If you did this activity again tomorrow, would you use a different strategy or the same one?Why?”

How many things did the VeryHungry Caterpillar

eat altogether?

Early Number SenseUNIT 1

N1.1 Representing,

comparing and

ordering numbers to

20 using effective

counting strategies.

We give aspecification of theproblem that canbe displayed in aprominent placefor all to refer to.

A link to the mostrelevant big ideahelps to provide a focus for observation of theactivity.

After theinvestigation, eachsituation leads onto the final part ofthe lesson – thereflection stage.

The activity guidemakes suggestionsfor running theactivity.

The resourceslist is a suggestiononly – it needs tobe tailored to yourclass’s needs.

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Introduction

introduction

Part3

In the busy classroom the end of the lesson approaches all too quickly and as a result thereflection is often neglected. Yet the reflection is the most important part of the lesson. It isthe time when the students use mathematical language to explain what they have done andthey see that there are many strategies for solving problems, and that some are more effectivethan others. It is also the time when the teacher can formalise a particular idea, concept orprocess and scaffold the students to the next level. In fact there are some who go so far as tosay that if you didn’t do a reflection then the students will probably retain nothing. The devel-opment of a community of learners who share, listen and learn from each other is at the heartof this approach to mathematics. The reflection time sets up the mathematical culture of theclassroom with its tight-knit community of learners. It allows for mathematical mind journeysand adds to the excitement of learning mathematics.

The principles of rigorous reflection are:

2 the identification of a range of strategies to share and discuss2 the use of one or more errors to show the value of checking results and of

developing a fix-up strategy2 celebrating risk-taking, inventiveness, mathematical reasoning and learning from

mistakes2 building on, extending and presenting more formal methods of recording as

students demonstrate readiness for them2 positive, constructive feedback with a focus on feed forward – what you will do next

time.Through the dialogues and participation of all students in the class, the reflection stagebecomes crucial to the development of a community of learners, through which active involve-ment in learning mathematics is successfully fostered.

Part 1:Mental

Routines


Situations

Part 3:Reflection

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xii © 2006 Blake Education – Natural Maths Strategies – Book 1

introduction


OTHER CONSIDERATIONSConcrete materials

It may seem like a contradiction to say that hands-on materials could be used as part of men-tal routines but let’s explore this idea a little further. It is our belief that mental maths caninvolve the use of concrete materials and does not have to be totally abstract. For instancewhen students are first becoming familiar with doubles and near doubles (doubles plus orminus one), the use of materials such as Unifix cubes can become a mental routine. The stu-dents can be asked closed questions such as “Show me a near double that makes seven.” Thestudents may hold up a stack of 3 and a stack of 4 to show double 3 plus 1 or they may holdup two stacks of 4 and then snap off 1 Unifix cube to show double 4 take 1. The beauty of thisis that the students can actually see the cubes and show how they match the strategies. Thismeans that language and visual imagery are combined to chunk the information into a mean-ingful whole.

So there are two points to keep in mind as we discuss the uses of tools: First, meaning isnot inherent in the tool; students construct meaning for it. Second, meaning developed fortools and meaning developed with tools both result from actively using tools. Teachersdon’t need to provide long demonstrations before allowing students to use tools; teachersjust need to be aware that when students are using tools, they are working on two frontssimultaneously: what the tool means and how it can be used effectively to understandsomething else.

Heibert, J.et al. 1997

This use of invented tools is equally important when the students are working on the problema-tised situations described below. You will notice that we have provided hands-on resources foreach mental routine. These can be photo-enlarged, reduced or copied as appropriate. Welaminate ours because we know that they will be used time and time again, and we want stu-dents to interact with them.

They are used repeatedly and have uses outside those initially presented. The students enjoyusing water-soluble highlighter pens and a tissue to clean them. The use of darker colouredpens means that we can see what the students record and also watch their thinking as they findtheir answers. The students can hold their cards up for everyone to see and this means thatthey see a broad range of possible answers during the open questions. The resources are alsoused to develop adaptive reasoning during the flip questioning.

