Cambridge Maths 7 Prelim Pages

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STUART PALMER | DAVID GREENWOOD

BRYN HUMBERSTONE | JUSTIN ROBINSONJENNY GOODMAN | JENNIFER VAUGHAN

NSW SYLLABUS FOR THE AUSTRALIAN CURRICULUM

CambridgeMATHS

YEAR

77

ISBN: 9781107626973

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© David Greenwood et al. 2013 Cambridge University Press



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It furthers the University’s mission by disseminating knowledge in the pursuit ofeducation, learning and research at the highest international levels of excellence.

www.cambridge.edu.auInformation on this title: www.cambridge.org/9781107626973

© Stuart Palmer, David Greenwood, Bryn Humberstone,Justin Robinson, Jenny Goodman, Jennifer Vaughan 2013

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First published 2013

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ISBN 978-1-107-62697-3 Paperback

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ISBN: 9781107626973





Table of ContentsTable of Contents

iii

Table of ContentsTable of Contents

iii

1 Number and Algebra

Computation with Integers

MA4–4NA

Strand and substrand

2 Measurement and Geometry

Angle Relationships

MA4–18MG

About the authors viii

Introduction and guide to this book ix

Acknowledgements xii

Computation with positive integers 2

Pre-test 4

1A Place value in ancient number systems FRINGE 51B Place value in Hindu-Arabic numbers REVISION 10

1C Adding and subtracting positive integers 14

1D Algorithms for adding and subtracting 19

1E Multiplying small positive integers 23

1F Multiplying large positive integers 27

1G Dividing positive integers and dealing with remainders 31

1H Estimating and rounding positive integers 36

1I Order of operations with positive integers 40

Investigation 44

Puzzles and challenges 46

Review: Chapter summary 47

Multiple-choice questions 48

Short-answer questions 49

Extended-response questions 50

Angle relationships 52

Pre-test 54

2A Points, lines, intervals and angles 55

2B Measuring and classifying angles REVISION 60

2C Adjacent angles and vertically opposite angles 66

2D Transversal lines and parallel lines 71

2E Solving geometry problems 78

2F Circles and constructions with ruler and compasses FRINGE 83

2G Constructions with dynamic geometry software EXTENSION 87

Investigation 90


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Computation with Integers

MA4–4NA


Fractions, Decimals and

Percentages

MA4–5NA





Computation with positive and negative integers 98

Pre-test 100

3A Working with negative integers 101

3B Adding or subtracting a positive integer 105

3C Adding or subtracting a negative integer 109

3D Multiplying or dividing by an integer 114

3E Order of operations with positive and negative integers 118

3F The Cartesian plane 122

Investigation 126






Understanding fractions, decimals and percentages 132

Pre-test 134

4A Factors and multiples 135

4B Highest common factor and lowest common multiple 1404C What are fractions? 144

4D Equivalent fractions and simplified fractions 150

4E Mixed numerals and improper fractions 156

4F Ordering positive and negative fractions 162

4G Place value in decimals and ordering decimals 168

4H Rounding decimals 172

4I Decimal and fraction conversions 176

4J Connecting percentages with fractions and decimals 181

4K Decimal and percentage conversions 188

4L Fraction and percentage conversions 193

4M Percentage of a quantity 198

4N Using fractions and percentages to compare two quantities 203

Investigation 208






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v



Fractions, Decimals and

Percentages

MA4–5NA

Time

MA4–13MG

5

Probability

MA4–21SP

Statistics and ProbabilityProbability 218

Pre-test 220

5A Describing probability 221

5B Theoretical probability in single-step experiments 228

5C Experimental probability in single-step experiments 233

5D Compound events in single-step experiments 238

5E Venn diagrams and two-way tables 243

5F Probability in two-step experiments EXTENSION 250

Investigation 255






Computation with decimals and fractions 264

Pre-test 266

6A Adding and subtracting decimals 267

6B Adding fractions 271

6C Subtracting fractions 278

6D Multiplying fractions 284

6E Multiplying and dividing decimals by 10, 100, 1000 etc. 292

6F Multiplying by a decimal 297

6G Dividing fractions 301

6H Dividing decimals 307

6I Computation with negative fractions EXTENSION 312

Investigation 318






Semester review 1 327

Time 336

Pre-test 338

7A Units of time 339

7B Working with time 344

7C Using time zones 349


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Extended-response question 359

Algebraic techniques 1 360

Pre-test 362

8A Introduction to formal algebra 363

8B Substituting positive numbers into algebraic expressions 368

8C Equivalent algebraic expressions 373

8D Like terms 377

8E Multiplying, dividing and mixed operations 381

8F Expanding brackets 385

8G Applying algebra EXTENSION 390

8H Substitution involving negative numbers and mixed operations 3958I Number patterns EXTENSION 398

