CAMBRIDGE PRIMARY

Mathematics TeacherĂ s Resource

Emma Low

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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/9781107658547 © 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 978-1-107-65854-7 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

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Term 1 1A: Number and problem solving 1 The number system (whole numbers) 1.1 Revisiting place value 1.2 Ordering and rounding 1.3 Sequences (1) 2 Mental and written strategies for addition and subtraction 2.1 Addition and subtraction (1) 2.2 Adding more numbers 3 Mental and written strategies for multiplication and division 3.1 Multiplication and division facts 3.2 Written methods of multiplication 3.3 More multiplication 3.4 Written methods of division 4 Multiples, square numbers and factors 4.1 Multiples and squares 4.2 Tests of divisibility 4.3 Factors 1B: Geometry and problem solving 5 Shapes and geometric reasoning 5.1 Parallel and perpendicular 5.2 Triangles 5.3 Cuboids

1 2 4 6 11 12 18 25 26 30 34 36 41 42 44 46 49 50 54 58

6 Position and movement 6.1 Coordinates 6.2 Translation and reflection 1C: Measure and problem solving 7 Mass 7.1 Mass 8 Time and timetables 8.1 Telling the time 8.2 Timetables 9 Area and perimeter (1) 9.1 Area (1) 9.2 Perimeter (1)

65 66 70 77 78 85 86 90 95 96 100

Term 2 2A: Number and problem solving 10 Number and number sequences 10.1 Sequences (2) 10.2 General statement 10.3 Positive and negative numbers 11 Decimal numbers 11.1 The decimal system 12 Mental strategies 12.1 Decimal facts 12.2 Multiplication strategies 12.3 Doubling and halving 13 Mental and written strategies for addition and subtraction 13.1 Subtraction 13.2 Addition 13.3 Adding and subtracting money

105 106 110 112 119 120 123 124 126 130 135 136 140 144

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14 Written methods for multiplication and division 14.1 Multiplication and division 2B: Handling data and problem solving 15 Handling data 15.1 Questions and surveys 15.2 Examining data 16 Probability 16.1 Probability 17 Line graphs 17.1 Line graphs 18 Finding the mode 18.1 Finding the mode 2C: Measure and problem solving 19 Length 19.1 Measuring and drawing lines 20 Time (2) 20.1 Measuring time 20.2 Using calendars 21 Area and perimeter (2) 21.1 Area (2) 21.2 Perimter (2)

149 150 153 154 158 163 164 167 168 173 174 177 178 183 184 188 197 198 200

Term 3 3A: Number and problem solving 22 Number: mental strategies 22.1 Using mental strategies 23 Working with decimals 23.1 Working with decimals 24 Fractions, decimals and percentages 24.1 Percentages 24.2 Equivalent fractions, decimals and percentages 24.3 Mixed numbers and improper fractions

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205 206 209 210 213 214 218 220

25 Calculation 25.1 Addition and subtraction (2) 25.2 Fractions and division 25.3 Using inverse operations and brackets 26 Ration and proportion 26.1 Ratio and proportion 3B: Geometry and problem solving 27 Angles 27.1 Angles 28 Shapes and geometric reasoning (2) 28.1 Triangles (2) 28.2 Symmetry in polygons 28.3 Symmetry in patterns 28.4 3D shapes and nets 29 Position and movement 29.1 Coordinates and transformation 3C: Measure and problem solving 30 Capacity 30.1 Volume, capacity and mass 31 Time (3) 31.1 More about time 32 Area and perimeter (3) 32.1 Area and perimeter

223 224 226 228 231 232 237 238 245 246 248 252 254 263 264 269 270 277 278 285 286

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 resources 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.

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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.

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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.

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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 5 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 Introduction

vii

remembering a technique. The investigations and questions are written 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 5, 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 5. 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. ∑

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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

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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.

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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.

1A

1 The number system (whole numbers)

Quick reference Core activity 1.1: Revisiting place value (Learner’s Book p2) Learners consolidate previous learning and extend their understanding of the number system to include larger whole numbers. They use place value to partition numbers with up to six digits and use their understanding of place value to multiply and divide by 10 and 100. Core activity 1.2: Ordering and rounding (Learner’s Book p4) Learners round numbers to the nearest 10, 100 or 1000 and compare and order whole numbers using the > and < signs. Core activity 1.3: Sequences (1) (Learner’s Book p6) Learners work with different sequences which all have a constant ‘jump’ size. They work with sequences presented as grid patterns and number sequences, and extend their understanding of multiples.

Number

Ordering and rounding

Place value

Here are five digit cards.

2

7

Use three of these cards to make the missing number on the number line. 0

2

3

4

2

Prior learning This chapter builds on work in Stage 4 with numbers with up to four digits: ∑ Ordering, rounding, partitioning and multiplying and dividing by 10 and 100. ∑ Counting on and back in steps of 1, 10, 100 and 1000.

Objectives* –

4 5

0

5505

40 000

60 000

1

0

0

0

T

U

0

0

5045

5500

9

Number rounded to the:

Add further divisions to the number line.

1

nearest 10

nearest 100

nearest 1000

A

5500

5500

6000

B

5050

5100

5000

C

5050

5000

5000

D

5460

5500

5000

E

5510

5500

6000

(a) 3509 2

Write these numbers in figures.

?

"3

"3

0 1 2 3 4 5 6 7 8 9 3, 6, 9, 12, ... are multiples of 3.

The fifth number is 7 more than the fourth number. 1

2000

2500

3000

2500

2550

2600

2500

2505

Identify the number sequences shown on these grids. (a)

(c) 4655

Look at the number line.

(a) 3509

(c) one hundred and twenty thousand, two hundred and two 3

(b)

2505

(b) 3499

(c) 4655

Look at the number line.

2

2505

2505 is 2510 when rounded to the nearest ten.

(c) 790 320

Round these numbers to the nearest ten:

What number needs to be added or subtracted to change:

(a) 3509

(a) 36 473 to 86 473 in one step? 4

(b) 206 070 to 204 070 in one step? (c) 47 098 to 54 098 in one step?

(b) 3499

The grids at the top of page 7 have been torn so you can only see part of them. (a) What multiples have been shaded?

2510

(b) How wide might the grids be?

(c) 4655

The highest point in the world is Mount Everest in Nepal. It is 8848 metres above sea level. Round 8848 to the nearest hundred metres.

Use a calculator to check your answers. Unit 1A: Core activity 1.1 Revisiting place value

?

2505

Round these numbers to the nearest hundred:

(b) one hundred and five thousand and fifty Write these numbers in words. (b) 577 006

?

The second number is equal to 10 " 6. The third number is half way between the second and fourth numbers.

2505 is 2500 when rounded to the nearest hundred.

(a) three hundred and thirty-five thousand, two hundred and seventy-one

(a) 307 201

(b) 3499

?

"3

Look at the number line. Round these numbers to the nearest thousand:

(a) What does the digit 9 represent? (b) What does the digit 5 represent?

?

The fourth number is equal to 3 ! 10.

You may find it easier to take the five starting numbers and round them to the nearest 10, 100 and 1000.

2505 is 3000 when rounded to the nearest thousand. Look at this number: 950 302

Vocabulary

Use the clues to find the sixth number in this sequence.

5050

Match each number to the correct letter A, B, C, D or E in the following table.

hundred thousand: is 100 times larger than one thousand (100 ! 1000 " 100 000). Hth Tth Th H

5455

multiple: a number that can be divided exactly by another number is a multiple of that number. Start at 0 and count up in equal steps and you will find numbers that are multiples of the step size.

Let’s investigate

Here are five numbers:

ten thousand: is 10 times larger than one thousand (10 ! 1000 " 10 000).

!100

20 000

1

0

Sequences (1)

Let’s investigate

Vocabulary

Let’s investigate

4

Unit 1A: Core activity 1.2 Ordering and rounding

6

Unit 1A: Core activity 1.3 Sequences (1)

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: Number and the number system Count on and back in steps of constant size, extending beyond zero. Know what each digit represents in five- and six-digit numbers. Partition any number up to one million into thousands, hundreds, tens and units. Multiply and divide any number from 1 to 10 000 by 10 or 100 and understand the effect. Round four-digit numbers to the nearest 10, 100 or 1000. Order and compare numbers up to a million using the > and < signs. Recognise and extend number sequences. Recognise odd and even numbers and multiples of 5, 10, 25, 50 and 100 up to 1000. 1A: Problem solving (Using understanding and strategies in solving problems) 5Ps3 – Explore and solve number problems and puzzles. 5Ps8 – Investigate a simple general statement by finding examples which do or do not satisfy it.

5Nn1 5Nn2 5Nn3 5Nn5 5Nn6 5Nn8 5Nn12 5Nn13

– – – – – – – –

Vocabulary

*for NRICH activities mapped to the Cambridge Primary objectives, please visit www.cie.org.uk/cambridgeprimarymaths

ten thousand ∑ hundred thousand ∑ multiple

Cambridge Primary Mathematics 5 © Cambridge University Press 2014

Unit 1A

1

Core activity 1.1: Revisiting place value

LB: p2

Resources: Place value chart: 1–100 000 photocopy master (CD-ROM); prepare large version for class display. Place value crossword photocopy master (CD-ROM). Calculators. (Optional: 0–9 spinners (CD-ROM).) Multiplying and dividing by 10 At the start of the session, briefly revise place value using the Place value chart 1–100 000 photocopy master, making sure to cover the hundred thousand column. Ask learners what changes and what stays the same when you multiply 6 by 10. (Answer: The digit 6 stays the same but the place value moves to the left.) Demonstrate this on the place value chart. Ensure that learners understand how zero can be used as a place holder by partitioning numbers of up to five digits. For example, 23 806 is 20 000 + 3000 + 800 + 6. In 23 806 the zero makes sure that the ‘8’ and the ‘6’ are in the correct place; without the zero we would read ‘2386’, which is not correct. Explain that, so far, they have learned about numbers up to tens of thousands. “Now we are going to look at hundreds of thousands, which is 10 times larger than tens of thousands, or a hundred times larger than a thousand.” Ask learners to write a single-digit number in the centre of a sheet of paper. Tell them to keep multiplying their chosen number by 10 and to record the result they get each time in words and figures. Learners should stop multiplying when they have generated a six-digit number. Explain that a number with six digits has a size of hundreds of thousands. Ensure that learners understand that multiplying by 100 is equivalent to multiplying by 10 and then by 10 again. Extend the work to dividing by 10 and 100. Use the Place value chart 1–100 000 photocopy master, with the hundred thousand column visible, to demonstrate that when dividing by 10 each digit moves one place to the right, and when dividing by 100 each digit moves two places to the right.

