maths connect 3r resourcebank-pack

ASSESSMENT ANDRESOURCEBANK

Sue Bright

Alexandra Hewitt

Dave Kirkby

Mavis Rayment

Catherine Roe

Bev Stanbridge

Unit testsJackie Fairchild

Tests A and BTessa Ford

HomeworksDavid Foster, Nicholas Georgiou,

Simon Gilbert, Ron Holt and Mary Pardoe

3

A Prelims i-iv.qxd 20/4/05 11:40 am Page i

Tasha

Heinemann Educational PublishersHalley Court, Jordan Hill, Oxford OX2 8EJPart of Harcourt Education

Heinemann is the registered trademark of Harcourt Education Limited

© Harcourt Education Limited 2005Thinking Maths Notesheets pp 47–61 © Mundher Adhami, David Johnson and Michael Shayer, 1998

First published 2005

09 08 07 06 0510 9 8 7 6 5 4 3 2 1

British Library Cataloguing in Publication Data is availablefrom the British Library on request.

ISBN 0 435 536265

© Harcourt Education Limited 2005. All rights reserved. The material in this publication is copyright. Theduplicating masters may be photocopied for one-time use as instructional material in a classroom by a teacher, butthey may not be copied in unlimited quantities, kept on behalf of others, passed on or sold to third parties, or storedfor future use in a retrieval system. If you wish to use the material in any way other than that specified you mustapply in writing to the publishers.

Designed and typeset by Tech-Set Ltd, Gateshead, Tyne and Wear

Original illustrations © Harcourt Education Limited 2005

Illustrated by Tech-Set Ltd and Bigtop Design

Cover design by mcc design ltdCover photo © Getty images

Printed in the UK by MFK Group Ltd, Stevenage, Herts.

AcknowledgementsEvery effort has been made to contact copyright holders of material reproduced in this book. Any omissions will berectified in subsequent printings if notice is given to the publishers.

A Prelims i-iv.qxd 20/4/05 11:40 am Page ii

ContentsHow to use this pack iv

Resource sheets 1

Thinking Maths Notesheets 47

Homework sheets 62

Homework answers 114

Year 9 Unit tests 129

Year 9 Unit test mark schemes 149

Test A (Levels 6–8) 153

Test A mark scheme 161

Test B (Levels 6–8) 163

Test B mark scheme 171

© Harcourt Education 2005 iii

A Prelims i-iv.qxd 20/4/05 11:40 am Page iii

How to use this packMaths Connect Assessment and Resourcebank contains OHTs andResource Sheets to support your classroom teaching. The links to these are provided in the Teacher Book wherever you see this .

Thinking Maths Notesheets to accompany the Thinking Mathsactivities can be found on pages 47–61.

Homework sheets for every lesson are included. These are providedin A5 format to save on photocopying costs. Each homework sheetprovides a short summary and a worked example based on thelesson objectives, followed by self-contained questions designed totake 25–30 minutes. The style of the homework sheets means thatpupils will not need to take their textbooks home if you do not wishthem to. The answers are on pages 114–128.

Assessment is provided by the following:● Unit Tests 1–12 and 14. These can be given to pupils as they

complete each Unit of the Pupil Book. Units 13 and 15 are projectbased, so there are no corresponding Unit tests. Approximately30–40 minutes should be allowed for each test. Detailed markschemes are provided for the tests.

● Levelled end of year tests. Test A (Non-calculator) and Test B(Calculator allowed) are both provided as write-on tests, alongwith mark schemes and a breakdown of marks by level. Both thesetests are targeted at Levels 6–8, but you will find tests for thedifferent tiers in the other Maths Connect Resourcebanks.

The enclosed Maths Connect 3R Assessment and Resourcebank CD-ROMcontains Acrobat files of all the material here. Further information isin the readme.txt file on the CD. If you experience any problems withthe CD, please phone our technical support line on 01865 888108.

© Harcourt Education 2005iv

A Prelims i-iv.qxd 20/4/05 11:40 am Page iv

© Harcourt Education 2005 1

3RResource sheet 1

Triangular numbers

Term number 1

Sequence

2 3 4 5

Number of dots

2 � Sequence

Number of dots

B Resource Sheets 001-046.qxd 20/4/05 11:40 am Page 1

© Harcourt Education 20052

3RResource sheet 2

x �5 �2 0 1 3

y

y

x

2

0

4

6

2 4�4 �2

�2

�4

�6

�8


Resource sheet 3


3R

A small open box is made from a sheet of metal 50 cm by 50 cm.Equal-size squares are cut from each corner, and the remaining pieces are folded up to makethe sides.The box needs to have a volume of 9000 cm3.What are the possible dimensions of the box?


Resource sheet 4


3RPythagoras’ theorem


Resource sheet 5


3R

4

9

6

5

11 12

20

63


Resource sheet 6


3RCongruent shapes

�5

�4

�3

�2

�1

1

2

3

4

5

y

0 x1 2 3 4 5�5 �4 �3 �2 �1

D

G

H

FC

B

A

E

Conditions for unique triangles

S

two sides andthe includedangle (SAS)

two angles andthe includedside (ASA)

right angle,hypotenuseand a side(RHS)

three sides (SSS)

S

S S

HSS

S

A A A


Resource sheet 7


3R

12.3 cm

9.7 cm7.9 cm

52° 40°

88°


Resource sheet 8a


3R

10

10

y

x

A (0, 0)

B(16, 12)

00

Resource sheet 8b

�10

10y

0 x10�10


Resource sheet 9


3R

order 2

order 2

order 2


Resource sheet 10


3RLoci

A B

3-D shapes


Resource sheet 11


3R

Inte

rpre

t and

dis

cuss

dat

aC

olle

ct d

ata

from

ava

riet

y of

sou

rces

Spec

ify

the

prob

lem

and

pla

n

Proc

ess

and

repr

esen

t dat

a

Data

han

dlin

g cy

cle


Resource sheet 12


3R

Survey on gym use

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%65�

15–640–14

Time spent in gym, T 15 � T � 30 30 � T � 45 45 � T � 60 65 � T � 75 75 � T � 90(minutes)

Frequency 2 7 18 15 3

Mid-point 22.5 37.5 52.5 67.5 82.5

Frequency � mid-point 45 262.5 945 1012.5 247.5

Age of child (years) 4–7 8–11 12�

Mid-point 6

Ages of children

Population of the UK, by age group


Resource sheet 13


3R

Time spent in the gym

Time (minutes)

Cu

mu

lati

ve f

req

uen

cy

0

5

10

15

20

25

30

35

40

45

15 30 45 60 75 90


Resource sheet 14


3R

Amount of time people spend in the gym

Time (minutes)

Freq

uen

cy

0

5

10

15

20

15 30 45 60 75

weekdays

90

weekends

Number of sisters

Nu

mb

er o

f ch

ild

ren

0

2

4

6

0 1 2 3 4

Time (minutes)

Freq

uen

cy0

5

10

15

20

3015 45 60 75 90


Resource sheet 15


3R

Height

Han

dsp

an

0 0 0

Premiership results

Scatter graphs

Number of matches wonagainst league position

League position

Nu

mb

er o

fm

atch

es w

on

005

1015202530

5 10 15 20

Number of matches drawnagainst league position

League position

Nu

mb

er o

fm

atch

es d

raw

n

005

1015202530

5 10 15 20

League position at Matches Matches Matches Goals Goalsend of season won drawn lost scored conceded

1 25 8 5 74 34

2 23 9 6 85 42

3 21 6 11 63 48

4 19 10 9 68 38

5 18 10 10 61 41

6 16 12 10 52 43

7 17 8 13 48 49

8 13 13 12 43 46

9 15 6 17 47 54

10 14 8 16 51 62

11 13 10 15 48 44

12 14 7 17 45 56

13 13 9 16 41 49

14 13 9 16 41 50

15 14 5 19 58 57

16 12 9 17 42 47

17 10 14 14 41 51

18 10 12 16 42 59

19 6 8 24 29 65

20 4 7 27 21 65


Resource sheet 16


3R

Table 1

Boy’s age (years) 1 4 6 10 12 14 18

Height (cm) 75 102 116 135 146 163 176

Number of tests 100 1000 2000 5000 10 000

Number of people who 80 720 1400 3250 6100cannot tell the difference

Table 2


Resource sheet 17


3R

130 ??

