Cambridge IGCSE Mathematics Extended Practice Book

Karen Morrison and Lucille Dunne

Cambridge IGCSE®

Mathematics

Extended Practice Book

cambridge university press
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
www.cambridge.org
Information on this title: www.cambridge.org/9781107672727

© Cambridge University Press 2013

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 2013

Printed and bound in the United Kingdom by the MPG Books Group

A catalogue record for this publication is available from the British Library

ISBN-13 978-1-107-67272-7 Paperback

Cover image: Seamus Ditmeyer/Alamy

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IGCSE® is the registered trademark of Cambridge International Examinations.

Contents 1 Chapter 3: Lines, angles and shapes v
1 3.1 Lines and angles
Introduction 2 3.2 Triangles 12
3 3.3 Quadrilaterals 12
Unit 1 3 3.4 Polygons 15
4 3.5 Circles 16
Chapter 1: Reviewing number concepts 4 3.6 Construction 18
1.1 Different types of numbers 5 Chapter 4: Collecting, organising and 19
1.2 Multiples and factors 7 displaying data 19
1.3 Prime numbers
1.4 Powers and roots 4.1 Collecting and classifying data 22
1.5 Working with directed numbers 7 4.2 Organising data 22
1.6 Order of operations 8 4.3 Using charts to display data 23
1.7 Rounding numbers 8 24
9
Chapter 2: Making sense of algebra 9
2.1 Using letters to represent

unknown values
2.2 Substitution
2.3 Simplifying expressions
2.4 Working with brackets
2.5 Indices

Unit 2 29 Chapter 7: Perimeter, area and volume 39

Chapter 5: Fractions 29 7.1 Perimeter and area in two dimensions 39
5.1 Equivalent fractions
5.2 Operations on fractions 29 7.2 Three-dimensional objects 43
5.3 Percentages
5.4 Standard form 30 7.3 Surface areas and volumes of solids 44
5.5 Estimation
32
Chapter 6: Equations and transforming formulae
6.1 Further expansions of brackets 33 Chapter 8: Introduction to probability 48
6.2 Solving linear equations
6.3 Factorising algebraic expressions 35 8.1 Basic probability 48
6.4 Transformation of a formula 35
35 8.2 Theoretical probability 49
36
37 8.3 The probability that an event does not

happen 50

8.4 Possibility diagrams 51

8.5 Combining independent and mutually exclusive

events 52

Unit 3

Chapter 9: Sequences and sets 54 Chapter 12: Averages and measures of spread 75
9.1 Sequences 54 12.1 Different types of average 75
9.2 Rational and irrational numbers 55 12.2 Making comparisons using averages
9.3 Sets 56 76
and ranges
Chapter 10: Straight lines and quadratic equations 59 12.3 Calculating averages and ranges for 77

10.1 Straight lines 59 frequency data 78
12.4 Calculating averages and ranges for 79
10.2 Quadratic expressions 61
grouped continuous data
Chapter 11: Pythagoras’ theorem and 66 12.5 Percentiles and quartiles
similar shapes 66
11.1 Pythagoras’ theorem 68
11.2 Understanding similar triangles 69
11.3 Understanding similar shapes 70
11.4 Understanding congruence

Contents iii

Unit 4

Chapter 13: Understanding measurement 81 Chapter 15: Scale drawings, bearings and 99
13.1 Understanding units 81 trigonometry 99
13.2 Time 82 15.1 Scale drawings 100
13.3 Upper and lower bounds 84 15.2 Bearings
13.4 Conversion graphs 85 15.3 Understanding the tangent, cosine 101
13.5 More money 86 105
and sine ratios 105
Chapter 14: Further solving of equations and 89 15.4 Solving problems using trigonometry 105
inequalities 89 15.5 Angles between 0° and 180° 108
14.1 Simultaneous linear equations 91 15.6 The sine and cosine rules 108
14.2 Linear inequalities 92 15.7 Area of a triangle
14.3 Regions in a plane 93 15.8 Trigonometry in three dimensions 111
14.4 Linear programming 94 111
14.5 Completing the square 95 Chapter 16: Scatter diagrams
14.6 Quadratic formula and correlation
14.7 Factorising quadratics where the coefficient of 96 16.1 Introduction to bivariate data
96
x2 is not 1
14.8 Algebraic fractions

Unit 5 114 Chapter 19: Symmetry and loci 126

Chapter 17: Managing money 114 19.1 Symmetry in two dimensions 126
17.1 Earning money
17.2 Borrowing and investing money 115 19.2 Symmetry in three dimensions 127
17.3 Buying and selling
116 19.3 Symmetry properties of circles 128
Chapter 18: Curved graphs
18.1 Plotting quadratic graphs (the parabola) 119 19.4 Angle relationships in circles 129
18.2 Plotting reciprocal graphs (the hyperbola) 119 19.5 Locus 131
18.3 Using graphs to solve quadratic equations
18.4 Using graphs to solve simultaneous linear 121 Chapter 20: Histograms and frequency distribution

and non-linear equations 122 diagrams 135
18.5 Other non-linear graphs
18.6 Finding the gradient of a curve 20.1 Histograms 135

122 20.2 Cumulative frequency 137

123

124

Unit 6 139 Chapter 23: Transformations and matrices 153
139 23.1 Simple plane transformations 153
Chapter 21: Ratio, rate and proportion 140 23.2 Vectors 158
21.1 Working with ratio 141 23.3 Further transformations 161
21.2 Ratio and scale 141 23.4 Matrices and matrix transformation 163
21.3 Rates 144 23.5 Matrices and transformations 164
21.4 Kinematic graphs
21.5 Proportion Chapter 24: Probability using tree diagrams 169
21.6 Direct and inverse proportion in algebraic 145 24.1 Using tree diagrams to show outcomes 169
169
terms 24.2 Calculating probability from tree diagrams
21.7 Increasing and decreasing amounts by a 146

given ratio 149
149
Chapter 22: More equations, formulae and 150
functions 151
22.1 Setting up equations to solve problems
22.2 Using and transforming formulae
22.3 Functions and function notation

Answers 171

Example practice papers can be found online, visit education.cambridge.org/extendedpracticebook

iv Contents

Introduction

This highly illustrated practice book has been written by experienced teachers to help students
revise the Cambridge IGCSE Mathematics (0580) Extended syllabus. Packed full of exercises, the
only narrative consists of helpful bulleted lists of key reminders and useful hints in the margins
for students needing more support.

There is plenty of practice offered via ‘drill’ exercises throughout each chapter. These consist of
progressive and repetitive questions that allow the student to practise methods applicable to
each subtopic. At the end of each chapter there are ‘Mixed exercises’ that bring together all the
subtopics of a chapter in such a way that students have to decide for themselves what methods to
use. The answers to all of these questions are supplied at the back of the book. This encourages
students to assess their progress as they go along, choosing to do more or less practice as
required.

The book has been written with a clear progression from start to finish, with some later chapters
requiring knowledge learned in earlier chapters. There are useful signposts throughout that link
the content of the chapters, allowing the individual to follow their own course through the book:
where the content in one chapter might require knowledge from a previous chapter, a comment
is included in a ‘Rewind’ box; and where content will be practised in more detail later on, a
comment is included in a ‘Fast forward’ box. Examples of both are included below:

REWIND FAST FORWARD

You learned how to plot lines from You will learn much more about
equations in chapter 10. sets in chapter 9. For now, just think
of a set as a list of numbers or other
items that are often placed inside
curly brackets.

Remember ‘coefficient’ is the Other helpful guides in the margin of the book are as follows:
number in the term.
Hints: these are general comments to remind students of important or key information that is
Tip useful when tackling an exercise, or simply useful to know. They often provide extra information
or support in potentially tricky topics.
It is essential that you
remember to work out Tip: these are tips that relate to good practice in examinations, and also just generally in
both unknowns. Every mathematics! They cover common pitfalls based on the authors’ experiences of their students,
pair of simultaneous linear and give students things to be wary of or to remember in order to score marks in the exam.
equations will have a pair
of solutions. The Extended Practice Book mirrors the chapters and subtopics of the Cambridge IGCSE
Mathematics Core and Extended Coursebook written by Karen Morrison and Nick Hamshaw
(9781107606272). However, this book has been written such that it can be used without the
coursebook; it can be used as a revision tool by any student regardless of what coursebook they
are using. Various aspects of the Core syllabus are also revised for complete coverage.

Also in the Cambridge IGCSE Mathematics series:
Cambridge IGCSE Mathematics Core and Extended Coursebook (9781107606272)
Cambridge IGCSE Mathematics Core Practice Book (9781107609884)
Cambridge IGCSE Mathematics Teacher’s Resource CD-ROM (9781107627529)

Introduction v



1 Reviewing number concepts

1.1 Different types of numbers

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JO UIJT DIBQUFS *SSBUJPOBM OVNCFST BSF DPWFSFE JO DIBQUFS

t 3BUJPOBM OVNCFST DBO CF XSJUUFO BT GSBDUJPOT JO UIF GPSN PG ab XIFSF a BOE b BSF JOUFHFST BOE b ≠
*OUFHFST BSF OFHBUJWF BOE QPTJUJWF XIPMF OVNCFST
BOE [FSP


t *OUFHFST
GSBDUJPOT BOE UFSNJOBUJOH EFDJNBMT BSF BMM SBUJPOBM OVNCFST

Tip Exercise 1.1

.BLF TVSF ZPV LOPX 1 5JDL UIF DPSSFDU DPMVNOT JO UIF UBCMF UP DMBTTJGZ FBDI OVNCFS
XIBU UIF GPMMPXJOH TFUT
PG OVNCFST BSF OBUVSBM Number Natural Integer Prime Fraction
OVNCFST
JOUFHFST
PEE ¦
BOE FWFO OVNCFST BOE −57
QSJNF OVNCFST

03





− 1
4

2
7



3 512

2 -JTU

(a) GPVS TRVBSF OVNCFST HSFBUFS UIBO
(b) GPVS SBUJPOBM OVNCFST TNBMMFS UIBO 13
(c) UXP QSJNF OVNCFST UIBU BSF >
(d) UIF QSJNF OVNCFST <

Unit 1: Number 1

1 Reviewing number concepts

1.2 Multiples and factors

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NVMUJQMF -$.
PG UXP PS NPSF OVNCFST JT UIF MPXFTU OVNCFS UIBU JT B NVMUJQMF PG CPUI PS BMM
PG UIF OVNCFST

t " GBDUPS PG B OVNCFS JT BOZ OVNCFS UIBU XJMM EJWJEF JOUP UIF OVNCFS FYBDUMZ
t ćF IJHIFTU DPNNPO GBDUPS )$'
PG UXP PS NPSF OVNCFST JT UIF IJHIFTU OVNCFS UIBU JT B GBDUPS PG BMM UIF

HJWFO OVNCFST

To find the LCM of a set of Exercise 1.2 A
numbers, you can list the multiples
of each number until you find the 1 'JOE UIF -$. PG UIF HJWFO OVNCFST
first multiple that is in the lists for
all of the numbers in the set. (a) BOE (b) BOE (c) BOE (d) BOE
(e) BOE (f)
BOE (g)
BOE (h)
BOE
FAST FORWARD
2 'JOE UIF )$' PG UIF HJWFO OVNCFST (c) BOE (d) BOE
You will use LCM again when (g) BOE (h) BOE
you work with fractions to find the (a) BOE (b) BOE
lowest common denominator (e) BOE (f) BOE
of two or more fractions. See
chapter 5. Exercise 1.2 B

You need to work out whether 1 "NJSB IBT UXP SPMMT PG DPUUPO GBCSJD 0OF SPMM IBT NFUSFT PO JU BOE UIF PUIFS IBT NFUSFT
to use LCM or HCF to find the PO JU 4IF XBOUT UP DVU UIF GBCSJD UP NBLF BT NBOZ FRVBM MFOHUI QJFDFT BT QPTTJCMF PG UIF
answers. Problems involving LCM MPOHFTU QPTTJCMF MFOHUI )PX MPOH TIPVME FBDI QJFDF CF
usually include repeating events.
Problems involving HCF usually 2 *O B TIPQQJOH NBMM QSPNPUJPO FWFSZ UI TIPQQFS HFUT B WPVDIFS BOE FWFSZ UI
involve splitting things into smaller TIPQQFS HFUT B GSFF NFBM )PX NBOZ TIPQQFST NVTU FOUFS UIF NBMM CFGPSF POF SFDFJWFT B
pieces or arranging things in equal WPVDIFS BOE B GSFF NFBM
groups or rows.
3 "NBOEB IBT QJFDFT PG GSVJU BOE TXFFUT UP TIBSF BNPOHTU UIF TUVEFOUT JO IFS DMBTT 4IF
JT BCMF UP HJWF FBDI TUVEFOU BO FRVBM OVNCFS PG QJFDFT PG GSVJU BOE BO FRVBM OVNCFS PG TXFFUT
8IBU JT UIF MBSHFTU QPTTJCMF OVNCFS PG TUVEFOUT JO IFS DMBTT

4 'SBODFTDB
"ZVCB BOE $MBJSF BSF 0MZNQJD BOE 1BSBMZNQJD DPOUFOEFST ćFZ TIBSF B USBJOJOH TMPU
PO B SVOOJOH USBDL 'SBODFTDB DZDMFT BOE DPNQMFUFT B MBQ JO TFDPOET
"ZVCB SVOT UIF MBQ JO
TFDPOET BOE $MBJSF
JO IFS XIFFMDIBJS
UBLFT TFDPOET ćFZ TUBSU USBJOJOH UPHFUIFS "ęFS
IPX MPOH XJMM BMM UISFF CF BU UIF TBNF QPJOU BHBJO BOE IPX NBOZ MBQT XJMM FBDI IBWF DPNQMFUFE

5 .S 4NJU XBOUT UP UJMF B SFDUBOHVMBS WFSBOEB XJUI EJNFOTJPOT N × N XJUI B XIPMF
OVNCFS PG JEFOUJDBM TRVBSF UJMFT .ST 4NJU XBOUT UIF UJMFT UP CF BT MBSHF BT QPTTJCMF
(a) 'JOE UIF BSFB PG UIF MBSHFTU QPTTJCMF UJMFT JO DN
(b) )PX NBOZ N × N UJMFT XJMM .S 4NJU OFFE UP UJMF UIF WFSBOEB

2 Unit 1: Number

1 Reviewing number concepts

1.3 Prime numbers

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t 1SJNF GBDUPST BSF GBDUPST PG B OVNCFS UIBU BSF BMTP QSJNF OVNCFST
t :PV DBO XSJUF BOZ OVNCFS BT B QSPEVDU PG QSJNF GBDUPST #VU SFNFNCFS UIF OVNCFS JUTFMG JT not B QSJNF OVNCFS

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t :PV DBO VTF UIF QSPEVDU PG QSJNF GBDUPST UP ĕOE UIF )$' PS -$. PG UXP PS NPSF OVNCFST

You can use a tree diagram or Exercise 1.3
division to find the prime factors
of a composite whole number. 1 *EFOUJGZ UIF QSJNF OVNCFST JO FBDI TFU

(a)









(b)










(c)











2 &YQSFTT UIF GPMMPXJOH OVNCFST BT B QSPEVDU PG UIFJS QSJNF GBDUPST

(a) (b) (c) (d)
(e) (f) (g) (h)

3 'JOE UIF -$. BOE UIF )$' PG UIF GPMMPXJOH OVNCFST CZ NFBOT PG QSJNF GBDUPST

(a) BOE (b) BOE (c) BOE (d) BOE
(e) BOE (f) BOE (g) BOE (h) BOE

1.4 Powers and roots

t " OVNCFS JT TRVBSFE n
XIFO JU JT NVMUJQMJFE CZ JUTFMG n × n

t ćF TRVBSF SPPU n
PG B OVNCFS JT UIF OVNCFS UIBU JT NVMUJQMJFE CZ JUTFMG UP HFU UIF OVNCFS
t " OVNCFS JT DVCFE n3
XIFO JU JT NVMUJQMJFE CZ JUTFMG BOE UIFO NVMUJQMJFE CZ JUTFMG BHBJO n × n × n

t ćF DVCF SPPU 3 n
PG B OVNCFS JT UIF OVNCFS UIBU JT NVMUJQMJFE CZ JUTFMG UXJDF UP HFU UIF OVNCFS

FAST FORWARD Exercise 1.4

Powers greater than 3 are dealt 1 'JOE BMM UIF TRVBSF BOE DVCF OVNCFST CFUXFFO BOE
with in chapter 2. See topic 2.5 2 4JNQMJGZ
indices.

