Cambridge Primary Mathematics Teacher's Resource 4 by Cambridge International Education

CAMBRIDGE PRIMARY

Mathematics TeacherĂ s Resource

Emma Low

CD-ROM Terms and conditions of use University Printing House, Cambridge cb2 8bs, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107692947 © Cambridge University Press 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in Poland by Opolgraf A catalogue record for this publication is available from the British Library isbn 978-1-107-69294-7 Paperback Cover artwork: Bill Bolton Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. notice to teachers in the uk It is illegal to reproduce any part of his work in material form (including photocopying and electronic storage) except under the following circumstances: (i) where you are abiding by a licence granted to your school or institution by the Copyright Licensing Agency; (ii) where no such licence exists, or where you wish to exceed the terms of a licence, and you have gained the written permission of Cambridge University Press; (iii) where you are allowed to reproduce without permission under the provisions of Chapter 3 of the Copyright, Designs and Patents Act 1988, which covers, for example, the reproduction for the purposes of setting examination questions. notice to teachers The photocopy masters in this publication may be photocopied or distributed [electronically] free of charge for classroom use within the school or institution that purchased the publication. Worksheets and copies of them remain in the copyright of Cambridge University Press, and such copies may not be distributed or used in any way outside the purchasing institution.

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Contents The ethos of the Cambridge Maths project Introduction Teaching approaches Talking mathematics Resources, including games

1C: Handling data and problem solving 7 Graphs, tables and charts (1) 7.1 Tally charts and bar charts 7.2 Pictograms 8 Carroll and Venn diagrams 8.1 Carroll diagrams 8.2 Venn diagrams

Term 1 1A: Number and problem solving 1 Numbers and the number system 1.1 Reading, writing and partitioning numbers 1.2 Ordering, comparing and rounding four-digit numbers 1.3 Multiplying and dividing by 10 and 100 2 Addition and subtraction (1) 2.1 Addition (1) 2.2 Subtraction (1) 2.3 Partitioning to add and subtract 3 Multiplication and division (1) 3.1 Learning and using multiplication facts 3.2 Using doubles 3.3 Multiplying a two-digit number by a single digit 1B: Measure and problem solving 4 Weight 4.1 Measuring weight 5 Time (1) 5.1 Telling the time (1) 5.2 Using timetables 6 Area and perimeter (1) 6.1 Area (1) 6.2 Perimeter (1)

1 2 4 8 15 16 20 24 31 32 36 38 41 42 47 48 52 57 58 62

69 70 74 79 80 82

Term 2 2A: Number and problem solving 9 The number system and properties of number 9.1 Decimal numbers in context 9.2 Positive and negative numbers 9.3 Odd and even numbers 10 Addition and subtraction (2) 10.1 Adding and subtracting near multiples of 10 10.2 Choosing the most efficient subtraction strategy 11 Multiplication and division (2) 11.1 More multiplication 11.2 Dividing two-digit numbers by single-digit numbers 2B: Geometry and problem solving 12 Angles, position and direction 12.1 Angles and turning 12.2 Position and direction 13 Symmetry 13.1 Shapes and symmetry 14 2D and 3D shapes 14.1 2D shapes 14.2 3D shapes

87 88 92 94 99 100 104 111 112 114 117 118 120 123 124 127 128 130

iii

2C: Measure and problem solving 15 Length 15.1 Meauring length 16 Time (2) 16.1 Telling the time (2) 16.2 Using calendars 17 Area and perimeter (2) 17.1 Area (2) 17.2 Perimeter (2)

135 136 143 144 148 155 156 160

Term 3 3A: Number and problem solving 18 Special numbers 18.1 Special numbers 19 Fractions and divisions 19.1 Exporing fractions 19.2 Fractions, decimals and mixed numbers 19.3 Fractions and division 20 Ratio and proportion 20.1 Ratio and proportion 3B: Measure and problem solving 21 Capacity 21.1 Measuring capacity 22 Time (3) 22.1 Measuring time 22.2 Calculating time 23 Area and perimeter (3) 23.1 Area and perimeter

163 164 167 168 172 174 181 182 187 188 193 194 198 209 210

3C: Handling data and problem solving 24 Graphs, tables and charts (2) 24.1 Tables and bar charts 24.2 Frequency tables and tree diagrams 25 Venn and Carroll diagrams 25.1 Carroll diagrams (2) 25.2 Venn diagrams (2)

215 216 219 227 228 230

The Ethos of the Cambridge Primary Maths project Cambridge Primary Maths is an innovative combination of curriculum and resources designed to support teachers and learners to succeed in primary mathematics through bestpractice international maths teaching and a problem-solving approach.

To get involved visit www.cie.org.uk/cambridgeprimarymaths 2 1

Cambridge Primary Maths brings together the world-class Cambridge Primary mathematics curriculum from Cambridge International Examinations, high-quality publishing from Cambridge University Press and expertise in engaging online eFment materials for the mathematics curriculum from NRICH. Cambridge Primary Maths offers teachers an online tool that maps resources and links to materials offered through the primary mathematics curriculum, NRICH and Cambridge Primary Mathematics textbooks and e-books. These resources include engaging online activities, best-practice guidance and examples of Cambridge Primary Maths in action. The Cambridge curriculum is dedicated to helping schools develop learners who are confident, responsible, reflective, innovative and engaged. It is designed to give learners the skills to problem solve effectively, apply mathematical knowledge and develop a holistic understanding of the subject. The Cambridge University Press series of Teacher’s Resource printed books and CD-ROMs provide best-in-class support for this problem-solving approach, based on pedagogical practice found in successful schools across the world. The engaging NRICH online resources help develop mathematical thinking and problem-solving skills. The benefits of being part of Cambridge Primary Maths are: ∑ the opportunity to explore a maths curriculum founded on the values of the University of Cambridge and best practice in schools ∑ access to an innovative package of online and print resources that can help bring the Cambridge Primary mathematics curriculum to life in the classroom.

1 You can explore the available resources on the Cambridge Primary Maths website by curriculum framework, scheme of work, or teacher resources. In this example, the ‘Teacher resources’ tab has been selected. 2 The drop-down menu allows selection of resources by Stage. 3 Following selection of the ‘Teacher resource’ and ‘Stage 1’, the chapters in the Cambridge University Press textbook ‘Teacher’s resource 1’ are listed. 4 Clicking on a chapter (‘2 Playing with 10’ in this example) reveals the list of curriculum framework objectives covered in that chapter. Clicking on a given objective (1Nc1 in this example) highlights the most relevant NRICH activity for that objective. 5 A list of relevant NRICH activities for the selected chapter are revealed. Clicking on a given NRICH activity will highlight the objectives that it covers. You can launch the NRICH activity from here.

The Cambridge Primary Maths project provides a complete support package for teachers. The Teacher's Resource is a standalone teaching textbook that can be used independently or together with Cambridge Primary Maths website. The free to access website maps the activities and games in the Teacher's Resource to the Cambridge Primary curriculum. It also highlights relevant online activities designed by the NRICH project team based at the University of Cambridge. The additional material that the Cambridge Primary Maths project provides can be accessed in the following ways: As a Cambridge Centre: If you are a registered Cambridge Centre, you get free access to all the available material by logging in using your existing Cambridge International Examinations log in details. Register as a visitor: If you are not a registered Cambridge Centre you can register to the site as a visitor, where you will be free to download a limited set of resources and online activities that can be searched by topic and learning objective. As an unregistered visitor: You are given free access an introductory video and some sample resources, and are able to read all about the scheme.

Introduction The Cambridge Primary Maths series of resources covers the entire content of the Cambridge Primary Mathematics curriculum framework from Cambridge International Examinations. The resources have been written based on a suggested teaching year of three, ten week terms. This can be amended to suit the number of weeks available in your school year.

of misconception. A section called ‘More activities’ provides you with suggestions for supplementary or extension activities.

The Cambridge Primary Mathematics framework provides a comprehensive set of learning objectives for mathematics. These objectives deal with what learners should know and be able to do. The framework is presented in five strands: the four content strands of Number (including mental strategies), Geometry, Measures and Handling Data are all underpinned by the fifth strand, Problem Solving. Problem solving is integrated throughout the four content strands. Whilst it is important to be able to identify the progression of objectives through the curriculum, it is also essential to bring together the different strands into a logical whole.

The Teacher’s Resource can be used on its own to completely cover the course. (The Learner’s Book and Games Book should not be used without the associated teacher resource, as they are not sufficient on their own to cover all the objectives.)

