Cambridge Primary Mathematics Teacher's Resource 6

CAMBRIDGE PRIMARY

Mathematics Teacher’s Resource

Emma Low

6

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/9781107694361 © 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 978-1-107-69436-1 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.

This End User License Agreement (‘EULA’) is a legal agreement between ‘You’ (which means the individual customer) and Cambridge University Press (‘the Licensor’) for Cambridge Primary Mathematics Teacher's Resource CD-ROM Stage 6 (‘the Product’). Please read this EULA carefully. By continuing to use the Product, You agree to the terms of this EULA. If You do not agree to this EULA, please do not use this Product and promptly return it to the place where you obtained it.

1. Licence The Licensor grants You the right to use the Product under the terms of this EULA as follows: (a) You may only install one copy of this Product (i) on a single computer or secure network server for use by one or more people at different times, or (ii) on one or more computers for use by a single person (provided the Product is only used on one computer at one time and is only used by that single person). (b) You may only use the Product for non-profit, educational purposes. (c) You shall not and shall not permit anyone else to: (i) copy or authorise copying of the Product, (ii) translate the Product, (iii) reverse-engineer, disassemble or decompile the Product, or (iv) transfer, sell, assign or otherwise convey any portion of the Product.

2. Copyright (a) All content provided as part of the Product (including text, images and ancillary material) and all software, code, and metadata related to the Product is the copyright of the Licensor or has been licensed to the Licensor, and is protected by copyright and all other applicable intellectual property laws and international treaties. (b) You may not copy the Product except for making one copy of the Product solely for backup or archival purposes.

You may not alter, remove or destroy any copyright notice or other material placed on or with this Product. (c) You may edit and make changes to any material provided in the Product in editable format (‘Editable Material’) and store copies of the resulting files (‘Edited Files’) for your own non-commercial, educational use, but You may not distribute Editable Materials or Edited Files to any third-party, or remove, alter, or destroy any copyright notices on Editable Materials or Edited Files, or copy any part of any Editable Material or Edited Files into any other file for any purpose whatsoever.

3. Liability and Indemnification (a) The Product is supplied ‘as-is’ with no express guarantee as to its suitability. To the extent permitted by applicable law, the Licensor is not liable for costs of procurement of substitute products, damages or losses of any kind whatsoever resulting from the use of this Product, or errors or faults therein, and in every case the Licensor’s liability shall be limited to the suggested list price or the amount actually paid by You for the Product, whichever is lower. (b) You accept that the Licensor is not responsible for the persistency, accuracy or availability of any URLs of external or third-party internet websites referred to on the Product and does not guarantee that any content on such websites is, or will remain, accurate, appropriate or available. The Licensor shall not be liable for any content made available from any websites and URLs outside the Product or for the data collection or business practices of any third-party internet website or URL referenced by the Product. (c) You agree to indemnify the Licensor and to keep indemnified the Licensor from and against any loss, cost, damage or expense (including without limitation damages paid to a third party and any reasonable legal costs) incurred by the Licensor as a result of your breach of any of the terms of this EULA. 4. Termination Without prejudice to any other rights, the Licensor may terminate this EULA if You fail to comply with any of its terms and conditions. In such event, You must destroy all copies of the Product in your possession.

5. Governing law This agreement is governed by the laws of England and Wales, without regard to its conflict of laws provision, and each party irrevocably submits to the exclusive jurisdiction of the English courts. The parties disclaim the application of the United Nations Convention on the International Sale of Goods.

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

v vii x x x

Term 1 1A: Number and problem solving 1 The number system (1) 1.1 Place value 1.2 Ordering, comparing and rounding numbers 2 Multiples, factors and primes 2.1 Multiples and factors 2.2 Odd and even numbers 2.3 Prime numbers 3 Multiplication and division (1) 3.1 Multiply and divide by 10, 100 and 1000 3.2 Mental strategies for multiplication 4 More on number 4.1 Addition of decimals 4.2 Dvision (1) 4.3 Number sequences 1B: Measure and problem solving 5 Length 5.1 Working with length 5.2 Drawing lines 6 Time (1) 6.1 Timetables 6.2 Calendars

1 2 4 9 10 14 16 21 22 24 27 28 32 34 37 38 42 49 50 55

7 Area and perimeter (1) 7.1 Making a model house 1C: Geometry and problem solving 8 2D and 3D shape (1) 8.1 Identifying polygons 8.2 Properties of 3D shapes and their cross-sections 8.3 Nets 9 Angles in a triangle 9.1 Angles in a triangle 10 Shapes and geometric reasoning 10.1 Describing translation 10.2 Reflecting shapes 10.3 Rotation on a grid

61 62 69 70 74 76 81 82 87 88 92 94

Term 2 2A: Number and problem solving 11 The number system (2) 11.1 The number system (1) 11.2 History of number (1) 12 Decimals 12.1 The decimal system 12.2 Operations with decimals 12.3 Decimals in context 13 Positive and negative numbers 13.1 Positive and negative numbers 14 14 Multiples, factors and mental strategies using them 14.1 Common multiples 14.2 Mental strategies for addition and subtraction 14.3 Mental strategies for multiplication

103 104 106 109 110 114 118 121 122 127 128 130 132

iii

15 15 Multiplication and division (2) 15.1 Divisibility rules 15.2 Multipication 15.3 Division (2) 16 Special numbers 16.1 Special numbers 2B: Measure and problem solving 17 Mass and capacity 17.1 Measuring mass and capacity (1) 17.2 Measuring mass and capacity (2) 18 Time (2) 18.1 Converting times 18.2 Time zones (1) 19 Area and perimeter (2) 19.1 Calculating area and perimeter 2C: Handling data and problem solving 20 Graphs, charts and tables 20.1 Tables and line graphs 20.2 Pie charts 21 Statistics 21.1 The three averages 21.2 Using statistics to persuade

137 138 140 144 149 150 153 154 156 165 166 168 173 174 181 182 184 189 190 194

Term 3 3A: Number and problem solving 22 Probability 22.1 Language of probability 23 The number system (3) 23.1 The number system (2) 23.2 History of number (2) 24 Mental strategies 24.1 Addition and subtraction (1) 24.2 Multiplication and division

iv

203 204 207 208 212 217 218 222

25 Addition and subtraction 25.1 Addition and subtraction (2) 26 Multiplication and division (3) 26.1 The laws of arithmetic 26.2 Fraction and division 27 Fractions 27.1 Fractions 27.2 Mixed numbers and improper fractions 28 Fractions, decimals and percentages 28.1 Fractions and decimals 28.2 Percentages 29 Ratio and proportion 29.1 Using ratio and proportion 3B: Measure and problem solving 30 Metric and imperial measures 30.1 Capacity and mass 30.2 Distance 31 Time (3) 31.1 Times zones (2) 31.2 Leap years 32 Area and preimeter (3) 32.1 Rectangles 32.2 Irregular shapes 3C: Geometry and problem solving 33 2D and 3D shape (2) 33.1 Quadtrilateral prisms 33.2 Regular polyhedra 34 Locating 2D shapes 34.1 Classifying shapes 34.2 Transforming polygons 35 Angles and triangles 35.1 Drawing and measuring angles

227 228 231 232 234 237 238 242 245 246 248 253 254 259 260 264 273 274 278 285 286 290 297 298 300 309 310 312 319 320

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.

3

4

5

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.

v

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.

vi

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.

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.

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 6 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 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

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

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.)

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.

• 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 6, 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 6. 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.

• 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

viii

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

ix

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.

x

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 (1)

NOTE: the objectives that form the basis of this chapter revise, consolidate and extend number work covered in Stage 5. We suggest that you ask your learners what they know already and use this information to target work appropriately. You could give the learners a list of the objectives covered in this chapter and ask them to produce a poster showing what they know. Alternatively, you could ask them to complete the The Number System photocopy master (CD-ROM) as means of formative assessment.

Quick reference

Number

Ordering, comparing and rounding numbers

Place value

Core activity 1.1: Place value (Learner’s Book p2) Learners revise and consolidate work on place value up to 1 000 000 and down to two decimal places.

Raphael has eight digit cards.

1

4 5

3

2

Let’s investigate

Vocabulary

There are 1187 students in a large city school.

million: equal to one thousand thousands and written as 1 000 000.

Let’s investigate

6

7

8

He uses the cards to make two four-digit numbers.

He uses each card only once.

He finds the difference between his two numbers.

Do not attempt to work out an accurate answer.

There are 42 classes in the school.

Approximately, how many students are in each class?

1 million ! 10 " 10 " 10 " 10 " 10 " 10

Explain to a friend how you made your decision.

1

Draw a line 10 centimetres long. Mark 0 and 10 000 at the end points.

Think about the largest and smallest numbers you can make.

0

10 000

Estimate the positions of the following numbers.

What is the largest difference he can make?

Mark each one with an arrow and its letter:

1

Core activity 1.2: Ordering, comparing and rounding numbers (Learner’s Book p4) Learners revise and consolidate work on ordering and rounding. Work in this chapter concentrates on whole numbers as the objectives are repeated later in the year when decimals are the main focus.

Prior learning This chapter builds on the work done in Stage 5 on five- and six-digit numbers.

6000 marked A

Write the numbers shown on these charts in words and figures. (a)

(b)

3500 marked B

100 000

200 000

300 000

400 000

500 000

600 000

700 000

800 000

900 000

10 000

20 000

30 000

40 000

50 000

60 000

70 000

80 000

90 000

1000

2000

3000

4000

5000

6000

7000

8000

9000

100

200

300

400

500

600

700

800

900

10

20

30

40

50

60

70

80

90

1

2

3

4

5

6

7

8

9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

100 000

200 000

300 000

400 000

500 000

600 000

700 000

800 000

900 000

10 000

20 000

30 000

40 000

50 000

60 000

70 000

80 000

90 000

1000

2000

3000

4000

5000

6000

7000

8000

9000

100

200

300

400

500

600

700

800

900

10

20

30

40

50

60

70

80

90

9050 marked C 2

Round these numbers to the nearest hundred.

3

Round these numbers to the nearest thousand.

4

Order the following sets of numbers from smallest to largest.