8J Spatial patterns EXTENSION 403

8K Tables and rules EXTENSION 411

8L The Cartesian plane and graphs EXTENSION 417

Investigation 424






Equations 1 432

Pre-test 434

9A Introduction to equations 435

9B Solving equations by inspection 440

9C Equivalent equations 444

9D Solving equations systematically 449

9E Equations with fractions 456

9F Equations with brackets 4619G Formulas and relationships EXTENSION 465

9H Using equations to solve problems EXTENSION 469

Investigation 474






Algebraic Techniques

MA4–8NA

Equations

MA4–10NA

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vii



Measurement and computation of length, perimeter and area 480

Pre-test 482

10A Measurement systems of the past and present FRINGE 483

10B Using and converting units of length REVISION 489

10C Perimeter of rectilinear figures 493

10D Pi and circumference of circles 498

10E Arc length and perimeter of sectors and composite figures 503

10F Units of area and area of rectangles 509

10G Area of triangles 515

10H Area of parallelograms 521

10I Area of composite figures 525

10J Mass and temperature REVISION 530

Investigation 536

Puzzles and challenges 537Review: Chapter summary 538




Introducing indices 544

Pre-test 546

11A Divisibility tests 547

11B Prime numbers 552

11C Using indices 555

11D Prime decomposition 560

11E Squares, square roots, cubes and cube roots 564

11F The zero index and index laws 569

Investigation 574





Extended-response question 580Semester review 2 581

Answers 589

Index 645

Length and Area

MA4–12MG, MA4–13MG

Indices

MA4–9NA

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Table of Contents

ii

About the authors

ii

Stuart Palmer was born and educated in NSW. He is a high school mathematics teacher

with more than 25 years’ experience teaching students from all walks of life in a variety of

schools. Stuart has taught all the current NSW Mathematics courses in Stages 4, 5 and 6

numerous times. He has been a head of department in two schools and is now an

educational consultant who conducts professional development workshops for teachers all

over NSW and beyond. He also works with pre-service teachers at The University of

Sydney and The University of Western Sydney.

David Greenwood is the Head of Mathematics at Trinity Grammar School in Melbourne

and has 19 years’ experience teaching mathematics from Years 7 to 12. He has runnumerous workshops within Australia and overseas regarding the implementation of the

Australian Curriculum and the use of technology for the teaching of mathematics. He

has written more than 20 mathematics titles and has a particular interest in the sequencing

of curriculum content and working with the Australian Curriculum proficiency strands.

Bryn Humberstone graduated from University of Melbourne with an Honours degree in

Pure Mathematics, and is currently teaching both junior and senior mathematics in

Victoria. Bryn is particularly passionate about writing engaging mathematical

investigations and effective assessment tasks for students with a variety of backgrounds

and ability levels.

Justin Robinson is Head of Positive Education and a mathematics teacher at Geelong

Grammar School. Prior to this, he spent 20 years teaching mathematics and was a key writer

of in-house maths material. He has a keen interest in engaging all students through a wide

variety of effective teaching methods and materials.

Jenny Goodman has worked for 20 years in comprehensive State and selective high schools

in NSW and has a keen interest in teaching students of differing ability levels. She was

awarded the Jones medal for education at Sydney University and the Bourke prize for

Mathematics. She has written for Cambridge NSW and was involved in the Spectrum and

Spectrum Gold series.

Jennifer Vaughan has taught secondary mathematics for over 30 years in NSW, WA,

Queensland and New Zealand and has tutored and lectured in mathematics at Queensland

University of Technology. She is passionate about providing students of all ability levels with

opportunities to understand and to have success in using mathematics. She has taught special

needs students and has had extensive experience in developing resources that make

mathematical concepts more accessible; hence, facilitating student confidence, achievement

and an enjoyment of maths.

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Table of Contents

ix

Introduction andguide to this book

This resource has been developed from an analysis of the NSW Syllabus for the Australian Curriculum

and the Australian Curriculum: Mathematics. It is structured on a detailed teaching program for the

implementation of the NSW Syllabus, and a comprehensive copy of the teaching program can be found

on the companion website.

The chapters are based on a logical teaching and learning sequence for the syllabus topic concerned,

so that chapter sections can be used as ready-prepared lessons. Exercises have questions graded by

level of difficulty, as indicated in the teaching program, and are grouped according to the WorkingMathematically components of the NSW Syllabus, as indicated by badges in the margin of the exercises.

This facilitates the management of differentiated learning and reporting on students’ achievement.

For certain topics the prerequisite knowledge has been given in sections marked as REVISION,

whereas EXTENSION marks a few sections that go beyond the Syllabus. Similarly, the word FRINGE

is used to mark a few topics treated in a way that lies at the edge of the Syllabus requirements, but which

provide variety and stimulus. Apart from these, all topics are aligned exactly to the NSW Syllabus, as

indicated at the start of each chapter and in the teaching program.

