Cambridge Igcse Mathematics Core and Extended Coursebook With CD Rom Cambridge Education Cambri Samples

8162019 Cambridge Igcse Mathematics Core and Extended Coursebook With CD Rom Cambridge Education Cambri Samples

httpslidepdfcomreaderfullcambridge-igcse-mathematics-core-and-extended-coursebook-with-cd-rom-cambridge 155

Endorsed by

University of Cambridge

International Examinations

Cambridge IGCSE

MathematicsCore and Extended

Coursebook



Karen Morrison and Nick Hamshaw

Cambridge IGCSE

Mathematics Core

and ExtendedCoursebook

copy Cambridge University Press 2012



983139 983137 983149 983138 983154 983145 983140 983143 983141 983157 983150 983145 983158 983141 983154 983155 983145 983156 983161 983152 983154 983141 983155 983155Cambridge New York Melbourne Madrid Cape own SingaporeSatildeo Paulo Delhi Dubai

Cambridge University PressTe Edinburgh Building Cambridge CB2 8RU UK

wwwcambridgeorg

Inormation on this title wwwcambridgeorg9781107606272


Tis publication is in copyright Subject to statutory exceptionand to the provisions o relevant collective licensing agreementsno reproduction o any part may take place without the writtenpermission o Cambridge University Press

First published 2012

Printed in the United Kingdom at the University Press Cambridge

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

ISBN-13 978-1-107-60627-2 Paperback with CD-ROM

Cover image Seamus DitmeyerAlamy

Cambridge University Press has no responsibility or the persistence oraccuracy o URLs or external or third-party Internet websites reerred to inthis publication and does not guarantee that any content on such websites isor will remain accurate or appropriate Inormation regarding prices traveltimetables and other actual inormation given in this work are correct atthe time o 1047297rst printing but Cambridge University Press does not guaranteethe accuracy o such inormation thereafer

983150983151983156983145983139983141 983156983151 983156983141983137983139983144983141983154983155Reerences to Activities contained in these resources are provided lsquoas isrsquo andinormation provided is on the understanding that teachers and techniciansshall undertake a thorough and appropriate risk assessment beoreundertaking any o the Activities listed Cambridge University Press makesno warranties representations or claims o any kind concerning the Activitieso the extent permitted by law Cambridge University Press will not be liableor any loss injury claim liability or damage o any kind resulting rom theuse o the Activities

IGCSEreg is the registered trademark o University o Cambridge International Examinations



httpslidepdfcomreaderfullcambridge-igcse-mathematics-core-and-extended-coursebook-with-cd-rom-cambridge 455Contents

ContentsIntroduction

Acknowledgements

Unit 1

Chapter 3 Lines angles and shapes31 Lines and angles 32 Triangles 33 Quadrilaterals 34 Polygons 35 Circles 36 Construction

Chapter 4 Collecting organising anddisplaying data41 Collecting and classifying data 42 Organising data 43 Using charts to display data

Unit 2

Unit 3

Chapter 7 Perimeter area and volume71 Perimeter and area in two-dimensions 72 Three-dimensional objects 73 Surface areas and volumes of solids

Chapter 8 Introduction to probability81 Basic probability 82 Theoretical probability

83 The probability that an event does nothappen 84 Possibility diagrams 85 Combining independent and mutually

exclusive events

113 Understanding similar shapes

114 Understanding congruence

Chapter 12 Averages and measures of spread

121 Different types of average

122 Making comparisons using averages

and ranges

123 Calculating averages and ranges forfrequency data

124 Calculating averages and ranges for grouped

continuous data

125 Percentiles and quartiles

Chapter 1 Reviewing number concepts11 Different types of numbers 212 Multiples and factors 313 Prime numbers 614 Powers and roots 1015 Working with directed numbers 1316 Order of operations 1417 Rounding numbers 18

Chapter 2 Making sense of algebra21 Using letters to represent

unknown values 2322 Substitution 25

23 Simplifying expressions 2724 Working with brackets 3125 Indices 33

Chapter 5 Fractions51 Equivalent fractions 9952 Operations on fractions 10053 Percentages 10554 Standard form 11055 Your calculator and standard form 11456 Estimation 115

Chapter 6 Equations and transforming formulae

61 Further expansions of brackets 11962 Solving linear equations 12163 Factorising algebraic expressions 12364 Transformation of a formula 124

Chapter 9 Sequences and sets91 Sequences 16592 Rational and irrational numbers 17093 Sets 172

Chapter 10 Straight lines and quadratic equations

101 Straight lines 184

102 Quadratic expressions 198Chapter 11 Pythagorasrsquo theorem andsimilar shapes

111 Pythagorasrsquo theorem 207

112 Understanding similar triangles 211

Examination practice structured question for Unit 1-3




Contentsiv

Chapter 13 Understanding measurement

131 Understanding units 251

132 Time 253

133 Upper and lower bounds 257

134 Conversion graphs 262

135 More money 264

Chapter 14 Further solving of equations andinequalities

141 Simultaneous linear equations 268

142 Linear inequalities 275

143 Regions in a plane 279

144 Linear programming 284

145 Completing the square 286

146 Quadratic formula 287

147 Factorising quadratics where the coefficient

of lsquo x 2lsquos is not 1 289

148 Algebraic fractions 291

Chapter 15 Scale drawings bearings andtrigonometry

151 Scale Drawings 3

152 Bearings 3

153 Understanding the tangent cosine

and sine ratios 3

154 Solving problems usingtrigonometry 3

155 Angles between 90deg and 180deg 3

156 The sine and cosine rules 3

157 Area of a triangle 3

158 Trigonometry in three-dimensions 3

Chapter 16 Scatter diagramsand correlation161 Introduction to bivariate data 3

Unit 4

Unit 5

Chapter 17 Managing money

171 Earning money 352

172 Borrowing and investing money 357

173 Buying and selling 363

Chapter 18 Curved graphs

181 Plotting quadratic graphs (the parabola) 371

182 Plotting reciprocal graphs (the hyperbola) 375

183 Using graphs to solve quadratic equations 377

184 Using graphs to solve simultaneous linear

and non-linear equations 379

185 Other non-linear graphs 381

186 Finding the gradient of a curve 390

Chapter 19 Symmetry and loci191 Symmetry in two-dimensions 3192 Symmetry in three-dimensions 4193 Symmetry properties of circles 4194 Angle relationships in circles 4195 Locus 4

Chapter 20 Histograms and frequency distributiondiagrams

201 Histograms 4

202 Cumulative frequency 4

Unit 6

Chapter 21 Ratio rate and proportion

211 Working with ratio 442

212 Ratio and scale 446

213 Rates 450

214 Kinematic graphs 452

215 Proportion 460

216 Direct and inverse proportion in

algebraic terms 463

217 Increasing and decreasing amounts

by a given ratio 467

Chapter 22 More equations formulae andfunctions

221 Setting up equations to solve problems 472

222 Using and transforming formulae 474

223 Functions and function notation 477

Chapter 23 Transformations and matrices

231 Simple plane transformations 4

232 Vectors 5

233 Further transformations 5

234 Matrices and matrix transformation 5

235 Matrices and transformations 5

Chapter 24 Probability using tree diagrams

241 Using tree diagrams to show outcomes 5

242 Calculating probability from tree diagrams 5

Examination practice structured question for Unit 4-6 5

Exercise Answers 5

Glossary 6

Index 6



httpslidepdfcomreaderfullcambridge-igcse-mathematics-core-and-extended-coursebook-with-cd-rom-cambridge 655Introduction

IntroductionTis highly illustrated coursebook covers the complete Cambridge IGCSE Mathematics (0580syllabus Core and Extended material is combined in one book offering a one-stop-shop or

all students and teachers Useul hints are included in the margins or students needing moresupport leaving the narrative clear and to the point Te material required or the Extendedcourse is clearly marked using colour panels and these students are given access to the parts othe Core syllabus they need without having to use an additional book

Te coursebook has been written with a clear progression rom start to 1047297nish with some laterchapters requiring knowledge learnt in earlier chapters Tere are useul signposts throughouthe coursebook that link the content o the chapters allowing the individual to ollow their owcourse through the book where the content in one chapter might require knowledge rom aprevious chapter a comment is included in a lsquoRewindrsquo box and where content will be coveredmore detail later on in the coursebook a comment is included in a lsquoFast orwardrsquo box Exampo both are included here

Worked examples are used throughout to demonstrate each method using typical workings a

thought processes Tese present the methods to the students in a practical and easy-to-ollowway that minimises the need or lengthy explanations

Tere is plenty o practice offered via lsquodrillrsquo exercises throughout each chapter Te exercisesare progressive questions which allow the student to practise methods that have just beenintroduced At the end o each chapter there are lsquoExam-stylersquo questions and lsquoPast paperrsquoquestions Te exam-style questions have been written by the authors in the style o questionson exam papers Te past paper questions are real questions taken rom past exam papers Tend o chapter questions typically re1047298ect the lsquoshortrsquo Paper 1 (Core) and Paper 2 (Extended)questions though you will 1047297nd some more structured ones in there as well Te answers to alo these questions are supplied at the back o the book allowing sel- andor class- assessmenStudents can assess their progress as they go along choosing to do more or less practise asrequired

Te lsquosuggestedrsquo progression through the coursebook is or Units 1-3 to be covered in the1047297rst year o both courses and Units 4-6 to be covered in the second year o both coursesOn this basis there is mixed exam practice at the end o Unit 3 and the end o Unit 6 Tis ishowever only a suggested structure and the course can be taught in various different waysthe signposting throughout the coursebook means that it can be used alongside any order oteaching Te end o Unit questions represent the longer answer lsquostructuredrsquo questions o Pap(Core) and Paper 4 (Extended) exam papers and will use a combination o methods rom acrall relevant chapters As with the end o chapter questions these are a mixture o lsquoExam-stylersquoand lsquoPast paperrsquo questions Te answers to these questions are on the eacherrsquos resource so thathey can be used in classroom tests or or homework i desired

Te coursebook also comes with a glossary to provide a de1047297nition or important tricky term

Helpul guides in the margin o the book include

Hints these are general comments to remind you o important or key inormation that is use

to tackle an exercise or simply useul to know Tey ofen provide extra inormation or suppoin potentially tricky topics

You learned how to plot lines from

equations in chapter 10

REWIND

You will learn much more about

sets in chapter 9 For now just think

of a set as a list of numbers or otheritems that are often placed inside

curly brackets

FAST FORWARD

Remember lsquocoefficientrsquo is the number in the term



httpslidepdfcomreaderfullcambridge-igcse-mathematics-core-and-extended-coursebook-with-cd-rom-cambridge 755Introduction vi

ip these are tips that relate to the exam Tey cover common pitalls based on the authorsrsquo experiences o their students and give you things to be wary o or to remember in order toscore marks in the exam Please note that this advice is not rom the University of CambridgeInternational Examinations Syndicate and they bear no responsibility or any such advice given

Te accompanying student CD-ROM at the back o the coursebook includes

A lsquocoverage gridrsquo to map the contents o the syllabus to the topics and chapters in thebullcoursebook

A lsquoCalculator supportrsquo chapter Tis chapter covers the main uses o calculators thatbullstudents seem to struggle with and includes some worksheets to provide practice at usingyour calculator in these situations

RevisionbullCore revision worksheets (and answers) provide extra exercises or each chapter o thebullbook Tese worksheets contain only content rom the Core syllabus

Extended revision worksheets (and answers) provide extra exercises or each chapterbullo the book Tese worksheets contain the same questions as the Core worksheets inaddition to some more challenging questions and questions to cover content unique

to the Extended syllabus Students are encouraged to do some (i not all) o the lsquoCorersquoquestions on these worksheets as well as the Extended ones (shaded) in order to ullyrevise the course I time is limited you might 1047297nd it easier to pick two or three lsquoCorersquoquestions to do beore moving on to the lsquoExtendedrsquo questions