(a) 45 678 (a) 147 950

(b) 24 055

(c) 157 846

(a) 54 754

55 475

55 547

54 775

55 447

(b) 45 054

45 540

45 504

45 045

45 500

9

(c) 456 065

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Use any of the numbers in part (c) to complete these inequalities.

1

2

3

4

5

6

7

8

450 566

?

2

(c) 50 505

(b) 65 507

Unit 1A: Core activity 1.1 Place value

4

455 656

!

456 565

?

450 666

?

"

?

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

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

1A: Numbers and the number system Know what each digit represents in whole numbers up to a million.

Know what each digit represents in one- and two-place decimal numbers.

Round whole numbers to the nearest 10, 100 or 1000.

Make and justify estimates and approximations of large numbers.

Use correctly the symbols for >, < and =.

Estimate where four-digit numbers lie on an empty 0–10 000 line.

1A: Problem solving (Using techniques and skills in solving mathematical problems) Estimate and approximate when calculating e.g. use rounding and check working.

1A: Problem solving (Using understanding and strategies in solving problems) 6Ps9 – Make, test and refine hypotheses, explain and justify methods, reasoning, strategies, results or conclusions orally.

6Nn2 6Nn3 6Nn8 6Nn10 6Nn12 6Nn13

– – – – – – – 6Pt5 –

Vocabulary

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

million

Cambridge Primary Mathematics 6 © Cambridge University Press 2014

Unit 1A

1

Core activity 1.1: Place value

LB: p2

Resources: Place value grid photocopy master (p7); large version for class display. (Optional: Match the numbers photocopy master (CD-ROM);

The Number System photocopy master (CD-ROM).)

Display the Place value grid photocopy master for the whole class to see, and write the number 2002.2 so that everyone can see it.

Mark it on the grid and say, “Two thousand and two point two.”

Ask questions about each digit in turn: • “What is the value of this digit?”

• “How many times larger or smaller is the value of this two than this two?”

When discussing place value in terms of how many times larger or smaller one digit in the number is relative to another digit, the learners may find it easier to visualise it as follows: × 1000 Th 2

H 0

T 0

U 2

• .

t 2

Vocabulary 200 000 20 000 2000 200 20 2 0.2 0.02

h

÷ 10

Repeat with other numbers up to two decimal places, for example 3003.33 and also with numbers that don’t have the same digit repeated e.g. 2450.12.

“What is the largest number I can show on this chart?” (Answer: 999 999.99) “How do I read this number?” (Answer: nine hundred and ninety nine thousand, nine hundred and ninety nine point nine nine.)

“What happens if I add 0.01 to this number?” (Answer: I get one million (accept ‘1 and six zeros’); learners might not be familiar with the term ‘million’ even if they can identify that the answer will be a 1 followed by six digits, if this is the case, introduce them to this terminology.)

“How do I write this number?” (Answer: 1 000 000)

2

Unit 1A

1 The number system (1)

million: equal to one thousand thousands; 1 000 000 = 10 × 10 × 10 × 10 × 10 × 10

Look out for!

• Learners who do not read decimal numbers correctly. Explain why the number, for example, 1.23, is read as ‘one point two three’ and NOT as ‘one point twenty three’; it is because the digits 2 and 3 represent two tenths and 3 hundredths NOT 2 tens and 3 units. Similarly, 1.02 is read as ‘one point zero two’.

• Learners who may use different notation for writing large numbers. For example, 1 million is written as 1 000 000 or 1,000,000 in different parts of the world.

Opportunities for display!

Learners collect examples of numbers from newspapers, magazines and other sources for display. Each number could be written in words and figures and displayed on a place value chart.

Summary • Learners use a place value chart for numbers up to a million and down to two decimal places.

• They read and write numbers and recognise a million written in figures.

• They understand that the position of a digit affects its value.

• They read decimal numbers correctly.

Check up!

• Read this number, 1234.05. “What is the value of the digit 1; and the digit 5?”

• “Write the number six hundred and fifty six point six in figures.”

Notes on the Learner’s Book Place value (p2): contains examples that provide practice in reading and writing numbers up to 1 million. Further work related specifically to decimals can be found on pages 109 to 119 (starting with The decimal system, (chapter 12).

More activities Match the numbers (pairs) You will need the Match the numbers photocopy master (CD-ROM).

Cut out the cards from the activity sheet and lay them face up on the table. Learners work in pairs. They take turns to pick up two matching cards and say the number. Their partner checks the answer. Repeat until all the cards have been used. This will act as a check to see that learners have understood how to say large numbers.

Self assessment (individual) You will need The Number System photocopy master (CD-ROM).

This is a self assessment sheet where learners practise the skills from the core activity. They identify skills that they can do and which they need help on.

They identify what they want to get better at.

Games Book (ISBN 9781107667815) Place value challenge (p1) is a game for two players. It gives practice writing (and reading) large numbers.

Core activity 1.1: Place value

3

Core activity 1.2: Ordering, comparing and rounding numbers Resources: Blank number lines photocopy

master (p8); large version for class display. (Optional: 0–9 spinner (CD-ROM).)

Display the Blank number lines photocopy master, so that the whole class can see the empty number line marked from 0 to 10 000.

0

Mark and label other divisions on the line: start with by marking and labelling every 1000.

Then ask learners to show the position of some four-digit numbers, for example 4500, 4200 and 4800.

4500 4000 5000

10 000

Ask learners to use two of the numbers from the list above to complete the number sentences below, then read them aloud:

> Repeat with other sets of numbers.

Unit 1A

1 The number system (1)

Look out for!

• Learners who do not know the conventions for rounding. To round to the nearest thousand look at the hundreds digit: • if it is less than 5 round down • if it is 5 or more round up.

213 241

Ask learners to order the following numbers starting with the smallest: 4300, 4500, 4800, 4100, 4200; they can use the number line for help if they need to. (Answer: 4100, 4200, 4300, 4500, 4800)

4

Always ask, “How did you decide?” giving learners the chance to explain their methods using correct mathematical vocabulary.

10 000

Ask learners what number goes in the middle of the line? (Estimate the halfway mark, and mark 5000 on the line.) “How did you work it out?” (Learners should know, or be able to reason from previous work, that 5000 is half of 10 000.)

0

LB: p4

<

to the nearest thousand

213 000

• To round to the nearest hundred look at the tens digit: • if it is less than 5 round down • if it is 5 or more round up.

213 241

to the nearest hundred

213 200

• To round to the nearest ten look at the units digit: • if it is less than 5 round down • if it is 5 or more round up.

213 241

to the nearest ten

213 240

Using the number line ask learners to round each number (4300, 4500, 4800, 4100, 4200) to the nearest thousand. (Answer: 4100, 4200, 4300 round to 4000 and 4500 and 4800 round to 5000.)

Emphasise how a number line can help learners to visualise, for example, 4200 is nearer to 4000 than to 5000. Repeat with other sets of numbers Ask where they would place the number 4155 on the number line. (Answer: just over half way between 4100 and 4200.)

Support learner to: â&#x20AC;˘ round 4155 to the nearest thousand (Answer: 4000) â&#x20AC;˘ round 4155 to the nearest hundred (Answer: 4200) â&#x20AC;˘ round 4155 to the nearest ten. (Answer: 4160) Repeat with other numbers.

Draw a new number line from 0 to 1000000. Tell learners that 1000000 is 1 million, which is 1 thousand thousands. Ask learners to discuss what number they think would be positioned in the middle of the number line and to justify their answer (Answer: 500000, 5 hundred thousand). Mark 5000000 on the number line. Ask learners to suggest some other six-digit numbers and to estimate where those numbers would be placed on the number line. Suggest the number 843791. Ask some of the learners to mark with a dot where they think the number would be positioned, and ask the other learners which dot they think most accurately places the number and justify their answer. Encourage learners to use rounding and approximation to help their estimate and reasoning.

Core activity 1.2: Ordering, comparing and rounding numbers

5

Summary • Learners confidently round numbers to the nearest 10, 100 or 1000 using mathematical conventions.

• They use a number line when appropriate to position numbers and understand that a number line may be useful when ordering or rounding numbers.

• They use the signs < and > to compare numbers.

Notes on the Learner’s Book Ordering, comparing and rounding numbers (p4): the investigation challenges learners to use their knowledge to answer a different type of question and explain their decision to a partner. They should round each number to work out an approximation, for example: 1200 ÷ 40 = 30

Check up!

• “Round 512 345 to the nearest thousand. How did you work out your answer?”

• “Order this set of numbers: 41 325 43 521 45 123 43 324 43 512. What did you look for when making your decisions?”

• “Use < or > to complete this inequality: 45 123 45 213”

Questions 1 to 7 provide practise related to the Core activity. Part (c) of question 8 requires learners to think about all the numbers that could round according to two criteria.

More activities Rounding up (pairs) You will need 0–9 spinner (CD-ROM).

Player one ‘spins the spinner’ five times to create a 5-digit number. The player chooses which of the five digits to put in each box.

Player two rounds the number to the nearest thousand.

• If the number ‘rounds up’ player two scores a point.

• If the number ‘rounds down’ player one scores a point.

• The first player to score five points is the winner.

Games Book (ISBN 9781107667815) More or less (p1) is a game for two players. It provides an opportunity to use the symbols < and > using one place decimals.

6

Unit 1A

1 The number system (1)

Place value grid 100 000

200 000

300 000

400 000

500 000

600 000

700 000

800 000

900 000

10 000

20 000

30 000

40 000

50 000

60 000

70 000

80 000

90 000

1 000

2 000

3 000

4 000

5 000

6 000

7 000

8 000

9 000

100

200

300

400

500

600

700

800

900

10

20

30

40

50

60

70

80

90

1

2

3

4

5

6

7

8

9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Instructions on page 2

Original Material Š Cambridge University Press, 2014

Blank number lines

0

10 000

0

10 000

0

10 000

Instructions on page 4

Original Material Š Cambridge University Press, 2014

1A

2 Multiples, factors and primes

Quick reference

Multiples and factors

Core activity 2.1: Factors and multiples (Learner’s Book p6) Learners consolidate previous learning related to multiples and factors.

The sequence below uses the numbers 1 to 4 so that each number is either a factor or a multiple of the previous number.