In Stage 5, separate textbooks are provided for 5.1/5.2 and 5.1/5.2/5.3. In addition the NSW Syllabus

allocates topics to certain pathways, which are designated 5.2◊, 5.3§ and 5.3#. These categories, together

with Stage 4, 5.1, 5.2 and 5.3, are indicated for each chapter section by ‘ladder icons’ in the Year 9 and 10textbooks and teaching programs.

Guide to this book Features:

NSW Syllabus for the Australian

Curriculum: strands, substrands and

content outcomes for chapter (see

teaching program for more detail)

What you will learn: an

overview of chapter contents

Chapter introduction: use

to set a context for students

Chapter

Computation withpositive integers1What you will learn

Placevaluein ancientnumbersystems

Placevalue in Hindu-Arabic numbers

Adding and subtracting positiveintegers

Algorithms foradding and subtracting

Multiplying small positiveintegers

Multiplying largepositive integers

Dividing positiveintegers and dealing with remainders

Estimating and rounding positiveintegers

Orderof operations with positiveintegers

1A

1B

1C

1D

1E

1F

1G

1H

1I

Whole numbers in the ancientworld and now

Wholenumbers and numbersystems havebeen used

forthousands ofyears tohelp countobjects and

record information.

In ancientEgyptin about3000 BCE,ifone

hundred and twenty-fourblocks ofstone wereneededtobuild a monument,this numberwould havebeen

represented as

Today,weus ewhole numbers tohelp deal with all

sorts ofsituations.Herearesomeexamples:

• Recording thenumberofpoints in agame

• Calculating thenumberof pavers required fora

terrace

• Counting the numberofitems purchased ata shop

• Tallying thenumberof votes in an election

• Calculating theapproximate distancebetween

twotowns.

NSW Syllabus for the AustralianCurriculumStrand: Number and Algebra

Substrand: COMPUTATION WITH INTEGERS

Outcome:A studentcompares,orders,

and calculates with integers,applying a

rangeof strategies toaid computation

(MA4 —4NA)

3

Pre-test: establishes prior knowledge

(also available as a printable worksheet)

i i i i i

1 For each of the following, match the word with the symbol.

a add A −

b subtract B ÷

c multiply C +

d divide D ×

2 Write each of the following as numbers.

P r e -

t e s t

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Working Mathematically badges

All exercises are divided into sections marked by Working Mathematically badges, such as this example:

Understanding &

Communicating

W O RK

I N G

M A

T H E M A T I C

A L L Y

U F

R PSC

Fluency &

Communicating

W O RK I N G

M A

T H E M A T I C

A L L Y

U F

R PSC

Problem-solving &

Communicating

W O RK I N G

M A

T H E M A T I C

A L L Y

U F

R PSC

Reasoning &

Communicating

W O RK I N G

M A

T H E M A T I C

A L L Y

U F

R PSC

The letters U (Understanding), F (Fluency), PS (Problem-solving), R (Reasoning) and C (Communication) are

highlighted in colour to indicate which of these components apply mainly to the questions in that section. Naturally

there is some overlap between the components.

Additional resources online

TM

HOTmathsinteractive maths online

The CambridgeMATHS/ HOTmaths integrated program for the Australian Curriculum offers the

best of textbook and interactive online resources. It can be used with a HOTmaths account

(eg for class demos via IWBs or data projectors) or with student accounts, which enable

access to the full range of features, including use at home.

The integrated program is linked from icons and the topic

numbers in the textbooks, as detailed in a document available

(free) from the Cambridge website.

Interactive

simulations &

programs

Practice quizzes with competitive scoring option

Tests with learning management

system

Maths dictionary with links to

content

Interactive

examples

Worksheets

(also available as

workbooks)

All HOTmaths features are included in the program, eg:

xi

ISBN: 9781107626973





Table of Contents

i

Acknowledgements

The author and publisher wish to thank the following sources for permission to reproduce material:

Cover: Shutterstock / Michael Woodruff

Images: Alamy / B.O’Kane, p.359(b); Dreamstime / Carmentianya, p.311(b) / Miffycat, p.474;

© istockphoto / Johanna Doorenbosch, p.167 / TokenPhoto, p.238 / Claudia Dewald, p.453; © Jastrow.

Creative Commons Attribution 3.0 Unported License, p.498; © Kham Tran – www. khamtran.com.