Hundreds of thousands Write the number 985 432 on the board and ask: ∑ “What does the digit 9 represent?” (Answer: 9 hundred thousand) ∑ “What does the digit 8 represent?” (Answer: 8 ten thousands (or 80 thousand)) ∑ “What does the digit 2 represent?” (Answer: 2 units (or ones))

2

Unit 1A

1 The number system (whole numbers)

Vocabulary ten thousand: ten thousand is 10 times larger than a thousand (10 × 1000). hundred thousand: hundred thousand is 100 times larger than a thousand (100 × 1000). hundred thousand

ten thousand

thousand

hundred

ten

unit

HTh

TTh

Th

H

T

U

1

0

0

0

0

0

1

0

0

0

0

1

0

0

0

× 10 × 100

Example: multiplying 7 by 10 to generate a six-digit number. seven 7 seventy 70 seven hundred 700 seven thousand 7000 seventy thousand 70 000 seven hundred thousand 700 000

Practise partitioning numbers up to one million into thousands, hundreds, tens and units in this way until learners are comfortable with the larger numbers.

Look out for!

Learners who say, ‘To multiply by 10 add a 0.’ This Ask these additional questions to give learners more practice in working with numbers with ‘rule’ does not extend to decimals and should not be used. up to six digits. Learners may use a calculator to check their answers. Instead, insist that learners use the rule: ‘To multiply by ∑ “What number needs to be added or subtracted to change 35 873 to 95 873 in one step?” 10, each digit moves one place value to the left.’ Answer: 60 000 needs to be added) ∑ “What needs to be added or subtracted to change 209 050 to 202 050 in one step?” (Answer: 7000 needs to be subtracted) ∑ “If we partition 305 469 into expanded form, what number must go in each box? How do you know?” 305 469 = + 5000 + + + 9 (Answer: 300 000, 400, 60; partitioning a number gives a single digit and zeros to give the correct place value, starting with the largest place value) Set learners a challenge. Ask: ∑ “Does anyone know what I mean by one million? How do you write one million in figures?” (Learners might be unfamiliar with this terminology, or recognise it as 1 and 6 zeros; the correct answer is one thousand thousands.) ∑ How do you write half a million in figures? (Answer: 500 000) ∑ How do you say this number? (Answer: five hundred thousand)

Summary Learners have extended their knowledge of the number system and can work with whole numbers with up to six digits.

Check up!

Notes on the Learner’s Book Place value (p2): consolidates previous learning and extends learners’ understanding of the number system to include larger whole numbers. Give learners a copy of the Place value crossword photocopy master for question 7.

∑ ∑

∑

“Why do 3000 × 100 and 30 000 × 10 give the same answer?” “I have the number 456 000 showing on my calculator. What single calculation can I do to make 956 000?” “How can I change 456 000 to 416 000 in one step?”

More activities Target 100 000 (class) You will need a 0–9 dice or a 0–9 spinner (CD-ROM). Generate six digits and write them for the whole class to see. Learners use these digits to make two three-digit numbers. For example: 3, 5, 9, 2, 4 and 1 can be used to make 921 and 543. Learners write down their chosen numbers and then decide whether to multiply each number by 10 or 100. The aim is to get a total that is as close to 100 000 as possible when the two products are added together. For example: 921 × 100 = 92 100 and 543 × 10 = 5430; 92 100 + 5430 = 97 530 and 100 000 − 97 530 = 2470. Remind learners that the number pairs for 10 are useful here because they can be used to estimate number pairs to 100 000. The learner(s) whose total is closest to 100 000 scores one point. Generate a new set of numbers and continue the game.

Games Book (ISBN 9781107667815) Remove a digit (p6) is a game for two players. It encourages learners to consider the position, and therefore the value, of each digit in a whole number with up to six digits. Core activity 1.1: Revisiting place value

3

Core activity 1.2: Ordering and rounding

LB: p4

Resources: A large sheet of paper big enough to create a poster. (Optional: 0–9 spinners (CD-ROM).) Rounding to the nearest thousand Remind learners of the rules of rounding from Stage 4 (Unit 1A, chapter 1): to round a number to the nearest hundred look at the tens digit, if it is less than 5 the hundreds digit remains unchanged and the number is rounded down to the nearest hundred, if it is 5 or more round up. (Look at the digit in the place value to the right of the chosen place value when determining to round up or down). Write the numbers shown in the column to the right, for the whole class to see. Ask, “Which number do you think is closest to 2549 rounded to the nearest hundred?” Give learners time to discuss the question in pairs. Take feedback.

(Answer: 2549 is 2500 when rounded to the nearest hundred) Discuss that 2549 could also be rounded in different ways, for example: 2549 is 2550 when rounded to the nearest ten 2549 is 3000 when rounded to the nearest thousand. Learners should understand, from what they know about rounding to the nearest 10 or 100, that to round a number to the nearest thousand they need to look at the hundreds digit. If it is 5 or more, round up. If it is less than 5, the thousands digit remains unchanged. Still displaying the circled numbers from the start of the session, ask learners to imagine a reporter writing about the attendance at the match for a newspaper. “Which number would they use, the actual attendance figure or a rounded number? Why?” There is no correct answer but suggestions might include that in this kind of report an exact number probably isn’t important and a rounded number might be easier to visualise. Also, the reporter might choose to use a rounded number to make the attendance appear bigger in order to make the article more dramatic. Ask learners to round the following numbers to the nearest thousand and then put the rounded numbers in order from largest to smallest: 3990, 4500, 3495, 4090, 9550 (Answer: 10000, 5000, 4000, 4000, 3000) “What do you notice about two of the rounded numbers?” (Answer: 3990 and 4010 both round to 4000 to the nearest thousand)

4

Unit 1A

1 The number system (whole numbers)

2550

3000

2000

2549 2500

2600

Remind learners how to round using a number line to show that 2549 rounded to the nearest thousand is 3000. Example: 2549 is closer to 3000 than 2000. 2549 2000

2500

3000

Opportunity for display Collect examples from newspapers and magazines of numbers that are likely to be rounded.

Ordering six-digit numbers The review of place value in Core activity 1.1 should help learners order six-digit numbers. Make sure that they understand that the first digit represents the largest part of the number. Inform learners that they can partition the number first if they need to. Invite learners to order these six numbers, starting with the smallest: 250 000 260 000 300 000 254 900 255 000 200 000 Prompt them to realise that the answer can be written using the less than (<) symbol, as shown here: (Answer: 200 000 < 250 000 < 254 900 < 255 000 < 260 000 < 300 000)

Summary ∑ ∑

Learners can confidently round numbers to the nearest 10, 100 and 1000; and order and compare numbers using the < and > signs. Learners can order and compare numbers up to one million.

Notes on the Learner’s Book Ordering and rounding (p4): provides practice in rounding numbers to the nearest 10, 100 and 1000. Learners are given some facts about five famous mathematicians. They are asked to draw a time line and arrange the mathematicians’ dates of birth and death on it. Encourage learners to investigate the famous mathematicians listed. Help them to display the information they find in the form of a time line.

Check up! ∑ ∑ ∑

∑ ∑

Round 3568 to the nearest 10, 100 and 1000. (Answer: 3570, 3600, 4000) Round 9384 to the nearest 10. Is it less than or more than 9379 to the nearest ten? (Answer: they are equal) “Put the following numbers in size order, using the ‘less than’ or ‘more than’ symbol: 3647, 9540, 234, 9990” (Answer: 234 < 3647 < 9540 < 9990; or 9990 > 9540 > 3647 > 234) What is bigger, 999 800 or 998 900? (Answer: 999 800) “A newspaper reported that 5000 people attended the match. The organisers said that 4672 people were there. Explain the difference in numbers.”

More activities Nearest hundred (pairs) You will need a 0–9 dice or 0–9 spinner (CD-ROM). Each player rolls the dice or spins the spinner four times. Players record the digits in the order that they are generated to make one four-digit number. Each player then rounds their number to the nearest 100 and scores that number of points for the round. For example: 6

5

2

1

to the nearest hundred 6521

6500

6500 is 65 hundreds, so scores 65 points

The player with the most points after 10 rounds is the winner. You can adapt the game to rounding numbers to the nearest 10 or the nearest 1000.

Core activity 1.2: Ordering and rounding

5

Core activity 1.3: Sequences (1)

LB: p6

Resources: Number sequences photocopy master (p9). 1 cm2 square paper (or use Square paper photocopy master (CD-ROM)). Check up! photocopy master (p10). (Optional: Sequence cards photocopy master (CD-ROM).) Ask learners to imagine a grid of squares that is four squares wide. The number 1 is in the top left-hand corner, 2 is next to 1, and the numbers continue in order.

Vocabulary

Now ask: ∑ “Where would 5 be?” (Answer: first square in the second row) ∑ “What about 7? How did you work it out?” (Answer: third square in the second row) Choose other numbers, encouraging learners to explain each time how they worked out the number’s position in the grid. Possible suggestions might include counting the squares from left to right; or using multiples of the first digit in a column to fill the rest of the column.

For example:

“Imagine that the numbers in the 2× table are shaded on the grid. What pattern would they make?” (Answer: a vertical strip pattern) Give learners the following general statement: ‘All grids produce a strip pattern for multiples of 2’. Ask them to investigate if this general statement is true. If necessary, prompt learners by asking, “What if the grid was a different width?” (Link to work on general statements from Stage 4 (Unit 2A, chapter 9)). Allow time for feedback. The learners should agree that the general statement is not true; they should show the understanding that grids with an odd number of columns will not give a strip pattern for multiples of 2. Now invite learners to imagine a grid of squares that is six squares wide. They should imagine colouring in numbers that are multiples of 3 (i.e. numbers in the 3× table). Ask: ∑ “What pattern do you get?” (Answer: a vertical strip pattern) ∑ “How did you work it out?” Invite learners to draw the grids for themselves if they cannot visualise the pattern easily. Compare the patterns for the multiples of 2 and 3, “What do you notice? ” Establish that there is one square between each shaded square in a row for the multiples of 2, and two squares

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Unit 1A

1 The number system (whole numbers)

multiple: the result of multiplying a number by a positive whole number. Start at zero and count up in steps of the same size and you will find numbers that are multiples of the step size.