4

1

2

Odd and even numbers

Vocabulary

Let’s investigate

factor: a whole number that divides exactly into another number. For example, 1, 2, 3 and 6 are the factors of 6.

1!6"6

3

2!3"6

Use cards that can be easily moved around.

Find a similar sequence that uses the numbers 1 to 6.

Core activity 2.2: Odd and even numbers (Learner’s Book p8) Learners extend their work on odd and even numbers as they explore rules related to addition, subtraction and multiplication.

1

Which of these numbers are multiples of 8?

18

2

4 3

24

48

56

68

72

Which of these numbers are factors of 30?

5

6

10

20

60

Use each of the digits 5, 6, 7 and 8 once to make a total that is a multiple of 5.

?

?

+

?

0

#3

3

#3

6

9

#3

Prior learning • This chapter builds on previous work on odd and even numbers, multiples and factors.

• Prime numbers have not been formally introduced in previous Stages, so this might be the first time learners have encountered the definition.

Objectives* –

6"3#3

My age this year is a multiple of 8.

(5 and 7 are prime numbers)

prime number: a prime number has exactly two different factors; itself and 1.

NOTE: 1 is not a prime number. It has only one factor (1).

Examples of prime numbers: 2, 3, 5, 7, 11 …

Can you find an even number that does not satisfy the rule? Try some numbers greater than 30.

11

26

33

57

187

1

List all the prime numbers between 10 and 20.

2

Identify these prime numbers from the clues. (a) It is less than 30. (b) It is between 30 and 60.

The sum of its digits is 10.

2002

Explain to a partner how you know.

12 2

3

Andre makes a three-digit number.

All the digits are odd.

What could Andre’s number be?

(c) 25.

Unit 1A: Core activity 2.1 Multiples and factors

(3 is a prime number)

Check if the statement is true for all the even numbers to 30.

Which of these numbers are even?

9

3

Ollie makes a three-digit number using the digits 2, 3 and 6.

4

His number is odd.

The hundreds digit is greater than 2.

How old am I?

6

two prime numbers.

The sum of its digits is 8.

1

My age next year is a multiple of 7.

Core activity 2.3: Prime numbers (Learner’s Book p10) Learners are introduced to prime numbers and the definition of a prime number, and can recite the prime numbers less than 20.

Every even number greater than 2 is the sum of

Here are two examples:

The sum of the digits is 7.

Find all the factors of:

5

(b) 32

#3

Look at this statement.

12 " 5 # 7

Place 13 counters on the grid so that there is an odd number of counters in each row, column and on both diagonals. Only one counter can be placed in each cell.

Place 10 counters on the grid so that there is an even number of counters in each row, There is more column and on both diagonals. Only one than one answer. counter can be placed in each cell.

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

?

4

(a) 24

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

Vocabulary

Let’s investigate

even: even numbers are divisible by 2.

They end in 2, 4, 6, 8 or 0. For example, 6578 is an even number.

factor factor factor factor

Each number is used once only.

Prime numbers

Vocabulary odd: odd numbers are not divisible by 2. They end in 1, 3, 5, 7 or 9. For example, 7689 is an odd number.

Let’s investigate

You need 13 counters and a 5 by 5 grid.

Unit 1A: Core activity 2.2 Odd and even numbers

?

!

?

!

?

" 30

?

!

?

!

?

" 50

?

!

?

!

?

" 70

Identify the prime numbers represented by ? and ? . (a) ?

What could Ollie’s number be?

8

Copy and complete these number sentence by placing a prime number in each box.

2

" 49

(b) ? # 1 " 2 ! 9

10

(c) ? # 2 " 52 (d) ? # ? " 20

Unit 1A: Core activity 2.3 Prime 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

1A: Numbers and the number system Find factors of two-digit numbers.

Find some common multiples (e.g. for 4 and 5).

Recognise odd and even numbers and multiples of 5, 10, 25, 50 and 100 up to 1000.

Make general statements about sums, differences and multiples of odd ad even numbers.

Recognise prime numbers up to 20 and find all prime numbers less than 100.

1A: Problem solving (Using understanding and strategies in solving problems) 6Ps3 – Use logical reasoning to explore and solve number problems and puzzles.

6Ps9 – Make, test and refine hypotheses, explain ad justify methods, reasoning, strategies, results or conclusions orally.

6Nn6 6Nn7 6Nn17 6Nn18 6Nn19

– – – – –

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

Vocabulary factor • multiple • odd • even • prime number

Cambridge Primary Mathematics 6 © Cambridge University Press 2014

Unit 1A

9

Core activity 2.1: Factors and multiples

LB: p6

Resources: There are no specific resources required for this activity.

Revise learners’ knowledge of factors and multiples by creating a whole class ‘Mathematical orchestra’. Sit five learners on chairs facing the class. Number these learners 1 to 5, and inform them that they are members of the mathematical orchestra. Each learner has their own part in the orchestra. Explain that you are the conductor and your job is to slowly count the beat with up/down movements of your hand.

Instruct learner ‘number 1’ to stand and immediately sit down on each beat, then instruct learner ‘number 2’ to stand and immediately sit down on beats 2, 4, 6 … (the multiples of 2).

Then tell learner ‘number 3’ to stand on beats 3, 6, 9 … (the multiples of 3), learner ‘number 4’ to do so on beats that are multiples of 4, and learner ‘number 5’ to do so on beats that are multiples of 5. Practise together first so that you are confident the learners understand.

Practise the count to 8 beats, or beyond. If you were counting 8 beats, the learners would stand/sit as per the diagram below on beat 1, 2, 3, 4, etc, to beat 8.

multiple: a number that can be divided exactly by another number; start at 0 and count up in steps of the same size and you will find numbers that are multiples of the step size. For example, +3

+3 0

3

+3 6

+3 9

12

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

1:

2:

3:

4:

factor: a whole number that divides exactly into another number. For example, 1, 2, 3 and 6 are the factors of 6.

1×6=6 2×3=6

5:

6:

7:

8:

factor

stand

stay seated

Direct questions to the rest of the class such as: • “Can you predict how many learners will stand on count 10?” (Answer: 3) • “Which numbers do they represent?” (Answer: 1, 2, and 5) • “What is the relationship between these numbers?” (Answer: they are factors of 10.)

Establish the definitions of multiple and factor, then issue a challenge, “When is the first time four learners will stand together?” (Answer: 1, 2, 3 and 4 will stand on 12.) Act out with the mathematical orchestra to demonstrate this. You could try similar challenges that are also possible to act out with the mathematical orchestra.

10

Vocabulary

Unit 1A

2 Multiples, factors and primes

factor

factor

factor

general statement: a statement that does not use particular examples, e.g. ‘Two odd numbers added together give an even number’. counter-example: an example that shows a general statement is wrong.

Ask learners from the orchestra to sit back down and ask the class to discuss in groups, “When is the first time all five learners will stand together? How do you know?” (Answer: 60; it is a multiple of 1, 2, 3, 4 and 5.) Discuss answers and reasons as a class.

Tell learners that it can be useful to know all the factors of a number. Ask learners what strategies they would use to find all the factors of 24. If the learners cannot come up with ideas of their own, then suggest the following: • “Start with the lowest number, 1 is always a factor of a whole number.”

• “What is the ‘partner’ of 1?” (Answer: 24) “The number itself will also always be a factor.”

• “What about 2?” (Answer: It must be a factor because 24 is an even number.)

• “What is 2’s ‘partner’?” (Answer: halve 24 to get 12.)

• “What about 3?” (Answer: 24 ÷ 3 = 8 so 3 and 8 is another factor pair.)

• “Continue in this way to find all possible pairs.” (Answer: 1 & 24, 2 &12, 3 & 8, 4 & 6.)

• “Is there any point in continuing any further? Why not?” (Answer: 5 is not a factor, 6 has already been found.)

• “Record the results systematically, for example, {1 24} then {1, 2 12, 24} and so on until you reach the ‘middle’.”

The ‘Mathematical orchestra’ activity can be adapted for use with the whole class by: • having more learners in the row.

• having groups of learners seated around tables with each table representing a number. An alternative to standing up and sitting down is to wave arms up and down.

Look out for!

Learners who confuse factors and multiples (a common error). Remind these learners of the definitions and give them some examples of factors and multiples, particularly where a factor is not also a multiple. For example, 2 is a factor of 8 but is not a multiple of 8.

If necessary, remind them of factor bugs (Stage 5, chapter 4).

Ask groups of learners to discuss the general statement, ‘The larger the number, the more factors it has.’ Ask them to consider some test pairs, for example 16 and 27, 12 and 20. Learners should first predict which number will have the most factors, and then they can calculate all the factors for each number to test the statement.

Discuss the results as a class. Is the general statement correct? Or did someone find a counter-example?

Opportunities for display!

Ensure that the words ‘multiple’ and ‘factor’ are clearly displayed with their definitions. Add the posters made in the More activities section to the display

Summary Learners revise and extend previous work on multiples and factors, using them to solve puzzles and problems.

Notes on the Learner’s Book Multiples and factors (p6): learners are presented with a selection of straight-forward questions and puzzles. The puzzles involve both multiples and factors together, so learners have to think about the definitions of the words. Useful links are made with data handling objectives, as both Carroll diagrams and Venn diagrams are used in questions 6, 7 and 9.

Check up!

• “Here are four numbers: 3, 4, 7 and 12. Which of these numbers are factors of 12?” (Answer: 3, 4 and 12.)

• “I am thinking of a number between 20 and 40. It is a multiple of 5 and a multiple of 7. What number am I thinking of?” (Answer: 35) • “How can you be sure you have found all the factors of a number?”

Core activity 2.1: Factors and multiples

11

More activities Make a poster (individual) Learners design a poster about factors and multiples, remembering to include the definitions of each word.

Puzzles (pairs) Solve these puzzles, then write similar puzzles for your partner to solve: 1. What is my number? It is even, a multiple of 4, a factor of 24, and between 10 and 20 (Answer: 12) 2. What is my number? It is a factor of 24, a factor of 40, and a factor of 52 but is not the number â&#x20AC;&#x2DC;2â&#x20AC;&#x2122; (Answer: 4)

Games Book (ISBN 9781107667815) Factors in a row (p5) is a game for two players. The game provides practice in finding all the factors of 2-digit numbers.