Quick revision tests ndash these are interactive questions in the orm o multiple choice drbulland drop or hide and reveal Tey are quick-1047297re questions to test yoursel in a differenmedium to pen and paper and to get you thinking on the spot Tey cover the Corecontent with only a ew additional screens being speci1047297c to the Extended course Teris at least one activity or each chapter Students are recommended to use the Revisioworksheets for a more comprehensive revision exercise

Worked solutions ndash these are interactive hide and reveal screens showing workedbullsolutions to some o the end o chapter examination practice questions Some o thesewill be lsquoExam-stylersquo and some will be lsquoPast paperrsquo questions but all will be taken romthe end o the chapter Tere will be at least one or each chapter Te screen includesthe question and the answer but also includes a series o lsquoCluersquo or lsquoiprsquo boxes Te lsquoCluboxes can be clicked on to reveal a clue to help the student i they are struggling withhow to approach the question Te lsquoiprsquo boxes contain tips relating to the exam just likthe lsquoiprsquo boxes in the coursebook

Also in the Cambridge IGCSE Mathematics series are two Practice Books ndash one or Core and onor Extended ndash to offer students targeted practice Tese ollow the chapters and topics o thecoursebook including additional exercises or those who want more practice Tese too includlsquoHintsrsquo and lsquoipsrsquo to help with tricky topics

It is essential that youremember to work outboth unknowns Every

pair o simultaneous linearequations will have a pairo solutions

Tip



httpslidepdfcomreaderfullcambridge-igcse-mathematics-core-and-extended-coursebook-with-cd-rom-cambridge 855Acknowledgements

AcknowledgementsTe authors and publishers acknowledge the following sources of copyright material and are grateful for the permissionsgranted

Past paper examination questions are reproduced by permission o University o Cambridge International Examinations

Cover image Seamus DitmeyerAlamy p 1 copy sanderderwildecom p 13 Dmitry LavruhinShutterstock p 13 HadriannShutterstop 13 Jason CoxShutterstock p 13 Ruslan NabiyevShutterstock p 46 IvangottShutterstock p 47 Claudio Baldini Shutterstock pakiyokoShutterstock p 73 INSAGOShutterstock p 98 copy Te rustees o the British Museum p 119 Wikipedia p 128 FrancescoDazziShutterstock p 164 North Wind Picture ArchiveAlamy p 148 Paolo GiantiShutterstock pp 153 341 487 533 iStockphotTinkstock p 155 Opachevsky IrinaShutterstock p 156 Chad LittlejohnShutterstock p 159 sahua dShutterstock p 206 PhotoscomTinkstock pp 223 419 428 Mike van der Wold p 235 Pics1047297veShutterstock p 250 Vladislav Gur1047297nkelShutterstock p 259Mike an CShutterstock p 261 SuzanShutterstock p 261 R-studioShutterstock p 264 Galyna AndrushkoShutterstock p 29Gustavo Miguel FernandesShutterstock p 351 Stephanie FrayShutterstock p 375 Kristina PostnikovaShutterstock p 397 ConnBrosShutterstock p 441 Philippe WojazerAPPress Association Images



httpslidepdfcomreaderfullcambridge-igcse-mathematics-core-and-extended-coursebook-with-cd-rom-cambridge 955copy Cambridge University Press 2012


httpslidepdfcomreaderfullcambridge-igcse-mathematics-core-and-extended-coursebook-with-cd-rom-cambridge 1055Unit 1 Number

1 Reviewing numberconcepts

In this chapter youwill learn how to

identify and classifybulldifferent types of numbers

find common factors andbullcommon multiples ofnumbers

write numbers as productsbullof their prime factors

calculate squares squarebullroots cubes and cube rootsof numbers

work with integers used inbullreal-life situations

revise the basic rules forbulloperating with numbers

perform basic calculationsbull using mental methods andwith a calculator

Natural numberbullIntegerbullPrime numberbullSymbolbullMultiplebullFactorbullComposite numbers

bull Prime factorbullSquare rootbullCubebullDirected numbersbullBODMASbull

Key words

Our modern number system is called the Hindu-Arabic system because it was developed byHindus and spread by Arab traders who brought it with them when they moved to differentplaces in the world Te Hindu-Arabic system is decimal Tis means it uses place value basedon powers o ten Any number at all including decimals and ractions can be written usingplace value and the digits rom 0 to 9

Tis statue is a replica of a 22 000-year-old bone found in the Congo Te real bone is only 10 cm long and

is carved with groups of notches that represent numbers One column lists the prime numbers from 10 to

It is one of the earliest examples of a number system using tallies




1 Reviewing number concepts

Unit 1 Number2

11 Different types of numbersMake sure you know the correct mathematical words or the types o numbers in the table

Number De1047297nition Example

Natural number Any whole number rom 1 to in1047297nitysometimes called lsquocounting numbersrsquo 0 is notincluded

1 2 3 4 5

Odd number A whole number that cannot be dividedexactly by 2

1 3 5 7

Even number A whole number that can be dividedexactly by 2

2 4 6 8

Integer Any o the negative and positive wholenumbers including zero

minus3 minus2 minus1 0 1 23

Prime number A whole number greater than 1 which hasonly two actors the number itsel and 1

2 3 5 7 11

Square number Te product obtained when an integer is

multiplied by itsel

1 4 9 16

Fraction A number representing parts o a wholenumber can be written as a common (vulgar)raction in the orm o a

b or as a decimal usingthe decimal point

05 02 008 17

Exercise 11 1 Here is a set o numbers minus4 minus1 0 075 3 4 6 11 16 19 25

List the numbers rom this set that are

(a) natural numbers (b) even numbers (c) odd numbers(d) integers (e) negative integers (f) ractions(g) square numbers (h) prime numbers (i) neither square nor prime

2 List

(a) the next our odd numbers afer 107(b) our consecutive even numbers between 2008 and 2030(c) all odd numbers between 993 and 1007(d) the 1047297rst 1047297ve square numbers(e) our decimal ractions that are smaller than 05(f) our vulgar ractions that are greater than but smaller than 4

3 State whether the ollowing will be odd or even

(a) the sum o two odd numbers(b) the sum o two even numbers(c) the sum o an odd and an even number(d) the square o an odd number(e) the square o an even number(f) an odd number multiplied by an even number



of a set as a list of numbers or other

items that are often placed insidecurly brackets

FAST FORWARD

Remember that a sum is the

result of an addition The term isoften used for any calculation inearly mathematics but its meaningis very specific at this level

You should already be familiarwith most of the concepts in thischapter It is included here so that

you can revise the concepts andcheck that you remember them

You will learn about the difference

between rational and irrational

numbers in chapter 9

FAST FORWARD

Find the lsquoproductrsquo means lsquomultiplyrsquoSo the product of 3 and 4 is 12ie 3 times 4 = 12




1 Reviewing number conce

Unit 1 Number

Living maths

4 Tere are many other types o numbers Find out what these numbers are and give anexample o each

(a) Perect numbers(b) Palindromic numbers(c) Narcissistic numbers (In other words numbers that love themselves)

Using symbols to link numbersMathematicians use numbers and symbols to write mathematical inormation in the shortestclearest way possible

You have used the operation symbols + minus times and divide since you started school Now you will alsouse the symbols given in the margin below to write mathematical statements

Exercise 12 1 Rewrite each o these statements using mathematical symbols

(a) 19 is less than 45(b) 12 plus 18 is equal to 30(c) 05 is equal to

(d) 08 is not equal to 80(e) minus34 is less than 2 times minus16(f) thereore the number x equals the square root o 72(g) a number (x ) is less than or equal to negative 45(h) π is approximately equal to 314(i) 51 is greater than 501(j) the sum o 3 and 4 is not equal to the product o 3 and 4(k) the difference between 12 and minus12 is greater than 12(l) the sum o minus12 and minus24 is less than 0(m) the product o 12 and a number (x ) is approximately minus40

2 Say whether these mathematical statements are true or alse

(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000

(c) 1 110= (d) 62 + 43 = 43 + 62(e) 20 times 9 ge 21 times 8 (f) 60 = 6(g) minus12 gt minus4 (h) 199 le 20(i) 1000 gt 199 times 5 (j) 16 4(k) 35 times 5 times 2 ne 350 (l) 20 divide 4 = 5 divide 20(m) 20 minus 4 ne 4 minus 20 (n) 20 times 4 ne 4 times 20

3 Work with a partner

(a) Look at the symbols used on the keys o your calculator Say what each one meansin words

(b) List any symbols that you do not know ry to 1047297nd out what each one means

12 Multiples and factorsYou can think o the multiples o a number as the lsquotimes tablersquo or that number For example tmultiples o 3 are 3 times 1 = 3 3 times 2 = 6 3 times 3 = 9 and so on

MultiplesA multiple o a number is ound when you multiply that number by a positive integer Te 1047297rmultiple o any number is the number itsel (the number multiplied by 1)

= is equal to

ne is not equal to

asymp

is approximately equal tolt is less than

le is less than or equal to

gt is greater than

ge is greater than or equal to

there4 therefore

the square root of

Remember that the differencebetween two numbers is the result

of a subtraction The order of thesubtraction matters





Unit 1 Number4

Worked example 1

(a) What are the first three multiples of 12

(b) Is 300 a multiple of 12

(a) 12 24 36 To find these multiply 12 by 1 2 and then 3

12 times 1 = 12

12 times 2 = 2412 times 3 = 36

(b) Yes 300 is a multiple of 12 To find out divide 300 by 12 If it goes exactly then 300 is a multiple of 12

300 divide 12 = 25

Exercise 13 1 List the 1047297rst 1047297ve multiples o

(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100

2 Use a calculator to 1047297nd and list the 1047297rst ten multiples o

(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List

(a) the multiples o 4 between 29 and 53(b) the multiples o 50 less than 400(c) the multiples o 100 between 4000 and 5000

4 Here are 1047297ve numbers 576 396 354 792 1164 Which o these are multiples o 12

5 Which o the ollowing numbers are not multiples o 27

(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116

The lowest common multiple (LCM)

Te lowest common multiple o two or more numbers is the smallest number that is a multipleo all the given numbers

Worked example 2

Find the lowest common multiple of 4 and 7

M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28

List several multiples of 4 (Note M4 means multiples of 4)

List several multiples of 7

Find the lowest number that appears in both sets This is the LCM

Exercise 14 1 Find the LCM o

Later in this chapter you will see

how prime factors can be used to

find LCMs

FAST FORWARD (a) 2 and 5 (b) 8 and 10 (c) 6 and 4(d) 3 and 9 (e) 35 and 55 (f) 6 and 11(g) 2 4 and 8 (h) 4 5 and 6 (i) 6 8 and 9(j) 1 3 and 7 (k) 4 5 and 8 (l) 3 4 and 18





Unit 1 Number

2 Is it possible to 1047297nd the highest common multiple o two or more numbersGive a reason or your answer

FactorsA factor is a number that divides exactly into another number with no remainder For examp2 is a actor o 16 because it goes into 16 exactly 8 times 1 is a actor o every number Telargest actor o any number is the number itsel

To list the factors in numerical ordergo down the left side and then upthe right side of the factor pairsRemember not to repeat factors

Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12

= 1 2 3 4 6 12 Find pairs of numbers that multiply to give 12

1 times 12

2 times 6

3 times 4

Write the factors in numerical order

(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12

means the factors of 12

Exercise 15 1 List all the actors o

(a) 4 (b) 5 (c) 8 (d) 11 (e) 18(f) 12 (g) 35 (h) 40 (i) 57 (j) 90(k) 100 (l) 132 (m) 160 (n) 153 (o) 360