!3

0 0"3

!3

!3

3 1"3

6 2"3

3, 6, 9, 12 . . . are all multiples of 3

!3

9 3"3

12 4"3

between shaded squares for the multiples of 3. Learners should realise that they need to check the number of squares between two shaded squares on a row to determine the multiple. The multiple is one more than the number of blank squares between shaded squares in a row. Now consider different sizes of grid. Can the same vertical strip pattern be formed on a different size grid? Give learners time to experiment with drawing and colouring different sized grids. Encourage learners to realise that the width of the grid can vary and still produce the same pattern, provided the width of the grid is a multiple of the number being coloured. For example, for multiples of 3, grids of squares that are 3, 6, 9, etc. squares wide will produce the vertical strip pattern.

Example: multiples of 3 will form a vertical strip pattern on different grids, for example:

Explain to learners that the patterns they have been looking at form number sequences. If necessary, remind them that a sequence is a list/pattern of numbers arranged according to a rule. The sequences they have looked at so far have the following rules to determine what square to colour next: – multiples of 2, or counting on in twos starting at 2 – multiples of 3, or counting on in threes starting at 3.

Note: The space between the shaded squares on a row is two squares (one less than the multiple).

Ask learners to investigate the grids on the Number sequences photocopy master. Learners can either write their answers on the sheet or in their notebooks.

Take feedback on learners’ results. Remind learners again that number patterns such as these are called sequences. Recap, “In all the number sequences we have looked at so far we have: ∑ counted in steps of equal size ∑ started the count at a multiple.”

Number sequences photocopy master answers 1. (a) Multiples of 6 (b) Multiples of 3 (c) Multiples of 4 (d) Multiples of 5 (e) Multiples of 3 2. (a) Grid 1: multiples of 4; Grid 2: multiples of 5 (b) Grid 1: width of grid could be 9, 13, 17 . . . squares; Grid 2: width of grid could be 6, 11, 16 . . . squares

Establish that we do not always start the count in this way. We could have steps of 4 but start counting from a number which is not a multiple of 4. For example: 3, 7, 11, 15, 19 . . . (counting on in 4s) 15, 11, 3, 3, −1 . . . (counting back in 4s) Give similar sequences and ask learners to find the next number in the sequence.

Extend to counting on or back in multiples of larger numbers. For example: 50, 100, 150, 200, 250 . . . 300, 200, 100, 0, −100 . . . Give similar sequences and ask learners to find the next number in the sequence.

Core activity 1.3: Sequences (1)

7

Summary ∑ ∑

Learners are able to explore patterns of multiples. They can count on and back in steps of constant size from any start number, including working with negative numbers. Notes on the Learner’s Book Sequences (1) (p6): gives the opportunity to work with different sequences which all have a constant ‘jump’ size. Many of the questions also provide valuable tables practice. ∑

∑

Check up! “Noura writes down a sequence of numbers starting with 100. She subtracts 55 each time. What are the next two numbers in the sequence?” Display the Check up! photocopy master (p10) and say, “This grid has been torn so you can see only part of it. What multiples have been shaded? ” How wide might the grid be?”

More activities Sequence cards (for individuals or pairs) You will need a set of sequence cards from the Sequence cards photocopy master (CD-ROM); per learner or pairs of learners. Learners sort the cards into four sets. Each set shows a different part of the sequence (the ‘jump’ size is the same but the starting number is different). Learners give the rule for the sequence. (Answers: Sequence A – counting back in 2s; Sequence B – counting on in 5s; Sequence C – counting on in 3s; Sequence D – counting on in 25s)

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Unit 1A

1 The number system (whole numbers)

Number sequences 1. Look at the grids and identify the number sequences. (a)

(b)

………………………

………………………

(c)

(d)

………………………

(e)

………………………

………………………

2. These grids have been torn so you can see only part of the grid. (a) What multiples have been shaded? (b) How wide might the grids be? Grid 1

Grid 2

………………………

………………………

Instructions on page 7

Original Material © Cambridge University Press, 2014

Check up!

Instructions on page 8

Original Material ÂŠ Cambridge University Press, 2014

1A

2 Mental and written strategies for addition and subtraction

Quick reference A

I will count on in tens from 903.

C

I will count on in hundreds from 3.

B

I will count back in thousands from 9003.

If all three children count at the same speed, who will say 1003 first? 1

Adding more numbers

Vocabulary addition: to combine more than one number to make a total, or sum.

Let’s investigate

This is a magic square. Two of the numbers in this square have been swapped.

subtraction: to take away, or find the difference between numbers.

Find the two numbers and swap them back so that the magic square works.

total: the result when numbers are added together.

1

37 32 36

28

38

39

25

Every row and column, and the two diagonals, should add up to 130.

Alyssa, Anish and Axel have been raising money by doing chores for four months. Anish’s earnings 52

52

50

50

50

48

48

48

46

46

46

44

44

44

42

42

42

40

40

40

38

38

38

36

36

34 32 30 28 26 24 22 20

10

Axel’s earnings

52

34 32 30 28 26 24 22

36 34 32 30 28 26 24 22

20

20

18

18

18

16

16

16

14

14

12

12

10

10

8

8

6

6

4

4

2

2

0

Unit 1A: Core activity 2.1 Addition and subtraction (1)

27 31 30

Alyssa’s earnings

, 72 , 42, 52, , 418, 428 (a) , 332.5, , (b) 388 132.5, 232.5, 1089 , , 1189, 7 (c) , 9, 8207, 830 2 (d) 148 , 8107, , 976 , (e) , 9782, (f) 9802,

8

26 35 34

These graphs show how much they raised each month.

difference: the result when a number is subtracted from another number.

Complete the number sequences to open the safe.

Core activity 2.2: Adding more numbers (Learner’s Book p10) Learners develop strategies for finding the total of more than three two-digit numbers.

40 29 33

Money raised in $

Let’s investigate

Money raised in $

Addition and subtraction (1)

Money raised in $

Core activity 2.1: Addition and subtraction (1) (Learner’s Book p8) Learners use counting on and back in 1000s, 100s, 10s and 1s to help them add or subtract. They develop their strategies for adding and subtracting pairs of two- and three-digit numbers. Learners begin to develop ways of adding and subtracting numbers with one decimal place.

14 12 10 8 6 4 2

0

0

Jan. Feb. Mar. Apr.

Jan. Feb. Mar. Apr.

Jan. Feb. Mar. Apr.

Months of the year

Months of the year

Months of the year

Unit 1A: Core activity 2.2 Adding more numbers

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

Objectives* –

∑

1A: Calculation (Mental strategies) 5Nc8 – Count on or back in thousands, hundreds, tens and ones to add or subtract. 5Nc10 – Use appropriate strategies to add or subtract pairs of two- and three-digit numbers and numbers with one decimal place, using jottings where necessary. 1A: Calculation (Addition and subtraction) 5Nc18 – Find the total of more than three two- or three-digit numbers using a written method. 1A: Problem solving (Using techniques and skills in solving mathematical problems) 5Pt3 – Check with a different order when adding several numbers or by using the inverse when adding or subtracting a pair of numbers. 1A: Problem solving (Using understanding and strategies in solving problems) 5Ps2 – Choose an appropriate strategy for a calculation and explain how they worked out the answer.

∑

Prior learning

∑ ∑

This chapter builds on work in Stage 4 on different strategies for adding and subtracting: finding pairs of small numbers that equal 10 or 20; adding and subtracting multiples of 10 and 100; adding and subtracting near multiples of 10 and 100. Learners should be able to choose a suitable strategy for a particular problem such as subtracting a small number crossing 100, for example, 304−8. Learners should be able to partition numbers including numbers with one decimal place. Learners should know some different methods used for checking the results of adding and subtracting numbers.

*for NRICH activities mapped to the Cambridge Primary objectives, please visit www.cie.org.uk/cambridgeprimarymaths

Vocabulary addition ∑ subtraction ∑ total ∑ difference

Cambridge Primary Mathematics 5 © Cambridge University Press 2014

Unit 1A

11

Core activity 2.1: Addition and subtraction (1)

LB: p8

Resources: Addition and subtraction crosses photocopy master (p22); large version for class display. Addition and subtraction strategies for whole numbers The first part of this Core Activity revises learning from Stage 4 before progressing to adding two three-digit numbers. Use this to assess learners’ understanding and fluency of addition and subtraction and use professional judgement about how quickly they should move onto Stage 5 objectives. Display the Addition and subtraction crosses photocopy master. Choose a two-digit number, e.g. 72 and write this in the centre of the first cross. Create four groups of learners and inform them that they are either +1, +10, +100 and +1000, and that we need to complete the grid by counting on (or adding) 1, 10, 100 and 1000 in each direction on the cross, as indicated on the photocopy master. Learners discuss the sequence of numbers that their group would generate if they counted on from the start number. One group at a time, and one learner at a time, each learner calls out their number in the sequence for you to write in the appropriate place on the grid. Choose a four-digit number greater than 4000, e.g. 7342, and use the second grid on the Addition and subtraction crosses photocopy master to subtract 1, 10, 100 and 1000. You can change the groups or keep them the same, and repeat the activity as before. This activity can be modified to add/subtract other multiples such as 2, 20, 200, 2000; 5, 50, 500, 5000 and so on. Pairs of learners discuss advice for how to add or subtract 1, 10, 100 or 1000 to any whole number. Give time for learners to present their advice to the class. If necessary, remind learners of the

different strategies for adding and subtracting that they learnt in Stage 4 (Units 1A and 2A, chapters 2 and 10 respectively). Create a bulleted list that the whole class agrees on. Demonstrate that adding 600 is the same as counting on six 100s, and subtracting 4000 is the same as counting back four 1000s. Remind learners that counting on and back is often useful but it is not always the most efficient way of adding and subtracting. Challenge learners to use any suitable method to add the following pairs of numbers: 32 + 60 = (92)

12

Unit 1A

254 + 40 = (294)

543 + 300 = (843)

2 Mental and written strategies for addition and subtraction

80 + 128 = (208)

Vocabulary addition: to combine two numbers to make a total, or sum. subtraction: to take away, or find the difference between two numbers. total: the result when numbers are added together. difference: the result when a number is subtracted from another number.

Opportunity for display Display advice as posters to support learners in remembering methods of adding 1, 10, 100 or 1000. Addition and subtraction strategies from Stage 4: ∑ Counting on/back in hundreds, tens and ones. ∑ Using near doubles and compensating. ∑ Using number pairs of 10 or 20. ∑ Partitioning into hundreds, tens and units. ∑ Rearranging the order of the addition, e.g. largest to smallest numbers. ∑ Adding or subtracting near multiples of 10 to or from a three-digit number. ∑ Adding three numbers where the sum of two of the numbers is a near multiple of 10. ∑ Subtraction by finding the difference.