12

Unit 1A

2 Multiples, factors and primes

Blank page

13

Core activity 2.2: Odd and even numbers

LB: p8

Resources: Blank 3 by 3 grid photocopy master (p18).

By Stage 6 learners should be confident of what is meant by odd and even numbers but if necessary, reinforce that odd numbers cannot be divided by 2 without leaving a remainder.

This point is important because some learners might be confused by previous work on division where odd numbers were divided by two to leave a remainder.

Show an example of how you can arrange the numbers 1, 2, 3 and 4 in a 2 by 2 grid so that the sum of the numbers horizontally and vertically is odd and the sum of the numbers diagonally is even. One solution is:

even

2

1

odd

3

4

odd

odd odd

even

Ask learners to find a different solution.

As a class, discuss the different solutions learners found. Ask, “How did you decide where to put the numbers? Can you give us any rules about adding odd and even numbers?” (Answer: either two evens or two odds need to go along the diagonal (based on the rules: even + even = even and odd + odd = even); then the other two numbers can be placed to fill the gaps so that in all rows and columns there will be one odd and one even number (based on the rule, even + odd = odd.)

Learners work in pairs on a different investigation. Ask them to place the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 in the 3 by 3 blank grid photocopy master so that each line horizontally, vertically and diagonally add up to an odd number.

Review work and arrive at the rules for adding three numbers that gives the following pattern for solutions to the problem: E O E

O O O

E O E

(Answer: either three odds have to add together or two evens and an odd.)

14

Unit 1A

2 Multiples, factors and primes

Vocabulary odd numbers: are not divisible by 2 without a remainder; they end in 1, 3, 5, 7 or 9. For example, 4689 is an odd number. even numbers: are divisible by 2, without a remainder; they end in 2, 4, 6, 8 or 0. For example, 7578 is an even number.

Opportunities for display!

Display the rules for: • adding odd and even numbers: even + even = even odd + odd = even even + odd = odd odd + even = odd • subtracting odd and even numbers: even - even = even odd - odd = even even - odd = odd odd - even = odd • multiplying odd and even numbers: even × even = even even × odd = even odd × even = even odd × odd = odd.

Move on to discussing multiplying odd and even numbers, asking, “What are the rules for multiplying odd and even numbers?”

Allow the learners thinking time then collect their ideas. (Answer: odd × odd = odd, even × odd = even, even × even = even.)

Summary • Learners work confidently with odd and even numbers.

• They find and use general statements and solve increasingly challenging puzzles.

Notes on the Learner’s Book Odd and even numbers (p8): learners are familiar with odd and even numbers so the questions are designed to encourage them to think about the properties of odd and even numbers (with the exception of question 1). Links are made with place value, multiples and calculation.

Learners who need support could work in pairs.

Check up!

Mira says, ‘I can add three odd numbers to get a total of 30.’

“Is she right? Explain your answer.”

More activities Squirrels nut store (pairs) Squirrels hide nuts to eat in the winter. Three squirrels hide 25 nuts altogether. Each of them hides a different odd number of nuts.

How many nuts did each squirrel hide? Find as many different ways as you can.

(Answer: There are 10 different solutions. [1, 3, 21] [1, 5, 19] [1, 7, 17] [1, 9, 15] [1, 11, 13] [3, 5, 17] [3, 7, 15] [3, 9, 13] [5, 7, 13] [5, 9, 11])

Core activity 2.2: Odd and even numbers

15

Core activity 2.3: Prime numbers

LB: p10

Resources: Sieve of Eratosthenes photocopy master (p19).

Remind learners of the definition of a multiple and a factor.

Vocabulary

Give each learner a copy of the Sieve of Eratosthenes photocopy master. Learners will use this to explore patterns for the 2, 3, 4, 5, 6 and 7 times tables. First they should cross out the number 1, then they should cross out the multiples of 2 except 2, the multiples of 3 except 3, the multiples of 4, the multiples of 5 except 5, the multiples of 6 and the multiples of 7 except 7. Before they cross out the multiples of each number, they should try to predict what patterns might emerge and try to explain the patterns they’ve found after doing the crossing out.

prime number: a prime number has exactly two different factors, 1 and the number itself. For example, 2, 3, 5, 7, 11 are all prime numbers.

NOTE: 1 is not a prime number; it has only one factor (1).

You could use the following questions as a guide:

Look out for!

• (2× table) “What do you notice? Can you explain what you see?” (Every other number is crossed out, because consecutive numbers have a pattern of odd-even-odd-even-odd etc. and all multiples of 2 are even numbers.)

• (3×table) “Can you predict what will happen? Will you shade any numbers that are already shaded? If so, which ones?” (Even multiples of 3 will already be crossed out as they are also multiples of 2.)

• (4× table) “Do you need to shade these multiples? Why not?” (No, because all multiples of 4 are also multiples of 2; they are all even numbers.)

• (5× table) “Can you explain the pattern?”

• (6× table) “Do you need to shade the multiples of 6? Why not?” (No, because all multiples of 6 are even and therefore are also multiples of 2.)

• (7× table) “Which numbers were not already shaded in?” (49, 77, 91)

Learners who think that 1 is a prime number.

Emphasise that prime numbers always have two different factors and 1 only has one factor.

At the end they should look at the grid and say what is special about the numbers that they haven’t crossed. Establish that they are the prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Explain what is meant by a prime number. Explain that the method they just carried out is called the ‘Sieve of Eratosthenes’ and was devised by a historical mathematician, called Eratosthenes, to identify all the prime numbers less than 100.

16

Unit 1A

2 Multiples, factors and primes

Repeat the mathematical orchestra activity from Core activity 2.1. You might wish to add extra multiples: 6, 7 . . . Ask learners which ‘beats’ had two people standing together (Answer: beats 2, 3, and 5) and what is special about these numbers? (Answer: prime numbers.)

Summary Learners know the definition of a prime number and can recite the prime numbers less than 20.

Notes on the Learner’s Book Prime numbers (p10): the investigation links work on odd and even numbers with prime numbers. This is followed by examples focusing on identification and use of prime numbers.

Check up!

“Which of these are prime numbers?”

11, 21, 31, 41, 51, 61

More activities Eratosthenes (individuals or pairs) The ‘Sieve of Eratosthenes’ is the 10 by 10 grid that learners used to find all the prime numbers less than 100. The mathematician who devised the sieve was Eratosthenes. Learners find out as much as they can about his life and work.

Core activity 2.3: Prime numbers

17

Instructions on page 14

Original Material Š Cambridge University Press, 2014

Blank 3 by 3 grid

81

71

61

51

41

31

21

11

1

92

82

72

62

52

42

32

22

12

2

93

83

73

63

53

43

33

23

13

3

94

84

74

64

54

44

34

24

14

4

95

85

75

65

55

45

35

25

15

5

96

86

76

66

56

46

36

26

16

6

97

87

77

67

57

47

37

27

17

7

98

88

78

68

58

48

38

28

18

8

99

89

79

69

59

49

39

29

19

9

100

90

80

70

60

50

40

30

20

10

Sieve of Eratosthenes

91

1. Cross out the number 1.

2. On the grid, cross out all the multiples of 2 except 2.

• What do you notice? Can you explain what you see?

3. On the same grid, cross out all the multiples of 3 except 3.

• Can you predict what will happen? Will you cross out any numbers that are already crossed out? If so, which ones?

4. Explore what happens for multiples of 4, 5, 6 and 7.

• (4× table) Do you need to cross out these multiples? Why not?

• (5× table) Cross out all the multiples of 5 except 5. Can you explain the pattern?

• (6× table) Do you need to cross out the multiples of 6? Why not?

• (7× table) Which numbers were not already crossed out?

Original Material © Cambridge University Press, 2014

Now look at your grid. What is special about the numbers that you haven’t crossed out?

Instructions on page 16

20

Blank page

1A

3 Multiplication and division (1)

Quick reference

Multiplying and dividing by 10, 100 and 1000

Core activity 3.1: Multiply and divide by 10, 100 and 1000 (Learner’s Book p11) Learners multiply and divide whole numbers by 10 and 100, extending to multiplying and dividing by 1000.

I multiply my number by 100, then divide by 10, then multiply by 1000. My answer is one hundred and seventy thousand.

1 1

?

?

! 10 " 2500

What is the missing number?

3

A decagon has 10 sides.

?

?

2 What is the perimeter of a regular decagon with sides 17 centimetres long?

4

?

!

?

!

?

" 24

Use the given fact to derive a new fact and then explain your method.

Fact

# 100 " 250

2

100 ! 10 " 10 000 #

Core activity 3.2: Mental strategies for multiplication (Learner’s Book p13) Learners develop and refine mental strategies for multiplication, including working with multiples and near multiples of 10, halving and doubling.

?

250 # 10 "

Let’s investigate

Find different ways of completing this calculation.

Copy and complete the table, the first one has been done for you.

Copy and complete this set of missing numbers.

25 ! 100 "

Mental strategies for multiplication

Vocabulary near multiple of 10: a number either side of a multiple of 10.

For example, 20 is a multiple of 10, so 19 and 21 are near multiples of 10.

Let’s investigate

Cheng is thinking of a number.

What number is Cheng thinking of?

3

Milly says, “Every multiple of 1000 is divisible by 100.”

Is she right?

Derived fact

a

7 ! 9 " 63

7 ! 18 " 126

b

7 ! 3 " 21

70 ! 3 "

c

5 ! 7 " 35

50 ! 70 "

d

6 ! 8 " 48

6 ! 16 "

e

8 ! 13 " 104

4 ! 13 "

f

6 ! 7 " 42

6 ! 70 "

g

5 ! 9 " 45

5 ! 91 "

6 ! 9 " 54

6 ! 89 "

h i

4 ! 7 " 28

j

3 ! 9 " 27

Method 18 is double 9 so double the answer

39 ! 7 " 30 ! 91 "

Use table facts to help you work out the following: (a) 30 × 70

(b) 50 × 9

(d) 50 × 80

(e) 8 × 90

(c) 20 × 6 (f) 70 × 60

Work out the following using a mental strategy: (a) 29 × 6

(b) 41 × 5

(c) 19 × 7

(d) 21 × 8

(e) 49 × 6

(f) 51 × 4

Explain to your partner how you worked out the answers.