2 Which number in each set is not a actor o the given number

(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14

Later in this chapter you will learn

more about divisibility tests and

how to use these to decide whetheror not one number is a factor of

another

FAST FORWARD3 State true or alse in each case

(a) 3 is a actor o 313 (b) 9 is a actor o 99

(c) 3 is a actor o 300 (d) 2 is a actor o 300(e) 2 is a actor o 122 488 (f) 12 is a actor o 60(g) 210 is a actor o 210 (h) 8 is a actor o 420

4 What is the smallest actor and the largest actor o any number





Unit 1 Number6

The highest common factor (HCF)

Te highest common actor o two or more numbers is the highest number that is a actor o althe given numbers

Worked example 4

Find the HCF of 8 and 24

F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8

List the factors of each number

Underline factors that appear in both sets

Pick out the highest underlined factor (HCF)

Exercise 16 1 Find the HCF o each pair o numbers

(a) 3 and 6 (b) 24 and 16 (c) 15 and 40 (d) 42 and 70(e) 32 and 36 (f) 26 and 36 (g) 22 and 44 (h) 42 and 48

2 Find the HCF o each group o numbers

(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121

3 Not including the actor provided 1047297nd two numbers that have

(a) an HCF o 2 (b) an HCF o 6

4 What is the HCF o two different prime numbers Give a reason or your answer

Living maths

5 Simeon has two lengths o rope One piece is 72 metres long and the other is 90 metres longHe wants to cut both lengths o rope into the longest pieces o equal length possible Howlong should the pieces be

6 Ms Sanchez has 40 canvases and 100 tubes o paint to give to the students in her art group

What is the largest number o students she can have i she gives each student an equalnumber o canvasses and an equal number o tubes o paint

7 Indira has 300 blue beads 750 red beads and 900 silver beads She threads these beads tomake wire bracelets Each bracelet must have the same number and colour o beads Whatis the maximum number o bracelets she can make with these beads

13 Prime numbersPrime numbers have exactly two actors one and the number itsel

Composite numbers have more than two actors

Te number 1 has only one actor so it is not prime and it is not composite

Finding prime numbersOver 2000 years ago a Greek mathematician called Eratosthenes made a simple tool or sortingout prime numbers Tis tool is called the lsquoSieve o Eratosthenesrsquo and the 1047297gure on page 7 showhow it works or prime numbers up to 100

You will learn how to find HCFs

by using prime factors later in the

chapter

FAST FORWARD

Word problems involving HCFusually involve splitting things intosmaller pieces or arranging thingsin equal groups or rows





Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100

Cross out 1 it is not prime

Circle 2 then cross out other

multiples of 2


multiples of 3

Circle the next available num

then cross out all its multiple

Repeat until all the numbers i

the table are either circled or

crossed out

The circled numbers are the

primes

You should try to memorisewhich numbers between 1 and100 are prime

Other mathematicians over the years have developed ways o 1047297nding larger and larger primenumbers Until 1955 the largest known prime number had less than 1000 digits Since the1970s and the invention o more and more powerul computers more and more prime numbhave been ound Te graph below shows the number o digits in the largest known primessince 1955

1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year

Number of digits in largest known prime number

against year found

Number

of digits

oday anyone can join the Great Internet Mersenne Prime Search Tis project links thousano home computers to search continuously or larger and larger prime numbers while thecomputer processors have spare capacity

Exercise 17 1 Which is the only even prime number

2 How many odd prime numbers are there less than 50

3 (a) List the composite numbers greater than our but less than 30

(b) ry to write each composite number on your list as the sum o two prime numbersFor example 6 = 3 + 3 and 8 = 3 + 5

4 win primes are pairs o prime numbers that differ by two List the twin prime pairs up to

A good knowledge of primes can

help when factorising quadratics in

chapter 10

FAST FORWARD





Unit 1 Number8

5 Is 149 a prime number Explain how you decided

6 Super-prime numbers are prime numbers that stay prime each time you remove a digit(starting with the units) So 59 is a super-prime because when you remove 9 you are lef with which is also prime 239 is also a super-prime because when you remove 9 you are lef with 23which is prime and when you remove 3 you are lef with 2 which is prime

(a) Find two three-digit super-prime numbers less than 400(b) Can you 1047297nd a our-digit super-prime number less than 3000(c) Sondrarsquos telephone number is the prime number 987-6413 Is her phone number a

super-prime

Prime factorsPrime factors are the actors o a number that are also prime numbers

Every composite whole number can be broken down and written as the product o its prime actorsYou can do this using tree diagrams or using division Both methods are shown in worked example

Prime numbers only have twofactors 1 and the number itselfAs 1 is not a prime number donot include it when expressinga number as a product of primefactors

Choose the method that worksbest for you and stick to it Alwaysshow your method when usingprime factors

Worked example 5

Write the following numbers as the product of prime factors

(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4

36 = 2 times 2 times 3 times 3

48

124

32 2

2 2

4

48 = 2 times 2 times 2 times 2 times 3

Write the number as two

factors

If a factor is a prime

number circle it

If a factor is a composite

number split it into two

factors

Keep splitting until you endup with two primes

Write the primes in

ascending order with times

signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3


Divide by the smallest

prime number that will go

into the number exactly

Continue dividing using

the smallest prime number

that will go into your newanswer each time

Stop when you reach 1

Write the prime factors in


signs

Whilst super-primenumbers are interestingthey are not on thesyllabus

Tip

Remember a product is the answerto a multiplication So if you write anumber as the product of its primefactors you are writing it usingmultiplication signs like this12 = 2 times 2 times 3





Unit 1 Number

Exercise 18 1 Express the ollowing numbers as the product o prime actors

(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240

Using prime factors to find the HCF and LCM

When you are working with larger numbers you can determine the HCF or LCM by expressineach number as a product o its prime actors

Worked example 6




2 times 2 times 3 = 12

HCF = 12

First express each number as a product of prime

factors Use tree diagrams or division to do this

Underline the factors common to both numbers

Multiply these out to find the HCF

Worked example 7

Find the LCM of 72 and 120



2 times 2 times 2 times 3 times 3 times 5 = 360

LCM = 360



Underline the largest set of multiples of each factor

List these and multiply them out to find the LCM

Exercise 19 1 Find the HCF o these numbers by means o prime actors


2 Use prime actorisation to determine the LCM o


3 Determine both the HCF and LCM o the ollowing numbers

(a) 72 and 108 (b) 25 and 200 (c) 95 and 120 (d) 84 and 60

Word problems involving LCMusually include repeating eventsYou may be asked how manyitems you need to lsquohave enoughrsquoor when something will happenagain at the same time

Living maths4 A radio station runs a phone-in competition or listeners Every 30th caller gets a ree airt

voucher and every 120th caller gets a ree mobile phone How many listeners must phonebeore one receives both an airtime voucher and a ree phone

5 Lee runs round a track in 12 minutes James runs round the same track in 18 minutes I tstart in the same place at the same time how many minutes will pass beore they both crothe start line together again

When you write your number asa product of primes group alloccurrences of the same primenumber together

You can also use prime factors to

find the square and cube roots

of numbers if you donrsquot have a

calculator You will deal with this in

more detail on page 11

FAST FORWARD





Unit 1 Number10

Divisibility tests to find factors easilySometimes you want to know i a smaller number will divide into a larger one with noremainder In other words is the larger number divisible by the smaller one

Tese simple divisibility tests are useul or working this out

A number is exactly divisible by

2 i it ends with 0 2 4 6 or 8 (in other words is even)

3 i the sum o its digits is a multiple o 3 (can be divided by 3)

4 i the last two digits can be divided by 4

5 i it ends with 0 or 5

6 i it is divisible by both 2 and 3

8 i the last three digits are divisible by 8

9 i the sum o the digits is a multiple o 9 (can be divided by 9)

10 i the number ends in 0

Tere is no simple test or divisibility by 7 although multiples o 7 do have some interestingproperties that you can investigate on the internet

Exercise 110 23 65 92 10 104 70 500 21 64 798 1223

1 Look at the box o numbers above Which o these numbers are

(a) divisible by 5 (b) divisible by 8 (c) divisible by 3

2 Say whether the ollowing are true or alse

(a) 625 is divisible by 5 (b) 88 is divisible by 3(c) 640 is divisible by 6 (d) 346 is divisible by 4(e) 476 is divisible by 8 (f) 2340 is divisible by 9(g) 2890 is divisible by 6 (h) 4562 is divisible by 3(i) 40 090 is divisible by 5 (j) 123 456 is divisible by 9

3 Can $3407 be divided equally among

(a) two people (b) three people (c) nine people4 A stadium has 202 008 seats Can these be divided equally into

(a) 1047297ve blocks (b) six blocks (c) nine blocks

5 (a) I a number is divisible by 12 what other numbers must it be divisible by(b) I a number is divisible by 36 what other numbers must it be divisible by(c) How could you test i a number is divisible by 12 15 or 24

14 Powers and roots

Square numbers and square rootsA number is squared when it is multiplied by itsel For example the square o 5 is 5 times 5 = 25 Tsymbol or squared is 2 So 5 times 5 can also be written as 52

Te square root o a number is the number that was multiplied by itsel to get the squarenumber Te symbol or square root is You know that 25 = 52 so 25 = 5

Cube numbers and cube rootsA number is cubed when it is multiplied by itsel and then multiplied by itsel again For examplthe cube o 2 is 2 times 2 times 2 = 8 Te symbol or cubed is 3 So 2 times 2 times 2 can also be written as 23

Divisibility tests are notpart o the syllabus Tey

are just useul to knowwhen you work withactors and prime numbers

Tip

In section 11 you learned that the

product obtained when an integeris multiplied by itself is a square

number

REWIND





Unit 1 Number

Te cube root o a number is the number that was multiplied by itsel to get the cube number

Te symbol or cube root is You know that 8 = 23 so = 2

2

2

a) Square numbers can be arranged to form a

square shape Tis is 22

2

2

2

b) Cube numbers can be arranged to form a sol

cube shape Tis is 23

Finding powers and roots

You can use your calculator to square or cube numbers quickly using the x 2 and x 3 key

or the x◻ key Use the or 3 keys to 1047297nd the roots I you donrsquot have a calculator yocan use the product o prime actors method to 1047297nd square and cube roots o numbers Both

methods are shown in the worked examples below

Worked example 8

Use your calculator to find

(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =

(b) 53 = 125 Enter 5 x 3 = If you do not have a x 3 button then enter

5 x◻ 3 = for this key you have to enter the power

(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9

If you do not have a calculator you can write the integer as a product of primes and group the prime factors into pairs or

threes Look again at parts (c) and (d) of worked example 8

(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18

Group the factors into pairs and write down the square root of each pair

Multiply the roots together to give you the square root of 324

(d)512 2 2

22 2

22 2

2


512 83

Group the factors into threes and write the cube root of each threesome

Multiply together to get the cube root of 512

Not all calculators have exactly the

same buttons x ◻ x y andand all mean the same thing on

different calculators





Unit 1 Number12

Exercise 111 1 Calculate

(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003

Learn the squares of all integersbetween 1 and 20 inclusiveYou will need to recognisethese quickly

3 Find a value o x to make each o these statements true

(a) x times x = 25 (b) x times x times x = 8 (c) x times x = 121(d) x times x times x = 729 (e) x times x = 324 (f) x times x = 400(g) x times x times x = 8000 (h) x times x = 225 (i) x times x times x = 1

(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =

4 Use a calculator to 1047297nd the ollowing roots

(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832

5 Use the product o prime actors given below to 1047297nd the square root o each numberShow your working

(a) 324 = 2 times 2 times 3 times 3 times 3 times 3 (b) 225 = 3 times 3 times 5 times 5(c) 784 = 2 times 2 times 2 times 2 times 7 times 7 (d) 2025 = 3 times 3 times 3 times 3 times 5 times 5(e) 19 600 = 2 times 2 times 2 times 2 times 5 times 5 times 7 times 7 (f) 250 000 = 2 times 2times 2 times 2times 5 times 5times 5 times 5times 5 times