This series of printed books and CD-ROMs published by Cambridge University Press is arranged to ensure that the curriculum is covered whilst allowing teachers flexibility in approach. The Scheme of Work for Stage 4 has been fully covered and follows in the same ‘Unit’ order as presented by Cambridge International Examinations (1A–C, 2A–C and then 3A–C) but the order of objective coverage may vary depending on a logical pedagogy and teaching approach.

The accompanying CD-ROM contains: a Word version of the entire printed book. This has been supplied so that you can copy and paste relevant chunks of the text into your own lesson plans if you do not want to use our book directly. You will be able to edit and print the Word files as required but different versions of Word used on different PCs and MACs will render the content slightly differently so you might have some formatting issues. Questioning – This document outlines some of the different types of question techniques for mathematics and how best to use them, providing support for teachers. Letters for parents – a template letter is supplied along with a mapping grid to help you to write a letter per Unit of material in order to inform parents what work their child is doing, and what they can do to support their child at home. Photocopy masters – resources are supplied as PDFs, and as Word files so that you can edit them as required.

The components of the printed series are as follows: ∑ Teacher’s Resource (printed book and CD-ROM) This resource covers all the objectives of the Cambridge framework through lessons referred to as ‘Core activities’. As a ‘lesson’ is a subjective term (taking more or less time depending on the school and the learners) we prefer to use the terms ‘Core activity’ and ‘session’ to reinforce that there is some flexibility. Each Core activity contains the instructions for you to lead the activity and cover the objectives, as well as providing expected outcomes, suggested dialogue for discussion, and likely areas ∑

Learner’s Book (printed book) This resource is supplementary to the course. As the ethos of the Cambridge Maths Project is to avoid rote learning and drill practice, there are no accompanying write-in workbooks. The Learner’s Book instead combines consolidation and support for the learner with investigations that allow freedom of thought, and questions that encourage the learner to apply their knowledge rather than just remembering a technique. The investigations and questions are written Introduction

vii

to assess the learner’s understanding of the learning outcomes of the Core activity. Learners can write down their answers to investigations and questions in an exercise book in order to inform assessment. The overall approach of the Teacher’s Resource accompanied by the Learner’s Book allows a simple way for you to assess how well a learner understands a topic, whilst also encouraging discussion, problemsolving and investigation skills. At Stage 4, each Learner’s Book page is designed to help learners to consolidate and apply knowledge. Each section associated with a Core activity starts with an introductory investigation called “Let’s investigate”, which is an open-ended question to get the learners thinking and investigating. These are often ‘low threshold, high ceiling’ so that learners can approach the question at many levels. This is followed by a series of questions and/or activities to develop problemsolving skills and support learning through discovery and discussion. New vocabulary is explained, and where possible this is done using illustrations as well as text in order to help visual learners and those with lower literacy levels. Hints and tips provide direct support throughout. Ideally, the session should be taught using the appropriate Core activity in the Teacher’s Resource with the Learner’s Book being used at the end of the session, or set as homework, to consolidate learning. There is generally a double page in the Learner’s Book for each associated Core activity in the Teacher’s Resource for Stage 4. The Teacher’s Resource will refer to the Learner’s Book page by title and page number, and the title of the Core activity will be at the bottom of the Learner’s Book page. Please note that the Learner’s Book does not cover all of the Cambridge objectives on its own; it is for supplementary use only. ∑

viii

Games Book (printed book and CD-ROM) This resource is complete in its own right as a source of engaging, informative maths games. It is also a supplementary resource to the

Introduction

series. It can be used alongside the Teacher’s Resource as a source of additional activities to support learners that need extra reinforcement, or to give to advanced learners as extension. Each game comes with a ‘Maths focus’ to highlight the intended learning/reinforcement outcome of the game, so that the book can be used independently of any other resource. For those who are using it as part of this series, relevant games are referred to by title and page number in the ‘More activities’ section of the Teacher’s Resource. The accompanying CD-ROM contains nets to make required resources; it also contains a mapping document that maps the games to the other resources in the series for those who require it. Please note that the Games Book does not cover all of the Cambridge objectives on its own; it is for supplementary use only.

∑

Each chapter in the Teacher’s Resource includes A Quick reference section to list the title of each of the Core activities contained within the chapter. It provides an outline of the learning outcome(s) of each Core activity. (See page vii and later in this list, for a reminder of what is meant by a Core activity.) A list of the Objectives from the Cambridge Primary Mathematics curriculum framework that are covered across the chapter as a whole. Please note that this means that not all of the listed objectives will be covered in each of the chapter’s Core activities; they are covered when the chapter is taken as a whole. The objectives are referenced using subheadings from the framework, for example ‘1A: Calculation (Mental strategies)’ and the code from the Scheme of Work, for example, ‘2Nc3’. Please be aware that the content of an objective is often split across different Core activities and/or different chapters for a logical progression of learning and development. Please be assured that provided you eventually cover all of the Core activities across the whole Teacher’s Resource, you will have covered all of the objectives in full. It should be clear from the nature of a Core activity when parts of an objective have not been fully covered. For example, a chapter on length

∑

will list ‘Measure’ objectives that also include weight, such as ‘1MI1’ (Compare lengths and weights by direct comparison…) but the weight aspect of the objective will not be covered in a chapter on length(!); that part of the objective will be covered in a chapter on weight. Or a chapter focussing on understanding teen numbers as ‘ten and some more’ might cover the action ‘recite numbers in order’ but only up to 20 and therefore only partially cover objective ‘1Nn1’ (Recite numbers in order … from 1 to 100…)). But please be reassured that, by the end of the Teacher’s Resource, all of objectives 1MI1 and 1Nn1 will have been covered in full; as will all objectives. The Summary bulleted list at the end of each Core activity lists the learning outcome of the activity and can add some clarity of coverage, if required. A list of key Prior learning topics is provided to ensure learners are ready to move on to the chapter, and to remind teachers of the need to build on previous learning. Important and/or new Vocabulary for the chapter as a whole is listed. Within the Core activity itself, relevant vocabulary will be repeated along with a helpful description to support teaching of new words. The Core activities (within each chapter) collectively provide a comprehensive teaching programme for the whole stage. Each Core activity includes: ∑ A list of required Resources to carry out the activity. This list includes resources provided as photocopy masters within the Teacher’s Resource printed book (indicated by ‘(pxx)’), and photocopy masters provided on the CD-ROM (indicated by ‘(CD-ROM)’), as well as resources found in the classroom or at home. ‘(Optional)’ resources are those that are required for the activities listed in the ‘More activities’ section and thus are optional. ∑ A main narrative that is split into two columns. The left-hand (wider) column provides instructions for how to deliver the activity, suggestions for dialogue to instigate discussions, possible responses and outcomes, as well as general support for teaching the objective. Differences in formatting in this section identify different types of interactivity:

Teacher-led whole class activity The main narrative represents work to be done as a whole class. Teacher-Learner discussion “Text that is set in italics within double-quotation marks represents suggested teacher dialogue to instigate Teacher-Learner disccusion.” Learner-Learner interaction

∑

Group and pair work between learners is encouraged throughout and is indicated using a grey panel behind the text and a change in font.

The right-hand (narrow) column provides, the vocabulary panel side-notes and examples a Look out for! panel that offers practical suggestions for identifying and addressing common difficulties and misconceptions, as well as how to spot advanced learners and ideas for extension tasks to give them an Opportunity for display panel to provide ideas for displays. A Summary at the end of each Core activity to list the learning outcomes/expectations following the activity. This is accompanied by a Check up! section that provides quick-fire probing questions useful for formative assessment; and a Notes on the Learner’s Book section that references the title and page number of the associated Learner’s Book page, as well as a brief summary of what the page involves. A More activities section that provides suggestions for further activities; these are not required to cover the objectives and therefore are optional activities that can be used for reinforcement and differentiation. The additional activities might include a reference to a game in the Games Book. You are encouraged to also look on the Cambridge Maths Project website to find NRICH activities linked to the Cambridge objectives. Together, these activities provide a wealth of material from which teachers can select those most appropriate to their circumstances both in class and for use of homework if this is set.

Introduction

We would recommend that you work through the chapters in the order they appear in this book as you might find that later chapters build on knowledge from earlier in the book. If possible, work with colleagues and share ideas and over time you will feel confident in modifying and adapting your plans.