FeedbackFeedback should be immediate and useful, and should create a win-win situation for all ratherthan the competitive win-lose situation that so many students are familiar with. By this we meanthat there is no place for the over-learning of number facts or for the stressful learning and test-ing practices that often typify mental arithmetic.

The intention is to replace this with a situation where students share their solutions and strate-gies, where they consider the benefits of different approaches to something as simple as 4 plus7 and, in so doing, receive valuable feedback for making comments such as the following:

“I did my rainbow fact to 10 and then added 1 on.”“I took 1 from the 7and put it with the 4 and knew I had to double 5 plus 1.”“I did a turnaround and counted on from 7 until I got to 11.”“I just knew that fact.”“I thought it was 10 because I took 1 from 7 and gave it to the 4, but then I forgotthe extra 1 that was left over.”

Teacher feedback is barely necessary, is it? The last example shows that the student, while lis-tening to the others, realised that an error had been made and wondered how and where.

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Introduction

introduction

There was no embarrassment about the error, just a willingness to share with others a commonerror in such methods. The final questions from the teacher enhance this feedback by requir-ing the students to reflect on what was said as they answer questions such as:

“Which strategy did you think worked best in this example and why?”

“Which strategy would you like to explore a little further?”

“When we do another one, will you stick with the strategy you just used or try adifferent one?”

Through this process the students have been engaged in the feedback process and have beenasked to self-reflect and use that reflection as a planning tool ready for next time.

EngagementThe ten or so minutes set aside for mental routines are fast and pacey. They may involve con-crete materials, number cards, 100 squares, dice, bottle tops, . . . The students engage withthe activities because they are different to the rest of the lesson. When we first began explor-ing the mental activities that we suggest here, we had no idea how much fun and of coursehow much learning would flow from them. We soon realised that we didn’t need to make upa new mental activity every day because the nature of the tasks and the students’ interest inthem meant that they could be used and easily adapted over several days, hence the term“mental routines”. We now use the routines for several consecutive days, all the time watchingto see the level of engagement and, of course, we switch to a new routine if we think the inter-est is dwindling.

As we introduce each routine, we use the meta-language of the strategies or process that gowith it. At first it was our intention to simply immerse the students in the meta-language butthey were so captivated by words, such as “subitise” and “unitise”, that they soon wanted touse them too.

Watching the students engage with the activities has been rewarding for us and for them too.When the students are having fun and are engaged, they seem to be hungry for more. Wehave seen even the switched-off learners re-engage through the mental routines.

Learning opportunitiesIt appears that there is little value in participating in mental activities which are too easy orwhich are already well developed. Activities, then, need to be just on the edge of the students’comfort zones, scaffolding them to the next level. If this is the case then obviously errors incomputation are going to occur from time to time. We found at first when we worked in thisway, students would be derisive and snicker at errors. We also found that some students wouldnot have a go for fear of failure. It was interesting however to see how quickly this mindsetturned around. The students seemed to embrace the idea of using errors as learning opportu-nities and were often heard saying, “Oh good, a learning opportunity.” Very often as a child isexplaining their strategy, they notice their own errors and are keen to fix them up on the spot.Other times, though, the error is not noticed. For instance, recently a 6-year-old responding toa reading of The Very Hungry Caterpillar showed how she had drawn two rows of foods thatthe caterpillar had eaten during the week. She counted in 2s but only touched one thing at atime. She was unperturbed by her error; she had counted by 1s earlier and written the num-ber 26 on her paper. One of the other students was very impressed by her ability to count sofast and so far by 2s, but another asked her why she only touched one thing at a time. Shestood perplexed looking at her page and asked the other student to come up and show whatshe meant. She laughed out loud when she saw what she should have done and proceededto repeat her counting correctly this time.

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Introduction

xiv © 2006 Blake Education – Natural Maths Strategies – Book 1

introduction

Introduction

We recall when children have wanted to share mistakes; after presenting their solutions,they explain their errors. The children also become very supportive of one another andunderstand that errors are a natural part of doing mathematics. Errors often can lead to newunderstandings about the concept.