6 Use the product o prime actors to 1047297nd the cube root o each number Show your working

(a) 27 = 3 times 3 times 3 (b) 729 = 3 times 3 times 3 times 3 times 3 times 3(c) 2197 = 13 times 13 times 13 (d) 1000 = 2 times 2 times 2 times 5 times 5 times 5

(e) 15 625=

5times

5times

5times

5times

5times

5(f) 32 768 = 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2 times 2

7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64

(i) 1 6minus (j) 1 36minus (k) 4times (l) 5 4times

(m) 4 (n) 4 (o) 36

4 (p)

36

8 Find the length o the edge o a cube with a volume o

(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3

9 I the symbol means lsquoadd the square o the 1047297rst number to the cube o the secondnumberrsquo calculate

(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5

Brackets act as grouping symbolsWork out any calculations insidebrackets before doing thecalculations outside the brackets

Root signs work in the same way

as a bracket If you have 25 9+ you must add 25 and 9 beforefinding the root





Unit 1 Number

15 Working with directed numbers

A negative sign is used to indicate that values are less than zero For example on a thermometer on a ban

statement or in an elevator

When you use numbers to represent real-lie situations like temperatures altitude depth belosea level pro1047297t or loss and directions (on a grid) you sometimes need to use the negative sign

indicate the direction o the number For example a temperature o three degrees belowzero can be shown as minus3 degC Numbers like these which have direction are called directed

numbers So i a point 25 m above sea level is at +25 m then a point 25 m below sea level isat minus25 m

Exercise 112 1 Express each o these situations using a directed number

(a) a pro1047297t o $100 (b) 25 km below sea level(c) a drop o 10 marks (d) a gain o 2 kg(e) a loss o 15 kg (f) 8000 m above sea level(g) a temperature o 10 degC below zero (h) a all o 24 m(i) a debt o $2000 (j) an increase o $250(k) a time two hours behind GM (l) a height o 400 m(m) a bank balance o $45000

Comparing and ordering directed numbersIn mathematics directed numbers are also known as integers You can represent the set ointegers on a number line like this

ndash5 ndash9 ndash7 ndash8 ndash10 ndash3 ndash2 ndash1 0 1 2 3 4 5 6 7 8 9 10 ndash4 ndash6

Te further to the right a number is on the number line the greater its value

Exercise 113 1 Copy the numbers and 1047297ll in lt or gt to make a true statement

(a) (b) 4 (c) 1

(d) 6 4 (e) minus 4 (f) minus2 4(g) minus 1minus 1 (h) minus minus1 (i) minus

(j) minus (k) minus minus1 4 (l) minus minus

(m) (n) minus 11 (o) 1 minus

2 Arrange each set o numbers in ascending order

(a) minus8 7 10 minus1 minus12 (b) 4 minus3 minus4 minus10 9 minus8(c) minus11 minus5 minus7 7 0 minus12 (d) minus94 minus50 minus83 minus90 0

Once a direction is chosen to bepositive the opposite direction istaken to be negative So

bull if up is positive down is negative

bull if right is positive left is negative

bull if north is positive south isnegative

bull if above 0 is positive below 0 isnegative

You will use similar number lines

when solving linear inequalities in

chapter 14

FAST FORWARD

It is important that you understandhow to work with directed numbersearly in your IGCSE course Manytopics depend upon them





Unit 1 Number14

Living maths

3 Study the temperature graph careully

ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21

M T W T F S M T W T F S Sunday

28Day of the week

Temperature (degC)

(a) What was the temperature on Sunday 14 January

(b) By how much did the temperature drop rom Sunday 14 to Monday 15(c) What was the lowest temperature recorded(d) What is the difference between the highest and lowest temperatures(e) On Monday 29 January the temperature changed by minus12 degrees What was the

temperature on that day

4 Matt has a bank balance o $4550 He deposits $1500 and then withdraws $3200 What ishis new balance

5 Mr Singhrsquos bank account is $420 overdrawn

(a) Express this as a directed number(b) How much money will he need to deposit to get his account to have a balance o $500(c) He deposits $200 What will his new balance be

6 A diver 27 m below the surace o the water rises 16 m At what depth is she then

7 On a cold day in New York the temperature at 6 am was minus5 degC By noon the temperaturehad risen to 8 degC By 7 pm the temperature had dropped by 11 degC rom its value at noonWhat was the temperature at 7 pm

8 Local time in Abu Dhabi is our hours ahead o Greenwich Mean ime Local time inRio de Janeiro is three hours behind Greenwich Mean ime

(a) I it is 4 pm at Greenwich what time is it in Abu Dhabi(b) I it is 3 am in Greenwich what time is it in Rio de Janiero(c) I it is 3 pm in Rio de Janeiro what time is it in Abu Dhabi(d) I it is 8 am in Abu Dhabi what time is it in Rio de Janeiro

16 Order of operationsAt this level o mathematics you are expected to do more complicated calculations involvingmore than one operation (+ minus times and divide) When you are carrying out more complicatedcalculations you have to ollow a sequence o rules so that there is no conusion about whatoperations you should do 1047297rst Te rules governing the order o operations are

complete operations in grouping symbols 1047297rst (see page 15)bulldo division and multiplication next working rom lef to rightbulldo addition and subtractions last working rom lef to rightbull

The difference between the highestand lowest temperature is alsocalled the range of temperatures





Unit 1 Number

Many people use the letters BODMAS to remember the order o operations Te letters stand o

Brackets

Of

D

ivide M

ultiply

dd S

ubtract

(Sometimes lsquoIrsquo or lsquoindicesrsquo is used instead o lsquoOrsquo or lsquoo rsquo)

BODMAS indicates that powers are considered afer brackets but beore all other operations

Grouping symbolsTe most common grouping symbols in mathematics are brackets Here are some examples othe different kinds o brackets used in mathematics

(4 + 9) times (10 divide 2)

[2(4 + 9) minus 4(3) minus 12]

2 minus [4(2 minus 7) minus 4(3 + 8)] minus 2 times 8

When you have more than one set o brackets in a calculation you work out the innermostset 1047297rst

Other symbols used to group operations are

raction bars egbull 5 12

root signs such as square roots and cube roots egbull 9 16

powers eg 5bull 2 or 43

Worked example 10Simplify

(a) 7 times (3 + 4) (b) (10 minus 4) times (4 + 9) (c) 45 minus [20 times (4 minus 3)]

(a) 7 times 7 = 49 (b) 6 times 13 = 78 (c) 45 minus [20 times 1] = 45 minus 20

= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus

(c)36 100 36divide + minus

(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16

Exercise 114 1 Calculate Show the steps in your working

(a) (4 + 7) times 3 (b) (20 minus 4) divide 4 (c) 50 divide (20 + 5) (d) 6 times (2 + 9)(e) (4 + 7) times 4 (f) (100 minus 40) times 3 (g) 16 + (25 divide 5) (h) 19 minus (12 + 2(i) 40 divide (12 minus 4) (j) 100 divide (4 + 16) (k) 121 divide (33 divide 3) (l) 15 times (15 minus 1

2 Calculate

(a) (4 + 8) times (16 minus 7) (b) (12 minus 4) times (6 + 3) (c) (9 + 4) minus (4 + 6)

(d) (33 + 17) divide (10 minus 5) (e) (4 times 2) + (8 times 3) (f) (9 times 7) divide (27 minus 20)(g) (105 minus 85) divide (16 divide 4) (h) (12 + 13) divide 52 (i) (56 minus 62) times (4 + 3)

3 Simpliy Remember to work rom the innermost grouping symbols to the outermost

(a) 4 + [12 minus (8 minus 5)] (b) 6 + [2 minus (2 times 0)](c) 8 + [60 minus (2 + 8)] (d) 200 minus [(4 + 12) minus (6 + 2)](e) 200100 minus [4 times (2 + 8)] (f) 6 + [5 times (2 + 30)] times 10(g) [(30 + 12) minus (7 + 9)] times 10 (h) 6 times [(20 divide 4) minus (6 minus 3) + 2](i) 1000 minus [6 times (4 + 20) minus 4 times (3 + 0)]

4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus

5 Insert brackets into the ollowing calculations to make them true

(a) 3 times 4 + 6 = 30 (b) 25 minus 15 times 9 = 90 (c) 40 minus 10 times 3 = 90(d) 14 minus 9 times 2 = 10 (e) 12 + 3 divide 5 = 3 (f) 19 minus 9 times 15 = 150(g) 10 + 10 divide 6 minus 2 = 5 (h) 3 + 8 times 15 minus 9 = 66 (i) 9 minus 4 times 7 + 2 = 45(j) 10 minus 4 times 5 = 30 (k) 6 divide 3 + 3 times 5 = 5 (l) 15 minus 6 divide 2 = 12(m) 1 + 4 times 20 divide 5 = 20 (n) 8 + 5 minus 3 times 2 = 20 (o) 36 divide 3 times 3 minus 3 = 6(p) 3 times 4 minus 2 divide 6 = 1 (q) 40 divide 4 + 1 = 11 (r) 6 + 2 times 8 + 2 = 24

Working in the correct orderNow that you know what to do with grouping symbols you are going to apply the rules or ordo operations to perorm calculations with numbers

Exercise 115 1 Simpliy Show the steps in your working

(a) 5 times 10 + 3 (b) 5 times (10 + 3) (c) 2 + 10 times 3(d) (2 + 10) times 3 (e) 23 + 7 times 2 (f) 6 times 2 divide (3 + 3)

(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus

(j) 17 + 3 times 21 (k) 48 minus (2 + 3) times 2 (l) 12 times 4 minus 4 times 8(m) 15 + 30 divide 3 + 6 (n) 20 minus 6 divide 3 + 3 (o) 10 minus 4 times 2 divide 2

2 Simpliy

(a) 18 minus 4 times 2 minus 3 (b) 14 minus (21 divide 3) (c) 24 divide 8 times (6 minus 5)(d) 42 divide 6 minus 3 minus 4 (e) 5 + 36 divide 6 minus 8 (f) (8 + 3) times (30 divide 3) divide 11

3 State whether the ollowing are true or alse

(a) (1 + 4) times 20 + 5 = 1 + (4 times 20) + 5 (b) 6 times (4 + 2) times 3 gt (6 times 4) divide 2 times 3(c) 8 + (5 minus 3) times 2 lt 8 + 5 minus (3 times 2) (d) 100 + 10 divide 10 gt (100 + 10) divide 10

A bracket lsquotypersquo is always twinnedwith another bracket of thesame typeshape This helpsmathematicians to understandthe order of calculations evenmore easily

You will apply the order of operationrules to fractions decimals and

algebraic expressions as you

progress through the course

FAST FORWARD





Unit 1 Number

4 Place the given numbers in the correct spaces to make a correct number sentence

(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus

Using your calculatorA calculator with algebraic logic will apply the rules or order o operations automatically Soyou enter 2 + 3 times 4 your calculator will do the multiplication 1047297rst and give you an answer o (Check that your calculator does this)

When the calculation contains brackets you must enter these to make sure your calculator dothe grouped sections 1047297rst

Experiment with your calculator bymaking several calculations withand without brackets For example3 times 2 + 6 and 3 times (2 + 6) Do youunderstand why these are different

Your calculator might only have one

type of bracket ( and )

If there are two different shapedbrackets in the calculation (such as[4 times (2 ndash 3)] enter the calculatorbracket symbol for each type

Worked example 12

Use a calculator to find

(a) 3 + 2 times 9 (b) (3 + 8) times 4 (c) (3 times 8 minus 4) minus (2 times 5 + 1)

(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =

(c) 9 Enter ( 3 times 8 minus 4 ) minus ( 2 times 5 + 1 ) =

Exercise 116 1 Use a calculator to 1047297nd the correct answer

(a) 10 minus 4 times 5 (b) 12 + 6 divide 7 minus 4(c) 3 + 4 times 5 minus 10 (d) 18 divide 3 times 5 minus 3 + 2