Ask pairs of learners to describe their methods to each other. They then try out each others’ strategies and decide which works the best for different problems.

Demonstrate how the same methods can be used to find the result of adding three-digit numbers that are not multiples of 10 or 100, because these numbers can be partitioned into multiples of 100, 10 and 1. Example: 543 + 342 can be calculated by partitioning into 543 + 300 + 40 + 2. This addition can be recorded using: ∑ a number line +300

543

+40

843

Look out for! Learners who have difficulty explaining their methods. Model how to say aloud the thought processes that you use when solving addition questions, include using known facts, counting on (perhaps using fingers to keep track of the counting), and checking that the answer is reasonable.

+2

883

885

∑

a vertical addition 543 +342 843 (+ 300) 883 (+ 40) 885 (+ 2) or with jottings in another way. Write this set of numbers in a circle for the whole class to see: 482 136 275 319 193 427 368 544 Ask learners to choose two numbers at a time to add. Remind them to partition one number to add to the other number using one of the methods shown. If appropriate, encourage learners to experiment with different ways of using jottings to support their calculations. Take feedback and check answers. If desired, record answers for the whole class to see. Now show how to use the same ideas of partitioning to work out the result of subtracting threedigit numbers that are not multiples of 10 or 100.

Support learners who wish to experiment with their own ways of jotting by asking them to consider whether their method is: ∑ Checkable ∑ Accurate ∑ Reliable ∑ Efficient.

Core activity 2.1: Addition and subtraction (1)

13

Example: 543 – 342 can be calculated by partitioning as 543 – 300 – 40 – 2 This subtraction can be recorded using: ∑ a number line −40

−2

201

∑

203

−300

243

543

∑

a vertical subtraction 543 −342 243 (−300) 203 (−40) 201 (−2) counting on from a smaller number (finding the difference) 342 + 8 = 350; + 100 = 450; + 90 = 540 + 3 = 543, 8 + 100 + 90 + 3 = 201 so 543 – 342 = 201

or with jottings in another way. Ask learners to choose two numbers from the set written in the circle and subtract the smaller from the larger. Remind learners to partition the smaller number each time. If appropriate, encourage learners to experiment with different ways of using jottings.

Look out for! Adding and subtracting decimal numbers Repeat the activity from the start of the session using the Adding and subtracting crosses photocopy master, but this time with a starting number that has one decimal place, e.g. 72.3, and 5283.9. Ask, “Do you use the same method you used for adding to a whole number? What is different when you add to a decimal number?” Discuss methods used as a class. Learners should realise that if they are adding a whole number then the decimal part of the original number stays the same and they can use the same methods they did before; they just need to remember to always write the decimal part in the answer.

14

Unit 1A

2 Mental and written strategies for addition and subtraction

Learners who are unsure about decimal numbers. Remind learners that the first decimal place is for ‘tenths’. If necessary use place value equipment to demonstrate what a ‘tenth’ is. Represent the starting number in the grid using a picture or equipment and show how to add 1s, 10s, 100s and 1000s to the number. Some learners may find it useful to think about this addition in context such as length in centimetres.

Ask, “What happens if I want to add 10.1 to 34.5?” Discuss suggestions as a class. Explain that here it is best to partition the number into tens, units and tenths, e.g. 10.1 = 10 + 0.1 34.5 = 30 + 4 + 0.5 First add up the tens (40), then add up the units (4), then add up the tenths (0.6). Then add the parts together: 44.6 “What if I wanted to subtract 10.1 from 34.5? How can I use partitioning to do this?” Discuss suggestions as a class. 34.5 = 30 + 4 + 0.5 10.1 = 10 + 0.1 Do subtractions in parts: 30 – 10 = 20; 4 – 0 = 4; 0.5 – 0.1 = 0.4. Then add these parts together: 24.4 Write this set of numbers in a circle for the whole class to see: 34.1 56.8 1034.6 4564.3 Ask learners to add 1, 10, 100 and 100 to each number. Then ask learners to choose just two numbers to add together. Encourage learners to use jottings to support their calculations. Now ask them to subtract a smaller number from a larger number. Choose learners to demonstrate to the class which two numbers they added together, or subtracted, and how. Discuss any errors. Make sure the learners know that it is useful to partition the numbers first and then add/subtract. Ask groups of learners to share the calculations and methods they have used, then set each other addition and subtraction questions to answer using jottings; these can be with whole numbers and/ or decimal numbers. Group members should check each others’ calculations by using the inverse operation.

Remind learners that addition and subtraction are inverse operations and can be used to check addition and subtraction solutions, e.g. ∑ check the results of adding numbers by subtracting one number from the total ∑ check subtraction by adding the answer to the smaller number in the original calculation.

Core activity 2.1: Addition and subtraction (1)

15

Summary ∑ ∑

Learners know that counting on and back in 1000s, 100s, 10s and 1s can help them add or subtract. Learners have developed strategies for adding and subtracting pairs of two- and three-digit numbers. Learners start to develop strategies for adding and subtracting numbers with one decimal place. ∑

Check up! “Here are two numbers: 176 and 438. Find the total and the difference. Explain your method.” (Answers: total 614, difference 262) Repeat for other three-digit numbers.

Notes on the Learner’s Book Addition and subtraction (1) (p8): provides practise with number patterns that can be completed by adding and subtracting 10 or 100. The number patterns are then used to solve problems and play games involving addition and subtraction of three-digit numbers.

More activities Total and difference (pairs) Learners choose two different three-digit numbers less than 500. They find the total and the difference of these numbers to give a new set of two numbers. They find the total and difference again to make a new set of numbers and repeat until one of the numbers is greater than 2000. They investigate which pairs of numbers get them to 2000 quickly and which take longer.

Games Book (ISBN 9781107667815) The counting game (p1) is a game for two or three players. Players move around a board adding or subtracting multiples of 10, 100 and 1000 to try to make a total as close as possible to 3000.

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Unit 1A

2 Mental and written strategies for addition and subtraction

Blank page

17

Core activity 2.2: Adding more numbers

LB: p10

Resources: Addition and subtraction dartboards photocopy master (p23); large version for class display. Coloured pencils. Camera. 0–100 number cards photocopy master (CD-ROM); with 0–9 removed. (Optional: calculators.)

Explain that we are going to extend what we did in the last session by adding together three or more two-digit numbers. Give each pair of learners a copy of the Addition and subtraction dartboards photocopy master and a set of differently coloured pencils (at least four different colours per learner). Pairs of learners take turns to close their eyes and make three marks with their coloured pencil on Dartboard (1). They then add up their score using a written method without showing their partner. Tell learners to say their calculation and method aloud, so that they can discuss it with their partner. Extension: Once the learners are confident, ask them to repeat the activity but this time making four marks so that they are adding up four two-digit numbers. Encourage learners to add sets of numbers in different ways to check their solutions.

Ask learners to feed back the strategies they used to add the numbers. These should include finding pairs of numbers that add to 10 or 20; as well as other strategies used to add two two-digit numbers in Stage 4. List useful strategies on the board. Ask learners which of the strategies could be used or adapted to help them add threedigit numbers. Discuss responses. Learners should discover that they can apply previous knowledge and methods to larger numbers. Learners repeat the activity with Dartboard (2), discussing their strategies so that they can feed back to the class. Again, learners start by making three marks and move onto making four marks if/when they are ready. Learners can start by using a written method and move to mental methods when they feel more confident. If they use mental methods, encourage them to check their solutions by adding the numbers again, in a different order, or using subtraction.

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Unit 1A

2 Mental and written strategies for addition and subtraction

Learners will need to remember the addition and subtraction strategies learned from Stage 4 for this session. (A reminder list in given in Core activity 2.1, p12).

Opportunity for display Display a large copy of ‘Dartboard (1)’ from the Addition and subtraction dartboards photocopy master. Ask learners to write down one of the sets of three two-digit numbers they found the total for, and the strategies they used, to display next to the dartboard.

Stock taking The context of this activity is a ‘stock take’ to find out how many pencils there are in the classroom or school so that the class can plan an order of new pencils. Explain that it is time to order new resources for the classroom or school and that this class will be ordering pencils. Ask for ideas for how they could find out how many should be ordered. Agree that as part of solving the problem, learners will need to find out how many pencils there currently are in the classroom or school. Organise the class into five or six groups. Each group has the responsibility of finding out how many pencils there are in a particular area of the classroom or school. Remind groups that they will need to find a way of counting efficiently and accurately. Learners organise themselves to find and count the pencils in their area.

Once all groups have found the total number of pencils in their area, ask each group to write this number on the class whiteboard (or equivalent large display at front of the class). Learners can write the number down anywhere they like; this might lead to a very random and disorganised arrangement of numbers. Ask learners for strategies to help them find the total of all the pencils. Challenge learners to use a strategy that they know to add the numbers together and to write down their solution. Explain that it is necessary to be very organised when adding so many numbers; if the numbers on the board are disorganised, explain how it would be easy to miss out a number, or count one more than once if there is no structure to how they’ve been recorded. Suggest that one way to be organised is to list the numbers vertically so that all the place values line up, e.g. 28 34 27 26 35

Adapt the activity to a context that is relevant to your classroom situation, e.g. books in different sections of the library or numbers of lunch boxes in each class, to create a realistic problem. The context chosen needs to provide approximately six sets of items that have between 20 and 100 objects in them.

As necessary, suggest strategies for efficient counting, such as grouping objects into tens.

Opportunity for display Take photographs of the learners counting and display with the addition that they have carried out. Save the photographs and calculation to refer to when the learners are developing more formal written addition strategies in chapter 13.

Look out for! Learners who find it challenging to add multiples of 10. Learners practised strategies for adding multiples of 10 in Stage 4. If learners find this difficult, revisit Stage 4 resources to support them.