Explain your answer.

For more questions, turn the page ...

Unit 1A: Core activity 3.1 Multiply and divide by 10, 100 and 1000

11

Unit 1A: Core activity 3.2 Mental strategies for multiplication

13

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

Prior learning

Objectives* –

This chapter revises work done in Stage 5 when learners multiplied and divided whole numbers by 10 and 100, and worked on mental strategies for multiplication.

1A: Numbers and the number system 6Nn4 – Multiply and divide any whole number from 1 to 10 000 by 10, 100 or 1000; explain the effect.

1A: Calculation (Multiplication and division) 6Nc8/6Nc14 – Multiply pairs of multiples of 10, e.g. 30 × 40, or multiples of 10 and 100, e.g. 600 × 40.

6Nc15 – Multiply near multiples of ten by multiplying the multiple of ten and adjusting.

6Nc16 – Multiply by halving one number and doubling the other, e.g. calculate 35 × 16 with 70 × 8.

1A: Problem solving (Using techniques and skills in solving mathematical problems) 6Pt1 – Choose appropriate and efficient mental or written strategies to carry out a calculation involving addition, subtraction, multiplication or division.

1A: Problem solving (Using understanding and strategies in solving problems) 6Ps1 – Explain why they choose a particular method to perform a calculation and show working.

6Ps4 – Use ordered lists or tables to help solve number problems systematically.

6Ps6 – Make sense of and solve word problems and represent them.

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

Vocabulary near multiple of 10

Cambridge Primary Mathematics 6 © Cambridge University Press 2014

Unit 1A

21

Core activity 3.1: Multiply and divide by 10, 100 and 1000

LB: p11

Resources: True or false multiplication and division cards photocopy master (p26).

Start the session by revising work on multiplying and dividing by 10 and 100.

Learners work in pairs using the True or false multiplication and division cards cut from the photocopy master. They take a card in turn and say whether the statement is true or false.

• If the statement is true they explain how they know.

• If the statement is false they give the correct answer.

Review work done, reminding learners of the following rules: To multiply by 10, move each digit one place value to the left. Zero may be needed as a place holder.

To multiple by 100, move each digit two place values to the left; 0 might be needed as a place holder.

Tth

Tth

3 “Following this pattern, how can I multiply by 1000?”

(Answer: move digits three places to the left e.g. 34 × 1000 = 34 000)

Th

Th

4

H

3 H

0

T

U

3

4

4 T

0 U

3

4

0

0

Emphasise that multiplying by 1000 is equivalent to multiplying by 10, then 10 again, then 10 again.

Repeat for division by 10, 100 and 1000. Ask, “Following this pattern, how can I divide by 1000?” (Answer: move digits three places to the right, e.g. 58 000 ÷ 1000 = 58) Emphasise that dividing by 1000 is equivalent to dividing by 10, then 10 again, then 10 again.

22

Unit 1A

3 Multiplication and division (1)

Example: • To divide by 10, move each digit one place value to the right.

• To divide by 100, move each digit two place values to the right.

Tth

Th

H

T

U

5

8

0

0

0

5

8

0

0

5

8

0

5

8

58 000 ÷ 10 5800 ÷ 10 (58 000 ÷ 100) 580 ÷ 10 (58 000 ÷ 1000)

Work with learners to complete these number sentences, asking each time for an explanation: 34 × 1000 = × 78 = 78 000 63 000 ÷ 1000 = 36 000 ÷

= 36

Summary Learners confidently multiply and divide any whole number by 10, 100 or 1000 and explain the effect.

Check up!

Notes on the Learner’s Book Multiplying and dividing by 10, 100 and 1000 (p11): learners apply their knowledge of multiplication and division by 10, 100 and 1000 to problems set in different contexts. Ensure that learners understand that to multiply by 1000 they can multiply by 10, then 10 again, then 10 again or they can multiply by 10 and 100 in either order. The investigation and questions 4 and 7 can be used to illustrate this relationship.

• “What is 48 000 ÷ 1000? How did you work it out?”

• “Complete these number sentences:” × 42 = 42 000

÷ 1000 = 6

More activities Make a poster (individual) Design a poster that shows how to multiply and divide by 10, 100 and 1000.

Illustrate it with examples, including drawings, pictures or photographs.

• 1 metre is 100 times as long as 1 centimetre.

• 1 cent is 100 times smaller than 1 dollar.

Core activity 3.1: Multiply and divide by 10, 100 and 1000

23

Core activity 3.2: Mental multiplication strategies

LB: p13

Resources: Large sheet of paper; one per pair of learners.

Ensure learners are confident with mental strategies for multiplication by revising methods they used in Stage 5.

“I’m going to start with a multiplication fact and use different strategies to find out other related facts.” Write 7 × 8 = 56 for the whole class to see.

Ask what other facts can be found.

Vocabulary

7 × 8 = 56

Take one response, for example, 7 × 4 = 28 (4 is half of 8 so halve the answer).

Learners work in pairs with a large sheet of paper to work out as many facts as they can for ‘7 × 8 = 56’ in a few minutes.

Review work done, recording facts on a ‘master’ diagram. Ensure that the following strategies are covered: • doubling Examples: 14 × 8 = 112 (double 7); 7 × 16 = 112 (double 8) • halving 7 × 4 = 28 (halve 8); 7 × 2 = 14 (halve 4) “What happens if I double one number and halve the other?”

(Answer: the answer stays the same, e.g 7 × 8 = 56 so 14 × 4 = 56) • using multiples of 10 7 × 80 = 560 (multiply 7 × 8 by 10) 70 × 8 =560 (multiply 7 × 8 by 10) 70 × 80 = 5600 (multiply 7 × 8 by 10 and 10 again, or by 100)

24

Unit 1A

3 Multiplication and division (1)

Spend a few minutes every day revising mental facts and developing mental strategies.

near multiple of 10: a number either side of a multiple of 10. For example, 20 is a multiple of 10 so 19 and 21 are near multiples of 20.

Opportunities for display!

Use ‘spider diagrams’ to show related facts.

• using multiples of 10 when the calculation involves near multiples of 10 “How could I use 7 × 80 = 560 to find 7 × 81?”

(Answer: add another 7 so 7 × 81 = 560 + 7 = 567) “How could I use 7 × 80 = 560 to find 7 × 79?”

(Answer: subtract 7 so 7 × 79 = 560 − 7 = 553) Say that, “We refer to numbers like 79 and 81 as near multiples of ten – in this case it means a ‘near to 80’, which is 10 × 8.”

Remind learners that building up a store of table facts and using mental strategies can often help work out multiplication calculations.

Summary • Learners revise their store of multiplication strategies to include doubling, halving and using multiples of 10.

• They understand how to adapt answers of multiplying by 10 to multiplying by a near multiple of 10.

Notes on the Learner’s Book Mental strategies for multiplication (p13): learners develop their mental strategies through oral work, best done frequently a little at a time. The learner book therefore contains a limited number of examples.

Check up!

• ‘“I know that 7 × 13 = 91. How can I work out 14 × 13?”

(Answer: double the answer as 14 is double 7.)

• “I know that 8 × 11 = 88. What is … ? How do you know?” … 80 × 11? (Answer: 880; multiply answer by 10 as 80 is ten times larger than 8.) … 8 × 110? (Answer: 880; multiply original answer by 10 because 110 is ten times larger than 11.) … 80 × 110? (Answer: 8800; multiply either of the previous answers by 10.)

• I know that 40 × 7 = 280. What is …? How do you know?”’ … 41 × 7? (Answer: 287; add 7 to the original answer, so 40 × 7 = 280 + 7 = 287) … 39 × 7? (Answer: 273; subtract 7 to the original answer, so 40 × 7 = 280 – 7 = 273)

More activities Mental mathematics (whole class) Ensure that you do frequent oral activities to revise and consolidate the various strategies of mental multiplication.

Create spider diagrams (individuals or pairs) As per the start of the core activity, start with a multiplication fact and derive other facts.

Games Book (ISBN 9781107667815) Domino multiplication (p5) is a game for two or four players. Learners practise multiplying multiples of 10 and 100.

Core activity 3.2: Mental multiplication strategies

25

True or false multiplication and division cards

789 × 10 = 7890

610 ÷ 10 = 61

407 × 100 = 40 070

4350 ÷ 10 = 435

675 × 10 = 6705

21 × 100 = 2100

4010 × 10 = 410

866 × 10 = 860

150 × 10 = 1050

1940 ÷ 10 = 194

4500 ÷ 10 = 45

6000 ÷ 100 = 6

5200 ÷ 100 = 520

302 × 100 = 30 200

45 600 ÷ 100 = 456

Instructions on page 22

Original Material © Cambridge University Press, 2014

1A

4 More on number

Quick reference Core activity 4.1: Addition of decimals (Learner’s Book p14) Learners revise previous learning on adding decimal numbers. They extend their knowledge to add decimals including with different numbers of decimal places.

Addition of decimals

Division (1)

Let’s investigate

Let’s investigate

Let’s investigate

Arrange the numbers 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6 in the circles so the sum along each side of the triangle is 1.2.

Abdul was asked how old he was.

Choose different starting numbers to make sequences that have a rule ‘add 5’.

If my age is divided by 2 or 3 or 4 there is 1 left over.

Is it possible to make a sequence where the rule is ‘add 5’ and the terms are:

If my age is divided by 7 there is no remainder.

Try using numbers on cards or small pieces of paper, that you can move around.

multiples of 5? multiples of 10?

List the multiples of 7.

How old is Abdul?

all odd? include 24 and 39?

1

Core activity 4.2: Division (1) (Learner’s Book p17) Learners revise division methods when there is a remainder by dividing two- and three-digit numbers by single-digit numbers.

Core activity 4.3: Number sequences (Learner’s Book p18) Learners consolidate previous learning using the vocabulary related to sequences with precision. They work increasingly with sequences involving fractions and decimals.