Trafton, P. & Thiessen, D. 1999

Community of learnersTo gain the most from these activities, the students need to become a community of learners.They need to really listen to the ideas of others, give positive feedback, ask questions, makesuggestions and comparisons, and finally to evaluate the strategies presented by others. Theyneed to feel safe to take a risk, present their ideas and to comment on the ideas of others.They need to learn to justify their viewpoints and stick with them. For instance, if after hearinghow near doubles can be used for an addition a child still prefers a count-on strategy then theyshould be able to explain why they prefer it. And at the end of the day if the response is,“Because I know it always gives me a correct answer”, then that justification has to be seen byall as valid for that student at that time, and as such should be respected.

Learning to be a member of a mathematical community means taking ownership of thegoals and accepting the norms of social interaction. Why is it important that classroomsbecome mathematical communities and that all students participate? Because such com-munities provide rich environments for developing deep understandings of mathematics.

Heibert, J. et al. 1997

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Introduction

introduction

The references given below are the key sources for our explanation of the developmentalsequence associated with topics at this level.

1. Checkley, K. (1999) Math in the early grades: Laying a foundation for later learning.Curriculum Magazine.

2. Copley, J.V. (2000) The Young Student and Mathematics. National Association for theEducation of Young Children, Washington DC.

3. Heibert, J. et al. (1997) Making Sense: Teaching and Learning Mathematics withUnderstanding. Heinemann, NH.

4. Hunt, J. (1999) “Maths in the Early Grades”, Curriculum Journal. ASCD.

5. Kamii, C., (1989) Young Children Continue to Reinvent Arithmetic: 2nd grade.Implications of Piaget’s Theory. Teacher’s College Press, N.Y.

6. Ladson-Billings, G. (1994) The Dreamkeepers: Successful Teachers of AfricanAmerican Students. Jossey-Bass, San Francisco.

7. National Research Council (2001) Adding It Up: Helping Children Learn Mathematics.The National Academies Press, Washington DC.

8. Russell, S.J. (2001) “Changing the Elementary Mathematics Curriculum: Obstaclesand Challenges”, in D. Zhang, T. Sawada & J.P. Becker (eds) Proceedings of theChina–Japan–U.S. Seminar on Mathematics Education.

9. Trafton, P.R. & Thiessen, D. (1999) Learning Through Problems: Number Sense andComputational Strategies. Heinemann, NH.

10. van Hiele, P.M. (1999) “Developing geometric thinking through activities that beginwith play”, Teaching Children Mathematics, 5 (6), pp. 310–16.

REFERENCES

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Using Patterns

Unit 11Unit 11Focus

Patterns occur everywhere in natural phe-nomena and in mathematics. They form the basis of

much of our everyday lives where we organise ourselvesaccording to naturally occurring patterns in time and natural

cycles such as the seasons and events for celebration. Patterns areused in music and dance, art and design, games and in everyday uses of

mathematics. The ability to spot patterns, create patterns and make predic-tions based on them is fundamental to mathematical and adaptive reasoning.

Change in patterns also forms part of this mathematical reasoning. For instanceknowing that you can count forwards to find a total but that you can count back in the

same way and arrive at the original starting point is a central part of understanding rela-tionships in and between numbers. In a similar way knowing that different pathways canbegin and finish at the same place is important to spatial knowledge. The activities in thisunit are intended to allow students to explore and engage with patterns and change inways that will build foundations for mathematical reasoning and visual memory.

ContextThe context chosen for this unit is that of playing Pattern detectives. Most studentsknow that detectives observe situations and look for clues. They know that detec-

tives then use that information to sort out muddles or puzzling situations. Aspattern detectives they will do the same. The students will observe places

where patterns are found, test the patterns and work on pattern puzzles.As detectives they will report on and present their findings.