Teaching approaches Learners have different learning styles and teachers need to appeal to all these styles. You will find references to group work, working in pairs and working individually within these materials. The grouping depends on the activity and the point reached within a series of sessions. It may be appropriate to teach the whole class, for example, at the beginning of a series of sessions when explaining, demonstrating or asking questions. After this initial stage, learners often benefit from opportunities to discuss and explain their thoughts to a partner or in a group. Such activities where learners are working collaboratively are highlighted in the main narrative as detailed in the previous section. High quality teaching is oral, interactive and lively and is a two-way process between teacher and learners. Learners play an active part by asking and answering questions, contributing to discussions and explaining and demonstrating their methods to the rest of the class or group. Teachers need to listen and use learner ideas to show that these are valued. Learners will make errors if they take risks but these are an important part of the learning process.

Talking mathematics We need to encourage learners to speak during a maths session in order to: ∑ communicate ∑ explain and try out ideas ∑ develop correct use of mathematical vocabulary ∑ develop mathematical thinking.

Introduction

It is important that learners develop mathematical language and communication in order to (using Bloom’s taxonomy): Explain mathematical thinking (I think that . . . because . . .) Develop understanding (I understand that . . .) Solve problems (I know that . . . so . . .) Explain solutions (This is how I found out that . . .) Ask and answer questions (What, why, how, when, if . . .) Justify answers (I think this because . . .) There is advice on the CD-ROM about the types of questioning you can use to get your students talking maths (Questioning).

Resources, including games Resources can support, assist and extend learning. The use of resources such as Ten frames, 100 squares, number lines, digit cards and arrow cards is promoted in the Teacher’s Resource. Games provide a useful way of reinforcing skills and practising and consolidating ideas. Learners gain confidence and are able to explore and discuss mathematical ideas whilst developing their mathematical language. Calculators should be used to help learners understand numbers and the number system including place value and properties of numbers. However, the calculator is not promoted as a calculation tool before Stage 5. NRICH have created an abundance of engaging and well-thought-out mathematical resources, which have been mapped to the Cambridge Primary scheme of work, and are available from the Cambridge Primary Maths website. Their interactive and downloadable activities can provide an alternative learning style or enrichment for some of the core concepts.

1 Numbers and the number system

Quick reference Number

Core activity 1.1: Reading, writing and partitioning numbers (Learner’s book: p2) Learners understand the place value of digits up to a four-digit number and use this knowledge to write numbers in figures, words and expanded form. Core activity 1.2: Ordering, comparing and rounding four-digit numbers (Learner’s book: p4) Learners compare numbers on a number line using the <, > and = notation. They round whole numbers to the nearest 10 or 100. Core activity 1.3: Multiplying and dividing by 10 and 100 (Learner’s book: p6) Learners practise multiplying and dividing by 10 and 100, including in the context of measures.

Prior learning

Objectives* –

This chapter builds on work done in Stage 3 where learners worked with numbers up to 1000 exploring place value, ordering, comparing and rounding.

4Nn1 4Nn2 4Nn3 4Nn9 4Nn10 4Nn11 4Nn12

– – – – – – –

4Nc15 – 4Nc25 – 4Nn7 – 4Ps4 – 4Ps5 – 4Ps9 – 4Ml2 –

Ordering and rounding

Reading, writing and partitioning numbers Pablo has these digit cards. He makes three-digit numbers with the cards. Write down all the numbers he could make. 1

100 10

2000 200 20

3000 300 30

4000

5000 500

400 40

7000

600

700 70

8000 800 80

900 90

2000

3000

4000

5000

6000

7000

8000

9000

200

300

400

500

600

700

800

900

Write each number in words.

Write these numbers in figures. (a) nine thousand and nine (b) four thousand and forty What is the value of 4 in these numbers? (a) 6423 (b) 4623 (c) 3409 (d) 9040 (e) 1234 (f) 4321

He writes the numbers in order of size, starting with the smallest. Write down all the numbers that could be in Ahmed’s list. Make sure you write them in order of size.

9000

100

(d) 7777 (e) 2816 (f) 9109

6000

Vocabulary

T ??

Let’s investigate

Use a calculator. Key in these numbers and signs.

U 00

1606

1660

(b) 9080

8990

9009

1666

9090

8999

What is the number shown by an arrow on each number line?

place value: the value of a digit determined by its position.

8000

Unit 1A: Core activity 1.1 Reading, writing and partitioning numbers

Calculate.

What is the missing number?

Write the missing digits. (a) ? ? ? ! 10 " 2320 (c) 34 ? 0 # 10 " ? ? 6

(d) 1007

1039

1100

980

6000

U 0

!10

fifty-eight

!100

(f) 35 ! 100 (i) 3900 # 100 #100

(b) 461 ? # 10 " ? 61 ? 60

(d) 31 ? ! 10 " ?

(b) 307 ! ? " 3070 (d) 3400 # ? " 34

Here are four number cards. A

? " 100

5400 # ?

five hundred and eighty

ed eight hundr and fifty five hundred and eight

Write down the letter of the card that is the answer to:

899

Unit 1A: Core activity 1.2 Ordering, comparing and rounding numbers

(e) 885

Which of these numbers is closest to 1000? 1050

(b) 7225

(h) 4700 # 10

(a) ? # 10 " 54 (c) ? ! 100 " 6000

Round these numbers to the nearest 100. (a) 1060

(e) 350 # 10

(g) 4100 # 100

Write the missing numbers.

6000

5500

8000

(b) 40 # 10

(d) 415 ! 10

(a) 67 ! 10

4 (d)

7000

thousand: is a four-digit number that is 10 times larger than a hundred. Th

5500

9000

(c)

value of 3 tens (30).

(b)

expanded form: 4567 ! 4000 " 500 " 60 " 7 partition: breaking up a number into its In 830, the 3 has a parts.

What happens when you press the equals (") sign? Try using different start numbers. Do you notice the same thing?

48 to the nearest ten, is 50.

1060

12 500 # 10 " " " …

830 to the nearest hundred, is 800.

Rounding numbers makes them easier to use.

(a) 1066

5 ! 10 " " " … 11 ! 100 " " " …

Write these numbers in order of size, starting with the smallest.

(a)

digit: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are digits.

#10

H 88

" 5 or # 5, round up

(b) 1000

(a) 2345 (b) 3030 (c) 2901 3

! 5, round down

Write each red number in figures, words and expanded form. (a) 1000

round to the nearest: to round to the nearest hundred, look at the tens digit and if it is … H T U

Ahmed writes a list of four-digit whole numbers. The digits in each number add up to 3.

1 8 7

Multiplying and dividing by 10 and 100

Vocabulary

Let’s investigate

(a) 85 ! 10

(b) 5800 # 10

(d) 8500 # 10

(e) 580 # 10

(f) 5080 # 10

Unit 1A: Core activity 1.3 Multiplying and dividing by 10 and 100

please note that listed objectives might only be partially covered within any given chapter but are covered fully across the book when taken as a whole

1A: Numbers and the number system Read and write numbers up to 10 000. Count on and back in ones, tens, hundreds and thousands from four-digit numbers. Understand what each digit represents in a three- or four-digit number and partition into thousands, hundreds, tens and units. Round three- and four-digit numbers to the nearest 10 or 100. Position accurately numbers up to 1000 on an empty number line or line marked off in multiples of 10 or 100. Estimate where three- and four-digit numbers lie on empty 0–1000 or 0–10 000 number lines. Compare pairs of three-digit or four-digit numbers, using the > and < signs, and find a number in between each pair. 1A: Calculation (Mental strategies) Understand the effect of multiplying and dividing three-digit numbers by 10. 1A: Calculation (Multiplication and division) Understand that multiplication and division are the inverse function of each other. 2A: Numbers and the number system Multiply and divide three-digit numbers by 10 (whole number answers) and understand the effect; begin to multiply numbers by 100 and perform related divisions. 1A: Problem-solving (Using understanding and strategies in solving problems) Explore and solve number problems and puzzles, e.g. logic problems. Use ordered lists and tables to help to solve problems systematically. Explain methods and reasoning orally and in writing; make hypotheses and test them out. 1B: Measure (Length, mass and capacity) Know and use the relationships between familiar units of length (m, cm and mm). *for NRICH activities mapped to the Cambridge Primary objectives,

Vocabulary

please visit www.cie.org.uk/cambridgeprimarymaths

digit ∑ expanded form ∑ partition ∑ place value ∑ thousand ∑ round to the nearest Cambridge Primary Mathematics 4 © Cambridge University Press 2014