Number sequences

The answers to the following questions are in the grid.

Find which answer goes with which question.

Which number is not one of the answers?

8.28

2 3

Vincent wants to put 75 photographs in an album.

A full page holds 6 photographs. What is the smallest number of pages Vincent uses?

2

Which number on the grid can be divided by 8 with a remainder of 1?

2.05

7.8

5.41

12.18

13.95

4.98

12.21

3

(a) 4.61 ! 0.8

(b) 0.45 ! 1.6

(c) 3.7 ! 4.58

(d) 6.1 ! 7.85

(e) 4.3 ! 0.68

(f) 7.5 ! 4.68

(g) 4.25 ! 7.96

(h) 3.45 ! 0.85

4

5.55

5.15

5.5

Unit 1A: Core activity 4.1 Addition of decimals

(b) 68 ! 7

Copy the grid. Shade the squares that have a remainder in the answer.

What letter have you made?

Find the sum of all the numbers less than 5.5 in this list.

5

67

72

51

42

73

64

60

20

69

are not whole numbers?

Complete these calculations: (a) 132 ÷ 6

(b) 146 ÷ 9

(c) 147 ÷ 2

(d) 107 ÷ 4

(e) 156 ÷ 8

(f) 148 ÷ 9

Unit 1A: Core activity 4.2 Division (1)

You will need to try different starting numbers.

step: the ‘jump size’.

For example, in the sequence 60 110 160 210 +50

1

Here is the beginning of a sequence of numbers: 8, 16, 24, 32, 40 ...

The sequence continues in the same way.

Will 88 be in the sequence? Explain how you know.

Complete these calculations: (a) 78 ! 4

Kiki has two pieces of rope. One piece is 93.7 metres long and the other piece is 125.9 metres long. What is the total length of her rope?

5.05

14

4.3

1

Vocabulary sequence: an ordered set of numbers, shapes or other mathematical objects arranged according to a rule.

For example, 3, 6, 9, 12, 15 …

1, 4, 9, 16, 25 … !"#!"#!

(c) 98 ! 6

2

41 ÷ 4

47 ÷ 9

48 ÷ 5

14 ÷ 4

25 ÷ 5

31 ÷ 3

55 ÷ 6

27 ÷ 6

50 ÷ 7

34 ÷ 7

48 ÷ 6

54 ÷ 9

60 ÷ 8

54 ÷ 6

49 ÷ 7

A sequence starts at 200 and 30 is subtracted each time.

200, 170, 140 ...

+50

+50

the step is ’!50’. term: one of the numbers in a sequence. rule: a rule tells you how things or numbers are connected. For example, the numbers 3, 7, 11, 15, 19 … are connected by the rule ‘add 4 to the previous number’.

What are the first two numbers in the sequence that are less than zero?

17

18

Unit 1A: Core activity 4.3 Number sequences

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

Prior learning

Objectives* –

This chapter builds on work in Stage 5 where learners worked on number sequences and continued to develop and refine mental and written strategies for the four operations: addition, subtraction, multiplication and division.

1A: Numbers and the number system 6Nn15 – Recognise and extend number sequences.

1A: Calculation (Mental strategies) 6Nc1 – Recall addition/subtraction facts for numbers to 20 and pairs of one-place decimals with a total of 1, e.g. 0.4 + 0.6.

6Nc10 – Divide two-digit numbers by single-digit numbers, including leaving a remainder.

1A: Calculation (Addition and subtraction) 6Nc11 – Add two and three-digit numbers with the same or different numbers of digits/decimal places.

1A: Problem solving (Using techniques and skills in solving mathematical problems) 6Pt1 – Choose appropriate / efficient mental or written strategies to carry out a calculation involving addition, subtraction, multiplication or division.

6Pt3 – Check addition with a different order when adding a long list of numbers; check when subtracting by using the inverse.

1A: Problem solving (Using understanding and strategies in solving problems) 6Ps1 – Explain why they choose a particular method to perform a calculation and show working.

6Ps6 – Make sense of and solve word problems and represent them.

Vocabulary

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

sequence • step • term • rule

Cambridge Primary Mathematics 6 © Cambridge University Press 2014

Unit 1A

27

Core activity 4.1: Addition of decimals

LB: p14

Resources: (Optional: 0–9 digit cards photocopy master (CD-ROM).)

Start by challenging the learners to solve the following number puzzle: “Complete the diagram (on the right) so that each line of numbers totals 1.”

Get feedback from the class about how learners solved the puzzle. Suggestions might include completing the bottom line first as a good starting place: they could add together the two existing numbers, then subtract them from 1, or use number pairs to 1 (for example 0.1 + 0.4 = 0.5; learners know that 5 and 5 is a number pair to 10 and therefore that 0.5 and 0.5 is a number pair to 1, so the third number must be 0.5). As they now have two numbers on both of the other lines, they can add and subtract from 1 (or use number pairs to 1) as before.

As a class, practise some decimal additions where both numbers have one decimal place, e.g.

“What is 0.3 + 0.7?” (Answer: 1) “What is 1 − 0.9?” (Answer: 0.1) (Again, number pairs can be useful here).

Move on to decimal addition where the numbers have different numbers of digits and decimal places, for example ask: “How could we work out 0.7 + 0.51?” (Answer: 1.21) Gather information from the learners on what we need to think about when we are adding numbers with different numbers of decimal places. Make sure this includes the importance of place value: tenths must be added to tenths, hundredths must be added to hundredths and so on.

Learners may suggest a variety of methods including: • partitioning (in this case into tenths and hundredths) 0.51 = 0.5 + 0.01 0.7 + 0.5 = 1.2 1.2 + 0.01 = 1.21

28

Unit 1A

4 More on number

0.2

0.1

0.4

Opportunities for display!

Posters with different addition methods shown for a given sum.

Addition and subtraction strategies from Stage 4 that could be adapted for adding decimal numbers: • 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; or adding the 20 to 30 first to make 50, then doubling 50.

• 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.

• using a number line +0.1

+0.1 0.51

0.61

+0.1 0.71

+0.1 0.81

+0.1 0.91

+0.1 1.01

+0.1 1.11

1.21

• using a vertical addition 0.70 + 0.51 1.20 (+ 0.5) 1.21 (+ 0.01) • using a standard written method: 0.70 + 0.51 1.21 1

When using either ‘vertical’ method, encourage learners to write 0.7 as 0.70 so that all numbers have the same number of decimal places.

Practise the methods discussed by asking learners to choose the best method to answer the following: 2.45 + 4.3 (Answer: 6.75) 34.61 + 1.92 (Answer: 36.53) 82.03 + 230.8 (Answer: 312.83)

In this standard written method, the learners work from right to left, starting at the hundredths. They add vertically down the place value column and write the answer below. In the tenths column 7 + 5 = 12, so they write ‘2’ down and carry the ‘1’ (a unit) over to the next column to the left.

Learners should use whatever method works best for them. Support learners by asking them to consider whether their method is: • Checkable • Accurate • Reliable • Efficient.

Gather answers from the class and ask learners which method they chose. Leaners should also tell you if they got the answer wrong in order to identify where it went wrong. Did other learners calculate the answer using a different method?

Core activity 4.1: Addition of decimals

29

Summary Learners add two- and three-digit numbers with the same or different numbers of decimal places using an appropriate mental or written strategy.

Notes on the Learner’s Book Addition (p14): a variety of questions focusing on the addition of decimals in and out of context. Question 5, 6, 7, 9 and 10 ask the learners to solve puzzles by applying what they know about adding decimals.

Check up!

• “Add 1.46 and 0.9. Explain your method.”

• “Ahmed has made a mistake in the calculation: 0.7 + 0.41 = 0.48 How would you help him avoid making the same mistake again?”

More activities Add the cards (pairs) You will need 0–9 digit cards photocopy master (CD-ROM).

Each player will need a set of 0–9 digit cards and two decimal points. Each player shuffles their cards and then deals out six digit cards together with the two decimal points. Arrange the cards like this: • +

•

Complete the calculation. The winner of the round is the player with the highest total. Play more rounds.

You can vary the game by choosing different arrangements for the cards, for example: • +

30

Unit 1A

• +

•

4 More on number

•

Blank page

31

Core activity 4.2: Division (1)

LB: p17

Resources: (Optional: 0–9 digit cards photocopy master (CD-ROM).)

Learners work in pairs to solve the three word problems in any way they like: 1. Three boys share 56 marbles. How many marbles does each boy get and how many are left over? (Answer: 18 r2) 2. 72 cubes are arranged in groups of five. How many groups are there and how many are left over? (Answer: 14 r2) 3. Zina needs 172 stickers. They are sold in packs of six. How many packs must she buy? How many extra stickers will she have?

(Answer: 29. She will have 2 extra) Review work done ensuring that the following points are covered: 1. Division can be thought of as sharing or grouping.

12 flowers shared between four people.

12 cubes in groups of four.

2. There are a number of ways of performing a division calculation, for example, 159 ÷ 7. (Answer: 22 r5) These include: • using a number line to count back from 159: −7 −7

−70

0 5 12 19

−70 89

159

• using a number line to count on from 0: +70 0

32

Unit 1A

+70 70

4 More on number

+7 +7 140 147 154 159

• using repeated subtraction: 159 70− 89 70− 19 14− 5

(10 × 7) (10 × 7) (2 × 7)

• using short division: 2 2r5 7 1519 3. Remainders can be thought of as ‘a number left over’ but in word problems the remainder must be dealt with in the context of the problem, and the learner must decide if they can keep the answer as a remainder or if they need to round the answer up or down. For example, for puzzle 3, Zina’s stickers, 172 ÷ 6 is 28 packs with a remainder of 4, so she must buy 29 packs.

4. Always use the most appropriate and efficient way to perform each calculation. (Teachers may need to discuss this with individuals, groups or the whole class.) (Is their method Checkable, Accurate, Reliable and Efficient?)

Summary • Learners confidently divide a two- or three- digit number by a single-digit number in and out of context.

• When dealing with contextual problems, learners treat any remainder appropriately; as a remainder or by rounding up or down to the nearest whole number.