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123© 2006 Blake Education – Natural Maths Strategies – Book 1

Developmental sequenceIn the early stages of working with patterns, students begin to:

1. identify the elements within a simple pattern2. identify the elements within more complex patterns

3. create simple patterns based on one property or characteristic (red orblue, square or round)

4. create more complex patterns using two or more properties or characteristics(square and red)

5. continue patterns begun by others6. translate patterns from one form to another (seeing the relationship between an

AB AB AB pattern made with shapes and the same pattern made with claps andclicks)

7. see the relationships between patterns in terms of number, pattern sequence,position, and so on

8. predict the effects of changes to a pattern.


Using Patterns

124

WWHHAATT’’SS IINN AA GGRRIIDD??

UNIT


11

Target strategies2 Language of position (next to, beside, above, below,

after, before)2 Language of direction (between, around, over, under,

up, down, left, right)2 Language of movement (forwards, backwards, sideways)2 Exploring pathways

Closed questionsThere is someone in the space above me. Who am I?

What number is in the space between the piglet and thesheep?

I was on Start and then I moved three spaces up the grid andfour spaces to the right. Who am I?

Open questionsI am in the middle row. Who might I be?

There is someone three spaces to my left/right. Who might I be?

There are one/two/three animals in the spaces around me. Who might I be?

I have made a path from Start to Finish just using empty spaces. What numbers did I meet?

Flip questionsI am thinking of an animal. You can ask me questions about position, direction or movementto find out who I am. What questions could you ask?

You may need to model this using simple position questions and showing the students how toeliminate animals from their boards. Each time you play, you will be able to introduce morecomplex language and pathways.

Mental

routines

Task Card 20

21 22 23 24 25

16 17 18 19 20

11 12 13 14 15

6 7 8 9 10

1 2 3 4 5

21 22 23 24 25

20 19 18 17 16

11 12 13 14 15

10 9 8 7 6

1 2 3 4 5

Start


task card 20

Where on a grid?


Using Patterns

125

PPAATTTTEERRNNSS TTOO 2200

UNIT


11

Target strategies2 Skip counting to 202 Exploring equal-sized jumps2 Looking for relationships in patterns

Closed questionsI began on Start and made three jumps of 3. What numberdid I finish on?

I started on 2 and made four jumps of 2. What number did Ifinish on?

I began on Start and finished on the number 9. I made threejumps. What size were my jumps?

Open questionsI began on Start and landed on 10. What size jumps might I have made?

I made three equal-sized jumps. Where might I have landed?

I jumped by 4s. Where might I have landed?

Flip questionsWe are going to play a game called Guess my Jump Pattern. You need to ask questions to findout where I started my pattern from and where I finished. Also ask about the size of the jumpand the number of jumps made. You may need to model the questions and the eliminationprocess for the students first.

Make sure the students have had experiences building up counting patterns to 20 by placingcoloured counters on the grids before introducing the next version of the flip.

I made equal-sized jumps to get to 20. You can ask me questions to find out how many jumpsI made and how big each jump was. The twist is that you can only ask me questions about thenumbers that I landed on.

Task Card 21 (5 number lines)

Mental

routines

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task card 21


126

PPAATTTTEERRNNSS IINN PPAATTHHSS


PROBLEMATISED

SITUATIONS

ResourcesPhotocopy the tiles and pathways from both taskcards onto different coloured card and cut out thesquare and rectangular tiles; glue.

Activity guideThis activity requires the students to investigate pat-terns that operate in more than one direction. Thepattern must repeat across the path and along it. Thiswill be a new challenge for many of the students.Although they will be covering the area, some tilesmay overlap the edges as would happen in real lifeand will need to be cut off or crossed off. This willallow the opportunity to talk about part and wholerelationships and the use of fractions. Some studentswill naturally include these ideas in their presentations,for instance by reporting that their path took 12 tiles and 5 part or half tiles. Show the studentshow their paths can be made permanent by gluing the tiles into position.

As the students make their patterns and count the tiles, select a variety for sharing. Look forsome complex patterns as well as some interesting methods of working out how many wereused.