10130 to the nearest 10

A B

35 ??

535 to the nearest 5

5.30 ??

0.015.30 to 2 d.p.

62.4 ??62.4 to 3 sig. fig.


Resource sheet 18


3R

Table 1

x x2 3x x3 x2 � 3x � x3

1

2

0

0.1

0.5�1�2

Table 2

x x3 3x2 y

0

1

2

3

3.5�1�2

y

x

4

8

12

16

20

�4

�8

�12

�16

�20

1 2 3 4 50�1�2�3�4�5


Resource sheet 19


3R

Graph Gradient of graph Gradient of perpendicular

y � x

y � 3x � 4

y � �2x � 1

y

x

2

4

6

8

10

�2

�4

�6

�8

�10

2 4 6 8 100�2�4�6�8�10


Resource sheet 20


3RGraphs

x �3 �2 �1 0 1 2 3

y

y

x

4

8

12

16

20

�4

�8

�12

�16

�20

1 2 3 4 50�1�2�3�4�5


Resource sheet 21


3RSimultaneous equationsEquation 1 Equation 2

x

y

x

y

y

x

2

4

6

8

10

�2

�4

�6

�8

�10

2 4 6 8 100�2�4�6�8�10


Resource sheet 22


3RSample space diagram

Dice 1

1 2 3 4 5 6

Dice 2 1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

Number of times coin spun 1 2 3 4

Required outcome H TH TTH TTTH

Probability �12� �

12� � �

12� � (�

12�)2 �

12� � �

12� � �

12� � (�

12�)3 �

12� � �

12� � �

12� � �

12� � (�

12�)4

Decimal 0.5 0.25 0.125 0.0625

Coin spinning


Resource sheet 23


3R

Number of deals 60 120 180 240

Frequency of 2 snaps

Relative frequency

1

6000

120 180 240

Number of deals

Rel

ativ

e fr

equ

ency

of 2

sn

aps

300 360 420 480

y

x

12


Resource sheet 24


3R

A

B

C

24

5

20

30

13

12

2010 x

x

x3


Resource sheet 25


3RTriangle enlargements

B

A

2.5 cm

5 cm

3 cm

CC�

B�

A�


Resource sheet 26


3RSquares and triangles

Cubes

Scale drawings and maps

1 cm represents 2 m 1 : 500

Lake

Scale: 1 : 25 000


Resource sheet 27


3R

x 3

2x

5

� x �3

2x

�5

�


Resource sheet 28


3R

�10 �8 �6 �4 �2

�10 �8 �6 �4 �2

�10 �8 �6 �4 �2

�10 �8 �6 �4 �2

�10 �8 �6 �4 �2

�10 �8 �6 �4 �2

�10 �8 �6 �4 �2

�10 �8 �6 �4 �2

�10 �8 �6 �4 �2

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10


Resource sheet 29


3R

y

x

2

4

6

8

10

�2

�4

�6

�8

�10

2 4 6 8 100�2�4�6�8�10


Resource sheet 30


3R

CD

E

BO

A70°

�EAB � 70°DE is a tangent to the circle at A.DE and CB are parallel.O is the centre of the circle.Find �ADO.

Angle facts


Resource sheet 31


3R

Year Winner Time(min:sec)

1946 Oxford 19:54

1947 Cambridge 23:01





1952 Oxford 20:23


1954 Oxford 20:23





1959 Oxford 18:52

1960 Oxford 18:59



1963 Oxford 20:47


1965 Oxford 18:07

1966 Oxford 19:12

1967 Oxford 18:52







1974 Oxford 17:35

Year Winner Time(min:sec)


1976 Oxford 16:58

1977 Oxford 19:28

1978 Oxford 18:58

1979 Oxford 20:33

1980 Oxford 19:02

1981 Oxford 18:11

1982 Oxford 18:21

1983 Oxford 19:07

1984 Oxford 16:45

1985 Oxford 17:11


1987 Oxford 19:59

1988 Oxford 17:35

1989 Oxford 18:27

1990 Oxford 17:22

1991 Oxford 16:59

1992 Oxford 17:44








2000 Oxford 18:04


2002 Oxford 16:54

2003 Oxford 18:06

Oxford and Cambridge University Boat Race


Resource sheet 32


3RSpreadsheet data

1 2 3 4 5 6 7 8 9 10

Tria

l 1

Tria

l 2

Tria

l 3

Sort

ed

data

Tria

l 1

Tria

l 2

Tria

l 3A

5.8

6.1

5.2

3.2

4.9

4.7B

9 9.3

6.2

5.3

5.9

5.2C

3.2

4.9

4.7

5.8

6.1

5.2D

7.8

7.8

7.2

6.7

7.2

6

E

9.4

9.1

8.3

7.3

7.5

6.1F

8.7

8.8

8.1

7.7

7.5

6.2G

6.7

8.2

6 7.8

7.6

6.2H

9.7

8.3

6.2

med

ian

8.2

7.8

6.8I

8.3

8.5

7.3

8.3

8.2

6.9J

9.8

7.5

8.3

8.3

8.3

7.1K

7.3

9.8

6.1

8.7

8.5

7.2L

8.2

7.5

7.1

9 8.8

7.3M

5.3

5.9

5.2

9.4

9.1

8.1N

7.7

7.2

6.8

9.7

9.3

8.3O

8.3

7.6

6.9

9.8

9.8

8.3P

mea

n

7.68

7.77

6.64Q

Siob

han’

s da

ta


Resource sheet 33


3REvaluating results

Interpret anddiscuss data

Collect data from avariety of sources

Specify theproblem and plan

Process andrepresent data

Evaluate results

Checklist● Are your results fair?

Did you take all the relevant steps to minimise bias when you collected the data?● Is the data representative of all the possible information that could have been

chosen?● Were there any problems with the data?

(Data that did not seem to ‘fit in’ with the rest, or ‘outliers’ that would haveaffected calculations.)

● How did you solve these problems?● How could you continue or extend your investigation?

(Any further hypotheses or areas of investigation suggested by the results.)● If you were doing the investigation again, what changes would you make to

your original plan?(Changes to improve the way the investigation was carried out, with the benefitof hindsight.)