Notes on the Learner’s Book Division (1) (p17): the investigation is set in a context of division but is really about using table facts, so it might provide an opportunity to remind learners why it is important to aim for recall of multiplication (and division) facts. The five questions provide practice of division, focusing on remainders.

Check up!

Ask learners questions such as: • “Write a word problem for 15 ÷ 4. What is an appropriate answer to your problem?”

• “What method would you use to divide 113 by 7? Explain why you chose this method.”

More activities Divide the cards (pairs) You will need 0–9 digit cards photocopy master (CD-ROM).

Each player needs a set of 0–9 digit cards. Shuffle the cards and deal four cards to each player. The players choose how to arrange their cards in to this layout: Each player works out the answer. The winner of the round is the player with the smallest answer.

÷

Core activity 4.2: Division (1)

33

Core activity 4.3: Number sequences

LB: p18

Resources: There are no specific resources required for this activity.

Ensure that learners are confident in finding the rule for number sequences by carrying out a series of activities.

Write the numbers 20.1, 20.3 and 20.5 for the whole class to see. Say, “These are three consecutive terms in a sequence.” Ask learners what you mean by the words ‘term’ and ‘sequence’, and remind them what we mean by ‘consecutive’ if you need to.

Continue the sequence forwards (20.7, 20.9, 21.1 . . .) and backwards (19.9, 19.7, 19.5 . . .)

“What is the step size?” (Answer: the ‘jump’ is 0.2) “What is the rule?” (Answer: add 0.2) Ask learners what you mean by ‘step’ and ‘rule’.

Repeat for other sequences, for example: 1

1

Ask, “If I start at 3, and my steps are 0.25 what will the 5th term of my sequence be?”

Allow thinking time, then ask learners to share their strategies (write the sequence out: 3, 3.25, 3.5, 3.75, 4 . . . ).

Ask learners to suggest a start number and step size, then find the sequence as a class.

Ask learners to work in pairs to complete the following sequences:

0.9

(add 15; 10, 25, 40, 55, 70, 85)

70 −6

(add 2; –10, –8, –6, –4,–2, 0)

−4 −2 1.1

1.3

(add 0.1; 0.8, 0.9, 1.0, 1.1, 1.2, 1.3)

Review the work done and then ask questions in a different format, for example: “These numbers form part of a sequence 1.4, 1.7, 2 . . .

• If 1.7 is the middle term in a sequence of five numbers, what are the start and finish numbers?

(Answer: 1.1 and 2.3) • If 1.4 is the third term what would the first term be? (Answer: 0.8) • If 1.4 is the fifth term what would the tenth term be?” (Answer: 2.9)

34

Unit 1A

consecutive: numbers increase from smallest to largest one after the other, without any gaps. For example, 1, 2, 3, 4, 5 . . . sequence: an ordered set of numbers, shapes or other mathematical objects arranged according to a rule.

For example, 3, 6, 9, 12, 15 . . .

1, 4, 9, 16, 25 . . . ∆ ∆ step: the ‘jump size’. For example, in this sequence the step is +50,

3

14, 12, 14, 2 . . .

1.25, 1.5, 1.75 . . .

10 25

Vocabulary

4 More on number

60 + 50 110 + 50 160

+ 50

210

+ 50

term: one of the numbers in a sequence. rule: tells you how things or numbers are connected.

For example, the numbers 3, 7, 11, 15, 19 . . . are connected by the rule ‘add 4 to the previous number’.

Summary • Learners revise work on sequences using ascending and descending sequences involving positive and negative numbers, fractions and decimals.

• They use the language (sequence, step, term, rule) with precision.

Notes on the Learner’s Book Number sequences (p18): learners find missing numbers in sequences or continue sequences.

They might be given the rule or have to work out the rule. Questions 4 to 7 use the vocabulary introduced in the unit (step, term, rule).

Check up!

• “Complete these sequences: first term 4, rule add 5 first term 0.5, rule subtract 2.”

• “A sequence starts at 100 and 40 is subtracted each time. What is the first term less than zero?”

More activities Generating sequences (small groups or whole class) The first player is given a starting number and the rest of the group take turns to continue the sequence, following a given rule with the leader choosing the next player. Start with a low number and a simple rule, for example, start at 5 and add on 2. Extend to include negative numbers, fractions and decimals.

Games Book (ISBN 9781107667815) Sequence trail (p8) is a game for individuals or pairs. It gives practice in finding missing terms in sequences.

Answers (start anywhere in the loop): −60 → −35 → 6 → 65 → 1210 → 1025 → 1046 → 8 → 22 → 17 → 120 → 12 → 15 → −2 → 7 → 162 → 3 → 1027 →

Core activity 4.3: Number sequences

35

36

Blank page

1B

5 Length

Quick reference

Measure

Drawing length

Measuring length

Core activity 5.1: Working with length (Learner’s Book p20) Learners select and use standard units of length. They convert between km, m, cm and mm. They use measuring instruments with different scales.

Nicola needs to put up bunting around a whole room for a party. The room is 4 m long and 3 m wide.

She has lots of 70 cm lengths of bunting. Every time she ties two pieces together she needs to use 50 mm of each piece of string for the knot. How many pieces of 70 cm long bunting does she need to go right around the room?

First work out the perimeter of the room.

Choose to do the calculation in mm, cm or m and make all the measurements have the same unit. You could take, or imagine, three or four pieces of string and tie them together to better understand the problem.

Core activity 5.2: Drawing lines (Learner’s Book p22) Learners practise drawing lines accurately to the nearest millimetre. They are reminded of the importance of using a sharpened pencil for greater accuracy.

1

Accurately draw straight lines that measure: (a) 9.6 cm

millimetre (mm): a unit for measuring length.

(b) 122 mm (c) 0.129 m (d) 5.1 cm

centimetre (cm): a unit for measuring length.There are 10 mm in 1 cm.

(e) 26 mm (f) 0.088 m 2

metre (m): a unit for measuring length.

There are 100 cm in 1 m.

Your pencil should be sharp.

Check the scale on your ruler. If necessary convert the length into the units shown on your ruler.

Find ‘0’ and the point on the scale you need to make the correct length before starting to draw the line.

Curves and patterns, such as those below, can be made by careful measuring and drawing straight lines.

Use your ruler to check that all the lines drawn on the designs are straight.

kilometre (km): a unit for measuring length.

There are 1000 m in 1 km.

Eric has lost his umbrella. At the Lost Property Office he looks at the list of umbrellas that have been handed in. The person who has filled in the Lost Property log seems to have confused the units of length.

Object

20

1

Vocabulary

Let’s investigate

Date Handed in

Colour

Length

umbrella

12 September

Black

218 m

umbrella

25 October

Blue (with flowers)

84.9 m

umbrella

26 October

Red

895 cm

umbrella

5 November

Black

97.2 mm

umbrella

19 November

Pink

527 cm

umbrella

20 November

Silver

1.05 cm

Unit 1B: Core activity 5.1 Working with length

22

Unit 1B: Core activity 5.2 Drawing lines

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

Prior learning

Objectives* –

• Read, choose, use and record standard units to estimate and measure length, mass and capacity to a suitable degree of accuracy.

• Convert larger to smaller metric (units decimals to one place) e.g. 2.6 kg to 2600 g.

• Order measurements in mixed units.

• Round measurements to the nearest whole unit.

• Interpret a reading that lies between two unnumbered divisions on a scale.

• Compare readings on different scales.

• Draw and measure lines to nearest centimetre and millimetre.

1B: Measure (Length, mass and capacity) 6Ml1 – Select and use standard units of measure. Read and write to two and three decimal places.

6Ml2 – Convert between units of measurement (kg and g, l and ml, km, m, cm and mm), using decimals to three places, e.g. recognising that 1.245 m is 1 m 24.5 cm.

6Ml3 – Interpret readings on different scales, on a range of measuring instruments.

6Ml4 – Draw and measure lines to nearest centimetre and millimetre.

2A: Numbers and the number system 6Nn16 – Recognise and use decimals with up to three places in the context of measurement.

1A: Problem solving (Using techniques and skills in solving mathematical problems) 6Pt2 – Understand everyday systems of measurement in length, weight, capacity, temperature and time and use these to perform simple calculations.

6Pt5 – Estimate and approximate when calculating, e.g. use rounding, and check working.

1B: Problem solving (Using understanding and strategies in solving problems) 6Ps4 – Use ordered lists or tables to help solve problems systematically.

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

Vocabulary millimetre • centimetre • metre • kilometre

Cambridge Primary Mathematics 6 © Cambridge University Press 2014

Unit 1B

37

Core activity 5.1: Working with length

LB: p20

Resources: Measuring length poster photocopy master (p44). Car insurance lengths resource sheet photocopy master (p45). Measuring straight lines group resource sheet photocopy master (p46). Measuring curved lines resource sheet photocopy master (p47). Tape-measures, metre sticks, rulers. Small pieces of paper, or sticky notes. String. Pens. Dice. (Optional: four cones (or other large place markers).)

Units of length In small groups, ask learners to list, or draw a diagram such as a cluster/cloud diagram, to show what they know about length and the equipment and units used for measuring and communicating length. Display the groups’ lists alongside the Measuring length poster. Encourage learners to refer to the poster and their lists during the following activities.

[Note: the poster only shows metric measures of length but some learners may be familiar with imperial measures. At this stage, don’t try to convert between the two systems but see if they know the following relationships: 12 inches = one foot (the length of a standard classroom ruler), three feet = one yard (one yard is a little bit less than a metre) and 1760 yards = one mile (one mile is about 1.6 km).]

Tell the learners that people often need to fill in some details about their car for the insurance company, or for taking it on a ferry. Show them the Car insurance lengths resource sheet. Look at the first example together. Ask learners to briefly discuss in pairs which pieces of information are unrealistic. Tell learners that some of the information has been given with the correct numbers, but the incorrect units. Ask them to say what the units should be. Use a tape measure to demonstrate the difference between values in metres, centimetres and millimetres, and the importance for using the correct units. Learners should correct the second example themselves.

(Answers: Ciorda; units of length and height should be m, distance units should be km. Mutsui; length should be 3.97 m, height units should be m, distance units should be km.)