Creative Commons Attribution 3.0 Unported License, p.191; © Mark Strozier. Creative Commons

Attribution 2.0 Generic, p.256; Used under license 2013 from Shutterstock.com / Vladimir

Korostyshevskiy, pp.2–3, 9 / Uroš Medved, p.5 / Rgien Paassen, p.8 / Johanna Goodyear, p.11 / Edd

Westmacott, p.13 / Harris Shiffman, p.14 / Eric Gevaert, p.16 / Phillip Minnis, pp.17, 78 / JinYoung Lee,

p.19 / Mel Brackstone, p.21 / Andreja Donko, p.26 / Robyn Mackenzie, pp.27, 171, 550 / paul Prescott,

pp.29(t), 580 / David Woolfenden, p.29(b) / Binkski, p.30 / Vladimir Mucibacic, p.34(t) / Dima Kalinin,

p.34(m) / Yegor Korzh, p.34(b) / Shawn Talbot, p.35 / arvzdix, p.36 / Mihai-Bogdan Lazar, p.38 / GDM,

p.40 / CandyBoxPhoto, pp.42, 299 / Monkey Business Images, pp.43(r), 70, 162, 270, 300, 398(b-r), 471,

554 / szefei, p.44(c) / Lipskiy, p.45 / marilyn barbone, p.50 / Anton Gvozdikov, p.51 / Anatolich, pp.52–

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Nicemonkey, p.82 / yuyangc, p.83 / Nigel Paul Monckton, p.86 / staskin, p.89 / Deklofenak, pp.90, 105 /

Dmytro Pylypenko, pp.98–99 / Alperium, p.104 / Armin Rose, p.107 / Tatiana Belova, p.111 / flashgun,

p.112 / Andrey Shadrin, p.113 / Shmeliova Natalia, p.120 / Anneka, p.121 / Diego Cervo, p.125 / Eder,

p.129 / Jason Maehl, p.131 / Jose Ignacio Soto, pp.132–133 / Nikola Bilic, p.135 / Elena Aliaga, p.138 /

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magicinfoto, p.198 / Dmitry Vinogradov, p.201 / sjgh, p.202 / Mark Schwettman, p.203 / Coprid, p.207

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Lye, p.290 / Dani Simmonds, p.291 / Heather Prosch-Jensen, p.292 / Paul Matthew Photography, p.295 /

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Victorovich, p.316 / Rafa Irusta, p.318(l) / BZH22, p.318(r) / Lisa S., p.319 / panyajam patong, p.325 /

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Kure, pp.360–361 / jgp-nyc, p.363 / Marc Dietrich, p.366 / Edw, p.367 / Vlue, p.372 / Gary Paul Lewis,

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p.392(b) / Elena Elisseeva, p.394 / Steven Coling, p.397 / visi.stock, p.398(l) / Alexander Raths, p.398(t-r) /

Andresr, pp.401, 443 / V.J. Matthew, p.410 / Malysh Falko, p.411 / S.john, p.413 / Rob Marmion, p.417 /

Sinitar, p.415 / Lars Hallstrom, p.422 / Weldon Scloneger, p.424 / Michael William, p.430 / yampi, p.431 /

Jaroslav Machacek, p.435 / terekhov igor, p.438 / MaszaS, p.440 / Marko Poplasen, p.442 / anweber,

p.448 / Darren Whitt, p.455 / Serghei Starus, p.459 / hfng, p.465 / vblinov, p.469 / theromb, p.473 /

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stephen mulcahey, p.488 / minik, p.489 / Noel Powell Schaumburg, p.492 / kwest, p.493 / Bork, p.501 /

aopsan, p.503 / Carlos Neto, p.580 / Terrance Emerson, p.509 / Fedorov Oleksiy, p.514 / haak78, p.515 /

C Berry Ottaway, p.518 / Salvador Garcia Gil, p.519 / Ljupco Smokovski, p.524 / Helen and Vlad Filatov,

p.525 / Gyukli Gyula, p.530(t) / Sebastian Duda, p.530(b) / Supri Suharjoto, p.533 / Andraž Cerar,

p.534(b) / darkgreenwolf, p.535 / Joern, p.536 / Tim Roberts Photography, p.542 / spirit of america,

p.543(t) / Edyta Pawlowska, p.543(b) / PL Petr Lerch, pp.544–545 / Konstantin Sutyagin, p.551 /

forestpath, p.552 / Ali Ender Birer, p.559 / fotohunter, p.563 / Marijus Auruskevicius, p.564 / Ryan Carter,

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Tsekhmister, p.575(6); Wikimedia Commons. Public domain, pp.10, 18, 44(t), 101, 122, 439.

All curriculum material taken from NSW Mathematics 7-10 Syllabus © Board of Studies NSW for and on

behalf of the Crown in right of the State of New South Wales, 2012.

Every effort has been made to trace and acknowledge copyright. The publisher apologises for any

accidental infringement and welcomes information that would redress this situation.

ISBN: 9781107626973





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