Resource sheet 34


3RReport checklistIntroduction

● Project title● An explanation of the problem● Your hypothesis

Planning

● What information you needed● The sample size (amount of data collected)● An explanation of why you chose to collect this amount of data● How you planned to minimise bias● Any special equipment you needed

Collecting the data

● How you collected the data● Where you collected the data● The accuracy of your data (e.g. to the nearest cm) and an explanation as to why

you chose this

Processing and representing the data

● Any necessary calculations● Relevant diagrams (with an explanation as to why a particular diagram was

appropriate)● Tables of results

Interpreting and discussing the data

● A link from your analysis back to your original hypothesis● Evidence to back up any statements that you make


Resource sheet 35


3RUnit circle

1

y

x1

60°

00

X

P

Sine and cosine curves

�1

1

y

x90°0 180° 270°

cosine

sine

360°


Resource sheet 36


3RTrigonometry practice

a) 0.537 b) 0.277 c) 0.105 d) 0.22e) 0.63 f) 0.894 g) 0.947 h) 0.999i) 0.002 j) 0.5 k) 0.005 l) 0.693


Resource sheet 37


3R3-D calculations

2

5

3 2d

d4

3d

2

4


Resource sheet 38


3R

2 3 2 2 3 3 2 3 2 2 3 4 2 3 2 2 2 31 2 4 4 2 4 4 3 2 2 3 1 3 2 2 2 3 22 3 3 4 2 2 2 1 2 2 3 4 4 2 2 2 3 32 3 3 1 3 3 1 3 2 3 3 2 3 3 4 2 2 33 3 3 3 4 4 3 2 3 2 3 2 3 4 3 2 3 32 4 4 3 2 1 4 1 2 4 3 4 2 3 3 4 2 21 2 3 1 1 2 4 2 3 1 2 2 3 3 2 3 1 34 1 2 1 2 3 3 1 2 1 1 4 2 3 2 1 3 22 2 3 3 2 3 2 1 3 4 3 3 3 3 3 2 4 33 1 2 2 2 3 2 4 1 3 1 1 3 3 3 4 2 23 3 3 3 1 1 4 2 1 2 2 3 2 3 4 3 3 42 3 4 4 1 2 4 2 4 2 1 2 1 3 2 2 4 22 4 3 4 2 1 2 2 4 2 1 2 4 3 4 1 4 13 3 4 2 1 4 3 2 1 2 1 1 2 3 3 2 2 23 4 3 4 2 3 3 2 3 1 1 1 1 4 2 3 1 14 3 2 4 1 3 4 4 3 2 1 3 3 3 3 2 3 24 4 4 2 3 2 2 2 4 2 3 2 2 1 2 3 1 43 3 4 1 2 2 2 4 2 3 1 2 4 2 3 2 3 22 3 2 4 4 2 1 4 2 3 4 1 3 3 2 3 3 11 2 2 2 1 1 2 1 3 4 1 3 2 3 1 1 4 33 4 3 3 3 3 2 2 3 1 2 1 2 2 3 3 3 22 2 3 3 2 3 3 2 3 3 2 4 3 3 2 3 2 13 3 1 2 3 2 2 2 4 3 1 3 3 3 3 2 3 12 2 2 2 1 3 3 2 4 2 1 4 2 1 2 4 3 24 2 3 3 2 1 2 1 2 1 1 3 4 1 4 2 4 3


Resource sheet 39


3RRavi’s results

Neil’s results

Number of heads 0 1 2 3 4

Theoretical frequency 10 40 60 40 10

Experimental frequency 9 43 58 38 12

Number of throws 20 40 60 80 100

Experimental probability 0.35 0.4 0.42 0.36 0.36

1.9

2

20 40 60 80 100 120 140 160

1.95

2.05

Nu

mb

erof

hea

ds

Number of trials

Mean number of heads


Resource sheet 40


3R


Resource sheet 41


3R


Resource sheet 42


3R

y

x

2

4

6

8

10

�2

�4

�6

�8

�10

1 20�1�2


Resource sheet 43


3R

0 1�1�2�3�4�5�6�7�8�9�10 2 3 4 5 6 7 8 9 10 x

1

�1�2�3�4�5�6�7�8�9

�10

23

456789

10y


Resource sheet 44


3R


Resource sheet 45


3R

C

F

E

B

D

A


Resource sheet 46


3RContainers

A B C

G

0

H

0

I

0

Graphs


© Mundher Adhami, David Johnson and Michael Shayer, 1998 47

3R

Tangent ratioThe tangent ratio is how the opposite and the adjacent sides of a right-angled triangle are related.You need paper, a protractor, a ruler and a calculator.

1 What is your angle? ______Draw a right-angled triangle with this angle so that the side adjacent to your angle and the rightangle is 5 cm.

2 Find the ratio of the opposite side to the adjacent side. ______(Round the ratio to one decimal place.)

3 Double the length of the adjacent side to make the triangle bigger.What happens to the opposite side?

What happens to the ratio of the opposite side to the adjacent side?

4 For this question, estimate or calculate before drawing.If you make the adjacent side 15 cm, what will the opposite side be?Explain.

Check your answer by drawing and measuring.

5 For this question, estimate or calculate before drawing.If you make the opposite side 6 cm, what will the adjacent side be?Explain.


6 Do you have a general way of finding the missing side of a right-angled triangle if you knowone other angle and either the adjacent or the opposite side?Explain.

Thinking Maths 1Triangle ratios Notesheet 1

C Think Maths 047-061.qxd 20/4/05 11:41 am Page 47

© Mundher Adhami, David Johnson and Michael Shayer, 199848

3R

Sine ratioThe sine ratio is how the opposite side and the hypotenuse of a right-angled triangle are related.You will need paper, a protractor, a ruler and a calculator.

1 Measure the opposite side and the hypotenuse of one of your triangles and find their ratio. This iscalled the sine ratio.Round the value to one decimal place and write it on the board.Complete the class table:

2 The angle 45° is not in the table.Use the table to estimate the tangent ratio and the sine ratio for the angle 45°.

tan 45° � ______

sin 45° � ______

Draw a triangle with 45° and a right angle.Check your estimates by measuring and calculating.

3 Estimate or calculate before drawing.If the hypotenuse in your 45° triangle is 12 cm, what will the opposite side be?Explain.


4 Look at the table for the tangent and sine ratios.What do you notice?

What do you expect for angles smaller than 10° and for angles larger than 80°?

Thinking Maths 1Triangle ratios Notesheet 2

Angle Tangent ratio Sine ratio

�o

ad

p

j

p

a

o

c

s

e

i

n

te

t� �

hy

o

p

p

o

p

t

o

e

s

n

i

u

te

se�

10°

20°

30°

40°

50°

60°

70°

80°



3R

Chains and pendantsA jewellers’ shop sells gold pendants with gold chains that can be of any length.

A pendant weighs 12 grams, while the chain weighs 0.5 grams per centimetre.

The price per gram of gold changes from day to day.Today the price is exactly £7 per gram.

1 a) Choose a length of chain in cm. ______Work out the total cost of gold in your pendant. ______

b) Write in words how you can work out the cost of any length chain.

2 Write the cost of a gold pendant with any length chain using symbols.

3 How many chunks are there in your expression?______Write the meaning of each chunk in your expression.(Use arrows, or rings, or colours.)

4 The cost of gold in one pendant with chain is £224. Work out its chain length. ______Explain your working, step by step. (You may want to make an equation.)

5 The jeweller charges £15 extra for carving your initials on the pendant.Calculate the price of your pendant and chain with your initials carved. ______Write a mathematical expression for the price for any pendant with initials carved.

6 The shop makes special pendants in groups of 10 with chain of length L.Write a mathematical expression for the total price of gold only for any group of 10 pendants.