Get permission from members of staff for learners to measure the length and height of their cars and, if possible, find out the distance travelled and the average fuel consumption.

(If necessary use an internet search or a conversion chart to convert miles into kilometres or, for a close approximation, multiply the miles by 1.6 to get the kilometres.) Before

38

Unit 1B

5 Length

Vocabulary millimetre (mm): a unit for measuring length. centimetre (cm): a unit for measuring length.

There are 10 mm in 1 cm. metre (m): a unit for measuring length.

There are 100 cm in 1 m. kilometre (km): a unit for measuring length.

There are 1000 m in 1 km.

Opportunities for display!

Display the learners’ diagrams/lists about lengths and measuring equipment.

Look out for!

Learners who don’t understand the difference between measuring in a unit and measuring to the nearest whole unit. Give an example, such as: a measurement given in metres could be measured to the nearest metre (2 m), the nearest centimetre (2.45 m), or the nearest millimetre (2.453 m). In the case of a car, measuring to the nearest metre would usually not be accurate enough and measuring to the nearest millimetre would be unnecessarily too accurate.

learners measure the cars, ask them to explain and discuss how accurate they think their measurements need to be, e.g. to the nearest metre, centimetre or millimetre. Learners can use the blank table to complete the information about the car they have measured.

Measuring Length

Use observations of this activity to inform teaching in the next activity, where learners will learn and practise measuring with greater accuracy to the nearest millimetre.

Tell learners that their challenge is to increase their accuracy in measuring and drawing lines to the nearest millimetre, which they started to do in Stage 5.

Give groups, of approximately six learners, the Measuring straight lines group resource sheet. Ask each member of the group to secretly measure line A. They should write their measurement to the nearest millimetre on a small piece of paper or sticky note. Once all the members of the group have measured the line they should compare their measurements and reach an agreement about the true length of the line, to the nearest millimetre. Ask the members of the group who have measured the line most accurately to support less accurate members to measure the line again, watching out for learners: • starting their measurement from the end of the ruler, rather than the start (0) of the scale • moving the ruler as they measure • reading the scale incorrectly • rounding incorrectly, to the wrong division.

Groups should carry out the same activity with the other lines.

Look out for!

Learners who make some of the common mistakes listed on the left. Whilst encouraging learners to self and peer diagnose problems with measuring, it could be useful to show some individuals what has gone wrong, for example, have two rulers that have different ‘gaps’ at the end.

Ask learners to reflect on whether they think that the group activity has improved their measuring, and on how they have improved their measuring technique.

Draw a large, simple curve on the board. Demonstrate to learners how to measure the line using a piece of string and a partner (one of the learners or an additional adult). Hold the end of the string at the start of the line and lay the string over the line until it reaches the end of the line. Mark the string with a pen where it meets the end of the line. Measure the string from the start to the marked point using a ruler or metre stick. Tell learners that when they measure a curve to make sure that they measure the piece of string that they used to measure the line, and not the end of the string.

Give pairs of learners the Measuring curved lines resource sheet, some string and a pen. Ask them to work together to measure each line.

Discuss together successful measuring techniques and advice for others trying to improve the accuracy of their measuring.

Core activity 5.1: Working with length

39

Summary • • • •

Learners have selected and used standard units of length.

They have converted between km, m, cm and mm.

They have used measuring instruments with different scales.

Learners have improved their accuracy in measuring lines to the nearest millimetre.

Notes on the Learner’s Book Length (measuring) (p20): learners correct measurements written in the wrong units and order lengths. They estimate the lengths of lines, to the nearest cm, and measure them to the nearest mm.

Check up!

Ask learners to measure the length of the classroom and to give the measurement in kilometres (to the nearest metre), metres (to the nearest centimetre), centimetres (to the nearest millimetre) and in millimetres. Ask them to suggest which units they think would be best for giving to someone laying a new floor for the classroom, and why.

More activities Circuit (whole class or small groups) You will need four cones (or other large place markers). Tape-measure.

Learners place four cones in an outside space to mark out a running circuit. Ask learners to measure the length of one complete circuit and calculate how many times they would need to run around the circuit to complete a 1 km run. Ask them to mark where the start and finishing lines would be on the circuit.

Unit conversion (individual) Ask learners to work out how many millimetres there are in a kilometre. Challenge them to write 1 mm in kilometres.

Games Book (ISBN 9781107667815) The ordering lengths game (p59) is a game for two to four players. Learners race to order lengths given in km, m, cm and mm.

40

Unit 1B

5 Length

Blank page

41

Core activity 5.2: Drawing lines

LB: p22

Resources: Dice.

Drawing lines Tell learners that they will be trying to improve the accuracy of their line drawing. Remind learners that when a pencil is sharpened it has a tip that is less than one millimetre in width, when it is blunt it could be more than two millimetres wide. Explain that the objective for this activity is for the learners to draw lines accurately to the nearest millimetre. If the learners’ pencils tips are wider than a millimetre they will not be able to draw the lines as accurately.

Tell the learners that they are going to draw a line that is 43 mm long, ask the learners how many centimetres that is. (Answer: 4.3 cm) All the learners will draw the line on plain paper.

Tell the learners to pass their line to a partner for them to check that the line measures 43 mm.

Ask learners what advice they can remember that helps them to accurately draw a line to the nearest millimetre. Ask the learners to apply this advice to draw a line of 38 mm/3.8 cm, which their partner will measure. Encourage learners to reflect on their accuracy and whether they need further support to improve their accurate drawing.

Remind learners that they are measuring and drawing lines to the nearest mm, not exactly that length as they should understand the continuous nature of measure. Ask learners to 1 mark a mm subdivision between 3 mm and 4 mm marks on a ruler with a sharp pencil. Ask 2 them what measurements would round, to the nearest half millimetre, to 3.5 mm. Discuss with learners what the range of measurements would be that would round to 3.5 mm (3.25 mm to 3.74 mm).

42

Unit 1B

5 Length

Look out for!

Learners who have difficulty drawing to the nearest millimetre. Gather these learners in a group and give instructions verbally to help them understand the process of accurately drawing lines, and avoiding common mistakes: 1. “Check whether your ruler measures in centimetres or millimetres.”

2. “Find ‘0’ on the ruler.”

3. “Find ‘4.3’ on the ruler if it measures in centimetres, or 43 if it measures in millimetres.”

4. “Place the ruler on the paper where you wish to draw the line.”

5. “Hold the ruler down firmly with the hand you do not write with.”

6. “Mark a dot on the paper next to the ruler at the positions ‘0’ and ‘43’ or ‘4.3’.”

7. “Still pressing firmly on the ruler with the nonwriting hand, put the tip of your pencil on the dot by the ‘0’ then run the pencil in one smooth movement along the ruler, stopping exactly on the other dot.”

Give pairs of learners two dice. They throw the dice and record the two 2-digit numbers made by the numbers shown on the dice, for example, if a 1 and a 4 are thrown the 2-digit numbers are 14 and 41. Each partner draws one line to match one of the 2-digit numbers made in millimetres.

Partners check the length of each other’s lines. They calculate the difference between the lengths of the two lines in centimetres, then mark the length of the shorter line on the longer line and measure the remainder of the line to check the accuracy of their drawing, measuring and calculation.

Opportunities for display!

Display the instructions in ‘Look out for!’ as a poster for learners to refer to when they are drawing lines.

Opportunities for display!

Ask learners to create a picture or pattern using one each of these lines: • 16 mm • 25 mm • 39 mm • 40 mm • 51 mm • 73 mm • 100 mm

Display the pictures and patterns made with the lines. Ask learners to identify which line is which.

Summary Learners will have improved their accuracy in drawing lines to the nearest millimetre.

Notes on the Learner’s Book Length (drawing) (p22): learners draw lines to the nearest mm, converting from other units of length. Learners use careful drawing of straight lines to produce interesting and colourful patterns.

Check up!

Ask learners to measure the length of an object, less than 20 cm, to the nearest mm. They should draw a line of the same length then place the object next to their line to check their accuracy in measuring and drawing. If their line does not match the length of the object, ask them to find out whether their measuring, drawing, or both, were not accurate.

More activities Spiral (individual) Learners can make a spiral pattern by drawing a 4 mm line, rotating the ruler slightly and adding an 8 mm line to the end of the original line at an angle, and then continuing by drawing a line 4 mm longer each time the ruler is rotated.

Games Book (ISBN 9781107667815) The length competition game (p59) is a game for three players. Players compete to draw the most accurate lines, to the nearest millimetre.

Core activity 5.2: Drawing lines

43

Measuring length poster Some of the units length is measured in are kilometres, metres, centimetres and millimetres.

There are 1000 metres (m) in 1 kilometre (km).

There are 100 centimetres (cm) in 1 metre (m).

There are 10 millimetres (mm) in 1 centimetre (cm).

There are 1000 millimetres (mm) in 1 metre (m).

You can use decimals to show part of a whole unit.

Examples: 1.69 km = 1690 m 2.85 m = 285 cm 0.78 m = 78 cm = 780 mm A tape measure for measuring around curves.

1.235 m = 123.5 cm

3.9 cm = 39 mm

A ruler for measuring straight lines.

Š l Nata-Lia/Shutterstock; r SmileStudio/Shutterstock

Instructions on page 38

Original Material Š Cambridge University Press, 2014

Colour

Height of car

Length of car

Number of seats

Type of car

Make of car

Average fuel consumption

Distance driven

Colour

Height of car

Length of car

Number of seats

Type of car

Make of car

58 000 mm

White

1.3 cm

397 km

2

Convertible

Mutsui

7.9 litres to 100 km

30 000 cm

Silver

1.5 km

4.272 mm

5

Hatchback

Ciorda

Car insurance lengths resource sheet

Distance driven

9.4 litres to 100 km

Original Material Š Cambridge University Press, 2014

Average fuel consumption Make of car Type of car Number of seats Length of car Height of car Colour Distance driven Average fuel consumption

Instructions on page 38

Instructions on page 39

G

E

B

A

D

Original Material Š Cambridge University Press, 2014

F

C

Measuring straight lines group resource sheet

B

A

C

D

Original Material Š Cambridge University Press, 2014

Measuring curved lines resource sheet

Instructions on page 39