Explain that a useful way to check a calculation is to first estimate the answer. One way to do this quickly is to add just the ‘tens’ part of the number from each group. As the numbers are arranged vertically it is easy to see which part is the ‘tens’ of each number and these can be added using whatever strategy the learner prefers: 20 + 30 + 20 + 20 + 30 = 120 Core activity 2.2: Adding more numbers

19

Tell learners that you know the total number of pencils will be greater than this, because the ‘units’ still need to be added on. Now ask learners to write down the numbers from the board in a vertical list and to estimate how many pencils there are. Then ask them to add up the numbers. Show learners how you could record the calculation vertically, e.g. add up the tens first, then the units (demonstrate the addition using methods that you have agreed are useful and efficient with the learners): 28 34 27 26 + 35 12 (tens) (=120) + 30 (units; which is 3 tens = 30) 150 total number of pencils (12 tens + 3 tens = 15 tens) Ask learners to discuss their estimate compared to the actual answer; in most cases the estimate should be lower than the real value. If the estimate is much lower than the real value, discuss how a better estimate might have been made by first rounding each number to the nearest 10 and then counting the decades. Ask learners to compare this written method and solution with the method they used and their solution. Remind learners that if they have added the same numbers in a different order and got the same solution, then this works as a check that the solution is correct. Explain that this vertical written method is useful because it makes it easy to estimate first (the answer will be around 120) and the numbers are organised so that it is easier to use place value to add them together.

Summary ∑ ∑

Learners have developed strategies for finding the total of more than three two-digit numbers. Learners begin to adapt strategies to find the total of more than three three-digit numbers. Notes on the Learner’s Book Adding more numbers (p10): learners solve problems involving adding more than three numbers in the context of data handling, money, mass and length.

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Unit 1A

2 Mental and written strategies for addition and subtraction

Check up! Shuffle a set of two-digit number cards and place the pile face down on a table. Ask learners to take four cards from the pile, then ask: ∑ “What is the total? How did you work it out? Why did you use this strategy?”

More activities Page numbers (class)

∑

You will need a book with at least 99 pages; or up to 300 pages (not too heavy).

∑

Ask learners to choose a suitable book in the classroom. As you flick through the pages ask them to stop you in four places. Give the learners the four page numbers that you stopped on and ask them to find the total of the numbers. Tell the learners the number on the last page in the book. Ask them to find four page numbers in the book that add up to the last page number.

Make my number (class) You will need a set of 0–100 number cards (CD-ROM). Choose a number between 100 and 300. Ask learners to find four two-digit number cards that add up to the given number.

Add 10 (individuals) You will need a set of calculators. Ask learners to add together 10 random two-digit numbers using a calculator. Allow learners to explore the memory function of the calculator and how it can be used to help with addition and subtraction.

Games Book (ISBN 9781107667815) Making 100 (p4) is a game for two or more players. Players choose and then add together four two-digit numbers to make a total close to 100. Making 1000 (p4) is a game for two or more players. Players choose and then add together four three-digit numbers to make a total close to 1000.

Core activity 2.2: Adding more numbers

21

Instructions on page 12

− 1000

+ 1000

− 100

+ 100

−1

+1

Original Material © Cambridge University Press, 2014

− 10

+ 10

Addition and subtraction crosses

Addition and subtraction dartboards Dartboard 1 25

20

21

Dartboard 2 834

18

503 42

235

13

29

745

325

24

12

200

823

14

26

154

45

11

10

43

543

15

28 16

22 13

27 19

Instructions on page 18

23

23

100 687

547 375

432 286

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Original Material ÂŠ Cambridge University Press, 2014

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1A

3 Mental and written strategies for multiplication and division

Quick reference Vocabulary

Core activity 3.1: Multiplication and division facts (Learner’s Book p13) Learners use problem-solving skills to answer a series of questions involving multiplication and division. Core activity 3.2: Written methods of multiplication (Learner’s Book p16) Learners practise multiplying three-digit whole numbers by single-digit numbers. They also multiply a number with one decimal place by a single-digit number.

operation: something you do to a number. !"#$ are all mathematical operations. inverse: having the opposite effect. ! 9 is the inverse of " 9 #5 is the inverse of $ 5. inverse operations: operations that ‘undo’ each other if applied to a number one after the other. For example,

Multiplication and division facts

Written methods of multiplication

Let’s investigate

Let’s investigate

These numbers follow a pattern. 63

8

? 4

56

121

?

3 " 363

Use the first triangle to find the rule.

Look at the numbers on each end of a line passing through the circle.

Then use the rule to complete the other two triangles.

9

363

777

121

3

482

9

10 " 2 ! 2 % 10

The bottom number in each pattern is the sum of the two middle blocks.

product

(c) 512 ! 7

!

30

9

(d) 936 ! 8

(e) 671 ! 9

(f) 384 ! 6

10 5

300 150

90 45

The last digit is 28 $ 7.

5

0.7

20

2.8

22.8 is close to 24, so 22.8 is a reasonable answer.

The sum of the middle two digits is 4. All the digits are in the 2# table. Two digits are multiples of 4.

Start with a subtraction.

Remember to show all your working

(a) 4.9 ! 5

(b) 6.3 ! 7

(c) 3.8 ! 8

(d) 5.7 ! 9

(e) 4.3 ! 6

(f) 4.5 ! 9

What is the code number?

3 Calculate double 15.5

Turn the page for more questions.

4 A packet contains 1.5 kg of rice.

Try using the inverse operation.

? # ? $ 136

so 124 # 5 $ 24 r 4

Remember to show all your working.

(a) 336 # 7

(b) 387 # 9

(c) 444 # 6

(d) 392 # 8

(e) 332 # 4

(f) 406 # 7

(a) 567 # 5 (b) 94 ! 35

(c) 87 ! 48

(d) 26 ! 56

(e) 58 ! 63

(f) 74 ! 42

1

?

?

?

?

6

0

(c) 515 # 9

160 # ? $ 8 He places the same number of stickers on each page. How many stickers does Hamid place on each page? 5 Plants are sold in trays. Each tray holds 12 plants.

!

0

(b) 396 # 7

3 What is the missing number? 4 Hamid has 104 stickers. He has 8 pages in his sticker album.

Discuss with your partner the most efficient way of working it out. 3 Use the digits 0, 2, 3 and 5 to complete this calculation.

How much rice is in five packets?

?

2 Estimate first, then calculate:

(a) 34 ! 27

2 Calculate 37 ! 25.

Use this method, or any other method, to work out the following:

The first digit is double the last digit.

61

1 Use any method to calculate: 20 # 2.8 " 22.8

?

124 100 ! 20 " 5 24 20 ! 4 " 5 4 24 " 5

1 Estimate first, then calculate:

450 " 135 # 585

last digit

You must use each digit only once.

A"B

The example below uses the grid method to find the product of 39 and 15.

(b) 426 ! 3

4

43

39

7

(a) 318 ! 2

!

Use the digits 2, 5, 7 and 9 to make a correct calculation. B

The example below uses repeated subtraction to divide 124 by 5. 25

Estimate: 5.7 ! 4 is approximately 6 ! 4 " 24 first digit

She has these clues:

A!B

A

Find the missing numbers.

2 This example shows one method to caluclate 5.7 ! 4

7#3$3%7

Let’s investigate

The top number in each pattern is the product of multiplying the two numbers in the middle blocks.

1 Estimate first, then calculate:

1 Sara is trying to find the code number to unlock a treasure chest.

Written methods of division

Let’s investigate

product: the answer you get when you multiply two or more numbers together.

7 ! 8 " 56

What number goes in the middle? Knowing your times tables will help you to solve the following questions.

More multiplication Vocabulary

The numbers in the triangles are connected by a rule.

28

Fatima needs 160 plants for her garden. How many trays must Fatima buy?

4 Calculate 13 ! 13 and 31 ! 31. What do you notice about the results? Unit 1A: Core activity 3.1 Multiplication and division facts

13

16

Unit 1A: Core activity 3.2 Written methods of multiplication

18

Unit 1A: Core activity 3.3 More multiplication

Unit 1A: Core activity 3.4 Written methods of division

19

Core activity 3.3: More multiplication (Learner’s Book p18) Learners use written methods to multiply pairs of two-digit numbers. Core activity 3.4: Written methods of division (Learner’s Book p19) Learners use methods of grouping and repeated subtraction for division and round answers with remainders according to context.

Prior learning ∑

∑

This chapter builds on work in Stage 4 on multiples and knowledge of multiplication and division facts for the 2, 3, 4, 5, 6, 7, 8, 9 and 10 times tables. The chapter extends earlier work on multiplication and division of two-digit numbers by singledigit numbers.

Objectives* – 5Nc3 – 5Nc5 – 5Nc20 5Nc21 5Nc22 5Nc23

– – – –

5Nc25 – 5Nc26 – 5Pt6 – 5Pt7 – 5Ps2 – 5Ps9 –

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: Calculation (Mental strategies) Know multiplication and division facts for the 2 × to 10 × tables. Recognise multiples of 6, 7 8 and 9 up to the tenth multiple. 1A: Calculation (Multiplication and division) Multiply or divide three-digit numbers by single-digit numbers. Multiply two-digit numbers by two-digit numbers. Multiply two-digit numbers with one decimal place by single-digit numbers, e.g. 3.7 × 7. Divide three-digit numbers by single-digit numbers, including those with a remainder (answers no greater than 30). Decide whether to group (using multiplication facts and multiples of the divisor) or to share (halving and quartering) to solve divisions. Decide whether to round an answer up or down after division, depending on the context. 1A: Problem solving (Using techniques and skills in solving mathematical problems) Estimate and approximate when calculating – using rounding and check working. Consider whether an answer is reasonable in the context of a problem. 1A: Problem solving (Using understanding and strategies in solving problems) Choose an appropriate strategy for a calculation and explain how they worked out the answer. Explain methods and justify reasoning orally and in writing; make hypotheses and test them out.

Vocabulary inverse ∑ operation ∑ inverse operations ∑ product ∑ divisor

*for NRICH activities mapped to the Cambridge Primary objectives, please visit www.cie.org.uk/cambridgeprimarymaths Cambridge Primary Mathematics 5 © Cambridge University Press 2014

Unit 1A

25

Core activity 3.1: Multiplication and division facts

LB: p13

Resources: Blank multiplication grid photocopy master (p39); large verion for class display. (Optional: Multiple maze photocopy master (CDROM). Counters. One-minute test photocopy master (CD-ROM).)

Display the Blank multiplication grid photocopy master for the whole class to see.

Vocabulary

Ask learners to help you fill in the grid. Help learners to discover that: ∑ All answers except square numbers appear in two places on the grid, so the grid is symmetrical (e.g. 3 × 4 = 4 × 3). ∑ You can use known facts to help you derive unknown facts (e.g. you know that 10 × 9 = 90, so halve it to get 5 × 9 = 45). ∑ You can use doubling facts (e.g. the 6× table is double the 3× table so if you know that 3 × 3 = 9 you know that 6 × 3 = 18).

operation: something you do to a number; adding, subtracting, multiplying and dividing are all mathematical operations.