ReflectionBased on the work selected, ask questions such as the fol-lowing:

“Which patterns are similar in some way?”

“Which pattern is trickiest to spot and why?”

“How did you work out how many of each tile was used?”

“Did all the pathways use the same number of tiles? Why/Why not?”

The tiler has two types of tiles, rectangles and

squares. He wants to tile a path like this pathway

but he can’t decide which pattern to do or how

many of each tile he will need. What would you

suggest?

Using Patterns Using PatternsUNIT 11

PA1.1 Creating and

describing patterns

based on simple

rules.


task card 23

Pathways

Task Cards 22, 23


task card 22

Pathway tiles



CCOOUUNNTTIINNGG PPAATTTTEERRNNSS

Using Patterns UNIT11

PROBLEMATISED

SITUATIONS

ResourcesLaminated copy of the number line 1 to 20 (card masters), some coloured pens or counters, acalculator for each group.

Activity guideFor this activity the students will be investigating the effects of different number patterns.Begin counting in 2s, 3s . . . and then encourage those who need more challenge to investi-gate what happens when they try mixed patterns such as count-on 4, count-back 1 type ofpatterns. Explain to the students that they need to find a away of keeping a record of whichnumbers are landed on in each pattern and that they can use coloured counters and/or pensto record that. Show the students the key sequence:

for investigating the counting patterns and suggest that one person in the group take respon-sibility for creating and or checking the pattern sequences. As the students are working, askthem to predict and then find out what happens when they count back in the same pattern asthey used for counting forward, starting from the last counted number of course. Observe thestudents as they work and select different approaches and patterns for the reflection.

ReflectionBegin the reflection by asking the students which numberswere landed on most frequently and which were landed onleast frequently. Ask them to offer reasons as to why thismight be the case.

Select two or three different methods of recording. Ask thestudents which methods showed the different patterns mostclearly as well as the number of times each number was landed on and why.

Which numbers on the number line are landed on

most often when you investigate counting

patterns? Which numbers do you land on least often?

2 + = =

PA1.1 Creating and

describing patterns

based on simple

rules.


128

GGRRIIDD PPAATTTTEERRNNSS


PROBLEMATISED

SITUATIONS

ResourcesLaminated copy of the 12 x 16 grid, counters, coloured pens.

Activity guideFor this activity the students will investigate different pathwaypatterns, such as “forward 2, right 3”, as they try to make acontinuous path from a selected start to a selected finish posi-tion. Allow time for the students to explore pattern movesbefore presenting the problem to them.

When you introduce the problem, demonstrate a pathwaypattern to them and then ask them to predict what would hap-pen if you backtracked by following the pattern steps inreverse. Tell the students that when they have found a path-way pattern that joins the two positions, they can write the pathway pattern down, mark thestart point but not the finish point onto their grid, and then ask a partner to follow theirpathway.

As an extra challenge the students can try marking boxes and try to find a pattern that will passthrough their marked boxes and still meet the finishing point.

Reflection Select a variety of pattern pathways to use for the reflectionincluding one that does not join start to finish correctly, andask the students to explain:

2 the repeating pattern2 which patterns are similar in some way and how two

patterns differ2 what happens when a pattern is reversed2 what problems they had when they were trying different pathways and what

strategies they used to fix them up.Then show the students the pathway that doesn’t work and ask for some fix-up strategies.

Which different pathway patterns on this grid will

take you from a starting point to a finishing

point, both chosen by you?

Which pathway was the longest/shortest?

Using Patterns Using PatternsUNIT 11

PA1.1 Creating and

describing patterns

based on simple

rules.

Task Card 24

Finish

Start


task card 24

12 x 16 grid



CCLLOOSSEEDD PPAATTHHWWAAYYSS


PROBLEMATISED

SITUATIONS

ResourcesLaminated copy of the 12 x 16 grid, coloured pens.