(a) 9 16 (b) 9 16 (c) 64 + 36 (d) 64 + 36
(e) 36 (h) 169 −144
( )2 (g) 9 (l) 16 × 3 27
4
(f) 25 16
(i) 3 27 − 3 1
(j) 100 ÷ 4 (k) 1+ 9
(m) × 16

( ) (n)1+ 1 2 (o) 3 1 − 3 −125
4 3


3 " DVCF IBT B WPMVNF PG DN3 $BMDVMBUF

(a) UIF IFJHIU PG UIF DVCF
(b) UIF BSFB PG POF GBDF PG UIF DVCF

Unit 1: Number 3

1 Reviewing number concepts

1.5 Working with directed numbers

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t /FHBUJWF JOUFHFST BSF XSJUUFO XJUI B NJOVT −
TJHO 1PTJUJWF JOUFHFST NBZ CF XSJUUFO XJUI B QMVT +
TJHO


CVU VTVBMMZ UIFZ BSF OPU

t *O SFBM MJGF
OFHBUJWF OVNCFST BSF VTFE UP SFQSFTFOU UFNQFSBUVSFT CFMPX [FSP NPWFNFOUT EPXOXBSET PS MFę EFQUIT
EJTUBODFT CFMPX TFB MFWFM CBOL XJUIESBXBMT BOE PWFSESBXO BNPVOUT
BOE NBOZ NPSF UIJOHT

Draw a number line to help you. Exercise 1.5

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XIBU XJMM UIF
UFNQFSBUVSF CF JO UIF NPSOJOH

2 8IJDI JT DPMEFS JO FBDI QBJS PG UFNQFSBUVSFT

(a) ¡$ PS − ¡$ (b) ¡$ PS − ¡$ (c) − ¡$ PS − ¡$

3 "O PďDF CMPDL IBT UISFF CBTFNFOU MFWFMT MFWFM −
− BOE −

B HSPVOE ĘPPS BOE ĘPPST
BCPWF UIF HSPVOE ĘPPS UP
8IFSF XJMM UIF MJę CF JO UIF GPMMPXJOH TJUVBUJPOT
(a) 4UBSUT PO HSPVOE BOE HPFT EPXO POF ĘPPS UIFO VQ ĕWF
(b) 4UBSUT PO MFWFM − BOE HPFT VQ ĘPPST
(c) 4UBSUT PO ĘPPS BOE HPFT EPXO ĘPPST
(d) 4UBSUT PO ĘPPS BOE HPFT EPXO ĘPPST
(e) 4UBSUT PO MFWFM −
HPFT VQ TFWFO ĘPPST BOE UIFO EPXO FJHIU

1.6 Order of operations

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ćFO EP BOZ EJWJTJPO PS NVMUJQMJDBUJPO GSPN MFę UP SJHIU
CFGPSF BEEJOH BOE TVCUSBDUJOH GSPN MFę UP SJHIU


t ćF XPSE APG NFBOT × BOE B GSBDUJPO MJOF NFBOT EJWJEF
t -POH GSBDUJPO MJOFT BOE TRVBSF PS DVCF SPPU TJHOT BDU MJLF CSBDLFUT
JOEJDBUJOH QBSUT PG UIF DBMDVMBUJPO UIBU IBWF

UP CF EPOF ĕSTU

Remember the order of operations Exercise 1.6
using BODMAS:
Tip
Brackets
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UIF
Add DBMDVMBUPS XJMM EJWJEF CZ UIF ĕSTU UFSN POMZ VOMFTT ZPV FOUFS CSBDLFUT
Subtract

FAST FORWARD 1 $BMDVMBUF BOE HJWF ZPVS BOTXFS DPSSFDU UP UXP EFDJNBM QMBDFT

The next section will remind you (a) + × (b) +
× (c) × − ÷
of the rules for rounding (f) ÷
+
numbers. (d) + × (e) × −

4 Unit 1: Number

1 Reviewing number concepts

(g) 1.453 + 7 6 (h) 5.34 3.315 (i) 6 54 −1 08
3 2 4 03 23

(j) 5 27 (k) 2 91−1.15.43 (l) 0.23 4.26
1.4 ×1.35 1.32 3.43

(m) 8 9 − 89 (n) 12.6 − 1 98 (o) −
10.4 83 4 62
( )(r) 2
16.8
(p) −
3 (q) − 93 − 1 01

4 072 1.4 19.23 ( )(u) 16 2
8.2 − 4.09 69 5
(s) (t) 6 8 + − 4.3 + 1.2 +

( )(v)6 1 + 2.1 2 (w) 6.4 (1.22 + 1 92 )2 ( )(x)4 8 − 1 × 43
2 8 16 96


1.7 Rounding numbers

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t 5P SPVOE UP B EFDJNBM QMBDF

o MPPL BU UIF WBMVF PG UIF EJHJU UP UIF SJHIU PG UIF QMBDF ZPV BSF SPVOEJOH UP

o JG UIJT WBMVF JT ≥ UIFO ZPV SPVOE VQ BEE UP UIF EJHJU ZPV BSF SPVOEJOH UP


o JG UIJT WBMVF JT ≤ UIFO MFBWF UIF EJHJU ZPV BSF SPVOEJOH UP BT JU JT

t 5P SPVOE UP B TJHOJĕDBOU ĕHVSF

o UIF ĕSTU OPO [FSP EJHJU CFGPSF PS BęFS UIF EFDJNBM QMBDF JO B OVNCFS
JT UIF ĕSTU TJHOJĕDBOU ĕHVSF

o ĕOE UIF DPSSFDU EJHJU BOE UIFO SPVOE PČ GSPN UIBU EJHJU VTJOH UIF SVMFT BCPWF

Exercise 1.7

FAST FORWARD 1 3PVOE UIFTF OVNCFST UP

Rounding is very useful when you (i) UXP EFDJNBM QMBDFT
have to estimate an answer. You (ii) POF EFDJNBM QMBDF
will deal with this in more detail in (iii) UIF OFBSFTU XIPMF OVNCFS
chapter 5.

(a) (b) (c)
(d) (e) (f)
(g) (h)

2 3PVOE FBDI PG UIFTF OVNCFST UP UISFF TJHOJĕDBOU ĕHVSFT

(a) (b) (c) (d)

3 3PVOE UIF GPMMPXJOH OVNCFST UP UXP TJHOJĕDBOU ĕHVSFT (d)
(h)
(a) (b) (c)
(e) (f) (g)

Unit 1: Number 5

1 Reviewing number concepts

Mixed exercise 1 4UBUF XIFUIFS FBDI OVNCFS JT OBUVSBM
SBUJPOBM
BO JOUFHFS BOE PS B QSJNF OVNCFS

− 3 − 3 1
2
4

2 -JTU UIF GBDUPST PG

(a) )PX NBOZ PG UIFTF GBDUPST BSF QSJNF OVNCFST
(b) &YQSFTT BT UIF QSPEVDU PG JUT QSJNF GBDUPST
(c) -JTU UXP OVNCFST UIBU BSF GBDUPST PG CPUI BOE
(e) 8IBU JT UIF IJHIFTU OVNCFS UIBU JT B GBDUPS PG CPUI BOE

3 8SJUF FBDI OVNCFS BT B QSPEVDU PG JUT QSJNF GBDUPST

(a) (b) (c)

4 "NJSB TUBSUT BO FYFSDJTF QSPHSBNNF PO UIF SE PG .BSDI 4IF EFDJEFT TIF XJMM TXJN FWFSZ
EBZT BOE DZDMF FWFSZ EBZT 0O XIJDI EBUFT JO .BSDI XJMM TIF TXJN BOE DZDMF PO UIF
TBNF EBZ

5 4UBUF XIFUIFS FBDI FRVBUJPO JT USVF PS GBMTF

(a) Ÿ ¨

(b) ¨ o

(c) 30 + 10 o

30
(d)


6 4JNQMJGZ ( )(c) 3
(a) 100 ÷ 4 (b) 100 ÷ 4 3 64 (d) 3 +


7 $BMDVMBUF (JWF ZPVS BOTXFS DPSSFDU UP UXP EFDJNBM QMBDFT

(a) 5.4 ×12.2 (b) 12.22 (c) 12.65 + 1.7 × 4.3
4.1 3 92 2 04

(d) 3.8 ×12.6 (e) 2.8 × 4.22 ( )(f) 05 2
4 35 3.32 × 6.22 5
2.5 − 3.1 +

8 3PVOE FBDI OVNCFS UP UISFF TJHOJĕDBOU ĕHVSFT

(a) (b) (c) (d)

9 " CVJMEJOH TVQQMZ TUPSF JT TFMMJOH UJMFT XJUI BO BSFB PG DN

(a) *T JU QPTTJCMF UP IBWF TRVBSF UJMFT XIPTF BSFB JT OPU B TRVBSF OVNCFS &YQMBJO
(b) 'JOE UIF MFOHUI PG FBDI TJEF PG UIF UJMF DPSSFDU UP TJHOJĕDBOU ĕHVSFT
(c) 8IBU JT UIF NJOJNVN OVNCFS PG UJMFT ZPV XPVME OFFE UP UJMF B SFDUBOHVMBS ĘPPS N

MPOH BOE N XJEF

6 Unit 1: Number

2 Making sense of algebra

2.1 Using letters to represent unknown values

t -FUUFST JO BMHFCSB BSF DBMMFE WBSJBCMFT CFDBVTF UIFZ DBO IBWF NBOZ EJČFSFOU WBMVFT UIF WBMVF WBSJFT

"OZ MFUUFS DBO CF VTFE BT B WBSJBCMF
CVU x BOE y BSF VTFE NPTU PęFO

t " OVNCFS PO JUT PXO JT DBMMFE B DPOTUBOU
t " UFSN JT B HSPVQ PG OVNCFST BOE PS WBSJBCMFT DPNCJOFE CZ UIF PQFSBUJPOT NVMUJQMZJOH BOE PS EJWJEJOH POMZ
t "O BMHFCSBJD FYQSFTTJPO MJOLT UFSNT CZ VTJOH UIF + BOE − PQFSBUJPO TJHOT "O FYQSFTTJPO EPFT OPU IBWF BO FRVBMT TJHO

VOMJLF BO FRVBUJPO
"O FYQSFTTJPO DPVME IBWF KVTU POF UFSN

Exercise 2.1

Tip 1 8SJUF FYQSFTTJPOT
JO UFSNT PG x, UP SFQSFTFOU

"O FYQSFTTJPO JO UFSNT (a) UJNFT UIF TVN PG B OVNCFS BOE
PG x NFBOT UIBU UIF
WBSJBCMF MFUUFS VTFE JO UIF (b) UJNFT UIF EJČFSFODF PG B OVNCFS BOE
FYQSFTTJPO JT x
(c) UXJDF UIF TVN PG BOE B OVNCFS

(d) B OVNCFS UJNFT UIF EJČFSFODF PG BOE o

(e) BEEFE UP UJNFT UIF TRVBSF PG B OVNCFS

(f) B OVNCFS TRVBSFE BEEFE UP UJNFT UIF EJČFSFODF PG BOE

(g) B OVNCFS TVCUSBDUFE GSPN UIF SFTVMU PG EJWJEFE CZ

(h) B OVNCFS BEEFE UP UIF SFTVMU PG EJWJEFE

(i) UIF TVN PG UJNFT 1 BOE B OVNCFS UJNFT
2

(j) UIF EJČFSFODF PG B OVNCFS UJNFT o BOE UJNFT o

2 " CPZ JT p ZFBST PME

(a) )PX PME XJMM UIF CPZ CF JO ĕWF ZFBST UJNF
(b) )PX PME XBT UIF CPZ GPVS ZFBST BHP
(c) )JT GBUIFS JT GPVS UJNFT UIF CPZ T BHF HPX PME JT UIF GBUIFS

3 ćSFF QFPQMF XJO B QSJ[F PG x

(a) *G UIFZ TIBSF UIF QSJ[F FRVBMMZ
IPX NVDI XJMM FBDI PG UIFN SFDFJWF
(b) * G UIF QSJ[F JT EJWJEFE TP UIBU UIF ĕSTU QFSTPO HFUT IBMG BT NVDI NPOFZ BT UIF TFDPOE

QFSTPO BOE UIF UIJSE QFSTPO HFUT UISFF UJNFT BT NVDI BT UIF TFDPOE QFSTPO
IPX NVDI
XJMM FBDI SFDFJWF

Unit 1: Algebra 7

2 Making sense of algebra

2.2 Substitution

t 4VCTUJUVUJPO JOWPMWFT SFQMBDJOH WBSJBCMFT XJUI HJWFO OVNCFST UP XPSL PVU UIF WBMVF PG BO FYQSFTTJPO
'PS FYBNQMF
ZPV NBZ CF UPME UP FWBMVBUF x XIFO x = − 5P EP UIJT ZPV XPSL PVU × −
= −

Exercise 2.2

REWIND 1 ćF GPSNVMB GPS ĕOEJOH UIF BSFB A
PG B USJBOHMF JT A = 1 bh
XIFSF b JT UIF MFOHUI PG UIF CBTF
Remember that the BODMAS rules
always apply in these calculations. 2

Take special care when substituting BOE h JT UIF QFSQFOEJDVMBS IFJHIU PG UIF USJBOHMF
negative numbers. If you replace x 'JOE UIF BSFB PG B USJBOHMF JG
with −3 in the expression 4x, you
will obtain 4 × −3 = −12, but in (a) UIF CBTF JT DN BOE UIF IFJHIU JT DN
the expression −4x, you will obtain (b) UIF CBTF JT N BOE UIF IFJHIU JT N
−4 × −3 = 12. (c) UIF CBTF JT DN BOE UIF IFJHIU JT IBMG BT MPOH BT UIF CBTF
(d) UIF IFJHIU JT DN BOE UIF CBTF JT UIF DVCF PG UIF IFJHIU

2 &WBMVBUF xy o x o y
XIFO x BOE y o

3 (JWFO UIBU a
b o BOE c o
FWBMVBUF a b
o c

4 8IFO m BOE n o
XIBU JT UIF WBMVF PG m3 o mn32 mn n


5 ćF OVNCFS PG HBNFT UIBU DBO CF QMBZFE BNPOH x DPNQFUJUPST JO B DIFTT UPVSOBNFOU JT HJWFO

CZ UIF FYQSFTTJPO 1 x o 12 x
2

(a) )PX NBOZ HBNFT XJMM CF QMBZFE JG UIFSF BSF DPNQFUJUPST

(b) )PX NBOZ HBNFT XJMM CF QMBZFE JG UIFSF BSF DPNQFUJUPST

2.3 Simplifying expressions

t 5P TJNQMJGZ BO FYQSFTTJPO ZPV BEE PS TVCUSBDU MJLF UFSNT
t -JLF UFSNT BSF UIPTF UIBU IBWF FYBDUMZ UIF TBNF WBSJBCMFT JODMVEJOH QPXFST PG WBSJBCMFT

t :PV DBO BMTP NVMUJQMZ BOE EJWJEF UP TJNQMJGZ FYQSFTTJPOT #PUI MJLF BOE VOMJLF UFSNT DBO CF NVMUJQMJFE PS EJWJEFE

Remember, like terms must have Exercise 2.3
exactly the same variables with
exactly the same indices. So 3x 1 4JNQMJGZ UIF GPMMPXJOH FYQSFTTJPOT
and 2x are like terms but 3x2 and
2x are not like terms. (a) 3x2 + 6x 8 3 (b) x2y + 3x2y 2 yx (c) 2ab − 4ac + 3ba
(d) x2 + 2x 4 + 3x2 y 3x −1
Remember, multiplication can be (e) −6m × 5n (f) 3xy 2x
done in any order so, although it is
better to put variable letters in a term (g) −2xy × −3y2 (h) −2xy × 2x2 (i) 12ab ÷ 3a (j) 12x ÷ 48xy
in alphabetical order, ab = ba. So,
3ab + 2ba can be simplified to 5ab. (k) 33abc (l) 4250mnn (m) 80xy2 (n) −36x3
11ca 12x2 y −12xy
Remember,
x × x = x2 (o) y × 2y (p) xy × y (q) 5a × 3a (r) 7 × −2 y
y × y × y = y3 x x 2 x 4 5
x÷x=1
(s) x × 2 (t) 3x × 9x
4 3y 5 2