Unit 1A

Core activity 1.1: Reading, writing and partitioning numbers

LB: p2

Resources: Numbers all around us photocopy master (p11); large version for class display. Place value chart: 1–9000 photocopy master (CD-ROM); large version for class display. (Optional: 0–9 dice, 0–9 spinners or 0–9 digit cards (CD-ROM).) Learners should recognise the place value of the digits in 103 as one ‘hundreds’, zero ‘tens’ and three ‘ones/units’ from Stage 3 (Unit 1A, chapter 1). Remind them how our number system is based on 10: ten lots of ones/units makes 10; ten lots of ten makes 100. Ask, “What do you think ten lots of one hundred makes?” Elicit that this makes one thousand, and this is a further column in the place value table. Explain that there are different ways we can describe a number using place value. For example, in 1830 we can say there is one ‘thousand’, eight ‘hundreds’, 3 ‘tens’ and zero ‘ones’. Or we could say that are 18 ‘hundreds’, three ‘tens’ and zero ‘ones’. Or, one ‘thousand’ and 83 ‘tens’. Similarly, seven hundreds (700) is the same as 70 tens; and one thousand and 10 (1010) is the same as 101 tens. Give learners some four-digit numbers and ask them to count on and back in ones, tens, hundreds and thousands using the place value table to help them. Then ask questions such as, “What is 1 less than 4000?” (Answer: 3999) “What is ten more than 2456?” (2466), encouraging them to think of the place value of each digit as they do so.

Reading numbers Display the Numbers all around us photocopy master. Learners discuss in pairs what they notice. Remind learners about place value by pointing to the picture of the car number plate (A357 NNM), and reading out these comments made by three learners: (1) “I think the 3 means three hundred.”; (2) “I think the value of the digit 3 on the number plate is three.”; (3) “I don’t think the 3 has a place value that is important. It’s just a number in a list of letters and numbers.” Discuss these comments with the class. Ensure that learners understand that groups of digits can have different purposes. For example, place value is important when recording a score of 3 or 30, but is irrelevant in a telephone number.

Vocabulary digit: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are digits; we use digits to make up the numbers we need. place value: the value of a digit is determined by its position. For example, in 1830 the 3 is worth 3 tens (30). Th

1 8 3 0 Each place value can only contain a single digit; if there are 11 units, then a 1 is placed in ‘U’ and the other 1 is carried over to ‘T’. For whole numbers, you must always have a digit in a smaller place value to the right in order to show the size of the number; zero is the placeholder. partition: breaking up a number into parts. expanded form: what you get when you partition a number according to place values, e.g. 4567 = 4000 + 500 + 60 + 7 thousand: a four-digit number that is 10 times longer than a hundred.

Reading, writing and partitioning numbers

Opportunity for display

Explain that a palindromic number reads the same when written forwards or backwards. Show how 343 is a palindromic number and ask learners to give other examples.

Display examples of numbers from newspaper articles, magazines and photographs.

Unit 1A

1 Numbers and the number system

Take examples, such as 9779 and: ∑ write the number on the board ∑ say the number, “Nine thousand, seven hundred and seventy-nine”. As you say ‘nine thousand’ point to it on the place value chart. Repeat for seven hundred, for seventy and for nine. ∑ partition the number into thousands, hundreds, tens and units: 9000 + 700 + 70 + 9.

Look out for! Learners who write what they hear, e.g. they write ‘60009’ when they hear “six thousand and nine”. Ensure that learners understand that the position of the digit determines its value.

Challenge pairs of learners to write down all the four-digit palindromic numbers where the sum of the digits is 10. Collect responses, ensuring that the numbers are written, said and partitioned (as above). Learners should be encouraged to work in an organised way. (Answer: 1441, 2332, 3223, 4114, 5005).

∑

Summary Check up!

∑

Learners are able to read and write numbers from 1 to 10 000. They understand the value of each digit and use this knowledge to write a number in expanded notation. For example: 9876 = 9000 + 800 + 70 + 6 Notes on the Learner’s Book Reading, writing and partitioning numbers (p2): learners investigate how many three-digit numbers they can make from the numbers 1, 8 and 7. They then practise reading and writing numbers in figures, words and expanded form.

∑

“What number is represented by 7000 + 40 + 2?” “Show me on the place value chart.” “How would you write it in words?”

More activities Boxes (for the whole class or groups) You will need a 0–9 dice, 0–9 spinner or set a of 0–9 digit cards (CD-ROM); templates for the dice and spinner can be found on the CD-ROM. Each learner draws four boxes in a row. They use a dice, a spinner or digit cards to generate numbers from 0 to 9. As each number is generated, learners decide which box to write the digit in. Once they have made a decision they cannot change it. The winner is the learner with the highest four-digit number. Learners should be asked to say their number in words. The game can be varied so that the learner with the smallest number is the winner.

Guess my number (for groups) One learner decides on a three- or four-digit number, for example, 471. They make up some sentences to define the number. For example: “My number has three digits.”, “The units digit is 1.”, “The tens digit is 3 more than the hundreds digit.”, “The sum of the digits is 12.” Other learners attempt to guess the number. The learner who gives the correct answer gains one point and defines the next number. Alternatively, one learner chooses a number and it is identified by other learners who ask questions with a yes/no answer. Demonstrate how learners might use questions to guess the number. For example: “Is the tens digit less than six?”

Core activity 1.1: Reading, writing and partitioning numbers

Core activity 1.2: Ordering, comparing and rounding four-digit numbers

LB: p4

Resources: Number line 0–1000 photocopy master (CD-ROM). (Optional: 0–9 dice; 0–9 spinners or 0–9 digit cards (CD-ROM); Volcanoes cards photocopy master (CD-ROM).) Generate a sequence (whole class mental activity) One learner is given a four-digit starting number and the rest of the class continues the sequence following a given rule. For example, add or subtract 1, 10, 100 or 1000.

Visualisation (whole class activity) Ask learners to close their eyes and visualise a blank number line. ∑ “Using the digits 1, 2, 3 and 4, what is the largest number you can make? Visualise this on your number line.” (Answer: 4321) ∑ “What is the smallest two-digit number you can make? Visualise this on your number line.” (Answer: 1234) ∑ “Use the four digits to make a number in between your two numbers. Is this number closer to the largest number or the smallest number?” Learners then draw a number line, mark their numbers on it and compare their results with a partner’s by stating if their number is greater than, less than or equal to their partner’s. Repeat with other digits.

Vocabulary round to the nearest: to round a number to the nearest hundred look at the tens digit: ∑ if it is less than 5 round down ∑ if it is 5 or more round up. (Look at the place value to the right when determining to round up or down; for example, if rounding to the nearest 10, you would look at the unit place value.)

Example:

1234

3241

4321

nearer to larger number

Rounding Challenge pairs of learners to discuss which of these numbers gives 50 when rounded to the nearest 10: 40, 54, 57, 42, 46, 60. (Answer: 46 & 54). Ask learners to explain their answers to the class.

Show a number line marked only from 3000 to 4000 (no other numbers marked). Ask learners: ∑ “Where would you position 3241 on this number line?” ∑ “Is the number nearer to 3000 or 4000?” ∑ “How do you know?” Explain that if 3241 was rounded to the nearest thousand the answer would be 3000 because 3241 is nearer to 3000 than to 4000. Show another number line marked only from 3200 to 3300. “What number is halfway between 3200 and 3300?” (Answer: 3250)

Unit 1A

1 Numbers and the number system

Example: 3000

3241

4000

“Where would you place 3241 on the number line? Why?” (Answer: less than the halfway mark because 3241 < 3250) “Is it nearer to 3200 or 3300?” (Answer: 3200) Explain that if 3241 was rounded to the nearest hundred the answer would be 3200 because 3241 is nearer to 3200 than to 3300. Ask: ∑ Look at the tens digit. ∑ If it is less than 5 round down. ∑ If it is 5 or more round up. Repeat, to round 3241 to the nearest ten. ∑ Look at the units digit. ∑ If it is less than 5 round down. ∑ If it is 5 or more round up. (Answer: 3200)

Example: 3200

3300

Example: to the nearest hundred 3200

3241

3300

Example: 3241

to the nearest ten

3240

Placing numbers on a number line Display the Number line: 0–1000 photocopy master for the whole class to see. The line is marked off in multiples of 100, (see right). Explain that sometimes we need to place numbers more accurately than we did earlier. ∑ “Where would I place 350 on the number line? How did you decide?” (Answer includes ‘estimating’ halfway between 300 and 400) ∑ “Where would I place 920? How did you decide?” (Answer includes close to 900) ∑ “How can I write a statement to show that 350 is less than 920?” (Answer: 350 < 920)

Example: 0

1000

Summary ∑

∑

Learners confidently use marked and unmarked number lines to locate the relative positions of numbers. They compare pairs of numbers using the notation <, > or =. When necessary, they round any whole number to the nearest 10 or the nearest 100. Notes on the Learner’s Book Ordering and rounding (p4): begins with an investigation involving ordering four-digit numbers with digits that add up to three. It then includes questions that feature rounding, number lines and the < and > symbols.