(e) 5 minus 3 times 8 minus 6 divide 2 (f) 7 + 3 divide 4 + 1(g) (1 + 4) times 20 divide 5 (h) 36 divide 6 times (3 minus 3)(i) (8 + 8) minus 6 times 2 (j) 100 minus 30 times (4 minus 3)(k) 24 divide (7 + 5) times 6 (l) [(60 minus 40) minus (53 minus 43)] times 2(m) [(12 + 6) divide 9] times 4 (n) [100 divide (4 + 16)] times 3(o) 4 times [25 divide (12 minus 7)]

2 Use your calculator to check whether the ollowing answers are correctI the answer is incorrect work out the correct answer

(a) 12 times 4 + 76 = 124 (b) 8 + 75 times 8 = 698(c) 12 times 18 minus 4 times 23 = 124 (d) (16 divide 4) times (7 + 3 times 4) = 76(e) (82 minus 36) times (2 + 6) = 16 (f) (3 times 7 minus 4) minus (4 + 6 divide 2) = 12

3 Each represents a missing operation Work out what it is

(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus

In this section you will use yourcalculator to perform operationsin the correct order However youwill need to remember the orderof operations rules and apply themthroughout the book as you domore complicated examples usingyour calculator

Some calculators have two lsquominusrsquo

buttonsminus

and ( minus

) Thefirst means lsquosubtractrsquo and is used tosubtract one number from anotherThe second means lsquomake negativersquoExperiment with the buttons andmake sure that your calculator isdoing what you expect it to do

The more effectively you are able touse your calculator the faster andmore accurate your calculations arelikely to be If you have difficultywith this you will find advice andpractice exercises on the CD-ROM





Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4

5 Use a calculator to 1047297nd the answer

(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4

6 Use your calculator to evaluate

(a) 64 125times (b) 62times

(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1

17 Rounding numbersIn many calculations particularly with decimals you will not need to 1047297nd an exact answerInstead you will be asked to give an answer to a stated level o accuracy For exampleyou may be asked to give an answer correct to 2 decimal places or an answer correct to 3signi1047297cant 1047297gures

o round a number to a given decimal place you look at the value o the digit to the right o thespeci1047297ed place I it is 5 or greater you round up i it less than 5 you round down

Worked example 13

Round 64839906 to

(a) the nearest whole number (b) 1 decimal place (c) 3 decimal places

(a) 64839906 4 is in the units place

64839906 The next digit is 8 so you will round up to get 5

= 65 (to nearest whole number) To the nearest whole number

(b) 64839906 8 is in the first decimal place

64839906 The next digit is 3 so the 8 will remain unchanged

= 648 (1 dp) Correct to 1 decimal place

(c) 64839906 9 is in the third decimal place

64839906 The next digit is 9 so you need to round upWhen you round 9 up you get 10 so carry one to the previous digit and write 0 in

the place of the 9

= 64840 (3 dp) Correct to 3 decimal places

In this chapter you are only dealing

with square and cube numbers

and the roots of square and cube

numbers When you work with

indices and standard form in

chapter 5 you will need to apply

these skills and use your calculator

effectively to solve problems

involving any powers or roots

FAST FORWARD





Unit 1 Number

o round to 3 signi1047297cant 1047297gures 1047297nd the third signi1047297cant digit and look at the value o thedigit to the right o it I it is 5 or greater add one to the third signi1047297cant digit and lose all o thother digits to the right I it is less than 5 leave the third signi1047297cant digit unchanged and loseall the other digits to the right as beore o round to a different number o signi1047297cant 1047297guresuse the same method but 1047297nd the appropriate signi1047297cant digit to start with the ourth or 4sthe seventh or 7s etc I you are rounding to a whole number write the appropriate number zeros afer the last signi1047297cant digit as place holders to keep the number the same size

Worked example 14

Round

(a) 1076 to 3 significant figures (b) 000736 to 1 significant figure

(a) 1076 The third significant figure is the 7 The next digit is 6 so round 7 up to get 8

= 108 (3sf) Correct to 3 significant figures

(b) 000736 The first significant figure is the 7 The next digit is 3 so 7 will not change

= 0007 (1sf) Correct to 1 significant figure

Exercise 117 1 Round each number to 2 decimal places


2 Express each number correct to

(i) 4 signi1047297cant 1047297gures (ii) 3 signi1047297cant 1047297gures (iii) 1 signi1047297cant 1047297gure

(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9

to a decimal using your calculator Express the answer correct to

(a) 3 decimal places (b) 2 decimal places (c) 1 decimal place(d) 3 signi1047297cant 1047297gures (e) 2 signi1047297cant 1047297gures (f) 1 signi1047297cant 1047297gure

The first significant digit of a numberis the first non-zero digit whenreading from left to right The nextdigit is the second significant digitthe next the third significant and soon All zeros after the first significantdigit are considered significant

Remember the first significantdigit in a number is the first non- zero digit reading from left toright Once you have read past thefirst non-zero digit all zeros thenbecome significant

You will use rounding to a given

number of decimal places andsignificant figures in almost all

of your work this year You will

also apply these skills to estimate

answers This is dealt with in more

detail in chapter 5

FAST FORWARD





Unit 1 Number20

SummaryDo you know the following

Numbers can be classi1047297ed as natural numbers integersbullprime numbers and square numbers

When you multiply an integer by itsel you get a squarebullnumber (x

2

) I you multiply it by itsel again you get acube number (x 3)

Te number you multiply to get a square is called thebullsquare root and the number you multiply to get a cubeis called the cube root Te symbol or square root is Te symbol or cube root is

A multiple is obtained by multiplying a number by abullnatural number Te LCM o two or more numbers isthe lowest multiple ound in all the sets o multiples

A actor o a number divides into it exactly Te HCF obulltwo or more numbers is the highest actor ound in allthe sets o actors

Prime numbers have only two actors 1 and the numberbull itsel Te number 1 is not a prime number

A prime actor is a number that is both a actor and abullprime number

All natural numbers that are not prime can be expressedbullas a product o prime actors

Integers are also called directed numbers Te sign obullan integer (minus or +) indicates whether its value is aboveor below 0

Mathematicians apply a standard set o rules to decidebullthe order in which operations must be carried outOperations in grouping symbols are worked out 1047297rst

then division and multiplication then addition andsubtraction

Are you able to

identiy natural numbers integers square numbers anbullprime numbers

1047297nd multiples and actors o numbers and identiy the

bull LCM and HCF

write numbers as products o their prime actors usingbulldivision and actor trees

calculate squares square roots cubes and cube roots obullnumbers

work with integers used in real-lie situationsbullapply the basic rules or operating with numbersbullperorm basic calculations using mental methods andbullwith a calculator





Unit 1 Number

Examination practiceExam-style questions1 Here is a set o numbers minus4 minus1 0 3 4 6 9 15 16 19 20 Which o these numbers are

(a) natural numbers (b) square numbers (c) negative integers(d) prime numbers (e) multiples o two (f) actors o 80

2 (a) List all the actors o 12 (b) List all the actors o 24 (c) Find the HCF o 12 and 24

3 Find the HCF o 64 and 144

4 List the 1047297rst 1047297ve multiples o

(a) 12 (b) 18 (c) 30 (d) 80

5 Find the LCM o 24 and 36

6 List all the prime numbers rom 0 to 40

7 (a) Use a actor tree to express 400 as a product o prime actors(b) Use the division method to express 1080 as a product o prime actors(c) Use your answers to 1047297nd

(i) the LCM o 400 and 1080 (ii) the HCF o 400 and 1080

(iii) 4 (iv) whether 1080 is a cube number how can you tell

8 Calculate

(a) 262 (b) 433

9 What is the smallest number greater than 100 that is

(a) divisible by two (b) divisible by ten (c) divisible by our

10 At noon one day the outside temperature is 4 degC By midnight the temperature is 8 degrees lowerWhat temperature is it at midnight

11 Simpliy

(a) 6 times 2 + 4 times 5 (b) 4 times (100 minus 15) (c) (5 + 6) times 2 + (15 minus 3 times 2) minus 6

12 Add brackets to this statement to make it true

7 + 14 divide 4 minus 1 times 2 = 14

Past paper questions1 Insert brackets to make this statement correct

2 times 3 minus 4 + 5 = 3 [1]

[Question 2 Cambridge IGCSE Mathematics 0580 Paper 1 (Core) OctoberNovember 2006]

2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]

[Question 2a Cambridge IGCSE Mathematics 0580 Paper 1 (Core) MayJune 2009]



httpslidepdfcomreaderfullcambridge-igcse-mathematics-core-and-extended-coursebook-with-cd-rom-cambridge 3155Unit 2 Shape space and measures128

7 Perimeter areaand volumePerimeterbullAreabullIrrational numberbullSectorbullArcbullSemi-circlebullSolid

bull NetbullVerticesbullFacebullSurface areabullVolumebullApexbullSlant heightbull

Key words

Te glass pyramid at the entrance to the Louvre Art Gallery in Paris Reaching to a height of 206 m it is a

beautiful example of a three-dimensional object A smaller pyramid ndash suspended upside down ndash acts as a

skylight in an underground mall in front of the museum

When runners begin a race around a track they do not start in the same place because theirroutes are not the same length Being able to calculate the perimeters o the various lanes allowthe offi cials to stagger the start so that each runner covers the same distance

A can o paint will state how much area it should cover so being able to calculate the areas owalls and doors is very useul to make sure you buy the correct size can

How much water do you use when you take a bath instead o a shower As more households armetered or their water being able to work out the volume used will help to control the budget


calculate areas andbull perimeters of two-dimensional shapes

calculate areas andbullperimeters of shapes thatcan be separated into twoor more simpler polygons

calculate areas andbullcircumferences of circles

calculate perimeters andbullareas of circular sectors

understand nets for three-bulldimensional solids

calculate volumes andbull surface areas of solids

calculate volumes andbullsurface area of pyramidscones and spheres

E X T E N D E D

E X T E N D E D



httpslidepdfcomreaderfullcambridge-igcse-mathematics-core-and-extended-coursebook-with-cd-rom-cambridge 3255 1Unit 2 Shape space and measures

7 Perimeter area and volu

71 Perimeter and area in two dimensions

PolygonsA polygon is a 1047298at (two-dimensional) shape with three or more straight sides Te perimeter

a polygon is the sum o the lengths o its sides Te perimeter measures the total distance arouthe outside o the polygon

Te area o a polygon measures how much space is contained inside it

wo-dimensional shapes Formula for area

Quadrilaterals with parallel sides

b

h

b

h

b

h

rhombus rectangle parallelogram

Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h

Here are some examples o other two-dimensional shapes

kite regular hexagon irregular pentagon

It is possible to 1047297nd areas oother polygons such as those othe lef by dividing the shapeinto triangles and quadrilatera




7 Perimeter area and volume

Unit 2 Shape space and measures130

At this point you may need to

remind yourself of the work you

did on transformation of formulae

in chapter 6

REWIND

Worked example 1

(a) Calculate the area of the shape shown in the diagram

5 cm

7 cm

6 cm

This shape can be divided into two simple polygons a rectangle and a triangle

Work out the area of each shape and then add them together

5 cm

7 cm

5 cm 6 cm

rectangle triangle

Area of rectangle = h = 2cm (substitute values in place of b and h )

Area of triangle = 1

2

1

25

1

215 2

h = times = times = cm

Total area = 35 + 15 = 50 cm2

(b) The area of a triangle is 40 cm2 If the base of the triangle is 5 cm find the height

A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm

Use the formula for the area of a

triangle

Substitute all values that you know

Rearrange the formula to make h the

subject

The formula for the area of atriangle can be written in differentways

1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2

Choose the way that works bestfor you but make sure you write itdown as part of your method

You do not usually have to redrawthe separate shapes but you mightfind it helpful