8 Unit 1: Algebra

2 Making sense of algebra

2.4 Working with brackets

t :PV DBO SFNPWF CSBDLFUT GSPN BO FYQSFTTJPO CZ NVMUJQMZJOH FWFSZUIJOH JOTJEF UIF CSBDLFUT CZ UIF WBMVF PS WBMVFT
JO
GSPOU PG UIF CSBDLFU

t 3FNPWJOH CSBDLFUT JT BMTP DBMMFE FYQBOEJOH UIF FYQSFTTJPO
t 8IFO ZPV SFNPWF CSBDLFUT JO QBSU PG BO FYQSFTTJPO ZPV NBZ FOE VQ XJUI MJLF UFSNT "EE PS TVCUSBDU BOZ MJLF UFSNT

UP TJNQMJGZ UIF FYQSFTTJPO GVMMZ

t *O HFOFSBM UFSNT a b + c
= ab + ac

Exercise 2.4

Remember the rules for multiplying 1 3FNPWF UIF CSBDLFUT BOE TJNQMJGZ XIFSF QPTTJCMF
integers:
(a) 2x(x 2) (b) ( y 3)x (c) (x ) 3x (d) −2x − ( − )
+×+=+
(e) (x )( x) (f) 2(x 1) (1 x) (g) x(x2 − 2x 1)
−×−=+
(h) −x( − ) + 2( + ) − 4
+×−=−
2 3FNPWF UIF CSBDLFUT BOE TJNQMJGZ XIFSF QPTTJCMF
If the quantity in front of a bracket
is negative, the signs of the terms (a) 2x (1 x + 1 ) ( )(b) −3 ( − y) − 2x( y − 2x) (c) −2x − −
inside the bracket will change when 2 4
the brackets are expanded.
(d) ( x + y ) − ( 1 x 1 y) (e) 5 (3 − 2x) (f) 2x (2x − 2) x (x 2)
2

(g) x (1 x) x (2x − 5) 2x (1 3x)

2.5 Indices

t "O JOEFY BMTP DBMMFE B QPXFS PS FYQPOFOU
TIPXT IPX NBOZ UJNFT UIF CBTF JT NVMUJQMJFE CZ JUTFMG
t x NFBOT x × x BOE y
4 NFBOT y × y × y × y.
t ćF MBXT PG JOEJDFT BSF VTFE UP TJNQMJGZ BMHFCSBJD UFSNT BOE FYQSFTTJPOT .BLF TVSF ZPV LOPX UIF MBXT BOE

VOEFSTUBOE IPX UIFZ XPSL TFF CFMPX


t 8IFO BO FYQSFTTJPO DPOUBJOT OFHBUJWF JOEJDFT ZPV BQQMZ UIF TBNF MBXT BT GPS PUIFS JOEJDFT UP TJNQMJGZ JU

Tip Exercise 2.5 A

.FNPSJTF UIJT TVNNBSZ 1 4JNQMJGZ

PG UIF index laws: (a) x4 y × y2x6 (b) 2x2 y4 3x3 y (c) 2x5 y4 2xy3
x4 y5 2xy4 2x2 y5 3x2 y3
xm xn = xm+n

xm xn = xm n

( )n xmn (d) x3 y7 × x2 y8 (e) 2x7 y2 × 10x8 y4 (f ) x9 y6 ÷ x3 y2
xy 4 x3 y 4x3 y7 2x3 y2 x4 y2 x5 y
x0 = 1
1
x−m = xm ( ) ( )x5 y2 2
10x5 y2 3x3 y 7y3x2 5x6 y2 x3 y4
9x6 y6 5x7 y4 5y5x4 7x5 y3 ×
(g) ÷ (h) ÷ (i)
( ) ( )m m nx m ( )x3 y3 3
1
xn =
xn

( ) ( )2x4 y2 3 x4 y4 2 3 2 ( )(l) 3 2
⎛ x2 ⎞ ⎛ x5 ⎞ 5x3 y2 ⎛ 2 xy 3 ⎞
( ) ( )(j) 3 × 2 (k) ⎝⎜ y4 ⎟⎠ × ⎝⎜ y2 ⎠⎟ 4x7 y6 ÷ ⎝⎜ 5x2 y4 ⎟⎠
3
y3x2 x2 y

Unit 1: Algebra 9

2 Making sense of algebra

Tip 2 4JNQMJGZ FBDI FYQSFTTJPO BOE HJWF ZPVS BOTXFS VTJOH QPTJUJWF JOEJDFT POMZ

4PNF FYBN RVFTUJPOT (a) x5 y 4 (b) x −4 y3 × x7 y −5 ( )2x−3 y 1 3
x −3 y 2 x2 y −1 x −4 y3
XJMM BDDFQU TJNQMJĕFE (c)

FYQSFTTJPOT XJUI OFHBUJWF ( )y2x 2 2

JOEJDFT
TVDI BT x−4 ( )(d)⎛ x ⎞ −1 x2 4 x −10 ⎛ y2 ⎞ −4 ( ( ))(f) ⎛ x4 y 1⎞2 x −2 y6 2
⎜⎝ y3 ⎟⎠ y −3 y− 2 ⎝⎜ x3 ⎟⎠ ⎜⎝ x5 y 3 ⎟⎠ 2 xy3 −2
*G
IPXFWFS
UIF RVFTUJPO ÷ ( )(e) ÷ ×

TUBUFT QPTJUJWF JOEJDFT

POMZ
ZPV DBO VTF UIF MBX 3 4JNQMJGZ

x−m = 1 TP UIBU 5x −4 = 5 x1 1 x1 x1 x1 x1 3( )1
xm x4 4 4 3 5 2 3
y (a) × x (b) × (c) 1 (d)
4JNJMBSMZ
x −2 = x2 y x3
1

( )1 ( )1 (g) xy8 (h) ⎛ x6 ⎞ 2
⎝⎜ y2 ⎠⎟
(e) 64x6 2 (f) 8x9 y 3

x2 11
x3 x2 y2
( )(i) x1 8 × ( ) ( )1 1 ( )(k) xy 3 1 xy 4
2
(j) x6 y3 3 × x 8 y 10 2 2×

4 4JNQMJGZ

( )(a) 1 ( )(b) 3 (c) ( ) ( )1 1x1y2 2× x y3 4 3
22 55 2
4
x × x3 2 x × xx2 1
x3 2

x y1 2 y1 x1 2 1 1
33 3 2 3
( )(d) 4 xy 2 ⎛ y ⎞ 2 1 ⎛ 3 ⎞ 4
x1 ⎜ x3 y 4 ⎟ ⎜ ⎟
21 × (e) 2 ÷ ⎝ ⎠ (f ) x4 × ⎝ xy 4 ⎠
x3 y2
x3 y3 3

y2

Tip Exercise 2.5 B

"QQMZ UIF JOEFY MBXT BOE 1 &WBMVBUF (b) −24 (c) ( 63 (d) 81
XPSL JO UIJT PSEFS (a) ( 4 )( )2 (−2)4 3)4 3

t TJNQMJGZ BOZ UFSNT JO 256− 1 125− 4 ( )(g) − 5 ( )(h) − 2
CSBDLFUT (e) 4 (f ) 3 1 2 1 3

t BQQMZ UIF NVMUJQMJDBUJPO 4 8
MBX UP OVNFSBUPST BOE
UIFO UP EFOPNJOBUPST ( )(i) − 1 ( )(j) − 1
8 3 8 2
t DBODFM OVNCFST JG
ZPV DBO 27 18

t BQQMZ UIF EJWJTJPO 2 $BMDVMBUF
MBX JG UIF TBNF
MFUUFS BQQFBST JO (a) o o
o Ÿ 3
UIF OVNFSBUPS BOE (b)
o o o
Ÿ o
EFOPNJOBUPS (c) o o
Ÿ o
3
(d) o
4 o o
6
t FYQSFTT ZPVS BOTXFS
VTJOH QPTJUJWF JOEJDFT 3 4PMWF GPS x

(a) x = (b) 3x = (c) 3x = 1 (d) x
(e) x = 81 (h) x + =

(f ) ox = 1 (g) 3 x − =
25

10 Unit 1: Algebra

2 Making sense of algebra

Mixed exercise 1 8SJUF FBDI PG UIF GPMMPXJOH BT BO BMHFCSBJD FYQSFTTJPO 6TF x UP SFQSFTFOU AUIF OVNCFS

(a) " OVNCFS JODSFBTFE CZ
(b) " OVNCFS EFDSFBTFE CZ GPVS
(c) 'JWF UJNFT B OVNCFS
(d) " OVNCFS EJWJEFE CZ UISFF
(e) ćF QSPEVDU PG B OVNCFS BOE GPVS
(f) " RVBSUFS PG B OVNCFS
(g) " OVNCFS TVCUSBDUFE GSPN
(h) ćF EJČFSFODF CFUXFFO B OVNCFS BOE JUT DVCF

2 %FUFSNJOF UIF WBMVF PG x o x JG

(a) x = (b) x = − (c) x = 1
3

3 &WBMVBUF FBDI FYQSFTTJPO JG a = −
b = BOE c =

(a) −2a + 3b b(c − a) (c) a b2
2ab c a2
(b) b a

(d) 3 2( −1) (e) a3b2 − 2a2 + a4b2 ac3

c − a(b −1)

4 4JNQMJGZ FBDI PG UIF GPMMPXJOH FYQSFTTJPOT BT GVMMZ BT QPTTJCMF

(a) 3a 4b + 6a 3b (b) x2 + 4x x 2 (c) −2a2b(2a2 − 3b2 )

(d) 2 ( 3) − (x 4) 2x 2 (e) 16x2 y ÷ 4 y2x (f ) 10x2 5xy
2x


5 &YQBOE BOE TJNQMJGZ JG QPTTJCMF

(a) 2(4x 3) 3(x 3+(1x) − ) (b) 3 (2x + 3) 2(4 3x)
(c) x(x + )+ 3x − (d) x2 (x + 3) 2x3 (x − 5)

6 4JNQMJGZ (JWF BMM BOTXFST XJUI QPTJUJWF JOEJDFT POMZ ( )x3 4

(a) 15x7 (b) 5x2 × 3x5 ( )(c)x2 8
18x2 x7

x2 y4 3
⎛ 4x3 ⎞ 3 ( ) (( ))(f) xy2 3
(d) (2xy2 )4 (e) ⎝⎜ y5 ⎠⎟ x3y 2 ×

( ) ( )(g) 2xy3 −2 3x2 y 3 ( ( ))(h) x− y 4 ÷ ⎛ x −3 y3 ⎞ 2
2 xy −3 ⎜⎝ x2 y −1 ⎠⎟

7 4JNQMJGZ FBDI PG UIF GPMMPXJOH FYQSFTTJPOT

( )1 ( )(b) 1 x y1 1 ( ) ( )1 5
x6 y3 2 × 22
(a) 125x3 y 3 (c) x2 y 3 2 × x −4 y 2
x3 y2

( )(d)⎛ x 1 y 2 ⎞2 14
3 3 ⎟⎠
x2 y
⎝⎜ xy 1 ÷ 2x3 y5
3

Unit 1: Algebra 11

3 Lines, angles and shapes

3.1 Lines and angles

t "OHMFT DBO CF DMBTTJĕFE BDDPSEJOH UP UIFJS TJ[F

o BDVUF BOHMFT BSF < °

o SJHIU BOHMFT BSF °

o PCUVTF BOHMFT BSF > ° CVU < °

o SFĘFY BOHMFT BSF > ° CVU < °.

t 5XP BOHMFT UIBU BEE VQ UP ° BSF DBMMFE DPNQMFNFOUBSZ BOHMFT 5XP BOHMFT UIBU BEE VQ UP ° BSF DBMMFE
TVQQMFNFOUBSZ BOHMFT

t ćF TVN PG BEKBDFOU BOHMFT PO B TUSBJHIU MJOF JT °.
t ćF TVN PG UIF BOHMFT BSPVOE B QPJOU JT °.
t 8IFO UXP MJOFT JOUFSTFDU DSPTT

UXP QBJST PG WFSUJDBMMZ PQQPTJUF BOHMFT BSF GPSNFE

7FSUJDBMMZ PQQPTJUF BOHMFT BSF FRVBM

t 8IFO UXP QBSBMMFM MJOFT BSF DVU CZ B USBOTWFSTBM
BMUFSOBUF BOHMFT BSF FRVBM
DPSSFTQPOEJOH BOHMFT BSF FRVBM BOE
DP JOUFSJPS BOHMFT BEE VQ UP °.

t 8IFO BMUFSOBUF PS DPSSFTQPOEJOH BOHMFT BSF FRVBM
PS XIFO DP JOUFSJPS BOHMFT BEE VQ UP °, UIF
MJOFT BSF QBSBMMFM

Exercise 3.1 A

1 -PPL BU UIF DMPDL GBDF PO UIF MFę $BMDVMBUF UIF GPMMPXJOH

11 12 1 (a) ćF TNBMMFTU BOHMF CFUXFFO UIF IBOET PG UIF DMPDL BU
10 2
(i) P DMPDL (ii) IPVST (iii) B N
93
(b) ćSPVHI IPX NBOZ EFHSFFT EPFT UIF IPVS IBOE NPWF CFUXFFO Q N BOE Q N
84
765 (c) ćSPVHI IPX NBOZ EFHSFFT EPFT UIF NJOVUF IBOE UVSO JO

(i) 2 1 IPVST (ii) NJOVUFT
4

(d) " DMPDL TIPXT OPPO 8IBU XJMM UIF UJNF CF XIFO UIF NJOVUF IBOE IBT NPWFE

° DMPDLXJTF

2 8JMM EPVCMJOH BO BDVUF BOHMF BMXBZT QSPEVDF BO PCUVTF BOHMF &YQMBJO ZPVS BOTXFS

3 8JMM IBMWJOH BO PCUVTF BOHMF BMXBZT QSPEVDF BO BDVUF BOHMF &YQMBJO ZPVS BOTXFS

4 8IBU JT UIF DPNQMFNFOU PG FBDI UIF GPMMPXJOH BOHMFT

(a) ° (b) x° (c) o x
°

12 Unit 1: Shape, space and measures

3 Lines, angles and shapes

5 8IBU JT UIF TVQQMFNFOU PG FBDI PG UIF GPMMPXJOH BOHMFT

(a) ° (b) ° (c) x°
(d) o x
° (e) o x
° (f) + x
°

Tip Exercise 3.1 B

:PV OFFE UP CF BCMF UP *O UIJT FYFSDJTF
DBMDVMBUF EP OPU NFBTVSF GSPN UIF EJBHSBNT
UIF WBMVFT PG UIF MFUUFSFE BOHMFT
VTF UIF SFMBUJPOTIJQT :PV TIPVME BMTP TUBUF ZPVS SFBTPOT
CFUXFFO MJOFT BOE BOHMFT
UP DBMDVMBUF UIF WBMVFT PG 1 *O UIF GPMMPXJOH EJBHSBN
MN BOE PQ BSF TUSBJHIU MJOFT 'JOE UIF TJ[F PG BOHMF a.
VOLOPXO BOHMFT Q

Remember, give reasons M aO
for statements. Use these 37° N
abbreviations:
Comp ∠s P
Supp ∠s R QOˆR = 85°
∠s on line
∠s round point 2 $BMDVMBUF UIF WBMVF PG x JO FBDI PG UIF GPMMPXJOH ĕHVSFT
Vertically opposite ∠s
(a) (b)
Remember, give reasons
for statements. Use these AE
abbreviations to refer to types of
angles: 27° 3x + 25°
Alt ∠s D
Corr ∠s 6x x
Co-int ∠s O 5x + 35°
x 112°

C

PB

Exercise 3.1 C

1 'JOE UIF WBMVFT PG UIF BOHMFT NBSLFE x BOE y JO FBDI EJBHSBN

(a) (b) E
M yB
M N O P 57°
x y F

Q 95° 72° P Cx 81° G N
R T Q

S H

(c) K A (d)