Thinking Maths 2Chunking and breaking up Notesheet 1



3R

One pair and two pairs of brackets

A workshop makes silk scarves for a shop.The shop decides the final length and width required for each scarf. Call these l and w.The workshop calculates the starting size of the rectangle of silk by adding 18 cm to the length (forthe tassels) and 4 cm to the width (for the edging).It charges the shop by the area of the starting size of the rectangle of silk used in the scarf.

1 Write an expression for the starting width for any size scarf.

Write an expression for the starting length.

2 Write a mathematical expression for the starting area for any size scarf.

3 Multiply out your answer to question 2 to give four chunks in a new expression.

Write down the meaning of each chunk. (Use arrows, or rings, or colours.)

4 The price of silk itself ranges between 25p and 90p for each 100 square cm, depending on itsquality.Estimate a price for silk of an average quality: ______ per 100 square cm.Choose a final width of scarf ______ and a final length of scarf ______.Calculate the cost for the starting size of the rectangle of the silk needed.

5 The workshop charges £8 per scarf for working, and adds this to the price of the silk.Write the expression (not the calculation) for the price of 12 of your chosen scarves.

6 Which of the numbers in your expression can be made into variables, so that they can bechanged from time to time, keeping the expression the same? (Make sure you keep track of whateach letter you use stands for.)

Thinking Maths 2Chunking and breaking up Notesheet 2

l?

w ?



3R

Steady rise and acceleration

The values from the two tables are plotted on the graph along with new values up to 12 seconds.

How would you describe the difference in the way the balloon and the rocket rise?Think about and discuss this with your partners, using words like speed and acceleration beforesharing with the class.

Thinking Maths 3Accelerating the acceleration Notesheet 1

Time Height Change in(seconds) (metres) height

0 0 –

1 6 6

2 12 6

3 18 6

4 24 6

5 30 6

6 36 6

Weather balloon height

Time Height Change in(seconds) (metres) height

0 0 –

1 1 1

2 4 3

3 9 5

4 16 7

5 25 9

6 36 11

Rocket height

00 1 2 3 4 5 6

Time (seconds)

Hei

ght (

met

res)

7 8 9 10 11 12

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

BalloonRocket



3R

Use a calculator. Use approximate answers using the first two digits followed by zeros.

1 Fill in the middle column in the table on Notesheet 3 by doubling each time until the weight ofthe grain reaches one tonne (2 000 000 grains).

2 The king’s grain store contained 1000 tonnes. The astronomer calculated that by the 31st square,the whole of the king’s grain would have been needed!By continuing the table, or by another method, find out how he did this calculation, and explainwhy his calculation was right.

What accelerating an acceleration means3 Fill in the fourth and fifth columns down to about square 8. Look at the pattern of increases.

Discuss with your partners how you can explain in words the difference between this and thepattern of increases for the rocket on Notesheet 1.




3R

The grain mountain grows!


Square Number of doubling Number of grains Increase Increase of the(2�) increase

1 0 1 – –

2 1 2 1 –

3 2 4 2 1

4 3 8 4 2

5 4 16

6 5 32(2 � 2 � 2 � 2 � 2)

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34



Thinking Maths 4Graph of the rotating arm Notesheet 1

3R

Sine of angle

1 Find the sine ratios for the angles from 0° to 90°. Write them in the table.

2 a) Plot the graph of the sine ratio for the angles 0°–90°.b) Describe how the sine ratio changes with the angle.

3 Find the sine ratios for the second quarter-turn. Plot them on the same graph.

4 Describe how the sine ratio changes for the angles 90°–180°.

5 Find the sine ratios for these angles.

6 a) Plot the graph of the sine ratio for the angles 180°–360°.b) Describe how the sine ratio changes with the angle for these values.

90°

270°

180° 0°

10°

20°

30°

Angle (°) 0 10 20 30 40 50 60 70 80 90

Sine

Angle (°) 100 110 120 130 140 150 160 170 180

Sine

Angle (°) 190 200 210 220 230 240 250 260 270

Sine

Angle (°) 280 290 300 310 320 330 340 350 360

Sine



3RThinking Maths 4Graph of the rotating arm Notesheet 2

7 Fill in this table.

What does an angle which is more than 360° mean?

8 Fill in this table.

What does the sine of �10° mean?

9 How would you find the sine of 1000°?

10 What angle has a sine of 0.5? ______ Find another. ______

Angle (°) 370 380 390 400

Sine

Angle (°) �10 �20 �30

Sine

�40 8040 120 160 200 240 280 320 360 400 Angle°

�1.0

�0.8

�0.6

�0.4

�0.2

0.2

0

0.4

0.6

0.8

1.0

Sine ratio



3RThinking Maths 5Straight-line graphs Notesheet 1

Name and number the two axes so that you can use them easily.

1 Mark the two points (2, 1) and (5, 1) and draw the line they lie on.Mark two other points on the line, which are not on a corner of a square.Your points are: (______, ______) and (______, ______)Make a short clear description of this line so someone could draw it without seeing it.

Use mathematical or ordinary language, without reference to the two points.

2 Mark the two points (4, 6) and (4, �2) and draw the line they lie on. Use a different colour foreach line.Two other points on this line are (______, ______) and (______, ______)Make a short clear description of this line so someone could draw it without seeing it.

Use mathematical or ordinary language, without reference to the two points.

3 Choose a point not too far from the origin (0, 0). (______, ______)Label this point P.Draw the vertical and horizontal lines that go through this point.Draw the two vertical and horizontal lines that are twice as far as these two lines from the origin(0, 0) in each direction.Where do the new lines meet?Label this point Q.How far is point Q from the origin?




Slope and intercept

1 a) Draw the line on which (�1, �1) and (5, 5) lie.Find two other points on it: (______, ______) and (______, ______)Write the best mathematical description you can for this line:

b) Draw a second line, on which (1, 3) and (5, 7) lie.Does the point (3, 5) also lie on it? ______Compare this line to the first line. What is the same and what is different?

c) Draw a third line, on which (2, 5) and (0, 3) lie.What is the same about the three lines? What is different?Try to describe it mathematically.

2 a) Go over the third line in a different colour. Continue with that colour for the next questions.A fourth line passes through (0, 3) and (2, 7).Compare this to the third line. What do you notice?

b) A fifth line passes through (2, 4) and (6, 6). Describe this line.

c) Draw your own sixth line through (0, 3).Compare the last three lines (questions 2a, 2b and 2c).




Extension: Shooting starsAn object follows a path described by the equation y � 0.5x � 1.Two straight-line shooters are placed on the y-axis at y � 3 and y � 7. They are to shoot the objectafter it passes the line x � 6 and before it passes the line x � 8.Sketch with pencil the lines you need and find equations of possible paths of the two shots.

If you are using a graphical calculator, you may be able to estimate and improve your graph withouthaving to calculate.

If your graphical calculator allows you to plot three or more graphs at one time then you could watchhow the three lines intercept each other and improve your values to make them cross at one point.



3RThinking Maths 6How do I handle the data? Notesheet 1

Here are the results of three investigations, to be analysed and compared in the same lesson.You need to handle the data to show the meaning of each set.First discuss each investigation and how best to deal with its data, without doing the work fully.After you have looked at all three investigations, carry out your plans.

1 GCSE English results for boys and girls

Newspapers say that girls are doing better than boys at GCSEs in English.Maybe this is true. Or maybe there is more to it than that.Here is a part of a school’s results, with the pupils’ names omitted.

How can you handle this data?Think of possible ways, then decide on one.Remember that you are checking what the newspapers are saying.