Units of areaI the dimensions o your shape are given in cm then the units o area are square centimetresand this is written cm2 For metres m2 is used and or kilometres km2 is used and so on Area ialways given in square units

TipYou should always giveunits or a 1047297nal answer i itis appropriate to do so Itcan however be conusingi you include units

throughout your working





1Unit 2 Shape space and measures

Exercise 71 1 By measuring the lengths o each side and adding them together 1047297nd the perimeter o eaco the ollowing shapes

(a) (b)

(c) (d)

2 Calculate the perimeter o each o the ollowing shapes

(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km






3 Calculate the area o each o the ollowing shapes

(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm

4 Te ollowing shapes can all be divided into simpler shapes In each case 1047297nd the total area

(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m

Draw the simpler shapes separatelyand then calculate the individual

areas as in worked example 1






(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm

5 For each o the ollowing shapes you are given the area and one other measurementFind the unknown length in each case

(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm

Write down the formula for the areain each case Substitute into theformula the values that you alreadyknow and then rearrange it to findthe unknown quantity

(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2

6 How many 20 cm by 30 cm rectangular tiles would you need to tile the outdoor area showbelow

17 m

48 m

09 m

26 m






7 Sanjay has a square mirror measuring 10 cm by 10 cm Silvie has a square mirror whichcovers twice the area o Sanjayrsquos mirror Determine the dimensions o Silviersquos mirror correctto 2 decimal places

8 For each o the ollowing draw rough sketches and give the dimensions

(a) two rectangles with the same perimeter but different areas(b) two rectangles with the same area but different perimeters

(c) two parallelograms with the same perimeter but different areas(d) two parallelograms with the same area but different perimeters

Circles

Archimedes worked out the formula for the area of a circle

by inscribing and circumscribing polygons with increasing

numbers of sides

Te circle seems to appear everywhere in our everyday lives Whether driving a car running on arace track or playing basketball this is one o a number o shapes that are absolutely essential to u

c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O

Finding the circumference of a circle

Circumerence is the word used to identiy the perimeter o a circle Note that the diameter = 2 times radius (2r ) Te Ancient Greeks knew that they could 1047297nd the circumerence o a circle by

multiplying the diameter by a particular number Tis number is now known as lsquoπ

rsquo (which is thGreek letter lsquoprsquo) pronounced lsquopirsquo (like apple pie) π is equal to 3141592654

Te circumerence o a circle can be ound using a number o ormulae that all mean the same thin

ircum erence iamete

==

ππr

lsquoInscribingrsquo here means to draw acircle inside a polygon so that it justtouches every edge lsquoCircumscribingrsquomeans to draw a circle outside apolygon that touches every vertex

π is an example of an irrational

number The properties of irrationalnumbers will be discussed later in

chapter 9

FAST FORWARD

(where d = diameter)

(where r = radius)

You learnt the names of the parts

of a circle in chapter 3 The diagram

on the right is a reminder of some

of the parts The diameter is the

line that passes through a circle and

splits it into two equal halves

REWIND






Finding the area of a circle

Tere is a simple ormula or calculating the area o a circle Here is a method that shows howthe ormula can be worked out

Consider the circle shown in the diagram below It has been divided into 12 equal parts andthese have been rearranged to give the diagram on the right

length asymp times =1

22 r r

height asymp r r

π π

Because the parts o the circle are narrow the shape almost orms a rectangle with height equto the radius o the circle and the length equal to hal o the circumerence

Now the ormula or the area o a rectangle is Area = bh so

Area o a circle asymp times

=

1

2π

π

I you try this yoursel with a greater number o even narrower parts inside a circle you willnotice that the right-hand diagram will look even more like a rectangle

Tis indicates (but does not prove) that the area o a circle is given by A πr

You will now look at some examples so that you can see how to apply these ormulae

(Using the values o b and h shown above)

(Simpliy)

BODMAS in chapter 1 tells you to

calculate the square of the radius

before multiplying by π

REWIND

Note that in (a) the diameter isgiven and in (b) only the radiusis given Make sure that you lookcarefully at which measurement youare given

Worked example 2

For each of the following circles calculate the circumference and the area Give each

answer to 3 significant figures

(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm

(b) Circumference diameter

cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78

TipYour calculator shouldhave a π button

I it does not use the

approximation 3142 butmake sure you write thisin your working Makesure you record the 1047297nalcalculator answer beorerounding and then statewhat level o accuracy yourounded to






Worked example 3

Calculate the area of the shaded region in the diagram

25 cm

20 cm

18 cm

O

Shaded area = area of triangle ndash area of circle

Area

cm

minus

= times times minus times

1

2

12

18 2 5

160 365

160

2

2

2

r π

π

Substitute in values of b h and r

Round the answer In this

case it has been rounded to

3 significant figures

Exercise 72 1 Calculate the area and circumerence in each o the ollowing

(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ

In some cases you may find ithelpful to find a decimal value forthe radius and diameter beforegoing any further though you canenter exact values easily on mostmodern calculators If you knowhow to do so then this is a goodway to avoid the introduction ofrounding errors






2 Calculate the area o the shaded region in each case

(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m

Te diagram shows a plan or a rectangular garden with a circular pond Te part o thegarden not covered by the pond is to be covered by grass One bag o grass seed covers 1047297vsquare metres o lawn Calculate the number o bags o seed needed or the work to be do






4

04 m

05 m 12 m

Te diagram shows a road sign I the triangle is to be painted white and the rest o the signwill be painted red calculate the area covered by each colour

5 Sixteen identical circles are to be cut rom a square sheet o abric whose sides are 04 m lonFind the area o the lefover abric i the circles are made as large as possible

6 Anna and her riend usually order a large pizza to share Te large pizza has a diameter o24 cm Tis week they want to eat different things on their pizzas so they decide to order tw

small pizzas Te small pizza has a diameter o 12 cm Tey want to know i there is the samamount o pizza in two small pizzas as in one large Work out the answer

Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h

Te diagram shows a circle with two radii (plural oradius) drawn rom the centre

Te region contained in-between the two radii isknown as a sector Notice that there is a major sectoand a minor sector

A section o the circumerence is known as an arc

Te Greek letter θ represents the angle subtended at

the centre

Notice that the minor sector is a raction o the ull circle It isθ

6 o the circle

Area o a circle is πr 2 Te sector isθ

60 o a circle so replace lsquoorsquo with lsquotimesrsquo to give

Sector area =θ

6 times πr 2

Circumerence o a circle is 2π

r I the sector is

θ

6 o a circle then the length o the arc o a

sector isθ

36 o the circumerence So

Arc length =θ

360 times 2πr

Make sure that you remember the ollowing two special cases

Ibull θ = 90deg then you have a quarter o a circle Tis is known as a quadrant

Ibull θ = 180deg then you have a hal o a circle Tis is known as a semi-circle






Exercise 73 1 Find the area o the coloured region and 1047297nd the arc length l in each o the ollowing

(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l

Note that for the perimeter you

need to add 5 cm twice Thishappens because you need toinclude the two straight edges

Note that the size of θ has notbeen given You need to calculate it

(θ = 360 minus 65)

Note that the base of the triangle is

the diameter of the circle

Worked example 4

Find the area and perimeter of shapes (a) and (b) and the area of shape (c)

Give your answer to 3 significant figures

(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm

Total area = area of triangle + area of a semi-circle

Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π

(Semi-circle is half of a circle

so divide circle area by 2)

You will be able to find the

perimeter of this third shape after

completing the work on Pythagorasrsquo

theorem in chapter 11

FAST FORWARD






2 For each o the ollowing shapes 1047297nd the area and perimeter

(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O

3 For each o the ollowing 1047297nd the area and perimeter o the coloured region

(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m

4 Each o the ollowing shapes can be split into simpler shapesIn each case 1047297nd the perimeter and area

(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm

72 Three-dimensional objectsWe now move into three dimensions but will use many o the ormulae or two-dimensionalshapes in our calculations A three-dimensional object is called a solid

Nets of solidsA net is a two-dimensional shape that can be drawn cut out and olded to orm a three-dimensional solid

You might be asked tocount the number o

vertices (corners) edgesand aces that a solid has

Tip






Te ollowing shape is the net o a solid that you should be quite amiliar with

A

A

B

A

B

C

D

C

B D

I you old along the dotted lines and join the points with the same letters then you will ormthis cube

A C

B D

You should try this yoursel and look careully at which edges (sides) and which vertices (thepoints or corners) join up

Exercise 74 1 Te diagram shows a cuboid Draw a net or the cuboid

a

b

c

2 Te diagram shows the net o a solid

(a) Describe the solid in as much detail as you can(b) Which two points will join with point M when the net is

olded(c) Which edges are certainly equal in length to PQ

S T

M Z

R U Q V

N Y

P

O






3 A teacher asked her class to draw the net o a cuboid cereal box Tese are the diagrams ththree students drew Which o them is correct

4 How could you make a cardboard model o this octahedral dice Draw labelled sketches tshow your solution

73 Surface areas and volumes of solidsTe 1047298at two-dimensional suraces on the outside o a solid are called faces Te area o each can be ound using the techniques rom earlier in this chapter Te total area o the aces will us the surface area o the solid

Te volume is the amount o space contained inside the solid I the units given are cm then volume is measured in cubic centimetres (cm3) and so on

Some well known ormulae or surace area and volume are shown below

Cuboids

A cuboid has six rectangular aces 12 edges andeight vertices

I the length breadth and height o the cuboidare a b and c (respectively) then the surace areacan be ound by thinking about the areas o eachrectangular ace

a

b

c

Notice that the surace area is exactly the same asthe area o the cuboidrsquos net

Surace area o cuboid=

2(ab +

ac +

bc)Volume o cuboid = a times b times c

c

ac

c

c

b

b

a times b

(b) (c)(a)

It can be helpful to draw the netof a solid when trying to find itssurface area








A cylinder can be lsquounwrappedrsquo to produce itsnet Te surace consists o two circular acesand a curved ace that can be 1047298attened to makea rectangle

Curved surace area o a cylinder = 2πrh

and

Volume = r h

2 r π

Exercise 75 1 Find the volume and surace area o the solid with the net shown in the diagram

3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm

2 Find (i) the volume and (ii) the surace area o the cuboids with the ollowing dimensions

(a) length = 5 cm breadth = 8 cm height = 18 cm(b) length = 12 mm breadth = 24 mm height = 48 mm

Living maths

3 Te diagram shows a bottle crate Find the volume o the crate

FIZZ ZZ

90 cm

60 cm

80 cm






4 Te diagram shows a pencil case in the shape o a triangular prism

8 cm

6 cm

10 cm

3 2 c m

Calculate(a) the volume and (b) the surace area o the pencil case

5 Te diagram shows a cylindrical drain Calculate the volume o the drain

12 m

3 m

6 Te diagram shows a tube containing chocolate sweets Calculate the total surace area o the tub

10 cm

22 cm

7 Te diagram shows the solid glass case or a clock Te case is a cuboid with a cylinderremoved (to 1047297t the clock mechanism) Calculate the volume o glass required to make theclock case

10 cm

8 cm

4 cm

5 cm

Donrsquot forget to include thecircular faces






8 A storage company has a rectangular storage area 20 m long 8 m wide and 28 m high

(a) Find the volume o the storage area(b) How many cardboard boxes o dimensions 1 mtimes 05 m times 25 m can 1047297t into this storage ar(c) What is the surace area o each cardboard box

9 Vuyo is moving to Brazil or his new job He has hired a shipping container o dimensions3 m times 4 m times 4 m to move his belongings

(a) Calculate the volume o the container(b) He is provided with crates to 1047297t the dimensions o the container He needs to move eio these crates each with a volume o 5 m3 Will they 1047297t into one container

Pyramids

A pyramid is a solid with a polygon-shaped baseand triangular aces that meet at a point calledthe apex

I you 1047297nd the area o the base and the area oeach o the triangles then you can add these up to1047297nd the total surace area o the pyramid