Bx E x xx
F 108° xx
I C 65° Gy J x xx

D
H

Unit 1: Shape, space and measures 13

3 Lines, angles and shapes

2 $BMDVMBUF UIF WBMVF PG x JO FBDI PG UIF GPMMPXJOH ĕHVSFT (JWF SFBTPOT GPS ZPVS BOTXFST

(a) E (b)

S 57° N

A 112° B Q
F

x MT
CG
D

H

(c) A 30° B (d) A C E

C D D 126°
x F

E 60° 108° x
B F

(e) A
B x

C 50° D E
45°

GF

14 Unit 1: Shape, space and measures

3 Lines, angles and shapes

3.2 Triangles

t 4DBMFOF USJBOHMFT IBWF OP FRVBM TJEFT BOE OP FRVBM BOHMFT
t *TPTDFMFT USJBOHMFT IBWF UXP FRVBM TJEFT ćF BOHMFT BU UIF CBTFT PG UIF FRVBM TJEFT BSF FRVBM JO TJ[F

ćF DPOWFSTF JT BMTP USVF o JG B USJBOHMF IBT UXP FRVBM BOHMFT
UIFO JU JT JTPTDFMFT

t &RVJMBUFSBM USJBOHMFT IBWF UISFF FRVBM TJEFT BOE UISFF FRVBM BOHMFT FBDI CFJOH °

t ćF TVN PG UIF JOUFSJPS BOHMFT PG BOZ USJBOHMF JT °.
t ćF FYUFSJPS BOHMF PG B USJBOHMF JT FRVBM UP UIF TVN PG UIF UXP PQQPTJUF JOUFSJPS BOHMFT

Exercise 3.2

REWIND 1 'JOE UIF BOHMFT NBSLFE XJUI MFUUFST (JWF SFBTPOT GPS BOZ TUBUFNFOUT

You may also need to apply the (a) 48° 29° (b) (c) c 37°
angle relationships for points, a 113° 31°
lines and parallel lines to find the
missing angles in triangles. 62° b

(d) 53° d (e) e (f )
61°
32°
xy

(g) (h) x (i) x
54° 30°
102° y
21° 57°
x

Unit 1: Shape, space and measures 15

3 Lines, angles and shapes

2 $BMDVMBUF UIF WBMVF PG x BOE IFODF ĕOE UIF TJ[F PG UIF NBSLFE BOHMFT

(a) A (b) A
x

B

B 2x C C
D

(c) M (d) M

Nx O
18°
O xN
P

P

Tip 3 *O 6ABC
∠A ¡
∠B x BOE ∠C x.
$BMDVMBUF UIF TJ[F PG BOHMFT B BOE C JO EFHSFFT
'PS XPSE QSPCMFNT
XIFSF ZPV BSF OPU HJWFO B
EJBHSBN
B SPVHI TLFUDI
NBZ IFMQ ZPV XPSL PVU UIF
BOTXFST

3.3 Quadrilaterals

t " RVBESJMBUFSBM JT B GPVS TJEFE TIBQF

o " USBQF[JVN IBT POF QBJS PG QBSBMMFM TJEFT

o " LJUF IBT UXP QBJST PG BEKBDFOU TJEFT FRVBM JO MFOHUI ćF EJBHPOBMT JOUFSTFDU BU ° BOE UIF MPOHFS EJBHPOBM
CJTFDUT UIF TIPSUFS POF 0OMZ POF QBJS PG PQQPTJUF BOHMFT JT FRVBM ćF EJBHPOBMT CJTFDU UIF PQQPTJUF BOHMFT

o " QBSBMMFMPHSBN IBT PQQPTJUF TJEFT FRVBM BOE QBSBMMFM ćF PQQPTJUF BOHMFT BSF FRVBM JO TJ[F BOE UIF EJBHPOBMT
CJTFDU FBDI PUIFS

o " SFDUBOHMF IBT PQQPTJUF TJEFT FRVBM BOE QBSBMMFM BOE JOUFSJPS BOHMFT FBDI FRVBM UP ° ćF EJBHPOBMT BSF FRVBM
JO MFOHUI BOE UIFZ CJTFDU FBDI PUIFS

o " SIPNCVT JT B QBSBMMFMPHSBN XJUI BMM GPVS TJEFT FRVBM JO MFOHUI ćF EJBHPOBMT CJTFDU FBDI PUIFS BU ° BOE CJTFDU
UIF PQQPTJUF BOHMFT

o " TRVBSF IBT GPVS FRVBM TJEFT BOE GPVS BOHMFT FBDI FRVBM UP ° ćF PQQPTJUF TJEFT BSF QBSBMMFM ćF EJBHPOBMT BSF
FRVBM JO MFOHUI
UIFZ CJTFDU FBDI PUIFS BU SJHIU BOHMFT BOE UIFZ CJTFDU UIF PQQPTJUF BOHMFT

t ćF TVN PG UIF JOUFSJPS BOHMFT PG B RVBESJMBUFSBM JT °.

16 Unit 1: Shape, space and measures

3 Lines, angles and shapes

Exercise 3.3

REWIND 1 &BDI PG UIF GPMMPXJOH TUBUFNFOUT BQQMJFT UP POF PS NPSF RVBESJMBUFSBMT 'PS FBDI POF
OBNF
UIF RVBESJMBUFSBM T
UP XIJDI JU BMXBZT BQQMJFT
The angle relationships for parallel
lines will apply when aquadrilateral (a) "MM TJEFT BSF FRVBM JO MFOHUI
has parallel sides. (b) "MM BOHMFT BSF FRVBM JO TJ[F
(c) ćF EJBHPOBMT BSF UIF TBNF MFOHUI
(d) ćF EJBHPOBMT CJTFDU FBDI PUIFS
(e) ćF BOHMFT BSF BMM ° BOE UIF EJBHPOBMT CJTFDU FBDI PUIFS
(f) 0QQPTJUF BOHMFT BSF FRVBM JO TJ[F
(g) ćF EJBHPOBMT JOUFSTFDU BU SJHIU BOHMFT
(h) ćF EJBHPOBMT CJTFDU UIF PQQPTJUF BOHMFT
(i) 0OF EJBHPOBM EJWJEFT UIF RVBESJMBUFSBM JOUP UXP JTPTDFMFT USJBOHMFT

2 $PQZ UIF EJBHSBNT CFMPX 'JMM JO UIF TJ[FT PG BMM UIF BOHMFT

a g 63° a
b f

fe c ed b
d c

3 $BMDVMBUF UIF TJ[F PG UIF NBSLFE BOHMFT JO UIF GPMMPXJOH ĕHVSFT (JWF SFBTPOT PS TUBUF UIF
QSPQFSUJFT ZPV BSF VTJOH

(a) A 79° (b) A B (c) A
D 58° B 32° D 38°

x B

DC

C EF
C

(d) M N O (e) 3x (f ) M c N O
Q 65° x d 68°

x

P 2x Re b a
P
Q

4 2VBESJMBUFSBM PQRS IBT ∠P ∠S ¡ BOE ∠R ∠Q.

$BMDVMBUF UIF TJ[F PG

(a) ∠R ∠Q (b) ∠R (c) ∠Q.

5 " LJUF PMNO IBT EJBHPOBMT PN BOE MO UIBU JOUFSTFDU BU Q ∠QMN ¡ BOE ∠PNO ¡
$BMDVMBUF UIF TJ[F PG

(a) ∠MNP (b) ∠MNO (c) ∠PON.

Unit 1: Shape, space and measures 17

3 Lines, angles and shapes

3.4 Polygons

t " QPMZHPO JT B UXP EJNFOTJPOBM TIBQF XJUI UISFF PS NPSF TJEFT 1PMZHPOT BSF OBNFE BDDPSEJOH UIF OVNCFS PG TJEFT
UIFZ IBWF

o USJBOHMF
o IFQUBHPO


o RVBESJMBUFSBM
o PDUBHPO


o QFOUBHPO
o OPOBHPO


o IFYBHPO
o EFDBHPO


t " SFHVMBS QPMZHPO IBT BMM JUT TJEFT FRVBM BOE BMM JUT BOHMFT FRVBM
t ćF JOUFSJPS BOHMF TVN PG BOZ QPMZHPO DBO CF XPSLFE PVU VTJOH UIF GPSNVMB n −
× ° XIFSF n JT UIF OVNCFS PG

TJEFT 0ODF ZPV IBWF UIF BOHMF TVN
ZPV DBO ĕOE UIF TJ[F PG POF BOHMF PG B SFHVMBS QPMZHPO CZ EJWJEJOH UIF UPUBM CZ

UIF OVNCFS PG BOHMFT

t ćF TVN PG UIF FYUFSJPS BOHMFT PG BOZ DPOWFY QPMZHPO JT °

Exercise 3.4

Tip 1 'PS FBDI PG UIF GPMMPXJOH ĕOE

*G ZPV DBO U SFNFNCFS UIF (i) UIF TVN PG UIF JOUFSJPS BOHMFT (ii) UIF TJ[F PG POF JOUFSJPS BOHMF
GPSNVMB
ZPV DBO ĕOE UIF
TJ[F PG POF JOUFSJPS BOHMF (a) B SFHVMBS PDUBHPO
PG B SFHVMBS QPMZHPO VTJOH (b) B SFHVMBS EFDBHPO
UIF GBDU UIBU UIF FYUFSJPS (c) B SFHVMBS TJEFE QPMZHPO
BOHMFT BEE VQ UP °
%JWJEF CZ UIF OVNCFS 2 " DPJO JT NBEF JO UIF TIBQF PG B SFHVMBS TJEFE QPMZHPO $BMDVMBUF UIF TJ[F PG FBDI
PG BOHMFT UP ĕOE UIF TJ[F JOUFSJPS BOHMF
PG POF FYUFSJPS BOHMF
ćFO VTF UIF GBDU UIBU 3 ćF JOUFSJPS BOHMF PG B SFHVMBS QPMZHPO JT ° )PX NBOZ TJEFT EPFT UIF QPMZHPO IBWF
UIF FYUFSJPS BOE JOUFSJPS
BOHMFT GPSN B TUSBJHIU MJOF 4 0OF FYUFSJPS BOHMF PG B SFHVMBS QPMZHPO JT °.
°
UP XPSL PVU UIF TJ[F
PG UIF JOUFSJPS BOHMF (a) 8IBU JT UIF TJ[F PG FBDI JOUFSJPS BOHMF
(b) )PX NBOZ TJEFT EPFT UIF QPMZHPO IBWF

5 $BMDVMBUF UIF WBMVF PG UIF BOHMFT NBSLFE XJUI MFUUFST JO FBDI PG UIFTF JSSFHVMBS QPMZHPOT

(a) 120° 125° (b) x (c) A
100° C 84°
BH
110° 161° x 56° 100° y
x – 10° 17°

130° x x – 50° D E xC
98° 44°
125° G

100°

F

18 Unit 1: Shape, space and measures

3 Lines, angles and shapes

3.5 Circles

t " DJSDMF JT B TFU PG QPJOUT FRVJEJTUBOU GSPN B ĕYFE DFOUSF )BMG B DJSDMF JT B TFNJ DJSDMF
t ćF QFSJNFUFS PG B DJSDMF JT DBMMFE JUT DJSDVNGFSFODF
t ćF EJTUBODF BDSPTT B DJSDMF UISPVHI UIF DFOUSF
JT DBMMFE JUT EJBNFUFS " SBEJVT JT IBMG B EJBNFUFS
t "O BSD JT QBSU PG UIF DJSDVNGFSFODF PG B DJSDMF
t " DIPSE JT B MJOF KPJOJOH UXP QPJOUT PO UIF DJSDVNGFSFODF " DIPSE DVUT UIF DJSDMF JOUP UXP TFHNFOUT
t " ATMJDF PG B DJSDMF
NBEF CZ UXP SBEJJ BOE UIF BSD CFUXFFO UIFN PO UIF DJSDVNGFSFODF
JT DBMMFE B TFDUPS
t " UBOHFOU JT B MJOF UIBU UPVDIFT B DJSDMF BU POMZ POF QPJOU

Exercise 3.5

1 %SBX B DJSDMF XJUI B SBEJVT PG DN BOE DFOUSF O #Z ESBXJOH UIF QBSUT BOE MBCFMMJOH UIFN

JOEJDBUF UIF GPMMPXJOH PO ZPVS EJBHSBN

(a) B TFDUPS XJUI BO BOHMF PG ¡
(b) DIPSE DE
(c) MON
UIF EJBNFUFS PG UIF DJSDMF
(d) B UBOHFOU UIBU UPVDIFT UIF DJSDMF BU M
(e) UIF NBKPS BSD MP.

3.6 Construction

t :PV OFFE UP CF BCMF UP VTF B SVMFS BOE B QBJS PG DPNQBTTFT UP DPOTUSVDU USJBOHMFT HJWFO UIF MFOHUIT PG UISFF TJEFT
BOE
CJTFDU MJOFT BOE BOHMFT :PV BMTP OFFE UP CF BCMF UP DPOTUSVDU PUIFS TJNQMF HFPNFUSJD ĕHVSFT GSPN HJWFO TQFDJĕDBUJPOT

t ćF EJBHSBNT CFMPX TIPX ZPV IPX UP CJTFDU BO BOHMF BOE IPX UP ESBX UIF QFSQFOEJDVMBS CJTFDUPS PG B MJOF

AB

How to bisect an angle. How to draw the perpendicular bisector of a line.

Tip Exercise 3.6

"MXBZT TUBSU XJUI B SPVHI 1 %SBX BOHMF ABC = ° "DDVSBUFMZ CJTFDU UIF BOHMF
TLFUDI -BCFM ZPVS SPVHI
TLFUDI TP ZPV LOPX 2 $POTUSVDU ΔABC XJUI AC = DN
CB = DN BOE AB = DN
XIBU MFOHUIT ZPV OFFE UP
NFBTVSF 3 $POTUSVDU ΔMNO XJUI MN = DN
NO = DN BOE MO = DN $POTUSVDU UIF
QFSQFOEJDVMBS CJTFDUPS PG NO UP DVU NO BU X BOE FYUFOE UIF MJOF UP DVU MO BU Y .FBTVSF UIF
MFOHUI PG XY UP UIF OFBSFTU NJMMJNFUSF

Unit 1: Shape, space and measures 19

3 Lines, angles and shapes

4 $POTUSVDU ΔDEF XJUI DE = NN
FE = NN BOE DF = NN
(a) 8IBU UZQF PG USJBOHMF JT DEF
(b) #JTFDU FBDI BOHMF PG UIF USJBOHMF %P UIF BOHMF CJTFDUPST NFFU BU UIF TBNF QPJOU

5 "DDVSBUFMZ DPOTUSVDU B TRVBSF PG TJEF DN 8IBU JT UIF MFOHUI PG B EJBHPOBM PG UIF TRVBSF

Mixed exercise 1 'JOE UIF WBMVF PG UIF NBSLFE BOHMFT JO FBDI PG UIF GPMMPXJOH

(a) G (b) A B (c) A E B
A 23° F C x 98° x 111° 68°

B D D 71° C
D
C x
E

(d) (e) (f) y
32° y 104°

x 110°

30° 16°

(g) 40° y (h) x y

z 110°
x

2 'PS FBDI TIBQF DPNCJOBUJPO ĕOE UIF TJ[F PG BOHMF x ćF TIBQFT JO QBSUT B
BOE C
BSF
SFHVMBS QPMZHPOT

(a) (b) (c) A B
x x x 50°

C 70°

50° D
120° E 110°

60°

20 Unit 1: Shape, space and measures

3 Lines, angles and shapes

3 6TF UIF EJBHSBN PG UIF DJSDMF XJUI DFOUSF O UP BOTXFS UIFTF RVFTUJPOT

A (a) 8IBU BSF UIF DPSSFDU NBUIFNBUJDBM OBNFT GPS

O (i) DO (ii) AB (iii) AC

D (b) 'PVS SBEJJ BSF TIPXO PO UIF EJBHSBN /BNF UIFN
C
(c) *G OB JT DN MPOH
IPX MPOH JT AC

(d) % SBX B DPQZ PG UIF DJSDMF BOE PO UP JU ESBX UIF UBOHFOU UP UIF DJSDMF UIBU QBTTFT UISPVHI

B QPJOU B.