Sex Grade Sex GradeG A G GG C G CG D G EG D G CG B G G

G E G GG A G EG C G DG D G FG C G F

G C G BG D G FG D G DG D G EG B G E

G D G EG C G FG F G DG u G AG D G D

G B G BG E G CG DG CG D

Sex Grade Sex GradeB D B BB F B DB A B DB F B FB F B B

B F B uB B B GB B B GB C B CB C B E

B D B EB B B EB C B EB E B FB E B F

B D B DB C B GB C B GB C B uB D B A

B E B FB E B AB EB BB D




2 A difficult day in the science lab

Your science teacher is demonstrating a chemical reaction.The reaction is between two chemicals and at different temperatures. When the two chemicals aremixed the liquid turns black.The reaction is complete when the mixture suddenly turns clear and colourless.You record the time the reaction takes, in seconds.

The teacher first mixes equal amounts of the two chemicals.Here is the data for different temperatures:

The teacher then doubles the amount of the first chemical, keeping the amount of the second thesame.Here is the data for this experiment:

How should you present the data for both experiments?

Remember that you want to show as much as possible of the meaning of the data, so you need tothink about both the temperature and the change in the amount of the first chemical.

Temperature (°C) 20 30 40 50 60 70

Time (seconds) 230 115 58 28 14.5 7

Temperature (°C) 20 30 40 50 60 70

Time (seconds) 116 59 29 15 6.5 3.5




3 ‘How good you are at English is a good predictor of how good you are in othersubjects.’ True or false?

Someone says ‘You can predict how well girls will do in science just by looking at how well they didin English.’ Check how true this statement is, then find how well it works for boys.

Look at the table below levels awarded to two groups of pupils at KS3.First think of possible ways of handling this, in discussion with your partners.

Girls English Sciencelevel level

3 13 34 24 34 5

5 25 45 45 45 5

5 55 76 36 46 4

6 56 56 56 76 7

6 87 77 88 8

Boys English Sciencelevel level

3 13 43 54 14 2

4 44 44 64 75 2

5 25 35 55 75 7

5 85 86 26 46 7

6 87 57 67 8


© Harcourt Education 2005 © Harcourt Education 2005

3R3R1.1 Quadratic sequencesExamplea) Find the first four terms of the sequence T(n) � 2n2 � 3.b) Use your answer to part a) to find the first four terms of the sequence

T(n) � 2n2 � 8.

Exercise� a) Find the first four terms of the sequence T(n) � 3n2 � 1.

b) Use your answer to part a) to find the first four terms of the sequenceT(n) � 3n2 � 4.

c) What are the first five terms of the sequence T(n) � 4n2?d) Use your answer in part c) to find the first four terms of the sequence

T(n) � 4n2 � 6.

� Find the 7th, 10th and 12th terms of the sequence given in part a) of Q1.

� If I knew the first five terms of the sequence T(n) � n3, what would I do to eachterm to find the following sequences?

a) T(n) � 5n3 b) T(n) � �n2

3

� c) T(n) � 2n3 � 4

� Copy and complete the following table for each value given.

� Write down the first four terms for each of the following sequences, then findthe differences between consecutive terms.a) T(n) � n2 b) T(n) � n2 � 3 c) T(n) � n2 � 5

1.2 The general termExample Find the general term of the sequence 3, 6, 11, 18, 27, …

Exercise� Find the general term for the following sequence: 14, 17, 22, 29, 38, …

� Which of the following sequences are linear and which are quadratic?a) 4, 7, 10, 13, 16b) 4, 7, 12, 19, 28c) 4, 6, 8, 10, 12

� Find the general term for each of the sequences given in Q2.

� Find the general term for each of the quadratic sequences given below.a) 1, 4, 9, 16, 25 b) 2, 5, 10, 17, 26c) 2, 8, 18, 32, 50 d) 3, 9, 19, 33, 51

� For each of the following sequences, find the first five terms and then calculatethe first and second row of differences.a) T(n) � n2 � 3 b) T(n) � 2n2 � 4c) T(n) � 3n2 d) T(n) � 4n2 � 1

� Compare the second row of differences in each of the parts in Q5 with thegeneral term and comment on what you notice.

Substitute n = 1 for the first term, n = 2 for thesecond term and so on. Remember to followthe correct order of operations.

a) T(1) � 2 � 12 � 3 � 2 � 3 � 5T(2) � 2 � 22 � 3 � 8 � 3 � 11T(3) � 2 � 32 � 3 � 18 � 3 � 21T(4) � 2 � 42 � 3 � 32 � 3 � 35

b) 10, 16, 26, 40 We just add 5 onto the values of T(n) � 2n2 � 3 to get T(n) � 2n2 � 8.

If the numbers in the first row of differences arenot the same, work out the second row ofdifferences.

As the second row of differences are all the samevalue we know this is a quadratic sequence.

Compare the quadratic sequence T(n) � n2 withthe given sequence.

Term 1 2 3 4

T(n) � n2

T(n) � 2n2 � n

T(n) � 3n2 � 3

The differences between theconsecutive numbers in the givensequence are: 3, 5, 7, 9The second row of differences is:2, 2, 2The sequence T(n) � n2 is: 1, 4, 9, 16, 25, …The given sequence is: 3, 6, 11, 18, 27, …Each term has 2 added, so thesequence has general termT(n) � n2 � 2

A linear sequence will have the same differencesbetween each consecutive term.

D Homework sheets 062-073.qxd 20/4/05 11:44 am Page 62

3R 3R


1.3 Special sequencesExample Find the general term of the following

sequence: �14�, �

47�, �1

90�, �

1163�

Exercise� Find the general terms of each of the following sequences.

a) �13�, �

24�, �

35�, �

46�, … b) �

12�, �

28�, �1

38�, �3

42�, … c) �

23�, �

26�, �1

21�, �1

28�, …

� Using the general terms you worked out in Q1, find the 12th term for each ofthe three sequences.

�

a) Draw the next two patterns in this sequence.b) Copy and complete the following sentences:

The sequence comprises of one block of 4 and ………… .

The general rule for the total number of dots in this sequence is ………… .

� In the first week of term Al got �12� of his homework questions correct. In the

second week he got �23� correct, in week 3 he got �

34� correct, and so on.

a) Find a general rule for the sequence.b) What fraction of his questions would he get correct in

i) week 4 ii) week 7 iii) week 9?c) Write your fractions in part b) as decimals.d) If this sequence continues, would Al ever get all of his homework right?

Explain your answer.

Study the numerators and thedenominators separately.

1.4 FormulaeExample a) Derive a formula for the cost C (in £) of n items if each item

costs 72p.b) Rearrange the formula to find n if the total cost is known.c) Use part b) to find how many items can be bought for £2.88.

Exercise� A taxicab charges £2 for the call-out plus 40p per mile.

a) Write a formula to find the total cost C (in £), using m for the number of miles.

b) Rearrange the formula to make m the subject.c) Betty wants to travel 16 miles. How much would the journey cost her?d) Abdul has £5. What is the maximum distance he could travel?

� Rearrange the formula y � 3x � 12 to make x the subject.

� The instructions for cooking a chicken give the cooking time as‘35 minutes per kg plus 25 minutes’.

a) Write this as a formula using T (total time) and K (mass of the chicken).b) Calculate the total cooking time of a 3.5 kg chicken in hours and minutes.c) Sue says it took 2 �

34� hours to cook her chicken.

What was the mass of Sue’s chicken i) in kilograms ii) in grams?