Te volume can be ound by using the ollowingormula

Volume = 1

times ptimes erpendicular heigh

Cones

A cone is a special pyramid with a circular base Te length l is known as the slant height h is the perpendicular height

l

r

h

Te curved surace o the cone can be opened out and 1047298attenedto orm a sector o a circle

Curved surace area = πrl

and

Volume = 1

πr

The perpendicular height is theshortest distance from the base tothe apex

The slant height can be

calculated by using Pythagorasrsquo

theorem which you will meet

in chapter 11

FAST FORWARD

If you are asked for the totalsurface area of a cone you mustwork out the area of the circularbase and add it to the curvedsurface area

l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres

Te diagram shows a sphere with radius r

Surace area = 2π

and

Volume= 4 πr

1 Te diagram shows a beach ball

(a) Find the surace area o the beach ball

(b) Find the volume o the beach ball

40 cm

2 Te diagram shows a metal ball bearing that iscompletely submerged in a cylinder o waterFind the volume o water in the cylinder

2 cm

15 cm

3 Te Great Pyramid at Giza has a square base o side 230 m and perpendicular height 146 m

Find the volume o the Pyramid

Exercise 76

r

The volume of the water is thevolume in the cylinder minus thedisplacement caused by the metalball The displacement is equal tothe volume of the metal ball






4 Te diagram shows a rocket that consistso a cone placed on top o a cylinder

(a) Find the surace area o the rocket

(b) Find the volume o the rocket

5 m 1 3

m

1 2 m

2 5

m

5 Te diagram shows a childrsquos toy thatconsists o a hemisphere (hal o a sphere)and a cone

(a) Find the volume o the toy(b) Find the surace area o the toy

8 cm

6 cm

10 cm

6 Te sphere and cone shown

in the diagram ha ve the same volume

Find the radius o the sphere

24 cm

83 cm

r

7 Te volume o the largersphere (o radius R) is twice the

volume o the smaller sphere(o radius r )

Find an equation connectingr to R

Rr






8 A 32 cm long cardboard postage tube has a radius o 25 cm

(a) What is the volume o the tube(b) For posting the tube is sealed at both ends What is the surace area o the sealed tube

9 A hollow metal tube is made using a 5 mm metal sheet Te tube is 35 cm long and has anexterior diameter o 104 cm

(a) Draw a rough sketch o the tube and add its dimensions

(b) Write down all the calculations you will have to make to 1047297nd the volume o metal inthe tube(c) Calculate the volume o metal in the tube(d) How could you 1047297nd the total surace area o the outside plus the inside o the tube


Te perimeter is the distance around the outside obulla two-dimensional shape and the area is the space

contained within the sidesCircumerence is the name or the perimeter o a circlebullI the units o length are given in cm then the units obullarea are cm2 and the units o volume are cm3 Tis is trueor any unit o length

A sector o a circle is the region contained in-betweenbulltwo radii o a circle Tis splits the circle into a minorsector and a major sector

An arc is a section o the circumerencebullPrisms pyramids spheres cubes and cuboids arebullexamples o three-dimensional objects (or solids)

A net is a two-dimensional shape that can be olded to

bull orm a solid

Te net o a solid can be useul when working out thebullsurace area o the solid

Are you able to

recognise different two-dimensional shapes and 1047297ndbulltheir areas

give the units o the areabullcalculate the areas o various two-dimensional shapesbulldivide a shape into simpler shapes and 1047297nd the areabull1047297nd unknown lengths when some lengths and an areabullare given

calculate the area and circumerence o a circlebullcalculate the perimeter arc length and area o a sectorbullrecognise nets o solidsbullold a net correctly to create its solidbull1047297nd the volumes and surace areas o a cuboid prismbulland cylinder

1047297nd the volumes o solids that can be broken intobullsimpler shapes

1047297nd the volumes and surace areas o a pyramid conebulland sphere







3 120 m

d

A 400 metre running track has two straight sections each o length 120 metres and two semicircular ends(a) Calculate the total length o the curved sections o the track [1](b) Calculate d the distance between the parallel sections o the track [2]

[Question 18 Cambridge IGCSE Mathematics 0548 Paper 1 (Core) Nov 2005]

4 A large conerence table is made rom our rectangular sections and our corner sections Each rectangularsection is 4 m long and 12 m wide Each corner section is a quarter circle o radius 12 m

4

NOT TO

SCALE

Each person sitting at the conerence table requires one metre o its outside perimeter Calculate the greatestnumber o people who can sit around the outside o the table Show all your working [3]

[Question 11 Cambridge IGCSE Mathematics 0548 Paper 2 (Extended) Nov 2005]

5

A B12 cm

NOT TO

SCALE

Te largest possible circle is drawn inside a semi-circle as shown in the diagram Te distance AB is 12 centimetres(a) Find the shaded area [4](b) Find the perimeter o the shaded area [2]

[Question 23 Cambridge IGCSE Mathematics Paper 2 (Extended) June 2007]



Karen Morrison and Nick Hamshaw

Cambridge IGCSE

Mathematics Core

and ExtendedCoursebook






wwwcambridgeorg
















Acknowledgements

Unit 1



Unit 2

Unit 3




exclusive events






and ranges



continuous data



















Contentsiv



132 Time 253



135 More money 264













152 Bearings 3


and sine ratios 3







Unit 4

Unit 5















201 Histograms 4


Unit 6




213 Rates 450


215 Proportion 460


algebraic terms 463









232 Vectors 5








Exercise Answers 5

Glossary 6

Index 6

















REWIND




curly brackets

FAST FORWARD

















Tip























Key words









Unit 1 Number2




1 2 3 4 5


1 3 5 7


2 4 6 8




2 3 5 7 11


multiplied by itsel

1 4 9 16



05 02 008 17




2 List








FAST FORWARD








FAST FORWARD






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE







wwwcambridgeorg
















Acknowledgements

Unit 1



Unit 2

Unit 3




exclusive events






and ranges



continuous data



















Contentsiv



132 Time 253



135 More money 264













152 Bearings 3


and sine ratios 3







Unit 4

Unit 5















201 Histograms 4


Unit 6




213 Rates 450


215 Proportion 460


algebraic terms 463









232 Vectors 5








Exercise Answers 5

Glossary 6

Index 6

















REWIND




curly brackets

FAST FORWARD

















Tip























Key words









Unit 1 Number2




1 2 3 4 5


1 3 5 7


2 4 6 8




2 3 5 7 11


multiplied by itsel

1 4 9 16



05 02 008 17




2 List








FAST FORWARD








FAST FORWARD






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Acknowledgements

Unit 1



Unit 2

Unit 3




exclusive events






and ranges



continuous data



















Contentsiv



132 Time 253



135 More money 264













152 Bearings 3


and sine ratios 3







Unit 4

Unit 5















201 Histograms 4


Unit 6




213 Rates 450


215 Proportion 460


algebraic terms 463









232 Vectors 5








Exercise Answers 5

Glossary 6

Index 6

















REWIND




curly brackets

FAST FORWARD

















Tip























Key words









Unit 1 Number2




1 2 3 4 5


1 3 5 7


2 4 6 8




2 3 5 7 11


multiplied by itsel

1 4 9 16



05 02 008 17




2 List








FAST FORWARD








FAST FORWARD






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE





Contentsiv



132 Time 253



135 More money 264













152 Bearings 3


and sine ratios 3







Unit 4

Unit 5















201 Histograms 4


Unit 6




213 Rates 450


215 Proportion 460


algebraic terms 463









232 Vectors 5








Exercise Answers 5

Glossary 6

Index 6

















REWIND




curly brackets

FAST FORWARD

















Tip























Key words









Unit 1 Number2




1 2 3 4 5


1 3 5 7


2 4 6 8




2 3 5 7 11


multiplied by itsel

1 4 9 16



05 02 008 17




2 List








FAST FORWARD








FAST FORWARD






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE


















REWIND




curly brackets

FAST FORWARD

















Tip























Key words









Unit 1 Number2




1 2 3 4 5


1 3 5 7


2 4 6 8




2 3 5 7 11


multiplied by itsel

1 4 9 16



05 02 008 17




2 List








FAST FORWARD








FAST FORWARD






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE

















Tip























Key words









Unit 1 Number2




1 2 3 4 5


1 3 5 7


2 4 6 8




2 3 5 7 11


multiplied by itsel

1 4 9 16



05 02 008 17




2 List








FAST FORWARD








FAST FORWARD






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE
























Key words









Unit 1 Number2




1 2 3 4 5


1 3 5 7


2 4 6 8




2 3 5 7 11


multiplied by itsel

1 4 9 16



05 02 008 17




2 List








FAST FORWARD








FAST FORWARD






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE


















Key words









Unit 1 Number2




1 2 3 4 5


1 3 5 7


2 4 6 8




2 3 5 7 11


multiplied by itsel

1 4 9 16



05 02 008 17




2 List








FAST FORWARD








FAST FORWARD






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE
















Key words









Unit 1 Number2




1 2 3 4 5


1 3 5 7


2 4 6 8




2 3 5 7 11


multiplied by itsel

1 4 9 16



05 02 008 17




2 List








FAST FORWARD








FAST FORWARD






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number2




1 2 3 4 5


1 3 5 7


2 4 6 8




2 3 5 7 11


multiplied by itsel

1 4 9 16



05 02 008 17




2 List








FAST FORWARD








FAST FORWARD






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number

Living maths









(a) 0599 gt 60 (b) 5 times 1999 asymp 10 000







= is equal to

ne is not equal to

asymp



gt is greater than


there4 therefore

the square root of







Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number4

Worked example 1




12 times 1 = 12

12 times 2 = 2412 times 3 = 36


300 divide 12 = 25


(a) 2 (b) 3 (c) 5 (d) 8(e) 9 (f) 10 (g) 12 (h) 100


(a) 29 (b) 44 (c) 75 (d) 114(e) 299 (f) 350 (g) 1012 (h) 9123

3 List




(a) 324 (b) 783 (c) 816 (d) 837 (e) 1116



Worked example 2


M4 = 4 8 12 16 20 24 28 32

M7 = 7 14 21 28 35 42

LCM = 28







find LCMs






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number




Worked example 3

Find the factors of

(a) 12 (b) 25 (c) 110

(a) F 12


1 times 12

2 times 6

3 times 4


(b) F 25 = 1 5 25 1 times 255 times 5

Do not repeat the 5

(c) F 110

= 1 2 5 10 11 22 55 110 1 times 110

2 times 55

5 times 22

10 times 11

F 12





(a) 14 1 2 4 7 14(b) 15 1 3 5 15 45(c) 21 1 3 7 14 21(d) 33 1 3 11 22 33(e) 42 3 6 7 8 14