Ask learners to work in pairs to draw and complete their own multiplication grids, checking that they agree on the numbers.

If you created a display of multiplication facts for the 2×, 3×, 4×, 5×, 6×, 9× and 10× tables during Stage 4 (Unit 1A, chapter 3), you might want to display this now and remind them of the facts they knew last year. For example: ∑ × 2 multiples are even numbers ∑ × 5 multiples end in 0 or 5 ∑ × 10 multiples end in 0 ∑ × 3 multiples have digits that add up to a multiple of 3. Challenge learners to tell you what they notice about this set of numbers: 28

36

40

12

48

4

Give pairs of learners time to discuss what they notice.

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Unit 1A

3 Mental and written strategies for multiplication and division

inverse: having the opposite effect; adding is the opposite of subtracting; multiplying is the opposite of dividing. inverse operations: operations that ‘undo’ each other if applied to a number one after the other. For example: 10 – 2 + 2 = 10 7×3∏ 3 = 7

Look out for! Learners who do not understand the meaning of ‘lots of’ or ‘groups of’ so do not make links to table facts. Learner might have come across this language outside of school. Try to find examples of these expressions in magazines, newspapers or in advertising and explain to them that they are another way of saying ‘multiply’.

Take learners’ responses. These may include: ∑ All the numbers are even. ∑ All the numbers are multiples of 2 and 4. Ensure that learners can give multiples from other tables: call out numbers and ask learners to tell you what they are multiples of; or ask learners to list two or three multiples of 2, 3, 4, 5, 6, 7, 8, 9 and 10. Once learners have completed their multiplication grid, ask, “How can you use the multiplication grid to help you work out division facts?” Try some examples with the class such as: ∑ all even numbers are divisible by 2 ∑ all numbers ending in 0 or 5 are divisible by 5 ∑ all numbers ending in 0 are divisible by 10 and so on, ∑ if 3 × 4 = 12, then we know that 12 is divisible by 3 and 4 ∑ if 20 ∏ 10 = 2 then 20 ∏ 5 = 4 (this relationship is a little more tricky because you need to double the answer rather than halving it). Remind learners of the word ‘inverse’ and that multiplication and division are inverse operations.

∑

Summary ∑

Learners have extended their knowledge of table facts up to the 10× table and are able to derive corresponding division facts. They can recognise multiples up to the tenth multiple. Notes on the Learner’s Book Multiplication and division facts (p13): encourages learners to remember and implement useful multiplication and division facts, that they should know, and implement their problem-solving skills.

∑

∑

Check up! “If you multiply me by 7, you will get 56. What number am I? How do you know?” (Answer: 8) “Tell me five multiples of 8 that are less than 80.” (Answer: any five of 8, 16, 24, 32, 40, 48, 56, 64, 72)

Core activity 3.1: Multiplication and division facts

27

More activities One-minute test (individuals) You will need the One-minute test photocopy master (CD-ROM); one per learner. Learners have one minute to complete as many calculations as they can. They use their multiplication grid to check their answers, then record their score. Repeating the tests over a period of time allows learners to try to improve their scores.

Multiple maze (individuals or pairs) You will need the Multiple maze photocopy master (CD-ROM); one per learner. Learners follow the instructions to find a route through the maze. Not all multiples are in the 10 Ă— 10 tables so learners will need to use strategies to find them. These strategies may include doubling and halving, using known facts to find other facts or continuing number sequences. (Answers: Alien 7 = D; Alien 6 = C; Alien 9 = A; Alien 8 = B)

Games Book (ISBN 9781107667815) Multiplication bingo (p7) is a game for the whole class. It provides players with an opportunity to recall table facts to 10 Ă— 10.

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3 Mental and written strategies for multiplication and division

Blank page

29

Core activity 3.2: Written methods of multiplication

LB: p16

Resources: (Optional: 0–9 digit cards (CD-ROM); with the zero card removed.)

Write the following calculation for the whole class to see: 639 × 5 Ask, “What if I were to ask you for the product of six hundred and thirty nine times five?” “What do I mean by ‘product’?” Collect responses from the class and agree/explain what is meant by product. Challenge learners to find the product of 639 and 5. Learners can use any method to work out the answer. They record their chosen method and the result.

Discuss the methods learners have used for this calculation. Ask, “What method did you use? How does it work?” Remind learners of the multiplication methods that they have used before: repeated addition; multiplying by 10 and halving; partitioning; the grid method, and standard written methods. Remind learners always to start by estimating the answer. For example: Estimate 639 × 5: 600 × 5 = 3000 700 × 5 = 3500 So the answer to 639 × 5 will be between 3000 and 3500.

30

Repeated addition 639 + 639 + 639 + 639 + 639 = 3195

Multiply by 10 and halve 639 × 10 = 6390 6390 ∏ 2 = 3195

Partitioning (600 × 5) + (30 × 5) + (9 × 5) = 3000 + 150 + 45 = 3195

Standard written methods 600 + 30 + 9 ×5 3000 600 × 5 150 30 × 5 45 9 × 5 3195

Unit 1A

3 Mental and written strategies for multiplication and division

639 ×5 3000 150 45 3195

639 ×5 3195

Vocabulary product: the answer you get when you multiply two or more numbers together. product 7 × 8 = 56

Grid method ×

600

30

9

5

3000

150

45

Look out for!

(The grid method is useful because it can be extended later to multiplication by larger numbers, decimals and algebraic expressions.) ∑

3000 + 150 + 45 = 3195

Make sure learners understand that some methods are only appropriate for certain calculations. “Would multiplying by 10 and then halving be useful for multiplying 639 by 7?” (Answer: multiplying by 10 and halving only works for multiplying by 5) Show how to use the grid method to multiply decimals. If necessary, use the place value chart from Stage 4 (Unit 2A, chapter 9) to remind learners about decimal numbers. For example: Estimate first: 4.9 × 3 is approximately 5 × 3 = 15 Then use the grid, x

4

0.9

3

12

2.7

∑

Learners who give unreasonable answers. Encourage learners to make an estimate before doing the calculation so they can check that their calculation ‘looks right’. Learners who use unsuitable methods, such as repeated addition of small multiples. Emphasise the importance of choosing the most efficient multiplication method.

Example: place value order table with decimal point. Th

H

T

U

.

tenths

hundredths

12 + 2.7 = 14.7

Discuss other possible methods. Give learners time to practise them with different numbers. Make sure learners are confident with partitioning into tenths and multiplying tenths by a single-digit number. Example: multiplying tenths by a single-digit number. 3 × 0.9 multiply 0.9 by 10 to get an easier 3 × 9 = 27 calculation 3 × 0.9 = 2.7 divide by 10 to get the answer

Core activity 3.2: Written methods of multiplication

31

Summary ∑ ∑

Learners can estimate and then find the product of a three-digit number and a single-digit number. Learners can multiply a number with one decimal place by a single-digit number. Notes on the Learner’s Book Written methods of multiplication (p16): provides practise in multiplying with whole numbers and decimals. Questions 3–5 are set in context and questions 6–10 provide an opportunity to solve problems. Learners can be encouraged to work in pairs to solve these problems.

Check up! ∑

“How could you calculate 146 × 7? Explain your method.” ∑ Provide worked examples that contain errors. For each example, ask: ∑ “Is this correct?” ∑ “How do you know?” ∑ “How could you put it right?” Possible examples: (1) 325 × 5 = 1300; estimate 300 × 5 = 1500 325 + 325 + 325 + 325 (Answer: incorrect; 325 has only been added four times) (2) 3.6 × 7 = 63; estimate 4 × 7 = 28 3 0.6 × 21 + 42 = 63 7 21 42 (Answer: incorrect; 42 needs to be divided by 10 first)

More activities Greatest product (pairs) You will need a set of 0–9 digit cards (CD-ROM) with the zero card removed; per pair of learners. One player shuffles the cards and deals out four cards to each player. Players arrange their cards like this (ask them to write a multiplication sign on a piece × of paper): Players work out the answer to their calculation. The player with the greatest product scores one point. The winner is the player with the most points after a set time or a set number of rounds.

Decimal multiplication (pairs) Adapt the game above to use three cards and a grid like this:

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3 Mental and written strategies for multiplication and division

∑

×

Blank page

33

Core activity 3.3: More multiplication

LB: p18

Resources: No specific resources are required for this session. Remind learners of the need to estimate before multiplying. Encourage them to always consider if their answer seems reasonable based on their original estimate. Show the grid method for working out 39 × 5. Suppose we now want to multiply 39 by 15. We start by estimating the answer: 40 × 10 = 400 40 × 20 = 800

Example: Grid method for 39 × 5 ×

30

9

5

150

45

150 + 45 = 195

So, we can assume that the answer to 39 × 15 will be between 400 and 800. Demonstrate how the grid method can be used to multiply 39 × 15. ×

30

9

10

300

90

5

150

45

450 + 135 = 585

Ask learners: ∑ “Does the answer seem reasonable?” (Yes, it’s within the estimate.) ∑ “Do you think the grid method we have used is the most efficient for multiplying these numbers?” ∑ “Can you think of other methods we could use?” Discuss learners’ suggestions, which may include: so

39 × 10 = 390 39 × 5 = 195 (determined by halving) 39 × 15 = 390 + 195 = 585

In pairs, challenge learners to use any suitable method to calculate: 48 × 24 35 × 19 27 × 16

Discuss the methods used, including any methods that are particularly suitable for these numbers.

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3 Mental and written strategies for multiplication and division

Example: Other methods to discuss For 35 × 19: 35 × 19 = (35 × 20) – (35 × 1) = 700 – 35 = 665 For 27 × 16: double 27 = 54 double 54 = 108 double 108 = 216 double 216 = 432

(× 2) (× 4) (× 8) (× 16)

Summary Learners can confidently find the product of two two-digit numbers. Notes on the Learner’s Book More multiplication (p18): requires learners to work through a series of questions that need them to find the product of two two-digit numbers. In all instances, learners should be encouraged to use the most efficient method for carrying out the calculations.

∑

Check up! ∑

“Talk through your method of calculating 28 × 49.” “How would it be different if I calculated 49 × 28?”

More activities Make 240 (pairs) Learners work in pairs to investigate which two consecutive numbers multiply together to give 240. Each learner then writes down a similar calculation involving consecutive numbers for their partner to answer.