4 6TF B SVMFS BOE B QSPUSBDUPS UP ESBX MJOF TFHNFOU AB DN MPOH $POTUSVDU XY
UIF
QFSQFOEJDVMBS CJTFDUPS PG AB XJUI QPJOU X PO AB #JTFDU BOHMF BXˆY.

5 $POTUSVDU ΔABC XJUI AB = BC = AC = DN $POTUSVDU UIF QFSQFOEJDVMBS CJTFDUPS PG AB
8IFSF EPFT UIJT JOUFSTFDU XJUI UIF USJBOHMF

Unit 1: Shape, space and measures 21

4 Collecting, organising
and displaying data

4.1 Collecting and classifying data

t %BUB JT B TFU PG GBDUT
OVNCFST PS PUIFS JOGPSNBUJPO
DPMMFDUFE UP USZ UP BOTXFS B RVFTUJPO
t 1SJNBSZ EBUB JT APSJHJOBM EBUB BOE DBO CF DPMMFDUFE CZ NFBTVSJOH
PCTFSWBUJPO
EPJOH FYQFSJNFOUT
DBSSZJOH PVU
TVSWFZT PS BTLJOH QFPQMF UP DPNQMFUF RVFTUJPOOBJSFT

t 4FDPOEBSZ EBUB JT EBUB ESBXO GSPN B OPO PSJHJOBM TPVSDF 'PS FYBNQMF ZPV DPVME ĕOE UIF BSFB PG FBDI PG UIF XPSME T
PDFBOT CZ SFGFSSJOH UP BO BUMBT

t :PV DBO DMBTTJGZ EBUB BT RVBMJUBUJWF PS RVBOUJUBUJWF
t 2VBMJUBUJWF EBUB JT OPO OVNFSJD TVDI BT DPMPVS
NBLF PG WFIJDMF PS GBWPVSJUF ĘBWPVS
t 2VBOUJUBUJWF EBUB JT OVNFSJDBM EBUB UIBU XBT DPVOUFE PS NFBTVSFE 'PS FYBNQMF BHF
NBSLT JO B UFTU
TIPF TJ[F
IFJHIU
t 2VBOUJUBUJWF EBUB DBO CF EJTDSFUF PS DPOUJOVPVT
t %JTDSFUF EBUB DBO POMZ UBLF DFSUBJO WBMVFT BOE JT VTVBMMZ TPNFUIJOH DPVOUFE 'PS FYBNQMF
UIF OVNCFS PG DIJMESFO JO
ZPVS GBNJMZ ćFSF BSF OP JO CFUXFFO WBMVFT ZPV DBO U IBWF 2 1 DIJMESFO JO B GBNJMZ
2

t $POUJOVPVT EBUB DBO UBLF BOZ WBMVF BOE JT VTVBMMZ TPNFUIJOH NFBTVSFE 'PS FYBNQMF UIF IFJHIUT PG USFFT JO B
SBJOGPSFTU DPVME SBOHF GSPN UP NFUSFT "OZ WBMVF JO CFUXFFO UIPTF UXP IFJHIUT JT QPTTJCMF

Exercise 4.1

ćF GPMMPXJOH UBCMF PG EBUB XBT DPMMFDUFE BCPVU UFO TUVEFOUT JO B IJHI TDIPPM 4UVEZ UIF UBCMF BOE
UIFO BOTXFS UIF RVFTUJPOT BCPVU UIF EBUB

Student 1 2 3 4 5 6 7 8 9 10

Gender ' ' MMM ' M ' ' M

Height (m)

Shoe size 7 7 8 7

Mass (kg)

Eye colour #S (S (S #S #S #S #S (S #M #S

Hair colour #M #M #MP #S #S #S #M #M #M #M

No. of brothers/ 1 1

sisters

(a) 8IJDI PG UIFTF EBUB DBUFHPSJFT BSF RVBMJUBUJWF
(b) 8IJDI PG UIFTF EBUB DBUFHPSJFT BSF RVBOUJUBUJWF
(c) 8IJDI TFUT PG OVNFSJDBM EBUB BSF EJTDSFUF EBUB
(d) 8IJDI TFUT PG OVNFSJDBM EBUB BSF DPOUJOVPVT EBUB
(e) )PX EP ZPV UIJOL FBDI TFU PG EBUB XBT DPMMFDUFE (JWF B SFBTPO GPS ZPVS BOTXFST

22 Unit 1: Data handling

4 Collecting, organising and displaying data

4.2 Organising data

t 0ODF EBUB IBT CFFO DPMMFDUFE
JU OFFET UP CF BSSBOHFE BOE PSHBOJTFE TP UIBU JU JT FBTJFS UP XPSL XJUI
JOUFSQSFU BOE
NBLF JOGFSFODFT BCPVU

t 5BMMZ UBCMFT BOE GSFRVFODZ UBCMFT BSF VTFE UP PSHBOJTF EBUB BOE UP TIPX UIF UPUBMT PG EJČFSFOU WBMVFT PS DBUFHPSJFT
t 8IFO ZPV IBWF B MBSHF TFU PG OVNFSJDBM EBUB
XJUI MPUT PG EJČFSFOU TDPSFT
ZPV DBO HSPVQ UIF EBUB JOUP JOUFSWBMT DBMMFE

DMBTT JOUFSWBMT $MBTT JOUFSWBMT TIPVME OPU PWFSMBQ

t " UXP XBZ UBCMF DBO CF VTFE UP TIPX UIF GSFRVFODZ PG SFTVMUT GPS UXP PS NPSF TFUT PG EBUB

Exercise 4.2

In data handling, the word 1 4FTI EJE B TVSWFZ UP ĕOE PVU IPX NBOZ QIPOF DBMMT B HSPVQ PG TUVEFOUT SFDFJWFE JO POF
frequency means the number IPVS ćFTF BSF IJT SFTVMUT
of times a score or observation
occurs.




$PQZ BOE DPNQMFUF UIJT UBMMZ UBCMF UP PSHBOJTF UIF EBUB

Phone calls Tally Frequency
1


2 /JLB DPVOUFE UIF OVNCFS PG NPTRVJUPFT JO IFS CFESPPN GPS OJHIUT JO B SPX BOE HPU UIFTF
SFTVMUT







(a) $PQZ BOE DPNQMFUF UIJT GSFRVFODZ UBCMF UP PSHBOJTF UIF EBUB

Number of 0 1 2 3 4 5 6

mosquitoes

Frequency

(b) %PFT /JLB IBWF B NPTRVJUP QSPCMFN (JWF B SFBTPO GPS ZPVS BOTXFS

Unit 1: Data handling 23

4 Collecting, organising and displaying data

Score Frequency 3 ćFTF BSF UIF QFSDFOUBHF TDPSFT PG TUVEFOUT JO BO FYBNJOBUJPO
o
o
o
o
o
o
o (a) $PQZ BOE DPNQMFUF UIJT HSPVQFE GSFRVFODZ UBCMF UP PSHBOJTF UIF SFTVMUT
(b) )PX NBOZ TUVEFOUT TDPSFE BU MFBTU
(c) )PX NBOZ TUVEFOUT TDPSFE MPXFS UIBO
(d) )PX NBOZ TUVEFOUT TDPSFE BU MFBTU CVU MFTT UIBO
(e) ć F ĕSTU BOE MBTU DMBTT JOUFSWBM JO UIF UBCMF BSF HSFBUFS UIBO UIF PUIFST 4VHHFTU XIZ UIJT JT

UIF DBTF

4 ćJT JT B TFDUJPO PG UIF UBCMF ZPV XPSLFE XJUI JO &YFSDJTF

Student 1 2 3 4 5 6 7 8 9 10

Gender ' ' MMM ' M ' ' M

Eye colour #S (S (S #S #S #S #S (S #M #S

Hair colour #M #M #MP #S #S #S #M #M #M #M

No. of siblings 1 1

(brothers/sisters)

(a) $PQZ BOE DPNQMFUF UIJT UXP XBZ UBCMF VTJOH EBUB GSPN UIF UBCMF

Eye colour Brown Blue Green
Male
Female

(b) %SBX BOE DPNQMFUF UXP TJNJMBS UXP XBZ UBCMFT PG ZPVS PXO UP TIPX UIF IBJS DPMPVS BOE
OVNCFS PG CSPUIFST PS TJTUFST CZ HFOEFS

(c) 8SJUF B TFOUFODF UP TVNNBSJTF XIBU ZPV GPVOE PVU GPS FBDI UBCMF

4.3 Using charts to display data

t $IBSUT VTVBMMZ IFMQ ZPV UP TFF QBUUFSOT BOE USFOET JO EBUB NPSF FBTJMZ UIBO JO UBCMFT
t 1JDUPHSBNT VTF TZNCPMT UP TIPX UIF GSFRVFODZ PG EBUB JO EJČFSFOU DBUFHPSJFT ćFZ BSF VTFGVM GPS EJTDSFUF
DBUFHPSJDBM

BOE VOHSPVQFE EBUB

t #BS DIBSUT BSF VTFGVM GPS DBUFHPSJDBM BOE VOHSPVQFE EBUB " CBS DIBSU IBT CBST PG FRVBM XJEUI XIJDI BSF FRVBMMZ TQBDFE
t #BS DIBSUT DBO CF ESBXO BT IPSJ[POUBM PS WFSUJDBM DIBSUT ćFZ DBO BMTP TIPX UXP PS NPSF TFUT PG EBUB PO UIF TBNF TFU

PG BYFT

t 1JF DIBSUT BSF DJSDVMBS HSBQIT UIBU VTF TFDUPST PG DJSDMF UP TIPX UIF QSPQPSUJPO PG EBUB JO FBDI DBUFHPSZ
t "MM DIBSUT TIPVME IBWF B IFBEJOH BOE DMFBSMZ MBCFMMFE TDBMFT
BYFT PS LFZT

24 Unit 1: Data handling

4 Collecting, organising and displaying data

Exercise 4.3

1 4UVEZ UIF EJBHSBN DBSFGVMMZ BOE BOTXFS UIF RVFTUJPOT BCPVU JU

Number of students in each year Key (a) 8IBU UZQF PG DIBSU JT UIJT
Year 8 = 30 students (b) 8IBU EPFT UIF DIBSU TIPX
Year 9 (c) 8IBU EPFT FBDI GVMM TZNCPM SFQSFTFOU
Year 10 (d) )PX BSF TUVEFOUT TIPXO PO UIF DIBSU
Year 11 (e) )PX NBOZ TUVEFOUT BSF UIFSF JO :FBS
Year 12 (f) 8 IJDI ZFBS HSPVQ IBT UIF NPTU TUVEFOUT )PX NBOZ BSF UIFSF JO UIJT

ZFBS HSPVQ
(g) %P ZPV UIJOL UIFTF BSF BDDVSBUF PS SPVOEFE ĕHVSFT 8IZ

Tip 2 ćF UBCMF TIPXT UIF QPQVMBUJPO JO NJMMJPOT
PG ĕWF PG UIF XPSME T MBSHFTU DJUJFT

$IPPTF TZNCPMT UIBU BSF City Tokyo Seoul Mexico City New York Mumbai
FBTZ UP ESBX BOE UP EJWJEF
JOUP QBSUT *G JU JT OPU HJWFO
Population (millions)
DIPPTF B TVJUBCMF TDBMF GPS
ZPVS TZNCPMT TP ZPV EPO U %SBX B QJDUPHSBN UP TIPX UIJT EBUB
IBWF UP ESBX UPP NBOZ

Tip 3 4UVEZ UIF UXP CBS DIBSUT CFMPX (e) 8IJDI TQPSU JT NPTU QPQVMBS XJUI CPZT
(f) 8IJDI TQPSU JT NPTU QPQVMBS XJUI HJSMT
$PNQPVOE CBS DIBSUT (a) 8IBU EPFT DIBSU " TIPX (g) ) PX NBOZ TUVEFOUT DIPTF CBTLFUCBMM
TIPX UXP PS NPSF TFUT PG (b) )PX NBOZ CPZT BSF UIFSF JO $MBTT "
EBUB PO UIF TBNF QBJS PG (c) )PX NBOZ TUVEFOUT BSF UIFSF BT UIFJS GBWPVSJUF TQPSU
BYFT " LFZ JT OFFEFE UP
TIPX XIJDI TFU FBDI CBS JO " BMUPHFUIFS
SFQSFTFOUT (d) 8IBU EPFT DIBSU # TIPX

A. Number of students in 10AFrequency B. Favourite sport of students in 10A Key
20 Frequency 14 boys
18 12 girls
16 10
14 8
12 6
10 4
8 2
6 0
4
2 soccer basketball athletics
Sport
0
boys girls
Gender

Unit 1: Data handling 25

4 Collecting, organising and displaying data

4 ćF UBCMF CFMPX TIPXT UIF UZQF PG GPPE UIBU B HSPVQ PG TUVEFOUT JO B IPTUFM DIPTF GPS CSFBLGBTU

Girls Cereal Hot porridge Bread
Boys 8


Tip (a) %SBX B TJOHMF CBS DIBSU UP TIPX UIF DIPJDF PG DFSFBM BHBJOTU CSFBE
(b) %SBX B DPNQPVOE CBS DIBSU UP TIPX UIF CSFBLGBTU GPPE DIPJDF GPS HJSMT BOE CPZT
5P XPSL PVU UIF
5 +ZPUJ SFDPSEFE UIF OVNCFS BOE UZQF PG WFIJDMFT QBTTJOH IFS IPNF JO #BOHBMPSF
4IF ESFX UIJT QJF DIBSU UP TIPX IFS SFTVMUT

QFSDFOUBHF UIBU BO BOHMF JO Traffic passing my home

B QJF DIBSU SFQSFTFOUT
VTF taxis handcarts (a) 8IJDI UZQF PG WFIJDMF XBT NPTU DPNNPO
bicycles (b) 8IBU QFSDFOUBHF PG UIF WFIJDMFT XFSF UVL UVLT
UIF GPSNVMB (c) )PX NBOZ USVDLT QBTTFE +ZPUJ T IPNF
motor- (d) 8IJDI UZQFT PG WFIJDMFT XFSF MFBTU DPNNPO
n × 100 cycles
360

XIFSF n JT UIF TJ[F PG UIF

BOHMF trucks cars

buses

tuk-tuks

6 *O BO *($4& FYBN UIF SFTVMUT GPS TUVEFOUT XFSF BUUBJOFE BO " HSBEF
BUUBJOFE
B # HSBEF
BUUBJOFE B $ HSBEF
BUUBJOFE B % HSBEF BOE UIF SFTU BUUBJOFE & HSBEF
PS‫ڀ‬MPXFS

(a) 3FQSFTFOU UIJT JOGPSNBUJPO PO B QJF DIBSU
(b) )PX NBOZ TUVEFOUT BUUBJOFE BO "
(c) )PX NBOZ TUVEFOUT BUUBJOFE B % PS MPXFS
(d) 8IJDI HSBEF XBT BUUBJOFE CZ NPTU PG UIF TUVEFOUT

7 ćF HSBQIT CFMPX SFQSFTFOU UIF BWFSBHF NPOUIMZ UFNQFSBUVSF BOE UIF BWFSBHF NPOUIMZ
SBJOGBMM JO UIF EFTFSU JO &HZQU

"WFSBHF NPOUIMZ UFNQFSBUVSF "WFSBHF NPOUIMZ SBJOGBMM

30 F MAM J J A S O N D 140
20 Month 120
10Temperature (°C) 100
0 Rainfall (mm) 80
–10 60
–20 40 F MAM J J A S OND
–30 20 Month
–40
J
J

(a) 8IBU JT UIF NBYJNVN UFNQFSBUVSF
(b) *O XIBU NPOUIT JT UIF BWFSBHF UFNQFSBUVSF BCPWF ¡$

26 Unit 1: Data handling

4 Collecting, organising and displaying data

(c) *T &HZQU JO UIF OPSUIFSO PS TPVUIFSO IFNJTQIFSF
(d) *T UIF UFNQFSBUVSF FWFS CFMPX GSFF[JOH QPJOU
(e) 8IBU JT UIF BWFSBHF SBJOGBMM JO /PWFNCFS
(f) *O XIJDI NPOUI JT UIF BWFSBHF SBJOGBMM NN
(g) -PPLJOH BU CPUI HSBQIT
XIBU DBO ZPV TBZ BCPVU UIF SBJOGBMM XIFO UIF UFNQFSBUVSFT