� A regular hexagon fits inside a circle of radius r cm.a) If a regular hexagon is split into six triangles

from the centre of this shape, what type of triangles will they be? Explain your answer.

b) Write a formula for the perimeter of the hexagon.

The numerators are 1, 4, 9, 16, which has the general rule T(n) � n2.The denominators are 4, 7, 10, 13, which has the general rule T(n) � 3n � 1.

The general term is T(n) � �3n

n�

2

1�.

a) C � �17020n

� b) n � �10

702

C� c) 4

Multiply the number of items n by the cost per item,i.e. 72p; then, divide by 100 to find the cost in £.

Make sure that yourmonetary units are the same.

Convert the time intominutes first.



3R3R1.6 GraphsExample a) Rearrange the equation y � 2x � 2 � 0 to make y the subject.

b) Graph your equation in part a), for values of x from �2 to �3.

Exercise� On graph paper, plot the following graphs, using values of x from �5 to �5.

a) y � 2x � 3 b) y � x � 4 � 0 c) y � �x2

� � 6

� Copy and complete this table of values for the equation y � �x3

� � 12:

� Rearrange these equations to make y the subject.a) y � 3x � 2 � 0 b) 2x � 3y � 7 c) �

y4

� � 3x � 2 � 0

� Find the gradient and y-intercept for each of the functions listed in Q3.

Choose at leastthree valuesfor x andcalculate thecorrespondingvalues for y.

a) y � 2x � 2

b) x �2 0 3

y � 2x � 2 y � 2x � 2 y � 2x � 2

y � (2 � �2) � 2 y � (2 � 0) � 2 y � (2 � 3) � 2

y � �4 � 2 y � 0 � 2 y � 6 � 2

y �6 �2 4

Use the values in the table todraw a suitable pair of axes.

y

x0�2�4�6

2

4

�3 �2 �1 1 2 3

6

y � 2x � 2

x 0 6

y 0 �3

1.5 The inverse of a linear functionExample a) Draw the function machine to illustrate y � 2x � 5.

b) What is the inverse of the function y � 2x � 5?

Exercise� Draw function machines to illustrate the following functions:

a) y � x � 5 b) y � 3x � 7

c) y � �x4

� � 1 d) y � 3 � 2x

� Find the inverses of each of the functions in Q1.

� To change an amount in pounds to an amount in US dollars the followingfunction is used:

US$ � 1.78 � £

a) Explain in words how you would convert an amount in US dollars intopounds.

b) Illustrate your answer to part a) as a function machine.c) Using your function, find which is the greater amount: £2.50 or US$4.50.

� To convert a temperature in °C (Celsius) into a temperature in °F (Fahrenheit),the following function is used:

T(°F) � �T(°C

5) � 9� � 32

a) Illustrate the function as a function machine.b) Use the function to find a temperature in °F if the temperature is 21°C.c) Find the inverse of the function. d) Using your inverse function in part c), find what temperature in °C is the

same as a temperature of 82°F, giving your answer correct to 1 d.p.

Remember that 3 � 2x isthe same as 2x � 3.

Here we mean that £1 is US$1.78.

Here T(°F) is the temperature in °F.

a) x → → → y

b) �y �

25

� � x

�5�2 Multiply the value of x by 2, then subtract 5.

To find the inverse, carry out the inverseoperations in reverse order.


3R 3R


2.1 Adding and subtracting fractionsExample Work out the following fractions:

a) �67� � �

27�

b) �35� � �

14� � �1

10�

c) �2bca� � �

3cda�

Exercise� Work out the following fractions:

a) �58� � �

78�

b) �23� � �

37�

c) 1�25� � �

47�

� What is the LCM of each of the following?a) 4 and 6 b) a and h c) 4 and dd) 4d and 2c e) 4de and 6ce f) 8y and 3y2

� Simplify these fractions:

a) �ba

� � �db

� b) �ba

� � �bcd� c) �

56c� � �

23c�

� Simplify the following:

a) �23xyy

� � �63yx

2� b) �46

xy� � �

32xxy2�

a) �67� � �

27� � �

87� � 1�7

1�

b) �35� � �4

1� � �10

1� � �2

120� � �2

50� � �2

20�

� �290�

c) �2bca� � �

3cd

a� � �

2bcadd

� � �3bc

adb

� ��2ad

b�

cd3ab

�

The Lowest Common Multiple (LCM)of 5, 4 and 10 is 20.

If the denominators are the same just add the numerators.

The LCM of bc and cd is bcd.

In c) change the 1�25� to an improper fraction first.

You can simplify a fraction bycancelling. Look for a factor thatis common to both numerator anddenominator.

2.2 Multiplying fractionsExample Multiply these fractions:

a) �56� � �

38� b) 2�

12� � 2�

13� c) �

4yx� � �

35y�

Exercise� Complete the following multiplications:

a) �34� � �

57� b) �1

30� � �

59� c) 1�

14� � 2�

45� d) �

34� � �1

85� � �

52�

� Simplify these fractions by cancelling:

a) �2306� b) �

4108� c) �

26acc

� d) �42xx

2

py

� e) �23

xx

3

2�

� Complete the following multiplications of algebraic fractions:

a) �2pa� � �

3rc� b) �

32raht

� � �9art�

c) 1�45� � �

x3

� d) �p3

3

� � �1p2q� � �

54q�

� The circumference of a circle may be calculated using the formula C � �272� � D,

where D is the diameter.Find the circumference of circles with the following diameters.

a) 4 cm b) r cm c) �131t� cm

Always change mixed numbers to improperfractions first when multiplying.

Cancel any numerator and any denominatorby the same number, in this case by 3.

The common factor y is cancelled.

a) �65

� � �83

� � �156�

b) 2�21

� � 2�31

� � �52� � �3

7�

� �365� � 5�6

5�

c) �4yx� � �

35y� � �

125

x�

2

1

Remember to cancel where possible.

Remember to show the units.



3R3R2.3 ReciprocalsExample Find the reciprocals of the following:

a) �35� b) 6 c) 0.3

Exercise� Find the reciprocals of the following numbers:

a) �34� b) 8 c) 1�

14� d) 1

� Find the reciprocals of these numbers:a) 0.4 b) 1.3 c) 2.87 d) 0.3

� a) Find the reciprocal of �170�.

b) Now multiply �170� by its reciprocal.

� Copy and complete the following:

a) �23� � � 1 b) �

cad� � � 1

c) 3�14� � � 1 d) � 2�

12� � 1

� a) Copy and complete the following table.

b) As the value of y increases, what happens to the value of �1y

�?

The reciprocal of �a

b� is �

b

a�.

Change decimals to fractions first; then,find the reciprocal of the fraction.

6 is the same as �61�.

a) �35

�

b) �61�

c) �130�

y 0.2 0.4 0.6 0.8 1

�1y

�

2.4 Dividing fractionsExample What is the value of

a) �35� � �

23�

b) 2�14� � 1�

23�?

Exercise� Calculate:

a) �15� � �

38� b) �

47� � �

34� c) �

58� � �

59� d) �

34� � �

14�

� Work out the following divisions.a) 1�

34� � 2�

15� b) 3�

34� � 2�

13� c) 4�

12� � 3�

34� d) 1�

14� � 2�

12�

� Complete the following divisions.

a) �acb� � �

dt

� b) �cah� � �

ahf� c) �

ac

2

dc

� � �aad� d) �

ba

� � �ba

�

� How many times does �aec� go into �

eahf�?