Core activity 3.3: More multiplication

35

Core activity 3.4: Written methods of division

LB: p19

Resources: Thinking about division photocopy master (p40). (Optional: 0–9 digit cards with the zero card removed (CD-ROM).) Write the following for the whole class to see: 12 ∏ 4 Ask learners what this expression means to them. Encourage learners to draw a picture or write something down. Give them time for this, then take feedback. Learners might think of this as dividing into quarters, sharing between 4, grouping in 4s or as repeated subtraction. Use the Thinking about division photocopy master to remind learners that grouping uses multiplication facts and multiples of the divisor to divide, and sharing involves halving, quartering or breaking into thirds etc. Demonstrate how grouping and repeated subtraction are related: repeated subtraction involves counting backwards in multiples of the divisor. Explain that we can use both these ideas to help us divide. Remind learners that these ideas were covered in Stage 4 and explain that now we are going to extend them to divide a threedigit number by a single-digit number. Stress the importance of estimating an answer before doing the calculation. Write the following for the whole class to see: 124 ÷ 4 Demonstrate how learners can use the division methods that they already know: ∑ 124 ÷ 2 = 62; 62 ÷ 2 = 31 (half and half again) ∑ 124 – 4 – 4 – 4 – 4 … = 31 (repeated subtraction) ∑

(number line) 0

4

8

12

112

116

120

124

Explain that when you divide a three-digit number by a single-digit number these methods might not always be appropriate. For example, repeated subtraction and a number line might get too time consuming. Some useful written methods for division use partitioning:

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3 Mental and written strategies for multiplication and division

Vocabulary divisor: the number that another number is divided by. 25 ∏ 5 divisor

Example: methods for division covered in Stage 4. ∑ Half and half again. ∑ Using a number line. ∑ Repeated subtraction.

∑

124 ÷ 4 100 + 20 + 4 ÷4 25 (100 ÷ 4) 5 (20 ÷ 4) 1 (4 ÷ 4) 25 + 5 +1 = 31

∑

124 ÷4 25 5 1 31

Ask learners to calculate 124 ∏ 5 using either the grouping method or the sharing method, and to explain why they chose that method. What do they notice? (Answer: 5 will not divide exactly into 124) It divides with a remainder. If learners struggled to get this solution, work them through the written method shown here on the right. Now put the calculation 124 ∏ 5 in context to discuss rounding the answer following a division. For example, “124 eggs are packed in boxes of 5. How many boxes are needed?” We know from the earlier calculation that the exact answer is 24 with remainder 4. Ask, “Should we round our answer up or down to decide how many boxes we need? Why?” We can fill 4 out of the 5 spaces in the last box, so it makes sense to round up, rather than throw the 4 eggs away. (Answer: 25 boxes – the last box would contain only 4 eggs) Explain to learners that sometimes we round up and sometimes we round down depending on the context. In this case, we were able to round up to 25 boxes because the question only asks for the number of boxes, not the number of full boxes. If the question had asked, ‘How many boxes could we fill with 124 eggs?’ then the answer would be 24 boxes; we would round down as the 4 eggs left over would not fill a box.

Example: method for dividing 124 ∏ 5. 124 100 – 24 20 – 4

20 × 5 4×5 24 × 5

so 124 ∏ 5 = 24 r 4

Core activity 3.4: Written methods of division

37

Summary ∑ ∑

Learners understand division as sharing and grouping. They can confidently divide a three-digit number by a single-digit number and round answers up or down depending on the context. Notes on the Learner’s Book Written methods of division (p19): provides further practise in division. Learners must round up their answer to question 5, which is set in the context of buying trays of plants.

Check up! Provide a calculation, then ask: ∑ “What answer do you expect? How did you make your estimate?” ∑ “Can you explain your method?”

More activities Smallest answer (pairs) You will need a set of 0–9 digit cards (CD-ROM). Ask learners to create a ∏ sign on a sheet of paper or in their books. One player shuffles the cards and deals out four cards to each player. Players arrange their cards like this: ∏ Players work out their calculation. The player with the smallest answer scores one point. The winner is the player with the most points after a set time or a set number of rounds.

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3 Mental and written strategies for multiplication and division

! 1 2 3 4 5 6 7 8 9 10

1

Instructions on page 26

2

4 5

6

7

Blank multiplication grid 3 8

9

10

Original Material ÂŠ Cambridge University Press, 2014

12 ÷ 4 = Grouping 12 grouped into 4s

Sharing 12 shared between 4

4

4

0

0

8

8

12

12

16

16

20

20

24

24

Instructions on page 36

Original Material © Cambridge University Press, 2014

196 ∏ 6 = 32 r 4

Some calculations do not work out exactly so there is a remainder. 196 4 4 4 − 60 10 × 6 13 ÷ 4 = 3 remainder 1 0 1 5 9 13 136 − 60 10 × 6 This method can be extended to working with − 76 larger numbers. − 60 10 × 6 For example, 196 ∏ 6 can be shown by repeatedly 16 subtracting 6 but it is more efficient to take off − 12 2×6 larger ‘chunks’. 4 32 × 6

6 jumps of 4 24 ÷ 4 = 6

Division as repeated subtraction Division can be thought of as repeated subtraction. To find how many 4s there are in 24 we could count on or back in jumps of 4.

Repeated subtraction 12 − 4 − 4 − 4 = 0 so 12 ÷ 4 = 3

12 is the same as 12 quarters 4

Images of 12 ∏ 4

Thinking about division

1A

4 Multiples, square numbers and factors

Quick reference Core activity 4.1: Multiples and squares (Learner’s Book p20) They learn square numbers up to 10 × 10 and find factors of two-digit numbers.

Multiples and squares 15 multiplied by itself gives a three-digit number. 1

Prior learning ∑

∑

∑

This chapter builds on work in Stage 4 on the 2, 3, 4, 5, 6 and 9 times tables and related division facts. It also builds on learners’ ability to recognise and begin to know multiples of 2, 3, 4, 5 and 10 up to the tenth multiple. Learners have previously covered table facts and multiples in Stage 5 (see chapter 2).

Objectives* –

1

"

5

2

!

2

5

What is the smallest two-digit number that can be multiplied by itself to give a three-digit number?

Core activity 4.2: Tests of divisibility (Learner’s Book p22) Learners work on divisibility by 2, 5, 10 and 100, then explain their answers and use their knowledge to solve problems. Core activity 4.3: Factors (Learner’s Book p24) Learners identify all the factors of a variety of numbers including two-digit numbers and square numbers.

5

?

?

?

"

?

?

!

?

Tests of divisibility

Vocabulary

Let’s investigate

square number: the number you get when you multiply a whole number by itself.

Let’s investigate

For example, 4 " 4 ! 16

250

502

1

These patterns of dots show the first four square numbers.

100

10 530

5

650

40

?

! ? ! ? " 12

Find four different factors of 12 that total 12. ?

! ? ! ? ! ? " 12

5 if the last digit is 5 or 0.

factor: a whole number that divides into another number without a remainder. For example, 6 # 2 " 3 and 6#3"2 so, 2 and 3 are factors of 6

2$3"6

Write down all the factors of 12.

10 if the last digit is 0. 100 if the last two digits are 00.

25

300

Find three different factors of 12 that will give a total of 12 when added together.

2 if the last digit is divisable by 2.

Look at this set of numbers.

16 is a square number.

Vocabulary

Let’s investigate

test of divisibility: a number can be divided by …

520

Write down two more numbers that are divisible by 5 but not by 2 or 10.

Think about square numbers.

1

205

Explain to your partner how you know.

?

Factors

Vocabulary divisible: can be divided without a remainder.

Which of these numbers is divisible by 5 but not by 2 or 10?

factor

700

1

factor

This is a factor bug for 24.

Write down: (a) the numbers that are divisible by 100. 2

Draw a dot pattern for the fifth square number.

(b) the numbers that are divisible by 10.

Look at these numbers. Write down the numbers which are:

Discuss your results with a partner.

35

14 9 24

3

30 100

64

7

16

(a) multiples of 6.

21

(b) multiples of 7.

25

36

2

(c) square numbers.

42

63

21

48

84

6

3

How do you know they are divisible by 2? 3

not a square number

24

2

Write down the numbers from the list below that are divisible by 2: 13

Copy the sorting diagram. Write a number between 50 and 100 in each space. square number

1

(c) the numbers that are divisible by 5.

90

Pair these numbers so that the difference between each pair is divisible by 5. The first one has been done for you:

24

4

12 8 6

74 ! 39 " 35 and 35 is divisible by 5 48

even number

89

74

66

23

39

64

91

not an even number

20

Unit 1A: Core activity 4.1 Multiples and squares

22

Unit 1A: Core activity 4.2 Tests of divisibility

24

Unit 1A: Core activity 4.3 Factors

1A: Calculation (Mental strategies) Know and apply tests of divisibility by 2, 5, 10 and 100. Recognise multiples of 6, 7, 8 and 9 up to the tenth multiple. Know squares of all numbers to 10 × 10. Find factors of two-digit numbers. Recognise odd and even numbers and multiples of 5, 10, 25, 50 and 100 up to 1000. 1A: Problem solving (Using understanding and strategies in solving problems) 5Ps4 – Deduce new information from existing information to solve problems. 5Ps9 – Explain methods and justify reasoning orally and in writing; make hypotheses and test them out. 5Ps10 – Solve a larger problem by breaking it down into sub-problems or represent it using diagrams.