Check up! ∑ ∑

∑

“I rounded a number to the nearest 10. The answer is 830. What number could I have started with?” “The news report stated that 1500 people attended the match. What is the smallest number that could have attended? What is the largest number?” “How could you choose numbers to make this number sentence correct? + ∆ < 10”

Core activity 1.2: Ordering, comparing and rounding four-digit numbers

More activities Boxes again (class or groups) You will need a 0–9 dice, 0–9 spinner or set of 0–9 digit cards (CD-ROM); templates for the dice and spinner can be found on the CD-ROM.

Each learner draws a grid including the ‘less than’ symbol, as shown on the right. Use the dice, spinner or digit cards to generate eight digits. Learners write each digit in their grid, aiming to make a true statement. Once the digit is placed, its position cannot be changed. When complete, ask: ∑ “Is your statement true? How do you know?” ∑ “How did you decide where to put the digits?” ∑ “Which spaces were the most important? Why?”

Volcano database (pairs) You will need the Volcano cards photocopy master (CD-ROM); one per pair of learners. The teacher or learners prepare, then answer, questions related to ordering, comparing and rounding numbers. For example: ∑ “Name the volcanoes that are 4000 metres high when rounded to the nearest 100 metres.” (Answer: Colima, Fuego, Fuji, Mauna Loa, Semeru, Tajumulco) ∑ “List in order of height, lowest first, all the volcanoes less than 1000 metres high.” (Answer: Surtsey 169 m, Krakatoa 813 m, Stromboli 926 m) ∑ “Place the volcanoes on a number line according to their height.”

Games Book (ISBN 9781107685420) Find the largest number (p1) is a game for two players. It can be used to practise ordering four-digit numbers.

Unit 1A

1 Numbers and the number system

Blank page

Core activity 1.3: Multiplying and dividing by 10 and 100

LB: p6

Resources: Place value chart: 1–9000 photocopy master (CD-ROM); large version for class display. Whole-number slider photocopy master and approximately 10 slider strips (p12 to 14) for each learner and a large version for demonstration. (Optional: 10s and 100s dominoes (CD-ROM).)

Multiplying by 10 and 100

Example: 700 × 10.

Show the statement ‘74 metres = centimetres’ and ask learners, “How might we find the answer?” Remind learners that 100 cm = 1 m (a fact to be memorised).

Display the Place value chart: 1–9000 photocopy master and ask, “What is 7 × 10?” Take answers, and then demonstrate how to model this on the chart by pointing to 7, and then to 70. Repeat for 70 × 10 and 700 × 10 and ask, “Can you describe what happens?” Guide them to the explanation that to multiply by 10, digits move one place value to the left; if this leaves any place values to the right without a digit, insert a zero. Show the results on a place value grid, for example 700 × 10. Provide each learner with a slider from the Whole-number slider photocopy master and demonstrate how to use it. Emphasise how the zero is placed in the unit column as a placevalue holder. Set these questions: ∑ “What is 70 × 10?” (Answer: 700) ∑ “What is 7 × 100?” (Answer: 700) ∑ “What is 70 × 100?” (Answer: 7000) ∑ “What is the missing number in this number sentence: 7 × = 700?” (Answer: 100) Model these calculations on a place value grid so learners find out that to multiply by 100, the digits move two places to the left. Demonstrate how the slider can be used to model 41 × 100 by moving digits two places to the left. 4

Unit 1A

1 Numbers and the number system

Example: how to calculate 410 × 10 using the slider (for full instructions see the Whole-number Slider photocopy master). To calculate 410 × 10.

Return to the question posed earlier: 74 metres = centimetres. Ask: “What is the answer? How do you know?” Learners should use their slider to find that 74 × 100 = 7400. Therefore, 74 metres = 7400 centimetres.

Look out for!

Dividing by 10 and 100

Learners who try to apply rules that do not always work, for example: ∑ To multiply by 10, add a zero. ∑ To divide by 10, cross out the last digit. Learners should be discouraged from applying any ‘rule’ that does not generalise, so 16.52 × 10 ≠ 16.520.

Show a similar missing number problem: 5300 centimetres = metres. Ask: “How can I work out the answer?” Establish that you must divide 5300 by 100. Learners use their sliders to ‘undo’ the process of multiplication to find the answer. Remind learners that multiplication and division are inverse operations, one undoes the effect of the other. Learners work on the next activity using their sliders. Show these six numbers: 4

400

4000

100

Ask learners to choose three numbers to make a multiplication statement:

× =

Ask them to use the same three numbers to make a division statement:

÷ =

Model answers. Repeat with different numbers.

∑

Summary Learners use a slider to model multiplying and dividing whole numbers by 10. They begin to multiply and divide by 100. Learners begin to perform calculations mentally. They understand that multiplication and division are inverse operations. ∑

∑

Notes on the Learner’s Book Multiplying and dividing (p6): learners practise multiplying and dividing by 10 and 100, including in the context of measures. Please note that the idea of the “Let’s investigate” is that repeatedly pressing the = sign will multiply the previous answer by 10 (or 100 etc) each time. So, learners should see the following pattern:

∑

Check up! “Fill in the missing numbers.” 608 ! = 6080 ÷ 10 = 68 “Copy and complete.” 43 metres = ____ centimetres 90 metres = ____ centimetres 7100 centimetres = ____ metres 1500 centimetres = ____ metres

5 × 10 = 50, 500, 5000, 50000 11 × 100 = 1100, 11000, 110000 12500 ÷ 10 = 1250, 125, 12,5 Please note that in order for this to work on some calculators, the learners would need to enter the numbers in the opposite order i.e., 10 × 5 = , rather than 5 × 10 =.

Core activity 1.3: Multiplying and dividing by 10 and 100

More activities Dominoes (pairs) You will need 18 dominoes from the Dominoes photocopy master (CD-ROM); per pair of learners. (You might also want to create a larger version for demonstration.) Attach or draw a large set of dominoes on the board in a random way. Place one domino in the centre of the board and ask, â&#x20AC;&#x153;Which domino fits on this end?â&#x20AC;? The correct domino is placed in position. 73 ! 10

730

25 ! 100

2500

500

The game continues until the dominoes form a complete ring.

Games Book (ISBN 9781107685420) Hexalines (p1), a game of strategy for two players, involves recall of number facts and multiplication by 10 and 100.

Unit 1A

1 Numbers and the number system

A357 NNM

DEPARTURES Time

Flight no.

Gate

Remarks

19 : 25

AMSTERDAM

TK2164

CANCELLED

19 : 30

BERLIN

GT4592

DEPARTURE

19 : 40

WASHINGTON

LX3100

DEPARTURE

19 : 45

MADRID

ZL6658

CANCELLED

19 : 50

AMSTERDAM

EH5810

DEPARTURE

19 : 55

BERLIN

KS3208

DEPARTURE

20 : 05

TOKYO

EK5528

DEPARTURE

Numbers all around us TOTAL WKTS

SPEED LIMIT 55

Instructions on page 2

OVERS LAST INNS

28 9 11 85

Original material © Cambridge University Press, 2014

Whole-number slider To make the slider, you will need: ∑ slider sleeve (see page 13) ∑ scissors ∑ sticky tape ∑ slider strip (see page 14). Instructions 1. To make the sleeve, cut along the dashed lines to create four square ‘windows’. 2. Fold the sleeve inwards along the solid lines, so that you create a top flap, and the zeros show through the windows. 3. Stick down the flap to the back of the slider sleeve. 4. Fit the slider strip into the sleeve, ensuring that it slides freely. 5. When using the slider, always place it in the start position, with the corner marked ‘S’ hidden behind the furthest square to the right on the sleeve. 0

Instructions on page 8

Original material © Cambridge University Press, 2014

Slider sleeves

Original Material ÂŠ Cambridge University Press, 2014

0 0 0 0 Instructions on page 8

Instructions on page 8

S Original Material ÂŠ Cambridge University Press, 2014

Slider strips

2 Addition and subtraction (1)

Quick reference Core activity 2.1: Addition (1) (Learner’s book: p8) Learners practise adding three and four small numbers using number pairs. They practise adding on in tens, then units and vice versa. They add two-digit numbers using their chosen method and explain their methods. Core activity 2.2: Subtraction (1) (Learner’s book: p10) Learners practise subtraction of two-digit numbers by counting back, finding the difference and other methods. They are encouraged to choose their own method and explain it. Core activity 2.3: Partitioning to add and subtract (Learner’s book: p12) Learners practise addition and subtraction of two-digit and three-digit numbers using partitioning. They solve word problems using the method of their choice.