BSF IJHI

Mixed exercise 1 .JLB DPMMFDUFE EBUB BCPVU IPX NBOZ DIJMESFO EJČFSFOU GBNJMJFT JO IFS DPNNVOJUZ IBE

ćFTF BSF IFS SFTVMUT






(a) )PX EP ZPV UIJOL .JLB DPMMFDUFE UIF EBUB
(b) *T UIJT EBUB EJTDSFUF PS DPOUJOVPVT 8IZ
(c) *T UIJT EBUB RVBMJUBUJWF PS RVBOUJUBUJWF 8IZ
(d) %SBX VQ B GSFRVFODZ UBCMF
XJUI UBMMJFT
UP PSHBOJTF UIF EBUB
(e) 3FQSFTFOU UIF EBUB PO B QJF DIBSU
(f) % SBX B CBS DIBSU UP DPNQBSF UIF OVNCFS PG GBNJMJFT UIBU IBWF UISFF PS GFXFS DIJMESFO

XJUI UIPTF UIBU IBWF GPVS PS NPSF DIJMESFO

2 .ST 4BODIF[ CBLFT BOE TFMMT DPPLJFT 0OF XFFL TIF TFMMT QFBOVU DSVODIJFT
DIPDPMBUF
DVQT BOE DPDPOVU NVODIJFT %SBX B QJDUPHSBN UP SFQSFTFOU UIJT EBUB

3 4UVEZ UIF DIBSU JO UIF NBSHJO

Number of phones per 100 people Mobile phones and land lines, per 100 people (a) 8IBU EP ZPV DBMM UIJT UZQF PG DIBSU
100 (b) 8IBU EPFT UIF DIBSU TIPX
90 (c) $ BO ZPV UFMM IPX NBOZ QFPQMF JO FBDI DPVOUSZ IBWF B NPCJMF QIPOF
80
70 GSPN UIJT DIBSU &YQMBJO ZPVS BOTXFS
60 (d) *O XIJDI DPVOUSJFT EP B HSFBUFS QSPQPSUJPO PG UIF QFPQMF IBWF B MBOE
50
40 MJOF UIBO B NPCJMF QIPOF
30 (e) *O XIJDI DPVOUSJFT EP NPSF QFPQMF IBWF NPCJMF QIPOFT UIBO
20
10 MBOE MJOFT
(f) *O XIJDI DPVOUSZ EP NPSF UIBO PG UIF QPQVMBUJPO IBWF B MBOE
0
Canada USA Germany Denmark UK Sweden Italy MJOF BOE B NPCJMF QIPOF
(g) 8IBU EP ZPV UIJOL UIF CBST XPVME MPPL MJLF GPS ZPVS DPVOUSZ 8IZ

Key Land lines
Mobile phones

Unit 1: Data handling 27

4 Collecting, organising and displaying data

4 6TF UIF UBCMF PG EBUB GSPN &YFSDJTF SFQFBUFE CFMPX
BCPVU UIF UFO TUVEFOUT GPS UIJT
RVFTUJPO

Student 1 7 8
Gender ' ' MMM' M' ' M
Height (m)
Shoe size 7 7 8 7
Mass (kg)
Eye colour #S (S (S #S #S #S #S (S #M #S
Hair colour #M #M #MP #S #S #S #M #M #M #M
No. of brothers/sisters 1 1

(a) %SBX B QJF DIBSU UP TIPX UIF EBUB BCPVU UIF OVNCFS PG TJCMJOHT
(b) 3FQSFTFOU UIF IFJHIU PG TUVEFOUT VTJOH BO BQQSPQSJBUF DIBSU
(c) %SBX B DPNQPVOE CBS DIBSU TIPXJOH FZF BOE IBJS DPMPVS CZ HFOEFS

5 "NZ CPVHIU B OFX 7BVYIBMM $PSTB JO *UT WBMVF JT TIPXO JO UIF UBCMF CFMPX

FAST FORWARD Year Value of car

In part (b), the ‘percentage
depreciation’ requires you to
first calculate how much the
car’s value decreased in the year
she had it, and then calculate
this as a percentage of the (a) %SBX B MJOF HSBQI UP SFQSFTFOU UIJT JOGPSNBUJPO
original value. You will see more (b) 8IBU JT UIF QFSDFOUBHF EFQSFDJBUJPO JO UIF ĕSTU ZFBS TIF PXOFE UIF DBS
about percentage decrease in (c) 6TF ZPVS HSBQI UP FTUJNBUF UIF WBMVF PG UIF DBS JO
chapter 5.

28 Unit 1: Data handling

5 Fractions

5.1 Equivalent fractions

t &RVJWBMFOU NFBOT
AIBT UIF TBNF WBMVF
t 5P ĕOE FRVJWBMFOU GSBDUJPOT FJUIFS NVMUJQMZ CPUI UIF OVNFSBUPS BOE EFOPNJOBUPS CZ UIF TBNF OVNCFS

PS EJWJEF CPUI UIF OVNFSBUPS BOE EFOPNJOBUPS CZ UIF TBNF OVNCFS

You can cross multiply to make Exercise 5.1
an equation and then solve it. For
1 'JOE UIF NJTTJOH WBMVF JO FBDI QBJS PG FRVJWBMFOU GSBDUJPOT
example:

1 = x (a) 2 = 26 (b) 5 = 120 (c) 6 = 66
2 28 5 x 7 x 5 x

2x = 28 (d) 11 = 143 (e) 5 = 80 (f ) 8 = x
x =14 9 x 3 x 12 156

5.2 Operations on fractions

t 5P NVMUJQMZ GSBDUJPOT
NVMUJQMZ OVNFSBUPST CZ OVNFSBUPST BOE EFOPNJOBUPST CZ EFOPNJOBUPST
.JYFE OVNCFST TIPVME CF SFXSJUUFO BT JNQSPQFS GSBDUJPOT CFGPSF NVMUJQMZJOH PS EJWJEJOH

t 5P BEE PS TVCUSBDU GSBDUJPOT DIBOHF UIFN UP FRVJWBMFOU GSBDUJPOT XJUI UIF TBNF EFOPNJOBUPS
UIFO BEE PS TVCUSBDU

UIF OVNFSBUPST POMZ

t 5P EJWJEF CZ B GSBDUJPO
JOWFSU UIF GSBDUJPO UVSO JU VQTJEF EPXO
BOE DIBOHF UIF ÷ TJHO UP B × TJHO
t 6OMFTT ZPV BSF TQFDJĕDBMMZ BTLFE GPS B NJYFE OVNCFS
HJWF BOTXFST UP DBMDVMBUJPOT XJUI GSBDUJPOT BT QSPQFS PS

JNQSPQFS GSBDUJPOT JO UIFJS TJNQMFTU GPSN

Tip Exercise 5.2

*G ZPV DBO TJNQMJGZ UIF 1 3FXSJUF FBDI NJYFE OVNCFS BT BO JNQSPQFS GSBDUJPO JO JUT TJNQMFTU GPSN
GSBDUJPO QBSU ĕSTU ZPV XJMM
IBWF TNBMMFS OVNCFST (a) 3 5 (b) 1 12 (c) 11 24 (d) 3 75
UP NVMUJQMZ UP HFU UIF 40 22 30
JNQSPQFS GSBDUJPO 100

Remember: you can cancel to (e) 14 3 (f) 2 35
simplify when you are multiplying
fractions; and the word ‘of’ means ×. 4 45

2 $BMDVMBUF (b) 9 × 7 (c) 3 1 4 (d) 2 1 2 2
(a) 1 4 12
13 2 35
5
(f) 1 × 12 × 2 1 (g) 1 of 360 (h) 3 of 2
(e) 2 4 1 × 1
5 19 2 3 47
23
(j) 2 of 4 1 (k) 1 of 9 16 (l) 3 of 2 1
(i) 8 of 81
32 2 50 43
9

Unit 2: Number 29

5 Fractions

Tip 3 $BMDVMBUF
HJWJOH ZPVS BOTXFS BT B GSBDUJPO JO TJNQMFTU GPSN

:PV DBO VTF BOZ DPNNPO (a) 1 + 3 (b) 9 − 7 (c) 4 + 1 (d) 2 1 3 1
EFOPNJOBUPS CVU JU JT 6 8
FBTJFS UP TJNQMJGZ JG ZPV 10 12 73 23
VTF UIF MPXFTU POF
(e) 2 1 11 (f) 4 3 3 3 (g) 1 1 − 4 (h) 3 9 2 7
REWIND
The order of operations rules 87 10 4 13 5 10 8
(BODMAS) that were covered in
chapter 1 apply here too. (i) 2 5 11 (j) 11 − 7 (k) 2 1 − 17 (l) 1 4 − 13

73 23 33 93

4 $BMDVMBUF

(a) 8 ÷ 1 (b) 12 ÷ 7 (c) 7 ÷12

3 8 8

(d) 2 ÷ 18 (e) 8 ÷ 4 (f) 1 3 2 2

9 30 95 79

(g) 114 ÷ 10 (h) 3 6 5 2 (i) 5 1 1 3

26 13 15 3 5 10

5 4JNQMJGZ UIF GPMMPXJOH

(a) 4 + 2 × 1 (b) 2 1 ⎛ 2 1 − 7 ⎞ (c) 3 × ⎛ 2 + 6 2⎞ 5× 2
⎝ 5 8 ⎠ 7 ⎝ 3 3⎠
33 8 7

(d) 2 7 ⎛ 8 1 − 6 3 ⎞ (e) 5 × 1 + 5 × 1 (f ) ⎛ 5 ÷ 3 − 5⎞ × 1
⎝ 4 8 ⎠ ⎝ 11 12 ⎠ 6
8 6483

(g) ⎛ 5 ÷ 15 ⎞ − ⎛ 5 × 1⎞ (h) ⎛ 2 2 4 − 3⎞ × 3 (i) ⎛ 7 ÷ 2 − 1⎞ × 2
⎝ 8 4⎠ ⎝ 6 5⎠ ⎝ 3 10 ⎠ ⎝ 9 3⎠ 3
17

6 .ST 8FTU IBT EPMMBST JO IFS BDDPVOU 4IF TQFOET 7 PG UIJT

12

(a) )PX NVDI EPFT TIF TQFOE

(b) )PX NVDI EPFT TIF IBWF MFę

7 *U UBLFT B CVJMEFS 3 PG BO IPVS UP MBZ UJMFT

4

(a) )PX NBOZ UJMFT XJMM IF MBZ JO 4 1 IPVST

2

(b) ) PX MPOH XJMM JU UBLF IJN UP DPNQMFUF B ĘPPS OFFEJOH UJMFT

5.3 Percentages

t 1FS DFOU NFBOT QFS IVOESFE " QFSDFOUBHF JT B GSBDUJPO XJUI B EFOPNJOBUPS PG
t 5P XSJUF POF RVBOUJUZ BT B QFSDFOUBHF PG BOPUIFS
FYQSFTT JU BT B GSBDUJPO BOE UIFO DPOWFSU UIF GSBDUJPO UP B QFSDFOUBHF
t 5P ĕOE B QFSDFOUBHF PG B RVBOUJUZ
NVMUJQMZ UIF QFSDFOUBHF CZ UIF RVBOUJUZ
t 5P JODSFBTF PS EFDSFBTF BO BNPVOU CZ B QFSDFOUBHF
ĕOE UIF QFSDFOUBHF BNPVOU BOE BEE PS TVCUSBDU JU GSPN UIF

PSJHJOBM BNPVOU

Exercise 5.3 A

1 &YQSFTT UIF GPMMPXJOH BT QFSDFOUBHFT 3PVOE ZPVS BOTXFST UP POF EFDJNBM QMBDF

(a) 1 (b) 5 (c) 93 (d)
6 8 (h)
312

(e) (f) (g)

30 Unit 2: Number

5 Fractions

2 &YQSFTT UIF GPMMPXJOH QFSDFOUBHFT BT DPNNPO GSBDUJPOT JO UIFJS TJNQMFTU GPSN

(a) (b) (c) (d) (e)

3 &YQSFTT UIF GPMMPXJOH EFDJNBMT BT QFSDFOUBHFT

(a) (b) (c) (d)
(e) (f)

Tip 4 $BMDVMBUF

8IFO ĕOEJOH B QFSDFOUBHF (a) PG LH (b) PG (c) PG MJUSFT (d) PG NM
PG B RVBOUJUZ
ZPVS BOTXFS (e) PG
XJMM IBWF B VOJU BOE OPU B (i) PG (f) PG b (g) PG LN (h) PG HSBNT
QFSDFOUBHF TJHO CFDBVTF
ZPV BSF XPSLJOH PVU BO (j) PG DVCJD NFUSFT
BNPVOU
5 $BMDVMBUF UIF QFSDFOUBHF JODSFBTF PS EFDSFBTF BOE DPQZ BOE DPNQMFUF UIF UBCMF
3PVOE ZPVS BOTXFST UP POF EFDJNBM QMBDF

Original amount New amount Percentage increase or
decrease
(a)
(b)
(c)
(d)
(e)
(f)
(g)

6 *ODSFBTF FBDI BNPVOU CZ UIF HJWFO QFSDFOUBHF

(a) JODSFBTFE CZ (b) JODSFBTFE CZ
(c) JODSFBTFE CZ (d) JODSFBTFE CZ
(e) JODSFBTFE CZ (f) JODSFBTFE CZ

7 %FDSFBTF FBDI BNPVOU CZ UIF HJWFO QFSDFOUBHF

(a) EFDSFBTFE CZ (b) EFDSFBTFE CZ
(c) EFDSFBTFE CZ (d) EFDSFBTFE CZ
(e) EFDSFBTFE CZ (f) EFDSFBTFE CZ

Exercise 5.3 B

1 UJDLFUT XFSF BWBJMBCMF GPS BO JOUFSOBUJPOBM DSJDLFU NBUDI PG UIF UJDLFUT XFSF TPME
XJUIJO B EBZ )PX NBOZ UJDLFUT BSF MFę

2 .ST 3BKBI PXOT PG B DPNQBOZ *G UIF DPNQBOZ JTTVFT TIBSFT
IPX NBOZ TIBSFT
TIPVME TIF HFU

3 " CVJMEJOH
XIJDI DPTU UP CVJME
JODSFBTFE JO WBMVF CZ 3 1 8IBU JT UIF CVJMEJOH

2

XPSUI OPX

4 " QMBZFS TDPSFE PVU PG UIF QPJOUT JO B CBTLFUCBMM NBUDI 8IBU QFSDFOUBHF PG UIF QPJOUT
EJE IF TDPSF

Unit 2: Number 31

5 Fractions

5 " DPNQBOZ IBT B CVEHFU PG GPS QSJOUJOH CSPDIVSFT ćF NBSLFUJOH EFQBSUNFOU IBT
TQFOU PG UIF CVEHFU BMSFBEZ )PX NVDI NPOFZ JT MFę JO UIF CVEHFU

6 +PTI DVSSFOUMZ FBSOT QFS NPOUI *G IF SFDFJWFT BO JODSFBTF PG
XIBU XJMM IJT OFX
NPOUIMZ FBSOJOHT CF

7 " DPNQBOZ BEWFSUJTFT UIBU JUT DPUUBHF DIFFTF JT GBU GSFF *G UIJT JT DPSSFDU
IPX NBOZ
HSBNT PG GBU XPVME UIFSF CF JO B HSBN UVC PG UIF DPUUBHF DIFFTF

8 4BMMZ FBSOT QFS TIJę )FS CPTT TBZT TIF DBO FJUIFS IBWF NPSF QFS TIJę PS B
JODSFBTF 8IJDI JT UIF CFUUFS PČFS

Tip Exercise 5.3 C

'JOEJOH BO PSJHJOBM 1 .JTIB QBJE GPS B %7% TFU BU B PČ TBMF 8IBU XBT UIF PSJHJOBM QSJDF PG UIF %7% TFU
BNPVOU JOWPMWFT SFWFSTF
QFSDFOUBHFT /PUF UIBU 2 *O B MBSHF TDIPPM TUVEFOUT BSF JO (SBEF ćJT JT PG UIF TDIPPM QPQVMBUJPO
UIFSF BSF EJČFSFOU XBZT PG (a) )PX NBOZ TUVEFOUT BSF UIFSF JO UPUBM JO UIF TDIPPM
TBZJOH APSJHJOBM BNPVOU
(b) )PX NBOZ TUVEFOUT BSF JO UIF SFTU PG UIF TDIPPM
TVDI BT PME BNPVOU