5Nc4 5Nc5 5Nc6 5Nc7 5Nn13

– – – – –

*for NRICH activities mapped to the Cambridge Primary objectives, please visit www.cie.org.uk/cambridgeprimarymaths

Vocabulary square number ∑ divisible ∑ test of divisibility ∑ factor

Cambridge Primary Mathematics 5 © Cambridge University Press 2014

Unit 1A

41

Core activity 4.1: Multiples and squares

LB: p20

Resources: Square numbers (CD-ROM). Completed multiplication grid (use the Blank multiplication grid that learner’s completed in Core activity 3.1). Multiples Ask the learners to look at their completed multiplication grids from Core activity 3.1. Explain to learners that if we read across (or down) the multiplication grid from 1 to 10, we can see sequences of numbers, e.g. 5, 10, 15, 20, 25, ... Ask learners: ∑ “What do we call these numbers in relation to the first number in the sequence?” (Answer: multiples) ∑ “Can you tell me what number the following sequence of numbers are multiples of? 6, 12, 18, 24, 30, 36, 42, 48, and 54? (Answer: they are multiples of 6) ∑ “What number is 36 a multiple of? (Answer: 6 and 9) ∑ “Can you give me an example of a multiple of 8? A multiple of 9?” Ask learners to use their multiplication grid to help them identify and learn multiples of 6, 7, 8 and 9 up to the tenth multiple. Tell learners that they can use what they know about other times tables to help them recognise and remember multiples of the 6×, 8× and 9× tables. For example, ∑ 6× multiples are double the 3× multiples, e.g. 3× 6× ∑

6 12

9 18

12 24

15 30

18 36

20 40

24 48

21 42

… …

30 60

8× multiples are double the 4× multiples 4× 8 ∑

4 8

8 16

12 24

16 32

28 56

… …

40 80

9× multiples have digits that add up to 9 (or the digits add up to a multiple of 9 for some larger numbers) 9× sum of digits

42

3 6

Unit 1A

9 9

18 9

4 Multiples, square numbers and factors

27 9

36 9

45 9

54 9

…

90 9

Vocabulary square number: the number you get when you multiply a whole number by itself. For example, 4 × 4 = 16 square number

Square numbers occur on the diagonals of a multiplication grid.

× 1 2 3 1

1

2

3

2

2

4

6

3

3

6

9

Square numbers Explain to learners that sequences of numbers can also be shown using pictures. Draw these arrangements made from squares for the whole class to see.

Invite learners to look at the arrangements, then ask these questions: ∑ “How can you describe the pattern?” (Possible answers include: a staircase; or each row is made up of consecutive odd numbers 1, 3, 5; or the number of squares total 1, 4 and 9.) ∑ “How would the pattern continue?” (Answer: with 16 squares, then 25 squares, then 36 squares . . . ) Display the Square numbers photocopy master and establish that the squares could also be rearranged to make a pattern of squares of increasing size as is shown on the Square numbers photocopy master. Explain that the numbers 1, 4 and 9 are examples of square numbers. Explain what is meant by a square number. Ask the learners to look at their completed multiplication grids again. Establish that there are other square numbers; show that they occur on the diagonal. Ask learners to write down the sequence of square numbers up to 100. (Answer: 2, 4, 9, 16, 25, 36, 49, 64, 81, 100) Inform the learners that they need to learn and remember these square numbers. Learners should recognise square numbers as a multiplication calculation, for example 100 = 10 × 10. There is no requirement for them to recognise or use the notation 10² at this stage.

Summary ∑ ∑

Learners have extended their understanding of multiples to include multiples of 6, 7, 8 and 9. They know the square numbers up to 10 × 10. Notes on the Learner’s Book Multiples and squares (p20): provides a variety of examples on multiples and square numbers. It is important that learners can recognise and use these numbers in different contexts so, for example, multiples have to be recognised in question 2, found in question 4, recognised in a sequence in question 5 and used in a problem-solving context in question 7.

Check up!

∑

∑

∑

“Multiples of 9 are all even.’ Is this statement true or false? Explain your answer.” “List all the multiples of 7 between 20 and 30.” “Give me an example of a square number that is less than 100.”

More activities Adding square numbers (class) Write the following for the whole class to see: 9 + 25 = 36. In this example, two square numbers are added to make a third square number. Challenge learners to find other examples like this. (Possible answers: 36 + 64 = 100 and 9 + 16 = 25)

Games Book (ISBN 9781107667815) Square numbers (p7) is a game for two players. The game gives practice in recognising square numbers to 10 × 10. Core activity 4.1: Multiples and squares

43

Core activity 4.2: Tests of divisibility

LB: p22

Resources: 100 square photocopy master (CD-ROM). (Optional: Divisibility game photocopy master (CD-ROM). 2/5/10 spinner photocopy master (CD-ROM).)

Write these numbers for the whole class to see. Challenge learners to find pairs that total 100.

42

93

23 61

divisible: can be divided without a remainder.

39

77

7

48

test of divisibility: a number can be divided ...

14

by

86

After a period of time, ask, “How did you find the pairs?” (Learners should respond by saying that they looked at the units digits, which must total 10, then at the tens digits, which must total 90.) Now invite learners to look at this set of numbers:

500 5100

700 1700 100

Ask, “What can you say about all these numbers?” (Answers include: they are all even; they all divide exactly by 10 and 100) Make it clear to learners that all the numbers are multiples of 100. Explain that sometimes we need to know if numbers will divide exactly by a particular number, so we look for patterns or rules that help us do this. These rules are called a test of divisibility. “How can we know if a number divides exactly by 100 if we don’t do the calculation?” (Answer: the units and tens digits are both 0.) We can say that ‘the last two digits are zero’ is a test of divisibility for 100. “Can you tell me a test of divisibility for 10?” (Answer: the units digit must be 0)

44

Unit 1A

Vocabulary

4 Multiples, square numbers and factors

if …

2

the last digit is divisible by 2 (an even number)

5

the last digit is 5 or 0

10 100

the last digit is 0 the last two digits are 00

Invite learners to colour the following multiples on the 100 square photocopy master: ∑ the multiples of 2 ∑ the multiples of 5. Ask, “What do you notice about the patterns?” Establish that: ∑ the multiples of 2 always end in 0, 2, 4, 6 or 8; they are always an even number (which means they can always be divided by 2) ∑ the multiples of 5 always end in 0 or 5. These are the tests of divisibility for 2 and 5. “What else do you notice?” If all numbers that are divisible by 100 end in 00, then they must also be divisible by 5 and 10 based on the tests of divisibility for 5 and 10. Similarly, as a number ending in 00 is even, then they are also divisible by 2. Make sure learners know the test of divisibility for 2, 5, 10 and 100.

Summary Learners have worked on divisibility by 2, 5, 10 and 100 and have begun to see the relationships between these. For example, they know that any number divisible by 100 is also divisible by 2, 5 and 10. Notes on the Learner’s Book Tests of divisibility (p22): gives learners opportunities to work on divisibility by 2, 5, 10 and 100, then explain their answers and use their knowledge to solve problems.

Check up!

∑

∑

“Give me three numbers which are divisible by 5. How do you know?” “Anton thinks that all multiples of 5 end in 5. Is he correct? How do you know?”

More activities Divisibility game (pairs) You will need the Divisibility game photocopy master (CD-ROM) and a spinner from the 2/5/10 spinner photocopy master (CD-ROM). Players aim to get four numbers in a row on a grid. Each number must be divisible by 2, 5 or 10, as shown on the spinner. Full instructions are on the photocopy master.

Core activity 4.2: Tests of divisibility

45

Core activity 4.3: Factors

LB: p24

Resources: Factors photocopy master (CD-ROM). Factor bugs photocopy master (p48). (Optional: Factor spinner photocopy master (CD-ROM).) Set the following challenge:

Vocabulary

“I have 12 small squares. How can I arrange them to make a rectangle?”

Vocabulary factor: a whole number that divides exactly into another number. For example,

Allow learners ‘thinking time’, then take responses. (Possible suggestions: 1 row of 12 squares; 2 rows of 6 squares; 3 rows of 4 squares and so on)

2 × 3 = 6 so 6 ∏ 2 = 3 and 6 ∏ 3 = 2 2 and 3 are factors of 6

Explain that 1, 12, 2, 6, 3 and 4 are factors of 12. If necessary, display the Factors photocopy master to support this statement and explain what is meant by factors. ∑ “What are the factors of 8?” (Answer: 1, 2, 4 and 8) ∑ “What are the factors of 15?” (Answer: 1, 3, 5 and 15)

Unit 1A

4 Multiples, square numbers and factors

factor

Learners who confuse multiples and factors. Ensure that the definitions are clearly displayed in the classroom: ∑ multiples are the product of multiplying a number by a positive whole number ∑ factors are whole numbers that divide exactly into another number. For example, 4 is a factor of 8 but not a multiple of 8. 15 is a multiple of 5 but is not a factor of 5. Example: Factors bug for factors of 24.

1

24

2

12

24 3 4

46

factor

Look out for!

Tell learners that sometimes it is important to find all the factors of a number, so we need to be organised in the way we work. Display the Factor bugs photocopy master and explain it is a way of listing all the factors of a number. “Here is a factor bug. I can write the factors of 24 on its legs. I start at 1 and if it is a factor, I write 1 on a leg on the left and the other factor on the matching right leg.” Demonstrate this as you say it, and Ask: ∑ “What happens if I try to divide 24 by 1?” (Answer: 24, so 1 is a factor) ∑ “What happens if I try to divide 24 by 2?” (Answer: 12, so 2 is a factor) ∑ “What happens if I try to divide 24 by 3?” (Answer: 8, so 3 is a factor) ∑ “What happens if I try to divide 24 by 4?” (Answer: 6, so 4 is a factor) ∑ “What happens if I try to divide 24 by 5?” (Answer: there is a remainder so 5 is not a factor) ∑ “What happens if I divide 24 by 6?” (Answer: the pair of factors 4 and 6 have already been written down) Explain that once you have reached this ‘repeat’ stage you can be sure you have found all the factors. Now write the factors in order: 1, 2, 3, 4, 6, 8, 12, 24.

2×3=6

8 6

Give pairs of learners time to draw factor bugs for 18, 28, 32 and 40.

Example. Factor bug for 9.

Review learners’ work, then ask: “Can we draw a factor bug for 9? What happens when we try to pair the numbers on its legs?” (Answer: we get 3 × 3 so we write this on a tail.)

1

9

9

“Is there another number where the factor bug would need a tail?” (Answer: any square number) 3

Summary ∑ ∑

Learners understand the meaning of ‘factor’ as a whole number that divides exactly into another number. They are able to find factors of two-digit numbers.

Notes on the Learner’s Book

∑

∑

Check up! “Which of these numbers have 8 as a factor? How did you work out your answer?” 48 53 40 28 “Give me two numbers that have 4 as a factor.”

Factors (p24): provides learners with the opportunity to find factors of two-digit numbers.

More activities Factor sum (pairs) You will need the spinner from the Factor spinner photocopy master (CD-ROM). Players take turns to spin the spinner. They work out all the factors of the number generated and add them together. For example: 20

24 4

18 16

6

15

8 14

1 + 2 + 3 + 6 + 9 + 18 = 39

9 12

10

Players keep a running total. The first player to reach 100 or more is the winner.

Core activity 4.3: Factors

47

Cambridge Primary Mathematics Teacher's Resource 5

Published on May 16, 2014

Preview Cambridge Primary Mathematics Teacher's Resource 5, Emma Low, Cambridge University Press.