Addition (1)

Subtraction (1)

Vocabulary

Let’s investigate

Make a route through the grid from Start to Finish.You can move horizontally or vertically. Add up the numbers on your route. Find the route that gives the lowest total. 4

Finish

Start

Some words that we use for addition: add, addition, more, plus, increase, sum, total, altogether.

Break the four-digit code to open the treasure chest. 65 ! 58 " (a)

Questions that ask us to add: How many are there altogether? What is the total number of …?

8 9

67 2 ! 22 8 " 900

86 ! 79 " (c)

(d) 4 ! 17 ! 2 ! 3 " ? (f) 3 ! 14 ! 9 ! 3 " ?

65 ! 89 "

128

94 ! 22 "

154

17 ! 68 "

104

43 ! 52 "

91 ! 77 "

40 ! 64 "

116

Unit 1A: Core activity 2.1 Addition (1)

For example, 67 2

228

355

601

109

589

437

322

814

465

Vocabulary partition: breaking up a number into its parts. For example, 608 " 600 ! 8.

! 22 8

...

Partitioning to add

Solve 62 – 11 using the ‘counting back’ method.

423 ! 589 "

400 ! 20 ! 3 and 500 ! 80 ! 9 400 ! 500 " 900 20 ! 80 " 100 3 ! 9 " 12

!10

Therefore, 423 ! 589 " 1012

61 62 63 64

168

28 ! 34 "

791

463

900 is a multiple of 10. Look for two numbers that add to make a multiple of 10. We can do this by looking for number pairs to 10 in the units digits. Then choose a method to add the two numbers together.

How much more is ? than ? ? How much less is ? than ? ?

13 is a small number. I will take away 10 then 3 more. The answer is 61.

Copy the addition number sentences below.Then copy the list of numbers on the right. Draw arrows to complete the number sentences. The first one has been done for you. 76 ! 52 "

238

96 672

How many more is ? than ? ? How many fewer is ? than ? ?

67 ! (d) 8 " 39 (a) (b) (c) (d)

(e) 13 ! 2 ! 1 ! 5 " ?

545

Sarah used the ‘counting back’ method to calculate 74 ! 13.

(b) 4 ! 19 ! 12 ! 1 " ?

Explain to your partner why you chose that method. If you think your partner could choose a better method, tell them why.

Find five pairs of numbers that add up to 900. One has been done for you.

Choose a method to solve these addition problems. (a) 5 ! 8 ! 5 ! 3 " ?

Let’s investigate

Questions that ask us to subtract: How many are left? How many are left over?

41 ! 2 (b) " 12

For example, 2 ! 8 ! 5 ! 1 ! 9 ! 4 ! 8 ! 2 " 34

Partitioning to add and subtract

Vocabulary Some words that we use for subtraction: subtract, subtraction, take, take away, minus, decrease, fewer, leave, difference.

Let’s investigate

74 1

Tim used ‘finding the difference’ to calculate 81 ! 76. 76 is quite close to 81. I will count up to find the difference between the numbers. The answer is 5. #1

(b) 237 ! 149 " ? (c) 821 ! 546 " ? (d) 271 ! 649 " ? (e) 362 ! 841 " ?

(f) 598 ! 613 " ?

76 77 78 79 80 81

Unit 1A: Core activity 2.2 Subtraction (1)

Partition each number into hundreds, tens and ones. Then calculate each answer. (a) 482 ! 213 " ?

Solve 62 – 58 by ‘finding the difference’ between the two numbers.

Unit 1A: Core activity 2.3 Partitioning to add and subtract

please note that listed objectives might only be partially covered within any given chapter but are covered fully across the book when taken as a whole

Prior learning

Objectives* –

This unit builds on learning in Stage 3: ∑ Adding and subtracting two-digit and three-digit numbers. ∑ Using the = sign to represent equality. ∑ Finding complements to 100, solving number equations. ∑ Reordering an addition to help with a calculation. ∑ Adding and subtracting multiples of 10 and multiples of 100 to and from two- and three-digit numbers.

1A: Addition and subtraction 4Nc17 – Add pairs of three-digit numbers. 4Nc18 – Subtract a two-digit number from a three-digit number. 4Nc19 – Subtract pairs of three-digit numbers. 1A: Calculation (Mental strategies) 4Nc6 – Add three or four small numbers, finding pairs that equal 10 or 20. 4Nc9 – Add any pair of two-digit numbers, choosing an appropriate strategy. 4Nc10 – Subtract any pair of two-digit numbers, choosing an appropriate strategy. 1A: Problem solving (Using techniques and skills in solving mathematical problems) 4Pt1 – Choose appropriate mental or written strategies to carry out calculations involving addition and subtraction. 4Pt3 – Check the results of adding numbers by adding them in a different order or by subtracting one number from the total. 4Pt4 – Check subtraction by adding the answer to the smaller number in the original calculation. 4Pt8 – Estimate and approximate when calculating, and check working. 1A: Problem solving (Using understanding and strategies in solving problems) 4Ps1 – Make up a number story for a calculation, including in the context of measures. 4Ps3 – Choose strategies to find answers to addition or subtraction problems; explain and show working. 4Ps9 – Explain methods and reasoning orally and in writing; make hypotheses and test them out. *for NRICH activities mapped to

Vocabulary

the Cambridge Primary objectives,

add • addition • plus • increase • sum • total • altogether • subtract • subtraction • take • take away • minus • decrease • fewer • leave • difference • partition • What is the total number of …?

please visit www.cie.org.uk/

cambridgeprimarymaths

Unit 1A

Core activity 2.1: Addition (1)

LB: p8

Resources: Complements-to-20 dominoes photocopy master (p28); Timers; one per learner, if available. Number sentence stories – Addition photocopy master (p29); large version for class display. (Optional: Blank dominoes photocopy master (CD-ROM); 0–9 digit cards, 0–9 dice or 0–9 spinner (CD-ROM)). Warm-ups (1) Each learner has a set of Complements-to-20 dominoes from the photocopy master. Explain that they need to create a loop of end-to-end dominoes by matching pairs of numbers that total 20. Explain ‘complement to 20’; it is another way of saying number pair to 20. Game 1. Each learner shuffles their dominoes. They find how quickly they can arrange the dominoes into a loop so that each pair of touching numbers adds up to 20. Use timers if available. Game 2. For 11 players plus one learner who times the others. The dominoes are shuffled and dealt out, one to each player. The players arrange themselves into a loop so that each pair of their touching numbers adds up to 20. Use a timer to record how quickly they can do this.

complement to 20: the number that needs to be added to an existing number to make 20. For example, the ‘complement to 20’ of 14 is 6. 'Complement to' is another way to describe ‘number pairs’. add, addition, plus, increase, sum, total, altogether: words we use for addition.

Example: using the Complements-to-20 dominoes. 5

10 14

Adding three or four small numbers

Vocabulary

(2) Choose a two-digit number. Ask learners to add 10 to this number and to say what new number they get. Then ask learners to add 10 to the new number and say what they get; repeat this process eight further times. Explain that this way of adding numbers helps us to add a multiple of 10 to a two-digit number. In this example, we have added 100 (10 × 10) to our original number. Demonstrate using the example: 47 + 70 = 117 by adding 10 seven times to a starting number of 47. Set similar questions that the whole class can answer using slates, small whiteboards or an ICT resource. Repeat with three-digit numbers.

For learners to be able to add and subtract larger numbers, it is important that: ∑ They are confident in their knowledge of pairs of numbers that add to make a multiple of 10. ∑ They can add a multiple of 10 to any number.

Draw these circles for the whole class to see:

Look out for! 12 9

7 8

1 4

8 6

13 15

5 7

Challenge learners to come up with a method for finding the total of the numbers inside each circle. Collect responses.

Unit 1A

2 Addition and subtraction (1)

Learners who do not understand that addition can be done in any order. Use physical objects in the classroom to demonstrate that the order in which the objects are added together does not alter the total number of objects.