QSFWJPVT BNPVOU
BNPVOU 3 4VLJ
UIF XBJUSFTT
IBT IFS XBHFT JODSFBTFE CZ )FS OFX XBHFT BSF 8IBU XBT
CFGPSF UIF JODSFBTF PS IFS XBHF CFGPSF UIF JODSFBTF
EFDSFBTF
BOE TP PO
4 ćJT TVNNFS
BO BNVTFNFOU QBSL JODSFBTFE JUT FOUSZ QSJDFT CZ UP ćJT TVNNFS

UIF OVNCFS PG QFPQMF FOUFSJOH UIF QBSL ESPQQFE CZ GSPN UIF QSFWJPVT TVNNFS UP
(a) 8IBU XBT UIF FOUSZ QSJDF UIF QSFWJPVT TVNNFS
(b) )PX NBOZ WJTJUPST EJE UIF QBSL IBWF UIF QSFWJPVT TVNNFS
(c) *G UIF SVOOJOH DPTUT PG UIF BNVTFNFOU QBSL SFNBJOFE UIF TBNF BT UIF QSFWJPVT TVNNFS
BOE UIFZ NBEF B QSPĕU PO UIF FOUSZ GFFT JO UIJT TVNNFS
IPX NVDI XBT UIFJS QSPĕU
BNPVOU JO EPMMBST

5.4 Standard form

t " OVNCFS JO TUBOEBSE GPSN JT XSJUUFO BT B OVNCFS CFUXFFO BOE NVMUJQMJFE CZ SBJTFE UP B QPXFS F H
a × k
t 4UBOEBSE GPSN JT BMTP DBMMFE TDJFOUJĕD OPUBUJPO
t 5P XSJUF B OVNCFS JO TUBOEBSE GPSN

o QMBDF B EFDJNBM QPJOU BęFS UIF ĕSTU TJHOJĕDBOU EJHJU

o DPVOU UIF OVNCFS PG QMBDF PSEFST UIF ĕSTU TJHOJĕDBOU EJHJU IBT UP NPWF UP HFU GSPN UIJT OFX OVNCFS UP UIF
PSJHJOBM OVNCFS
UIJT HJWFT UIF QPXFS PG

o JG UIF TJHOJĕDBOU EJHJU IBT NPWFE UP UIF MFę OPUF UIJT looks MJLF UIF EFDJNBM QPJOU IBT NPWFE UP UIF SJHIU

UIF
QPXFS PG JT QPTJUJWF
CVU JG UIF TJHOJĕDBOU EJHJU IBT NPWFE UP UIF SJHIU PS EFDJNBM UP UIF MFę

UIF QPXFS PG JT
OFHBUJWF

t 5P XSJUF B OVNCFS JO TUBOEBSE GPSN BT BO PSEJOBSZ OVNCFS
NVMUJQMZ UIF EFDJNBM GSBDUJPO CZ UP UIF HJWFO QPXFS

Tip Exercise 5.4 A

.BLF TVSF ZPV LOPX IPX 1 8SJUF UIF GPMMPXJOH OVNCFST JO TUBOEBSE GPSN
ZPVS DBMDVMBUPS EFBMT XJUI
TUBOEBSE GPSN (a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k) (l)

32 Unit 2: Number

5 Fractions

2 8SJUF UIF GPMMPXJOH BT PSEJOBSZ OVNCFST

If the number part of your (a) × (b) × (c) × (d) × −
standard form answer is a whole (e) × − (f) × − (g) × − (h) ×
number, there is no need to add a (i) × −
decimal point.
Exercise 5.4 B
REWIND
Remember, the first significant 1 $BMDVMBUF
HJWJOH ZPVS BOTXFST JO TUBOEBSE GPSN DPSSFDU UP UISFF TJHOJĕDBOU ĕHVSFT
figure is the first non-zero digit
from the left. (a)
(b) ÷ (c) ÷
(d)
×
(e) ×
(f) 4525 × 8760

(g) 9500 (h) 5 25 108 0.00002
0.00054 (i) 3 9 1 × 10−8

2 4JNQMJGZ FBDI PG UIF GPMMPXJOH (JWF ZPVS BOTXFS JO TUBOEBSE GPSN

(a) ×
× ×
(b) ×
× ×
(c) ×

(d) × −
× × −
(e) ×
× ×
(f) ×
÷ ×

(g) ×
÷ ×
(h) × −
÷ × −
(i) 3 9 1 ×10−8

3 ćF SVO IBT B NBTT PG BQQSPYJNBUFMZ × UPOOFT ćF QMBOFU .FSDVSZ IBT B NBTT PG
BQQSPYJNBUFMZ × UPOOFT

(a) 8IJDI IBT UIF HSFBUFS NBTT
(b) )PX NBOZ UJNFT IFBWJFS JT UIF HSFBUFS NBTT DPNQBSFE XJUI UIF TNBMMFS NBTT

4 -JHIU USBWFMT BU B TQFFE PG × NFUSFT QFS TFDPOE ćF &BSUI JT BO BWFSBHF EJTUBODF PG
× N GSPN UIF 4VO BOE 1MVUP JT BO BWFSBHF × N GSPN UIF 4VO

(a) 8 PSL PVU IPX MPOH JU UBLFT MJHIU GSPN UIF 4VO UP SFBDI &BSUI JO TFDPOET
(JWF ZPVS
BOTXFS JO CPUI PSEJOBSZ OVNCFST BOE TUBOEBSE GPSN

(b) )PX NVDI MPOHFS EPFT JU UBLF GPS UIF MJHIU UP SFBDI 1MVUP (JWF ZPVS BOTXFS JO CPUI
PSEJOBSZ OVNCFST BOE TUBOEBSE GPSN

5.5 Estimation

t &TUJNBUJOH JOWPMWFT SPVOEJOH WBMVFT JO B DBMDVMBUJPO UP OVNCFST UIBU BSF FBTZ UP XPSL XJUI VTVBMMZ XJUIPVU
UIF OFFE GPS B DBMDVMBUPS


t "O FTUJNBUF BMMPXT ZPV UP DIFDL UIBU ZPVS DBMDVMBUJPOT NBLF TFOTF

Exercise 5.5

Remember, the symbol ≈ means 1 6TF XIPMF OVNCFST UP TIPX XIZ UIFTF FTUJNBUFT BSF DPSSFDU
‘is approximately equal to’.
(a) × ≈ (b) × ≈
(c) × ≈ (d) ÷ ≈

2 &TUJNBUF UIF BOTXFST UP FBDI PG UIFTF DBMDVMBUJPOT UP UIF OFBSFTU XIPMF OVNCFS

(a) + o + (b) +
÷ o

(d) 8 92 × 8 98
(c) 9 3 × 7.6
5 9 × 0 95

Unit 2: Number 33

5 Fractions

Mixed exercise 1 &TUJNBUF UIF BOTXFS UP FBDI PG UIFTF DBMDVMBUJPOT UP UIF OFBSFTU XIPMF OVNCFS

(a) × (b) ÷ (c) 36.4 6.32 (d) 64.25 × 3.0982
2 4JNQMJGZ 9.987

(a) 160 (b) 48 (c) 36

200 72 54

3 $BMDVMBUF

(a) 4 × 3 (b) 84 × 3 (c) 5 ÷ 1 (d) 9 − 3 (e) 5 + 7

98 4 93 11 4 24 16

(f) 2 1 9 1 ⎛ 3 ⎞ 2 (h) 9 1 17 (i) 9 1 17 × 2 5 4 18 ⎛ 2⎞ 2
⎝ 4 ⎠ 5 9 5 25 ⎝ 3⎠
32 (g) 4 59 8 (j) ÷ +

4 +PTIVB JT QBJE QFS IPVS )F OPSNBMMZ XPSLT B IPVS XFFL

(a) &TUJNBUF IJT XFFLMZ FBSOJOHT UP UIF OFBSFTU EPMMBS
(b) &TUJNBUF IJT BOOVBM FBSOJOHT

5 ćF WBMVF PG B QMPU PG MBOE JODSFBTFE CZ UP 8IBU XBT JUT QSFWJPVT WBMVF

6 *O BO FMFDUJPO
PG UIF CBMMPU QBQFST XFSF SFKFDUFE BT TQPJMFE WPUFT

(a) )PX NBOZ WPUFT XFSF TQPJMFE
(b) 0 G UIF SFTU PG UIF WPUFT
XFSF GPS $BOEJEBUF " )PX NBOZ WPUFT EJE $BOEJEBUF "

SFDFJWF

7 " CBCZ IBE B NBTT PG LH XIFO TIF XBT CPSO "ęFS XFFLT
IFS NBTT IBE JODSFBTFE UP
LH &YQSFTT UIJT BT B QFSDFOUBHF JODSFBTF
DPSSFDU UP POF EFDJNBM QMBDF

8 1MVUP JT × N GSPN UIF 4VO

(a) &YQSFTT UIJT JO LJMPNFUSFT
HJWJOH ZPVS BOTXFS JO TUBOEBSE GPSN
(b) * O B DFSUBJO QPTJUJPO
UIF &BSUI JT × LN GSPN UIF 4VO *G 1MVUP
UIF &BSUI BOE

UIF 4VO BSF JO B TUSBJHIU MJOF JO UIJT QPTJUJPO BOE CPUI QMBOFUT BSF UIF TBNF TJEF PG UIF
SVO

DBMDVMBUF UIF BQQSPYJNBUF EJTUBODF
JO LN
CFUXFFO UIF &BSUI BOE 1MVUP (JWF ZPVS
BOTXFS JO TUBOEBSE GPSN

9 " MJHIU ZFBS JT UIF EJTUBODF UIBU MJHIU USBWFMT JO POF ZFBS
LN

(a) 8SJUF POF MJHIU ZFBS JO TUBOEBSE GPSN
(b) ć F 4VO JT MJHIU ZFBST GSPN &BSUI &YQSFTT UIJT EJTUBODF JO MJHIU ZFBST JO

TUBOEBSE GPSN
(c) 1 SPYJNB DFOUBVSJ
B TUBS
JT MJHIU ZFBST GSPN &BSUI )PX NBOZ LJMPNFUSFT JT UIJT

(JWF ZPVS BOTXFS JO TUBOEBSE GPSN

34 Unit 2: Number

6 Equations and transforming
formulae

6.1 Further expansions of brackets

t &YQBOE NFBOT UP SFNPWF UIF CSBDLFUT CZ NVMUJQMZJOH PVU
t &BDI UFSN JOTJEF UIF CSBDLFU NVTU CF NVMUJQMJFE CZ UIF UFSN PVUTJEF UIF CSBDLFU
t /FHBUJWF UFSNT JO GSPOU PG UIF CSBDLFUT XJMM BČFDU UIF TJHOT PG UIF FYQBOEFE UFSNT

Remember: Exercise 6.1
+×−=−
−×+=− 1 &YQBOE BOE TJNQMJGZ JG QPTTJCMF
+×+=+
−+−=+ (a) − x + y
(b) − a − b
(c) − − x + y)
(d) x − y
(e) − x x + y
(f) − x −

(g) − a
(h) − x + y
(i) x − x −

(j) − x +
(k) x x − y
(l) − x x − y)

2 &YQBOE BOE TJNQMJGZ BT GBS BT QPTTJCMF

(a) x − y
+ x −
(b) − x y −
− x + xy
(c) − x + y
− x y −

(e) xy − + y
− − x
(f) x − y
− y − x )
(d) − 1 x ( − ) − 2y ( + ) (h) − x − y
+ x x − y

2 (i) − 1 ( − ) + 3 − ( + 7)
2
(g) − 1 x ( − ) + 2 − ( 2 − 3)

4

6.2 Solving linear equations

t 5P TPMWF BO FRVBUJPO
ZPV ĕOE UIF WBMVF PG UIF VOLOPXO MFUUFS WBSJBCMF
UIBU NBLFT UIF FRVBUJPO USVF
t *G ZPV BEE PS TVCUSBDU UIF TBNF OVNCFS PS UFSN
UP CPUI TJEFT PG UIF FRVBUJPO
ZPV QSPEVDF BO FRVJWBMFOU FRVBUJPO

BOE UIF TPMVUJPO SFNBJOT VODIBOHFE

t *G ZPV NVMUJQMZ PS EJWJEF FBDI UFSN PO CPUI TJEFT PG UIF FRVBUJPO CZ UIF TBNF OVNCFS PS UFSN

ZPV QSPEVDF BO
FRVJWBMFOU FRVBUJPO BOE UIF TPMVUJPO SFNBJOT VODIBOHFE

Exercise 6.2

In this exercise leave answers as 1 4PMWF UIFTF FRVBUJPOT GPS x 4IPX UIF TUFQT JO ZPVS XPSLJOH
fractions rather than decimals,
where necessary. (a) x + = x + (b) x + = x + (c) x − = x + (d) x − = x +
(e) x − = x + (f) x − = − x (g) − x = x + (h) + x = x −
(i) x + = x − (j) x − = x − (k) x + = x − (l) x + = x −

Unit 2: Algebra 35

6 Equations and transforming formulae

When an equation has brackets it 2 4PMWF UIFTF FRVBUJPOT GPS x
is usually best to expand them first.
(a) x −
= (b) x +
= (c) x +
= (d) x −
=
To remove the denominators of (e) − x −
= − (f) − x
=
fractions in an equation, multiply (i) x + = x −
(j) x − = x +
(g) x +
= x (h) x +
= x
each term on both sides by the (l) x −
− x −
= x −

common denominator. (k) x +
− x −
=

3 4PMWF UIFTF FRVBUJPOT GPS x

(a) x − 3 6 (b) x + 2 = 11 (c) 4x = 16 (d) 28 − x = 12
2 3 6 6

(e) x − 2 = 5 (f ) 12x − 1 = 9 (g) 5 3 2 = −1 (h) 5 2x = −1
3 5 4

(i) 2x 1 = x (j) 2 5 3 = x − 6 (k) 10x + 2 = 6 − x (l) x − x = 3
5 3 2 5

(m) 2x − x = 7 (n) −2 ( + 4) = x + 7
3 2 2

6.3 Factorising algebraic expressions

t ćF ĕSTU TUFQ JO GBDUPSJTJOH JT UP JEFOUJGZ BOE AUBLF PVU "-- DPNNPO GBDUPST
t $PNNPO GBDUPST DBO CF OVNCFST
WBSJBCMFT
CSBDLFUT PS B DPNCJOBUJPO PG UIFTF
t 'BDUPSJTJOH JT UIF PQQPTJUF PG FYQBOEJOH o XIFO ZPV GBDUPSJTF ZPV QVU CSBDLFUT CBDL JOUP UIF FYQSFTTJPO

Exercise 6.3

Remember, x2 means x × x, so x is 1 'JOE UIF IJHIFTU DPNNPO GBDUPS PG FBDI QBJS
a factor of x2
(a) x BOE (b) BOE x (c) a BOE b
Find the HCF of the numbers first. (f) a b BOE ab
Then find the HCF of the variables, (d) a BOE ab (e) xy BOE yz (i) abc BOE a b
if there is one, in alphabetical (l) x y BOE xy
order. (g) xy BOE xyz (h) pq BOE p q

Remember, if one of the terms is (j) x y z BOE xy z (k) a b BOE ab
exactly the same as the common
factor, you must put a 1 where the 2 'BDUPSJTF BT GVMMZ BT QPTTJCMF
term would appear in the bracket.
(a) x + (b) + y (c) a − (d) x − xy
(e) ab + a (f) x − y (g) xyz − xz (h) ab − bc
(i) xy − yz (j) x − xy

3 'BDUPSJTF UIF GPMMPXJOH

(a) x + x (b) a − a (c) x + x (d) x − x
(e) ab + b (f) xy − x y (g) x − x (h) x y − xy
(i) abc − a b c (j) x − xy (k) ab − b D (l) a b − ab