� Find the missing expression in the following equations.

a) �acef� � �

�ae

� � �kc

b) �12

51

xy

� � �5�xz� � �

3zy2

�

Convert mixed numbers to improperfractions first when dividing.

To divide by a fraction, simply multiply by its inverse.

a) �35� � �3

2� � �

35� � �

32�

� �190�

b) 2�41

� � 1�32

� � �94� � �3

5�

� �94� � �

35�

� �2207�

� 1�270�


3R 3R


2.5 Mental methodsExample a) Multiply 0.27 by 0.4.

b) How many items costing 80p each can be bought for £25.60?

Exercise� Calculate the following mentally.

a) 0.46 � 0.5 b) 0.23 � 0.3 c) 0.048 � 0.8 d) 0.72 � 0.6

� Calculate the following mentally.a) 28 � 0.7 b) 8.48 � 0.8 c) 20.4 � 0.6 d) 12 � 0.4

� Estimate the following mentally.a) 0.497 � 0.58b) 0.742 � 0.37c) 0.421 � 0.78d) 2.41 � 0.392

� Given that 7.9 � 2.8 � 22.12, what are the following?a) 7.9 � 28 b) 0.28 � 7.9 c) 0.79 � 0.28 d) 28 � 79

� Given 12.6 � 0.3 � 42, what are the results of the following calculations?a) 126 � 0.3 b) 126 � 3 c) 0.126 � 30 d) 12.6 � 42

a) 0.27 � 0.4 � 27 � 4 � 1000� 108 � 1000� 0.108

b) 25.60 � 0.80 � 25�160� � �1

80�

� �21506

� � �180�

� �25

86

� � 32

1

1

Change decimals to whole numbers.

Change decimals to fractions, thenmultiply, remembering to cancel.

As this is an estimate, round the numbersto 1 d.p., then work out mentally.

2.6 PercentagesExamplea) In a ‘15% off’ sale an item is priced at £6.80. Find the original price.b) My £400 savings earn 8% interest each year. Find the total value of my

savings, including interest, after i) one year ii) two years.

Exercise� The value of a house has risen by 10% in one year. If the value of the house at

the end of the year is £104 500, what was the value at the start of the year?

� Change these amounts by the given percentages.a) Increase £350 by 7%. b) Decrease £4.20 by 20%.c) Decrease £82 by 6%. d) Increase £75 by 11%.

� Copy and complete this table, stating both the percentage change and whetherthis was an increase or a decrease.

� Find the value after 3 years of £5000 earning 6% annual compound interest.

� A pack of soap powder costs £2.45. Another pack of the same powder contains20% more and costs £2.80. Which of the two packs is the best buy?

An item reduced by 15% willhave a value of 85% of theoriginal price. Similarly, anitem increased by 15% willhave a value of 115% of theoriginal price.

Use the value at the end ofone year as the startingvalue for the next year.

Original Price 640 15 160 150

New Price 448 18 140 195

% change

a) £6.80 represents 85% of the original value.£6.80 � 85 represents 1% of the original value.£6.80 ÷ 85 � 100 represents 100% of the original value.£6.80 � 85 � 100 � £8.00

b) Total value of savings

i) after one year: £400 � 1.08 � £432.

ii) after two years: £432 � 1.08 � £466.56.



3R3R2.7 Direct proportionExampleBen returns from his holiday in the USA. He has US$171 left and changes it intopounds. He receives £95.a) What is £1 worth in US dollars? b) What is US$1 worth in pounds?c) Write the relationship between pounds and US dollars in the form y � kx.

Here y represents US dollar amounts and x represents amounts in pounds.

Exercise� In preparation for his holiday, Zak changes £140 into euros, receiving €203.

a) What is £1 worth in euros?b) Write this in the form y � kx, where y represents the amount in euros and

x represents the amount in pounds.c) Is the number of euros directly proportional to the number of pounds?


� This table shows the cost of packets containing different numbers of brackets.

a) Is the cost in direct proportion to the number of brackets?b) What is the value of the constant in y � kx, where y is the amount paid in

pounds and x is the number of brackets?

� Ashok and Shivi are studying a map with a scale of 5 cm to 20 km.a) Ashok says that this is a ratio of 1:40 000. Is he correct?b) Shivi calculates that a distance of 4 km would measure 0.5 cm on the map.

Ashok disagrees. Who is right? Explain your answer.c) Two towns are 65 mm apart on the map. How far is this in km?

a) £1 � US$(171 � 95) � US$1.80

b) US$1 � £(95 � 171) � £0.56c) y � 1.8x k is a constant value.

We say that x is directly proportionalto y. This is written x � y.

Number of brackets 100 200 500 1000

Cost (£) 3.50 7 17.50 35

2.8 Inverse proportionExampleA team of 6 bricklayers can complete the walls of a new house in 5 days.a) To complete the same job, how long would it take a team of

i) 2 bricklayers ii) 3 bricklayers iii) 10 bricklayers?b) What is the constant of inverse proportion in this problem?

Exercise� A hotel has enough breakfast cereal to feed 20 guests for six days.

a) If they had 30 guests, how long would the breakfast cereal last?b) If they had 50 guests, how long would the breakfast cereal last?c) Do they have enough cereal to last 15 guests eight days?d) What does the constant of inverse proportion represent?

� The Year 9 mathematics exams are to be marked by a team of six teachers.They will each spend six hours on the work. One of the teachers is absent.How long will it take the remaining five to complete all the marking?

� A rectangle has an area of 20 cm2. Copy and complete the following table toshow the missing lengths and widths.

� Jim drives from Newcastle to Edinburgh in 2 hours, averaging 80 km/h.a) How long would the journey take at an average speed of 60 km/h?b) If the journey takes 2�

12� hours, what is his average speed?

c) If the journey was represented using y � k � �1x

�, where y is the speed and x

is the time, what would the value of k be?

The number of days is inversely proportional to thenumber of bricklayers. (As the number of bricklayersincreases, the number of days decreases.)

a) i) 15 days ii) 10 days iii) 3 daysb) 30

In inverse proportion

y � k � �1

x�, where k is

constant. In this case, yrepresents the number ofdays and x the number ofbricklayers.

Length 5 cm 1 cm

Width 10 cm 2.5 cm


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2.9 RatioExamplea) In Ben’s class at school there are 14 boys and 12 girls, but in Julie’s class there

are 12 boys and 10 girls. Which class has a greater proportion of girls?b) The areas of these two similar

triangles are 16 cm2 and 25 cm2.i) What is the ratio of the side

lengths?ii) What is the scale factor of

enlargement from the smaller triangle to the larger one?

Exercise� A picture measures 24 cm by 16 cm. It is enlarged by a scale factor of 1.5.

a) Give the ratio of the lengths of the original and the enlargement.b) Find the ratio of the areas of the original and the enlargement.

� Change these ratios into the form 1 : n.a) 3 : 2 b) 4 : 5 c) 5 : 1 d) 512 : 620

� Judy’s baby cousin has two sets of cubes, which have side lengths as shown.a) What is the ratio of the surface areas of the

two cubes?b) What is the ratio of the two volumes?

� Two cubical boxes have volumes 120 cm3 and 60 cm3

respectively.a) Work out the ratio of the lengths of the two boxes?b) What is the ratio of the areas of a corresponding face of the two boxes?c) By what scale factor is the larger box an enlargement of the smaller one?

a) In Ben’s class, boys : girls � 14 : 12 � 1 : 0.857In Julie’s class, boys : girls � 12 : 10 � 1 : 0.83Ben’s class has a greater proportion of girls.

b) i) Length ratio � �16� : �25� � 4 : 5

ii) 4 : 5 � 1 : 1.25, so the scale factor of enlargement is 1.25.