Share useful ways of adding numbers without using a calculator, such as: ∑ adding three numbers where the sum of two of the numbers is a near multiple of 10: “I need to add 13, 15 and 8. I know that 13 add 7 is equal to 20, so 13 add 8 must equal 21. I know that 20 add 15 is equal to 35, so 21 add 15 must equal 36.” ∑ using near doubles and compensating: “I need to add 6, 7 and 4. I know that double 6 is equal to 12, so 6 add 7 must equal 13. 13 add 4 equals 17.” ∑ using pairs of numbers that equal 10 or 20: “I need to add 2, 8 and 2. I know that 2 add 8 equals 10. 10 add 2 equals 12.”

If learners find this activity difficult, ensure they understand that they only need to add two numbers at a time. For example, 4 + 6 + 7 is solved by adding 4 and 6, then adding the total of that (10) to 7. If learners find it hard to remember the intermediate totals, encourage them to write totals down on paper.

Show the numbers in the circles as number sentences, e.g. 12 + 9 + 8 = ? Remind learners that addition can always be done in any order and that they should look for efficient ways of working rather than always tackling a problem in the order it is given. For example, they could recognise that if they add 12 and 8 first they get 20, then they can add 9.

Addition by adding on tens then units Read the following problem to the class: “The first part of my journey was 58 km. The second part of my journey was 25 km. How far did I travel altogether?” Ask: “How can we solve this problem?” Collect responses. Then display this number sentence (and answer) on the Number sentence stories photocopy master. Explain that one way of adding numbers is to add the tens part of one number to the whole of the other number, then add on the units. Show how this works using the example and number lines below. Counting on in tens and ones 10

21 + 32 = 53 21

52 53

Demonstrate the same strategy for three-digit numbers. Add the hundreds, then the tens and then the units of one number to the whole of the other number. Counting on in hundreds, tens and ones 100

200

210

220

230

231 232

100

346 + 232 = 578 346

446

546

556

566

576 577 578

346

446

546

556

566

Encourage learners to suggest ways to make calculations more efficient. ∑ For 21 + 32, it would be more efficient to calculate by starting with the larger number, 32, and adding on 21. ∑ For both examples, the last two jumps of 1 could be put together to make one jump of 2. Number lines can be annotated in different ways to support the calculation. Each of these number lines has been shown in two ways: ∑ one shows the jumps with a running total of what value has been added after each jump ∑ the other shows the separate value of each jump. Choose the method that supports the learners’ understanding.

576 577 578

Core activity 2.1: Addition (1)

Addition by adding on units then tens Read out this problem: “This time the first part of my journey was 83 km. The second part of my journey was 67 km. How far did I travel altogether?” Ask, “How can we solve this problem?” Collect responses. Then display this number sentence story (and answer) from the Number sentence stories photocopy master. Explain that another way of adding two numbers is to add the units part of one number to the whole of the other number, then add on the tens. This is particularly useful if the units of both numbers add up to 10, because it is easier to count on in tens from a starting number that is a multiple of 10. As a class, count on together to find the solution to 88 + 42. Then, show the class one of these examples and number lines: Counting on in ones and tens 1

68 + 42 = 110 68

100

110

100

110

Explain that this method can also be used if the units add up to more than 10. First add on enough units to make the number up to a multiple of 10, then add on the tens, then add on the rest of the units. Example: 1

77 + 45 = 122 77

Unit 1A

2 Addition and subtraction (1)

100

110

120

122

77 + 3 = 80 80 + 40 = 120 120 + 2 = 122

One number line shows the jumps with a running total of what value has been added after each jump; the other shows the separate value of each jump.

Summary ∑ ∑

Learners are able to add together three or four small numbers by finding pairs that equal 10 or 20. They can add any pair of two-digit numbers, choosing an appropriate strategy.

Notes on the Learner’s Book Addition (1) (p8): learners practise adding three and four small numbers together. They add two-digit numbers and explain their methods. ∑

∑

Check up! “Throw a dice four times and write down the four numbers generated. Add the numbers together. Which method did you use to add the numbers? Why?” “Use your four numbers to make two two-digit numbers. Add these numbers. Which method did you use to add the numbers? Why?”

More activities Domino pairs (pairs) You will need the Blank dominoes photocopy master (CD-ROM). Pairs of learners can make sets of dominoes, similar to those in the Complement-to-20 dominoes game, for other two-digit totals. They will need to work in an organised way to make sure that the dominoes form a continuous loop.

Reverse and add (pairs) You will need a set of 0–9 digit cards (CD-ROM). Ask learners to generate pairs of two-digit numbers by selecting two cards at a time. For example, the cards 6 and 7 make the numbers 67 and 76. Learners should add the pairs of numbers together and investigate the totals they find.

Number sentence stories (individuals) Ask learners to write two number sentence stories for addition facts. One should be in the context of measures. Use the Number sentence stories photocopy master for examples if you need to.

Games Book (ISBN 9781107685420) Add to 500 (p4) is an addition game for up to six players. Players practise adding pairs of two-digit numbers, and adding two-digit numbers to three-digit numbers.

Core activity 2.1: Addition (1)

Core activity 2.2: Subtraction (1)

LB: p10

Resources: Number sentence stories – Subtraction photocopy master (p30); large version for class display. (Optional: 0–9 digit cards, 0–9 dice or 0–9 digit spinner (CD-ROM).) Warm-up Choose a two-digit number greater than 60, for example, 87. Ask learners to subtract 10 from this number five times and to say what number they get each time. Use this method of subtraction to help learners subtract a multiple of 10 from a two-digit number. Set similar questions that the whole class can answer using slates, small whiteboards or an ICT resource. Repeat with three-digit numbers.

Subtraction by counting back

Example: 87 − 50 = 37 87 − 10 = 77 77 − 10 = 67 subtract 10 five times 67 − 10 = 57 57 − 10 = 47 47 − 10 = 37

Read out this problem: “I have 85 ml of oil in a bottle, then I use 15 ml in a recipe. How much oil is left in the bottle?” Challenge learners to explain how they might solve this problem. Collect and discuss responses. Then display this story from the Number sentence stories photocopy master. Explain that when the number to be subtracted is small (for example 85 − 15), or a multiple of 10, it is efficient to count back to find the answer. Show the following number lines to demonstrate this method:

Vocabulary subtract, subtraction, take, take away, minus, decrease, fewer, leave, difference: some words that we use for subtraction.

Counting back in ones 8

52 − 8 = 44

Explain to learners that, in this example, it can be useful to break the 8 into 2 and 6. 2 is subtracted so that learners reach a multiple of 10, then the remaining 6 is subtracted. Counting back in tens 10

82 − 60 = 22

Unit 1A

2 Addition and subtraction (1)

Draw these two circles of numbers for the whole class to see. Ask learners to subtract numbers in the second circle from numbers in the first circle, e.g. 72 − 6 = 66.

72 89

68 93

Learners can solve the subtraction problems mentally using the counting back method and with the help of a number line if appropriate.

6 30

Subtraction by finding the difference

Look out for!

Read out this problem: “I have 78 metres of rope, but I only need 72 metres. How many metres should I cut from the rope?” Ask learners to explain how they might solve the problem. Share and praise all valid methods. Then display this story from the Number sentence stories photocopy master. Explain that when the number to be subtracted is close in value to the amount it is to be subtracted from (e.g. 78 − 72) it can be more efficient to count up from one number to the other. This is called finding the difference. Show the following number lines to demonstrate this method.

Learners who continue to use number lines for questions they can easily answer without one, e.g. 67 − 65 = 2. Encourage learners to do bigger jumps on the number line, or to develop their own ways of recording their working and intermediate steps in their calculation. These must be efficient and reliable.

Counting up in ones 1

52 − 45 = 7 45

Counting up in tens and ones 10

53 − 21 = 32

Draw these two circles of numbers for the whole class to see. 48 60

55 53 58

647

43 25 36

38 31 27

Get learners to try the ‘finding the difference’ method by subtracting any number in the second circle from any number in the first circle, e.g. 58 − 31. Learners can solve the subtraction problems mentally, with the support of a number line if appropriate. Core activity 2.2: Subtraction (1)

Summary Learners can do a subtraction using any pair of two-digit numbers, choosing an appropriate strategy. Notes on the Learner’s Book Subtraction (1) (p10): learners practise subtraction by choosing and using different methods, then write a story to illustrate a subtraction number sentence. ∑

∑

Check up! “Our class had 39 pencils at the start of the year. Now we have 32. How many have we lost? Which method did you use to subtract the numbers? Why?” “Our class had 28 pens at the start of the year, but we lost 4. How many do we have now? Which method did you use to subtract the numbers? Why?”