4 3FNPWF B DPNNPO GBDUPS UP GBDUPSJTF FBDI PG UIF GPMMPXJOH FYQSFTTJPOT

(a) x + y
+ + y
(b) x y −
+ y −

(c) a + b
− a a + b
(d) a a − b
− a − b)
(e) x − y
+ − y
(f) x x −
+ x −

(g) + y
− x y +
(h) a b − c
− c − b)
(i) x x −
− x −
(j) x x − y
− x − y)
(k) x x +
+ y + x
(l) x − y
− x x − y)

36 Unit 2: Algebra

6 Equations and transforming formulae

6.4 Transformation of a formula

t " GPSNVMB JT B HFOFSBM SVMF
VTVBMMZ JOWPMWJOH TFWFSBM WBSJBCMFT
GPS FYBNQMF
UIF BSFB PG B SFDUBOHMF
A = bh.
t " WBSJBCMF JT DBMMFE UIF TVCKFDU PG UIF GPSNVMB XIFO JU JT PO JUT PXO PO POF TJEF PG UIF FRVBMT TJHO
t :PV DBO USBOTGPSN B GPSNVMB UP NBLF BOZ WBSJBCMF UIF TVCKFDU :PV VTF UIF TBNF SVMFT UIBU ZPV VTFE UP TPMWF

FRVBUJPOT

Exercise 6.4 A

1 .BLF m UIF TVCKFDU JG D = km

2 .BLF c UIF TVCKFDU JG y = mx + c

3 (JWFO UIBU P = ab − c
NBLF b UIF TVCKFDU PG UIF GPSNVMB

4 (JWFO UIBU a = bx + c
NBLF b UIF TVCKFDU PG UIF GPSNVMB

Tip 5 .BLF a UIF TVCKFDU PG FBDI GPSNVMB

1BZ BUUFOUJPO UP UIF TJHOT (a) a + b = c (b) a − b = c (c) ab − c = d (d) ab + c = d
XIFO ZPV USBOTGPSN B
GPSNVMB (e) bc − a = d (f) bc − a = −d (g) 2a b = d (h) c ba = e
c d
Tip
(i) abc − d = e (j) cab + d = ef (k) ab + de = f (l) c + ab = e
*G ZPV BSF HJWFO B WBMVF GPS (m) c a − b
= d (n) d a + b
= c c d
ɀ
ZPV NVTU VTF UIF HJWFO
WBMVF UP BWPJE DBMDVMBUPS Exercise 6.4 B
BOE SPVOEJOH FSSPST
1 ćF QFSJNFUFS PG B SFDUBOHMF DBO CF HJWFO BT P = l + b

XIFSF P JT UIF QFSJNFUFS
l JT UIF
Tip MFOHUI BOE b JT UIF CSFBEUI

*O RVFTUJPOT TVDI BT (a) .BLF b UIF TVCKFDU PG UIF GPSNVMB
RVFTUJPO
JU NBZ CF (b) 'JOE b JG UIF SFDUBOHMF IBT B MFOHUI PG DN BOE B QFSJNFUFS PG DN
IFMQGVM UP ESBX B EJBHSBN
UP TIPX XIBU UIF QBSUT PG 2 ćF DJSDVNGFSFODF PG B DJSDMF DBO CF GPVOE VTJOH UIF GPSNVMB C = ɀr
XIFSF r JT UIF SBEJVT PG
UIF GPSNVMB SFQSFTFOU UIF DJSDMF

(a) .BLF r UIF TVCKFDU PG UIF GPSNVMB
(b) 'JOE UIF SBEJVT PG B DJSDMF PG DJSDVNGFSFODF DN 6TF ɀ =
(c) 'JOE UIF EJBNFUFS PG B DJSDMF PG DJSDVNGFSFODF DN 6TF ɀ =

3 ćF BSFB PG B USBQF[JVN DBO CF GPVOE VTJOH UIF GPSNVMB A = h (a b)

2
XIFSF h JT UIF EJTUBODF

CFUXFFO UIF QBSBMMFM TJEFT BOE a BOE b BSF UIF MFOHUIT PG UIF QBSBMMFM TJEFT #Z USBOTGPSNJOH

UIF GPSNVMB BOE TVCTUJUVUJPO
ĕOE UIF MFOHUI PG b
JO B USBQF[JVN PG BSFB DN XJUI

a = DN BOE h = DN

4 "O BJSMJOF VTFT UIF GPSNVMB T P B UP SPVHIMZ FTUJNBUF UIF UPUBM NBTT PG QBTTFOHFST
BOE DIFDLFE CBHT QFS ĘJHIU JO LJMPHSBNT T JT UIF UPUBM NBTT
P JT UIF OVNCFS PG QBTTFOHFST
BOE B JT UIF OVNCFS PG CBHT

(a) 8IBU NBTT EPFT UIF BJSMJOF BTTVNF GPS
(i) B QBTTFOHFS (ii) B DIFDLFE CBH

Unit 2: Algebra 37

6 Equations and transforming formulae

(b) &TUJNBUF UIF UPUBM NBTT GPS QBTTFOHFST FBDI XJUI UXP DIFDLFE CBHT
(c) .BLF B UIF TVCKFDU PG UIF GPSNVMB
(d) $BMDVMBUF UIF UPUBM NBTT PG UIF CBHT JG UIF UPUBM NBTT PG QBTTFOHFST BOE DIFDLFE CBHT

PO B ĘJHIU JT UPOOFT

5 8IFO BO PCKFDU JT ESPQQFE GSPN B IFJHIU
UIF EJTUBODF m
JO NFUSFT UIBU JU IBT GBMMFO DBO CF
SFMBUFE UP UIF UJNF JU UBLFT GPS JU UP GBMM t
JO TFDPOET CZ UIF GPSNVMB m t

(a) .BLF t UIF TVCKFDU PG UIF GPSNVMB
(b) $BMDVMBUF UIF UJNF JU UBLFT GPS BO PCKFDU UP GBMM GSPN B EJTUBODF PG N

Mixed exercise 1 4PMWF GPS x

Remember to inspect your answer (a) x − = − (b) x + = − (c) 2x 4 = 2 (d) 5=1 4x
to see if there are any like terms. 7 5
If there are, add and/or subtract
them to simplify the expression. (e) x − = − x (f ) x − = x +
(g) 3x 7=1 4x (h) 3(2x − 5) = x +1
4 8 5 2

2 .BLF x UIF TVCKFDU PG FBDI GPSNVMB

(a) m = nxp − r (b) m = nx + p
q

3 &YQBOE BOE TJNQMJGZ XIFSF QPTTJCMF

(a) x −
+ (b) − x x −
(c) − x − y +

(d) − y − y
− y (e) x −
+ x +
(f) x x −
+ x −

(g) − x x −
+ x (h) x x +
− x x −


4 'BDUPSJTF GVMMZ

(a) x − (b) x − y (c) − x −
(d) xy − x (e) x y + xy (f) x − y
+ x x − y

(g) x + x
− x +
(h) x x + y
− x x + y)

5 (JWFO UIBU
GPS B SFDUBOHMF
BSFB = MFOHUI × CSFBEUI
XSJUF BO FYQSFTTJPO GPS UIF BSFB PG FBDI
SFDUBOHMF &YQBOE FBDI FYQSFTTJPO GVMMZ

(a) x − 7 (b) 2x (c) (d)
4 x+9 19x

REWIND 6 6TF UIF JOGPSNBUJPO JO FBDI EJBHSBN UP NBLF BO FRVBUJPO BOE TPMWF JU UP ĕOE UIF TJ[F
PG FBDI BOHMF
Use geometric properties to make
the equations (see chapter 3).

12x – 45 2x + 8 x 4x + 15
180 – 3x 4x – 10 6x – 45

38 Unit 2: Algebra

7 Perimeter, area and volume

7.1 Perimeter and area in two dimensions

t 1FSJNFUFS JT UIF UPUBM EJTUBODF BSPVOE UIF PVUTJEF PG B TIBQF :PV DBO ĕOE UIF QFSJNFUFS PG BOZ TIBQF
CZ BEEJOH VQ UIF MFOHUIT PG UIF TJEFT

t ćF QFSJNFUFS PG B DJSDMF JT DBMMFE UIF DJSDVNGFSFODF 6TF UIF GPSNVMB C = πd PS C = 2πr UP ĕOE UIF DJSDVNGFSFODF
PG B DJSDMF

t "SFB JT UIF UPUBM TQBDF DPOUBJOFE XJUIJO B TIBQF 6TF UIFTF GPSNVMBF UP DBMDVMBUF UIF BSFB PG EJČFSFOU TIBQFT
bh
o USJBOHMF A = 2

o TRVBSF A = s2

o SFDUBOHMF A = bh

o QBSBMMFMPHSBN A = bh

o SIPNCVT A = bh

o LJUF A = 1 (produc f diagonals)
2
f p ll l d )h
( 2
o USBQF[JVN A =

o DJSDMF A = πr2

t :PV DBO XPSL PVU UIF BSFB PG DPNQMFY TIBQFT JO B GFX TUFQT %JWJEF DPNQMFY TIBQFT JOUP LOPXO TIBQFT
8PSL PVU UIF BSFB PG FBDI QBSU BOE UIFO BEE UIF BSFBT UPHFUIFS UP ĕOE UIF UPUBM BSFB

Exercise 7.1 A 11.25 cm (c) 19 mm
45 mm
1 'JOE UIF QFSJNFUFS PG FBDI TIBQF
(a) 32 mm (b)
28 mm

(d) 21 mm (e) 1.5 cm (f ) 92 mm
6.8 cm
14 mm 5.3 cm 7.2 cm

3.4 cm 69 mm
4.9 cm

Unit 2: Shape, space and measures 39

7 Perimeter, area and volume

Tip 2 'JOE UIF QFSJNFUFS PG UIF TIBEFE BSFB JO FBDI PG UIFTF TIBQFT 6TF π = JO ZPVS DBMDVMBUJPOT

'PS B TFNJ DJSDMF
UIF (a) (b) (c) (d) 4.5 m
QFSJNFUFS JODMVEFT IBMG
UIF DJSDVNGFSFODF QMVT 5m 7 cm 21 mm
UIF MFOHUI PG UIF EJBNFUFS
*G ZPV BSF OPU HJWFO UIF 4m
WBMVF PG π
VTF UIF π LFZ
PO ZPVS DBMDVMBUPS 3PVOE (e) (f) (g)
ZPVS BOTXFST UP UISFF 3m 8 mm
TJHOJĕDBOU ĕHVSFT
60° 8 cm

Tip 3 " TRVBSF ĕFME IBT B QFSJNFUFS PG N 8IBU JT UIF MFOHUI PG POF PG JUT TJEFT

.BLF TVSF BMM 4 'JOE UIF DPTU PG GFODJOH B SFDUBOHVMBS QMPU N MPOH BOE N XJEF JG UIF DPTU PG GFODJOH JT
NFBTVSFNFOUT BSF JO UIF QFS NFUSF
TBNF VOJUT CFGPSF ZPV EP
BOZ DBMDVMBUJPOT 5 "O JTPTDFMFT USJBOHMF IBT B QFSJNFUFS PG DN $BMDVMBUF UIF MFOHUI PG FBDI PG UIF FRVBM TJEFT
JG UIF SFNBJOJOH TJEF JT NN MPOH
Remember, give your answer in
square units. 6 )PX NVDI TUSJOH XPVME ZPV OFFE UP GPSN B DJSDVMBS MPPQ XJUI B EJBNFUFS PG DN

7 ćF SJN PG B CJDZDMF XIFFM IBT B SBEJVT PG DN
(a) 8IBU JT UIF DJSDVNGFSFODF PG UIF SJN
(b) ćF UZSF UIBU HPFT POUP UIF SJN JT DN UIJDL $BMDVMBUF UIF DJSDVNGFSFODF PG UIF XIFFM
XIFO UIF UZSF JT ĕUUFE UP JU

Exercise 7.1 B

1 'JOE UIF BSFB PG FBDI PG UIFTF TIBQFT

(a) 12.5 cm (b) 1.7 m (c) 21 cm (d) (e)
90 cm 19 cm 15 cm 25 cm
17 cm 14 cm 19 cm 11 cm
20 cm 7 cm

35 cm 17 cm

(f ) 21 cm (g) 112 mm (h) (i) 7 cm cm
11 cm 12 cm 72 mm 10
41 mm 8 cm
5 cm 13 cm 67 mm 15 cm 12 cm
10 cm

40 Unit 2: Shape, space and measures

7 Perimeter, area and volume

2 'JOE UIF BSFB PG FBDI TIBQF 6TF π = JO ZPVS DBMDVMBUJPOT (JWF ZPVS BOTXFST DPSSFDU UP
UXP EFDJNBM QMBDFT

(a) (b) (c) (d) (e)
100 mm 14 mm
140 mm 20° 10 cm 300°
8 cm

Tip 3 'JOE UIF BSFB PG UIF TIBEFE QBSU JO FBDI PG UIFTF ĕHVSFT 4IPX ZPVS XPSLJOH DMFBSMZ JO FBDI
DBTF "MM EJNFOTJPOT BSF HJWFO JO DFOUJNFUSFT
8PSL PVU BOZ NJTTJOH
EJNFOTJPOT PO UIF (a) 6 (b) 2 (c) 22 (d) 12
ĕHVSF VTJOH UIF HJWFO 6 3 6 13 9 5
EJNFOTJPOT BOE UIF
QSPQFSUJFT PG TIBQFT 21 2 18 20 35
6
5 7 20
8

(e) 9 12 (g) 14 (h) 20 (i)
13 14 (f) 1 25 30
30 45
14 36 95
15 63 90 90
2 40

Tip 4 'JOE UIF BSFB PG UIF GPMMPXJOH ĕHVSFT HJWJOH ZPVS BOTXFST DPSSFDU UP UXP EFDJNBM QMBDFT
6TF π = GPS QBSU (f)

%JWJEF JSSFHVMBS TIBQFT
JOUP LOPXO TIBQFT BOE (a) 2.5 cm (b) (c) 2 cm
DPNCJOF UIF BSFBT UP HFU 6 cm 9 cm 4.5 cm
UIF UPUBM BSFB 11 cm

7 cm 4.8 cm

5 cm

(d) (e) 15 cm (f )

4 cm 32.4 cm 40 mm 50 mm
30 mm
6.5 cm 3 cm 40 cm
5.5 cm
6.3 cm 8 cm

20 cm

5 " N × N SFDUBOHVMBS SVH JT QMBDFE PO UIF ĘPPS JO B N × N SFDUBOHVMBS SPPN
)PX NVDI PG UIF ĘPPS JT OPU DPWFSFE CZ UIF SVH

6 ćF BSFB PG B SIPNCVT PG TJEF DN JT NN2 %FUFSNJOF UIF IFJHIU PG UIF SIPNCVT

Unit 2: Shape, space and measures 41

7 Perimeter, area and volume

Exercise 7.1 C

6TF ɀ GPS UIFTF RVFTUJPOT


1 $BMDVMBUF UIF MFOHUI PG UIF BSD AB
TVCUFOEFE CZ UIF HJWFO BOHMF
JO FBDI PG UIFTF DJSDMFT

(a) A (b) A (c)

21 mm O B O
120° 10 cm 12 mm

O 40°
A
B
B

2 ćF EJBHSBN TIPXT B DSPTT TFDUJPO PG UIF &BSUI 5XP DJUJFT
X BOE Y
MJF PO UIF TBNF
MPOHJUVEF (JWFO UIBU UIF SBEJVT PG UIF &BSUI JT LN
DBMDVMBUF UIF EJTUBODF
XY
CFUXFFO
UIF UXP DJUJFT

6371 km
60°

Y

X

3 $BMDVMBUF UIF TIBEFE BSFB PG FBDI DJSDMF

(a) (b) (c)
12 cm 20 cm 18 mm
60° 150°

4 " MBSHF DJSDVMBS QJ[[B IBT B EJBNFUFS PG DN ćF QJ[[B SFTUBVSBOU DVUT JUT QJ[[BT JOUP FJHIU
FRVBM TMJDFT $BMDVMBUF UIF TJ[F PG FBDI TMJDF JO DN2

42 Unit 2: Shape, space and measures