another









Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number6



Worked example 4


F 8 = 1 2 4 8

F 24

= 1 2 3 4 6 8 12 24

HCF = 8







(a) 3 9 and 15 (b) 36 63 and 84 (c) 22 33 and 121




Living maths











chapter

FAST FORWARD






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number

11 12

21

31

41

5161

71

81

91

3

13

23

33

43

5363

73

83

93

4

14

24

34

44

5464

74

84

94

5

15

25

35

45

5565

75

85

95

6

16

26

36

46

5666

76

86

96

2

22

32

42

5262

72

82

92

7

17

27

37

47

5767

77

87

97

1 8

18

28

38

48

5868

78

88

98

9

19

29

39

49

5969

79

89

99

10

20

30

40

50

6070

80

90

100



multiples of 2


multiples of 3





crossed out


primes



1955 1965 1975 1985 1995 2005 2015100

1000

10 000

100 000

10 000 000

1 000 000

100 000 000

Year


against year found

Number

of digits









chapter 10

FAST FORWARD





Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number8




super-prime





Worked example 5


(a) 36 (b) 48

Using a factor tree

36

123

3

2 2

4


48

124

32 2

2 2

4



factors


number circle it



factors


Write the primes in


signs

Using division

3618931

2233


48

24

12

6

31

2

2

2

2

3











signs


Tip






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number


(a) 30 (b) 24 (c) 100 (d) 225 (e) 360(f) 504 (g) 650 (h) 1125 (i) 756 (j) 9240



Worked example 6





HCF = 12





Worked example 7





LCM = 360





















FAST FORWARD





Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number10













Exercise 110 23 65 92 10 104 70 500 21 64 798 1223









14 Powers and roots






Tip



number

REWIND





Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number



2

2



2

2

2







Worked example 8


(a) 132 (b) 53 (c) 324 (d) 5123

(a) 132 = 169 Enter 1 3 x 2 =



(c) 324 18Enter

3

2

4

=

(d) 512 83 Enter 3 5 1 2 =

Worked example 9



(c) 3 (d) 5123

(c)324 2 2

23 3

33 3

3times


324 18



(d)512 2 2

22 2

22 2

2


512 83










Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number12


(a) 32 (b) 72 (c) 112 (d) 122 (e) 212

(f) 192 (g) 322 (h) 1002 (i) 142 (j) 682

2 Calculate

(a) 13 (b) 33 (c) 43 (d) 63 (e) 93

(f) 103 (g) 1003 (h) 183 (i) 303 (j) 2003




(j) (k) 1 (l) = 1

(m) = (n) x 3 1= (o) 64 =


(a) (b) 64 (c) 1 (d) (e) 1

(f) (g) 1 (h) 4 (i) 1296 (j) 1 64

(k)3

(l) 1 (m) 27 (n) 64 (o) 1000(p) 216 (q) 512 (r) 2 (s) 17283 (t) 5832





(e) 15 625=

5times

5times

5times

5times

5times


7 Calculate

(a) ( )2 (b) ( )2 (c) ( ) (d) ( )

(e) 16 (f) 16 (g) 6 64+ (h) 6 64


(m) 4 (n) 4 (o) 36

4 (p)

36


(a) 1000 cm3 (b) 19 683 cm3 (c) 68 921 mm3 (d) 64 000 cm3


(a) 2 3 (b) 3 2 (c) 1 4 (d) 4 1 (e) 2 4(f) 4 2 (g) 1 9 (h) 9 1 (i) 5 2 (j) 2 5








Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number













(a) (b) 4 (c) 1













chapter 14

FAST FORWARD






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number14

Living maths


ndash4

ndash2

0

2

4

6

8

10

Sunday

14

Sunday

21


28Day of the week

Temperature (degC)


















Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number


Brackets

Of

D

ivide M

ultiply

dd S

ubtract





[2(4 + 9) minus 4(3) minus 12]










= 25

Worked example 11

Calculate

(a)3+

8

2 (b)4 2817 9minus


(a) 3

6

67

=

)8times (b) ( )

8

4

= divide

(c) 36 100 36

64

8

11

divide + minus

=

=





Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number16



2 Calculate





4 Calculate

(a) 6 + 72 (b) 29 minus 23 (c) 8 times 42

(d) 20minus

4divide

2 (e)

31 10

14 (f)

100 40

4

minus

(g) 1 36 (h) (i) 0 minus






(g)15

2 5

minus (h) (17 + 1 ) divide 9 + 2 (i)

16

1

minus


2 Simpliy








FAST FORWARD





Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number


(a) 0 2 5 10

(b) 9 11 13 18

(c) 1 3 8 14 16 =minus

(d) 4 5 6 9 12 ( (minus







Worked example 12



(a) 21 Enter 3 + 2 times 9 =

(b) 44 Enter ( 3 + 8 ) times 4 =








(a) 12 (28 24) = 3 (b) 84 10 8 = 4(c) 3 7(07 13) = 17 (d) 23 11 22 11 = 11(e) 40 5 (7 5) = 4 (f) 9 15 (3 2) = 12

4 Calculate

(a)16

1minus

(b) 4

1 1minus

(c) 1 5

2

times minus



buttonsminus

and ( minus







Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number18

(d)6 11

2 )17 4 (e)

1 (f)

6

4 5

+

(g)6 16

15 3

minus (h)

minus

5 8 3

+

minus

(divide 4


(a)0 345

1

(b)

1

16 8 05

times

+

(c) 16 0 087

09

times (d)

19 087

4



(c) (d) 41 minus

(e) (f) 145 minus3

(g) 1

4

1

4

1

4

1

4+

(h) 75 minus times1



Worked example 13

Round 64839906 to










the place of the 9











FAST FORWARD





Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number


Worked example 14

Round










(a) 4512 (b) 12 305 (c) 65 238 (d) 32055(e) 25716 (f) 0000765 (g) 10087 (h) 734876(i) 000998 (j) 002814 (k) 310077 (l) 00064735

3 Change9










detail in chapter 5

FAST FORWARD





Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number20




2











Are you able to



bull LCM and HCF








Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






Unit 1 Number






(a) 12 (b) 18 (c) 30 (d) 80






8 Calculate

(a) 262 (b) 433




11 Simpliy





2 times 3 minus 4 + 5 = 3 [1]


2 Calculate

(a)0548

1 6

(b)

0 0763

1

[2]







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE







Key words















E X T E N D E D

E X T E N D E D











b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE












b

h

b

h

b

h


Area = bh

riangles

b

h

b

h

b

h

Area =1

2 or

2

rapezium

b

h

b

h

a a

Area = 1

( ) or( )h












in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE










in chapter 6

REWIND

Worked example 1


5 cm

7 cm

6 cm



5 cm

7 cm

5 cm 6 cm

rectangle triangle



2

1

25

1

215 2


Total area = 35 + 15 = 50 cm2


A b htimes2

01

25

2

40 2

5

80

516

= times times

rArr times times

rArr =

h

h

cm


triangle



subject


1

2 2

1

2

times =

=

htimes bh

b htimesOR

OR htimes2












(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE








(a) (b)

(c) (d)


(a) 25 cm

55 cm

(b)

4 cm

3 cm

5 cm

(c) 7 cm

10 cm

4 cm4 cm

(d) 2 cm

2 cm

10 cm

9 cm

(e)

84 m

19 m

28 m25 m

72 m

(f)

9 km8 km

3 km3 km







(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE








(a)

11 cm

5 cm

(b)

5 m

3 m

(c)

5 m

4 m

(d)

32 cm

14 cm

(e)

8 m

2 m

(f)

28 cm

(g) 6 cm

5 cm

10 cm

(h)

6 m

6 m

8 m

(i)

4 cm

4 cm

(j)

12 cm

6 cm

6 cm


(a)

5 m

8 m

4 m

8 m

(b)

72 m

45 m

12 m

21 m

51 m








(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE







(c) 49 cm

53 cm

82 cm

(d)

78 cm

72 cm 734 cm

54 cm

21 cm

(e)

18 cm

24 cm

12 cm 12 cm

(f)

191 cm

382 cm

38 cm

(g)

371 cm

182 cm

784 cm

853 cm


(a)

8 cm

h

242

cm

(b)

17 cm

b

289 2 cm


(c)

16 cm

a

14 cm132 2 cm

(d)

75 cm2

15 cm

b

(e) 6 cm

6 cm

18 cm

h

200 cm2


17 m

48 m

09 m

26 m










Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE











Circles



numbers of sides


c i r c u mf e

r e n c e

O is the

centre

r a d i u s

diameter

O






ircum erence iamete

==

ππr




chapter 9

FAST FORWARD


(where r = radius)







REWIND










22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE











22 r r

height asymp r r

π π




=

1

2π

π





(Simpliy)




REWIND


Worked example 2



(a)

O

8 mm

(a) diameter

mm

= times= times 8

25 1327

25 1

Area

mm

times

= times= times

r

r d

2

2

2

16

50 265

50 3

(b)

O

5 cm


cm

= times

= times10

31 41531 4

( )

= times

Area = times

times

times

r 2

2

2

5

2578 539

78









Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE







Worked example 3


25 cm

20 cm

18 cm

O


Area

cm

minus


1

2

12

18 2 5

160 365

160

2

2

2

r π

π






(a)

O

4 m

(b)

O

31 mm

(c)

O

08 m

(d)

O1

2cm

(e)

O

2 km

(f)

O

2

mπ








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE








(a)

2 cm

18 cm

(b)

8 cm

8 cm

O

8 cm8 cm

(c)

12 m

7 m

2 m

1 m

O

O

(d)

5 cm

O

15 cm

10 cm

(e)

5 cm

O15 cm

12 cm

19 cm

(f)

3 cm

12 c

Living maths

3

pond

3 m10 m

12 m







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE







4

04 m

05 m 12 m





Arcs and sectors

major sector

O

minor

sector

r

r

θ

a r c l e n g t h





the centre


6 o the circle



Sector area =θ

6 times πr 2


r I the sector is

θ


sector isθ


Arc length =θ

360 times 2πr










(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE








(a)

70deg

l

O

18 cm

(b)

120deg 82 cm

l

O

(c)

64 cm95deg

O

l (d)

175deg

3 m

O

l




(θ = 360 minus 65)



Worked example 4



(a)

30deg

O

5 m

Area

m

= times

times

θ

36030

3605

6 544

6 5

2

2

2

π

π

r Perimeter

m

= θ

3602

30360

2 5 5times

12 617

12 6

r +

(b)

O

4 cm65deg

Area

cm

=

minus360360 65

360295

36016

41 189

41 2

2

2

π

π

r

2

Perimeter

cm

=

+

θ

3602

295

3602 4 4times

28 594

28 6

r +

(c)

O

6 cm

4 cm


Area

cm

= +

= times + times

1

2

1

21

2 times

1

249 132

49 1

2

2

2

r π

π







FAST FORWARD







(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE








(a)

O

40deg 6 cm

(b)

O45deg

8 cm

(c) O

15deg32 cm

(d)

O

75deg5 m

(e)

O 172 cm

(f)

O

154 m

(g)

028 cm

O

(h)

43 cm

O

(i)

62deg

62deg

15 cm (j)

OO

6 m

14 m

O


(a)

5 cm

8 cm

(b)

9 cm

10 cm

6 cm






(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE







(c) 30deg

30deg

5 m

12 m

(d)

84 cm

84 cm

(e)

18 m


(a)

28 cm

(b)

100deg

32 c

(c)

15 cm

13 cm

(d)

3 cm

7 cm

11 cm





Tip







A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE








A

A

B

A

B

C

D

C

B D


A C

B D



a

b

c




S T

M Z

R U Q V

N Y

P

O











Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE












Cuboids



a

b

c



2(ab +

ac +


c

ac

c

c

b

b

a times b

(b) (c)(a)











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE











and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE









and

Volume = r h

2 r π


3 cm

3 cm4 cm

5 cm

5 cm

11 cm

5 cm

5 cm

4 cm3 cm

3 cm



Living maths


FIZZ ZZ

90 cm

60 cm

80 cm







8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE








8 cm

6 cm

10 cm

3 2 c m



12 m

3 m


10 cm

22 cm


10 cm

8 cm

4 cm

5 cm











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE











Pyramids




Volume = 1


Cones


l

r

h



and

Volume = 1

πr





in chapter 11

FAST FORWARD


l

l e n g

t h

o f

a r c

o f

s e c t

o r

=

l e ng t h

of c i r c u m

f e r e n

c e o

f b

l

r

h






Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE







Spheres


Surace area = 2π

and

Volume= 4 πr




40 cm


2 cm

15 cm



Exercise 76

r










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE










5 m 1 3

m

1 2 m

2 5

m



8 cm

6 cm

10 cm




24 cm

83 cm

r




Rr

















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE


















bull orm a solid


Are you able to












3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE








3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE






3 120 m

d




4

NOT TO

SCALE



5

A B12 cm

NOT TO

SCALE



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Cambridge Igcse Mathematics Core and Extended Coursebook With CD Rom Cambridge Education Cambri Samples