More activities Subtracting numbers (whole class) You will need 0–9 digit cards, a 0–9 dice or a 0–9 spinner (CD-ROM). Learners use a dice/spinner/digit cards to generate two-digit numbers. Ask learners to subtract one number from the other using an appropriate strategy.

100 down game (for two or more players) The players each start with 100. They take turns to choose whether to throw one, two or three dice. They subtract the total on the dice from their total number. The winner is the player who stops playing the game when their number is closest to 0, without going past 0.

Number sentence stories (for individuals) Use the Number Sentence Stories photocopy master for examples. Ask learners to each write two number sentence stories for subtraction facts. One should be in the context of measures.

Games Book (ISBN 9781107685420) Difference bingo (p4) is a game for six players and one leader. Players find the difference between two two-digit numbers.

Unit 1A

2 Addition and subtraction (1)

Blank page

Core activity 2.3: Partitioning to add and subtract

LB: p12

Resources: Place value chart: 1 to 9000 photocopy master (CD-ROM); large version for class display. Number sentence stories photocopy master (p29 and 30); large version for class display. Materials to demostrate place value, such as blocks or straws bundled in groups of hundreds, tens and ones. (Optional: 0–9 digit cards, 0–9 dice or 0–9 spinner (CD-ROM).) Warm-ups (1) Learners pick two two-digit multiples of 10. They find the total of the numbers and find the difference between the numbers. The total and difference are the two new numbers. Learners repeat the process of finding the total and difference using the two new numbers. They continue until both numbers are greater than 100, (see example, right). (2) As per Warm-up 1, but with two three-digit numbers. Learners continue until both numbers are greater than 1000. Remind learners about the work they have done on place value (chapter 1) and partitioning numbers into hundreds, tens and ones. Display the Place value chart: 1 − 9000 for learners to refer to during the next activity. Ask learners to partition some three-digit numbers such as 239, 932 and 392. The whole class can respond in number sentences such as 239 = 200 + 30 + 9, using slates, small whiteboards or an ICT resource.

Addition by partitioning Read out this problem: I had 783 ml of juice in the jug. I poured in another 419 ml. How much juice is in the jug in total? Encourage learners to make jottings to help solve this problem. (Note that learners are not expected to change the measure of liquid from millilitres to litres. Make a notes of any that do, to inform planning for Unit 1B). Ask them to explain their method for working out the solution and discuss all methods as a class. Then display this number sentence story from the Number sentence stories photocopy master, which includes the answer. Tell learners that as numbers in a calculation get larger, it is harder to keep track of the different parts of the calculation. A written method, rather than mental strategies, is often needed. Explain that if two numbers are partitioned into hundreds, tens and units they can be recombined in different ways and always produce the same total. Explain that this is useful because it is easier to add together multiples of 100 or 10, as learners did in the warmup. Demonstrate this using place value materials and the sum 732 + 548 partitioned into 700 + 500 + 30 + 40 + 2 + 8. Show how the method can be recorded.

Unit 1A

2 Addition and subtraction (1)

Example: 20 and 80 20 + 80 = 100 80 − 20 = 60 100 + 60 = 160 100 − 60 = 40 160 + 40 = 200 160 − 40 = 120 end.

Vocabulary partition: breaking up a number into its parts. For example, 608 = 6 hundreds, 0 tens, 8 ones = 600 + 8

Example: (1) written partitioning method 700 + 500 = 1200 30 + 40 = 70 2 + 8 = 10 1200 + 70 + 10 = 1280, therefore 732 + 548 = 1280

or Example (2): written partitioning method 2 + 8 = 10 10 + 40 + 30 = 80 500 + 700 = 1200 1200 + 80 = 1280 therefore, 732 + 548 = 1280

Give learners time to practise this method of addition: draw these circles for the whole class to see. Ask learners to add numbers in the first circle to numbers in the second circle, e.g. 619 + 438.

348 170 619

165 503 587

283 924 862

438 191 607

Subtraction by partitioning Give learners this problem: I had 783 ml of juice in the jug, I drank 419 ml. How much juice is left in the jug? Encourage learners to write down intermediate stages to help solve this problem. Ask them to explain their method for working out the solution. Discuss all methods as a class. Then display this problem from the Number sentence stories photocopy master, which includes the answer. Tell learners that, like in addition, when the numbers in a calculation are larger it is harder to keep track of the different parts of the calculation. For these types of problem a written method, rather than mental arithmetic, is often needed. Explain that some subtraction calculations can be approached using simple partitioning into hundreds, tens and units in the same way as with addition calculations. Write the following sequence and demonstrate it with place value materials: 746 – 232 = 700 – 200 + 40 – 30 + 6 – 2 Write the method for the whole class to see: 700 − 200 = 500 40 − 30 = 10 6−2=4 500 + 10 + 4 = 514, therefore 746 − 232 = 514 Core activity 2.3: Partitioning to add and subtract

Tell learners that sometimes we need to be more creative when we partition the numbers. For example, for the sum 728 – 552, when we partition each number separately we get: 728 = 700 + 20 + 8

and

552 = 500 + 50 + 2

700 − 500 = 200 But 20 − 50 = …? Here there is a problem because 20 − 50 is something that learners have not learnt how to do. Many learners will be tempted to find the difference between 20 and 50 and so will suggest the answer 30. Show learners 728 as 700 + 20 + 8 using place value equipment. Demonstrate that they have already started taking away (counting back) in this calculation, so it does not work to start counting on (finding the difference) halfway through. Point out that when we partition 728 into hundreds, tens and units, there are not enough tens to subtract 50, but we can create more tens by partitioning the number in a different way. Take one of the hundreds from 700 and break into bundles of 10, then put these bundles of 10 with the original two in 20. This makes twelve tens. Explain that they can do this by relating it back to place value: twelve tens is the same as 120. Allow learners to reflect on the fact that the number is still 728, but that it is now rearranged as 600 + 120 + 8. Refer back to the original question and ask a learner to now subtract 552 from the place value equipment. Now show the calculation as: 728 = 600 + 120 + 8 and 552 = 500 + 50 + 2 600 − 500 = 100 120 − 50 = 70 8−2=6 100 + 70 + 6 = 176, therefore 728 − 552 = 176 Give learners further examples where they will need to exchange and partition differently. Allow learners to use the place value equipment, or draw representations of the place value equipment to support them in consolidating this method of subtraction. Further examples should include calculations: ∑ where learners need to partition so that they break a ten into ones, e.g. 843 − 528; ∑ where they need to break hundreds into tens, and ten, into ones, e.g. 724 − 347; ∑ with 0 in the tens or units place value.

Unit 1A

2 Addition and subtraction (1)

Look out for! Learners who think it is impossible to subtract a larger number from a smaller number. Learners will have to ‘unlearn’ this idea when they start to calculate with negative numbers later. Try not to say “You cannot subtract 50 from 20” when explaining this method.

Look out for! Confident learners with excellent understanding of place value and of negative numbers. They might like to explore this further using the following method: 754 − 562 = ? 754 = 700 + 50 + 4

and

562 = 500 + 60 + 2

700 − 500 = 200 50 − 60 = −10 4−2=2 200 − 10 + 2 = 192, therefore 754 − 562 = 192

∑

Summary ∑

∑

Learners can add pairs of three-digit numbers. They can subtract a two-digit number from a three-digit number. They subtract pairs of three-digit numbers. ∑

∑

Check up! “Show me how to add 507 and 784.” “Show me how to subtract 365 from 598.”

Notes on the Learner’s Book Partitioning (p12): learners practise addition and subtraction using partitioning. They then solve worded problems using the method of their choice.

More activities Reverse and subtract (pairs) You will need 0–9 dice. Learners use a dice to generate three single digits. They arrange the digits to make a three-digit number then reverse the digits to make another three-digit number, e.g. 364 and 463. Subtract the smaller number from the larger. Try this with different pairs of three-digit numbers. What do they notice?

Partition poster (pairs) Learners make a poster showing how to partition in different ways to help with subtraction of three-digit numbers.

Games Book (ISBN 9781107685420) Subtract three in a row (p8) is a game for two players. The players calculate the difference between three-digit numbers to make numbers on a game board. They use subtraction and strategy skills to cover a row of three numbers.

Core activity 2.3: Partitioning to add and subtract

Complements-to-20 dominoes

Instructions on page 16

Original Material ÂŠ Cambridge University Press, 2014