Activity guideThis activity asks the students to investigate pathways thatmake closed shapes. As the students investigate pathways,encourage them to consider the possibility of them ever clos-ing up. While they may not be able to make these predictionsat first, encouraging them to think and plan ahead willimprove their visualisation. Ask the students to record theones they do find, giving each one a different colour.Challenge the students to investigate the types of moves thatwill result in larger/smaller shapes than the ones alreadyfound.

ReflectionAs the students report and show their findings, remind themthat they are pattern detectives and that they should havedrawn some conclusions from their observations. Forinstance, ask them what they have concluded about:

2 paths that will never close and why2 how to make a really wiggly path that closes2 how many different types of moves can be included in a pattern that still closes2 how to make the smallest possible closed shape with a path2 how to make the longest closed path.

What different pattern pathways on the grid

make a closed shape?

What is the largest/smallest closed pathway that

you can find?

PA1.1 Creating and

describing patterns

based on simple

rules.

Task Card 24

Finish

Start


task card 24

12 x 16 grid


Using PatternsUsing Patterns

130

UNIT


11

Investigation

PPAAPPEERR FFOOLLDDIINNGG

ResourcesA5 paper works best for this investigation.

The investigationDemonstrate the process of counting as you fold a piece of paper three times, explaining thatfolds can be made in any direction across the paper. Open out your paper and demonstratehow to draw in pencil along the fold lines to make the shapes clear. Begin counting how manyof each type of shape you have and construct a simple table or graph to show comparisonsbetween the numbers of each shape.

Allow time for the students to fold their paper and colour the shapes before they find their ownway of recording how many of each shape.

ReflectionInvite one or two students to draw one or two of their shapes on the board, making sure thatthe number of sides in each one is written underneath. Ask students if any shapes have beenmissed and, if so, which ones.

Students can compare how many shapes they made. They may be able to give reasons as towhy there are different numbers of shapes as well as a range of different shapes. The studentsmight like to call out how many of a particular shape they found and, with the help of a calcu-lator, can create a class record to find out which was the most common shape.

Which shapes can you make when you fold a

piece of paper three or even four times? Colour

shapes that have the same number of sides in

the same colour, even if they are smaller or

larger.Find a way of showing how many of each shape

you have made. How many shapes altogether?


Using Patterns

131

PPAATTTTEERRNN FFUUNN

UNIT


11

games

A game for 2 or more players

Resources Pattern and action dice made from the laminated task card.

How to play (competitively) Players take turns to throw the pattern die, which tells themthe pattern sequence. The action die is then thrown two orthree times to decide the pattern actions. The player then hasto create the pattern sequence in actions, and repeat it fivetimes without an error or without laughing so much that theycannot continue. The other players are pattern detectiveswho watch to check that the pattern actions and its repeatsare played correctly.

A correct completed pattern scores one point. At the end ofthree games, the player with the most points wins.

How to play (cooperatively)Make this a cooperative game by stating that a team only wins if all its members have fullmarks, in which case the pattern detectives call out the pattern to ensure that it is correctly performed.

Variations The students can make their own dice to match their own ideas and interests.

Task Card 25

Jump

Skip

Clap

Click

Wiggle

Shake

AABB

ABAB

AABCC

ABBA

ABC

ABBC


task card 25

Pattern and action dice


Using Patterns

132

NNUUMMBBEERRSS OONN PPEEGGSS


ASSESSMENT

ACTIVITIES

Using PatternsUNIT 11

Resources Pencils, scissors, numbered counters, glue stick.

Prior experiencesThe students will be ready for this activity if they have hadexperiences with:

2 exploring counting patterns2 sequencing and ordering numbers 1 to 102 identifying missing numbers in sequences2 continuing sequences already begun

Observer’s guideThe activity presented on the sheet requires the students to identify each of the countingsequences presented and to complete them using the numbers at the bottom of the sheet.There are different ways in which this can be done, for example:

2 The students can cut out the numbers at the bottom of the page and glue them inposition.

2 The students can place numbered counters on the numbers at the bottom of thepage and then move the counters into position on the number sequences.

2 The students can cross off the numbers at the bottom of the page as they draw inthe numbers needed to complete the sequences.

By providing pencils, scissors and numbered counters, the students make their own choice asto the approach they want to take, and in doing so will demonstrate their level ofdevelopment.

Activity Sheet 23

Numbers on pegs

These children are putting numbers on pegs to make numbersequences.

Cut out these numbers and put them on the pegs to completethe sequences.

NameActivity Sheet 23

170 © 2006 Blake Education Natural Maths Strategies – Book 1

PA1.1 Creating and

describing patterns

based on simple

rules.



PPYYTTHHOONN PPAATTTTEERRNNSS


ASSESSMENT

ACTIVITIES

Prior experiencesThe students will be ready for this page if they have hadexperiences with:

2 creating patterns and continuing patterns2 labelling patterns with numbers2 comparing lengths

Observer’s guideThe activity presented on the sheet requires the students toidentify the pattern unit presented and then to write thenumber labels for the pattern. The process is then reversedwith the students making a pattern to match the given num-ber labels. Finally the students can bring their own level ofcomplexity to the task as they create and label a pattern oftheir own.

This activity encourages the students to:

2 move fluently between pictures and symbols2 understand that a pattern can be shown in a number of ways.When creating their own patterns, some students will first use the cards to make the patternand then label the pattern, whereas, others will write the number labels first and then createthe pattern. Both approaches are perfectly acceptable. The second approach however mightindicate a higher degree of confidence with numbers and with the development of abstractthinking.

Activity Sheet 24

Python patterns


Name Activity Sheet 24

Jim and Tina made this python

pattern.

Copy their python and make it longer.

Write the number pattern for your python.

Make a python pattern that matches this number

Make your own python pattern and

write its number pattern.

PA1.1 Creating and

describing patterns

based on simple

rules.


Numbers on pegs

These children are putting numbers on pegs to make numbersequences.

Cut out these numbers and put them on the pegs to completethe sequences.

NameActivity Sheet 23


7343_Natural Maths BLMs.qxd 13/6/06 10:26 AM Page 170

Python patterns

Name Activity Sheet 24


Jim and Tina made this pythonpattern.

Copy their python and make it longer. Write the number pattern for your python.

Make a python pattern that matches this number.

Make your own python pattern and write its number pattern.


task card 20


task card 21task card 20

Where on a grid?


task card 21


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task card 2120


task card 22



Pathway tiles


task card 23


task card 23

Pathways

22


task card 24



12 x 16 grid


task card 25


Jump

Skip

Clap

Click

Wiggle

Shake

AABB

ABAB

AABCC

ABBA

ABC

ABBC

task card 25

Pattern and action dice

24


Based on world’s best practice! This series provides the core knowledge and understanding of the“big ideas” or concepts students require to become confident and enthusiastic maths users. Thisbook is organised into twelve units of work based on the current research into the developmentalsequence in which students generally acquire those concepts. Each unit is divided into fivesections:

Mental routines – 10-minute lesson starters with suggested closed and openquestions designed to engage students and arouse their enthusiasm

Problematised situations – challenges that encourage students to workmathematically with open-ended “real-life” situations and construct their own ideas.These lessons include a reflection session where mathematical language is used todescribe successful strategies and more formal methods are introduced anddemonstrated.

Games – fun activities designed to reinforce the strategies developed in each unit.

Investigations – open-ended investigations to encourage students to test and extendtheir skills.

Assessment activities – consolidation activities that students should readilyaccomplish at the end of each unit.

The series encourages the use of readily available concrete materials and is supported by over 50 photocopiable activity sheets and task cards. The CD-ROM included with this book is designedto help teachers to plan and personalise their maths program and to record individual student’sprogress.

The Natural Maths Strategies series is a complete school program, which also encourages the useof supplemental resources to ensure a variety of maths teaching and learning experiences.

CUTNMS Book 1 Cover.qxd 6/05/2006 5:09 PM Page 1

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Mental routines Problematised situations Games ... · unit containing mental routines, problematised situations, ... activities has been the research that suggests ... certain conditions