There are 0.857 girls toeach boy in Ben’s class,but just 0.83 girls to eachboy in Julie’s class.

If lengths are in the ratio a : b,areas are in the ratio a2 : b2 andvolumes in the ratio a3 : b3.

2 cm

5 cm

3.1 Constructing and solving equations 1ExampleThere are 820 pupils at Nina’s school; there are 64 more girls than boys.a) Write an expression for the total number of pupils at Nina’s school in terms of

b, where b represents the number of boys.b) How many boys and girls are there at Nina’s school?

Exercise� It costs £17.50 for two adults and three children to go to the cinema. Each

adult is charged £2.50 more than each child.a) Write an expression for the cost per adult if c is the cost per child.b) Write an expression for the total cost for all five people, in terms of c.c) Calculate the value of c, and hence state the price of admission for

i) each childii) each adult.

� These two shapes have the same perimeter.a) Find the value of x.b) Give the length and

width of the rectangle.c) Find the area of the

rectangle.

� Fenji has £20 less in savings than Chi. In total they have £300.How much does Chi have in savings?

a) b � b � 64� 2b � 64

b) 2b � 64 � 8202b � 756b � 378 so the number of boys � 378 and the number of girls � 442

The number of boys � the number of girls � 820.

b is the number of boys and b � 64 is the number of girls.

Check that the total cost � £17.50.

4x

3x

3x � 1

5x2x � 2



3R3R3.2 Solving more complex equationsExample Solve the equation x3 + 2 = 16.5 using trial and improvement

techniques, giving your answer correct to 1 d.p.

Exercise� Solve the equation x3 � 11 � 50 using trial and improvement techniques,

giving your answer correct to 1 d.p.

� a) Use trial and improvement techniques to solve the equation 3x2 � 2 � 24to 1 d.p.

b) Now solve the same equation to 2 d.p.

� Which of the following questions would you choose to solve using algebraicmethods and which using trial and improvement techniques?a) 6a � 3 � 7 b) 6a2 � 150 c) 6a2 � 0.75 � 3 d) 6a3 � 11 � 25

� Solve the equations given in Q3 using the methods you chose, giving youranswers to 1 d.p.

� The volume of this cuboid is 40 cm3.Write an equation in terms of w for this volume.Find the value of w correct to 1 d.p.

x � 2 23 � 2 � 10 Too smallx � 3 33 � 2 � 29 Too largex � 2.5 (2.5)3 � 2 � 17.63 Too largex � 2.4 (2.4)3 � 2 � 15.82 Too small

x � 2.45 (2.45)3 � 2 � 16.71 Too largex � 2.4 to the nearest 1 d.p.

Choose a value for x. Calculate for that value. Decide whether the result is toobig or too small. This helps youchoose the next value to try.

We now know that the value isbetween 2.4 and 2.5, so try thecentre of the interval todetermine the better answer.

4w

2w

w

3.3 Constructing and solving equations 2Example Jo and Betty are sisters. Jo is four years younger than her sister and

their total age is 28 years. What are their ages?

Exercise� The sum of three consecutive numbers is 78.

What are the three numbers?

� The length and width of this rectangle are x � 3 cm and x cm, respectively.If the area of the rectangle is 28 cm2, what are its length and its width?

� Jane says that when she squares a number and adds three the total is 28. Find the number Jane chose to start with.

� A cricketer has an average of 24 runs after his first three games.In the next game he scores 38 runs. What is his average over the four games?

� Chris gets 624 by multiplying two consecutive even numbers.a) If x is the smaller number, write an expression for the other number.b) What are the two numbers?

Betty’s age � Jo’s age � 28x � 4 � x � 28

2x � 4 � 282x � 24

x � 12So Jo’s age is 12 and Betty’s age is 16.

If x represents Jo’s age, Betty’s age wouldbe x � 4.

x � 3

x

Check: 12 � 16 � 28.

Decide whether to usealgebraic methods or trial andimprovement for this question.


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3.4 More than one unknownExample Solve the following pair of simultaneous equations.

r � c � 14 (1)c � 3 � r (2)

Exercise� Solve the following pairs of simultaneous equations.

a) a � e � 12 b) 2f � g � 17 c) 3h � k � 6e � a � 6 g � f � 5 k � h � 2

� Find the values of variables in these simultaneous equations:a) a � e � 5 b) 2m � n � 13 c) 3m � n � 14

e � 3 � a n � 2 � m n � 6 � 2m

� Solve these simultaneous equations:a) 3m � p � 30 b) 2r � t � 12

4m � 5 � p 3r � 13 � tc) 4m � 2e � 22 d) 3r � 4e � 27

2e � m � 2 4e � 2 � 2r

� It is quite possible for simultaneous equations to have solutions which may benegative, decimals or fractions. Solve:a) 4a � 3r � 14 b) 4e � 3t � 5

r � a � 3.5 t � 5 � 2e

c � 3 � rc � r � 3

r � c � 14r � r � 3 � 14

2r � 3 � 142r � 11

r � 5.5 and c � 8.5

Rearrange the second equation to find what cis in terms of r, which can then be substituted.

Now substitute r � 3 for c in the first equation.

3.5 Eliminating one unknownExample Solve this pair of simultaneous equations:

5x � 3y � 252x � 3y � 19

Exercise� Solve these pairs of simultaneous equations:

a) 3p � 4r � 26 b) 8a � 3b � 28p � 4r � 22 6a � 3b � 24

c) 12r � 2t � 483r � 2t � 3

� a) 3h � 2r � 342h � 2r � 6

b) 5x � 2t � 34 c) 3q � 2r � 113x � 2t � 30 14q � 2r � 6

� Sometimes answers may be fractions, decimals or negative. Solve:a) 4e � 3f � 22 b) 5h � 3k � 15 c) 3e � 2h � 16

2e � 3f � 12 9h � 3k � 21 5e � 2h � 16� Paul buys two oranges and three apples; he spends a total of 48p.

Alice buys two oranges and one apple; she spends a total of 32p.Using c to represent the cost of 1 orange and y to represent 1 apple:a) Write an equation showing Paul’s spending.b) Write an equation showing Alice’s spending.c) Solve the two equations in parts a) and b) to find the cost of

i) an orange ii) an apple.

3x � 6

x � 2

5x � 3y � 2510 � 3y � 25

3y � 15y � 5

So x � 2 and y � 5.

In part c) we have the same value of t in both equations (�2t)so again we can subtract the second equation from the first.

Subtract the second equation from the first equation.

This has eliminated the y’s, so we can find x.

Having found x, substitute this value into the first equation to find y.

This time we have 2r and �2r. To eliminatethese, the two equations are added.


4.1 ConventionsExampleFind the missing angles in this diagram, stating any angle facts that you use.

Exercise� Calculate the sizes of the missing angles, stating any angle facts that you use.

� Julie cuts the corners off a triangle and places them together to form a straightline. Is thisa) a proof that the angles of a triangle sum to 180°b) a practical demonstration that the angles of a triangle sum to 180°?

� a) Sketch at least six regular hexagons to show that this shape tessellates.b) Sketch a tessellating pattern using regular octagons and another regular

shape.

� Find all the equal angles in the following shapes.


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f � 60°(Alternate angle to the angle marked 60°.)t � 120°(Interior angle is 60°; angles on straightline sum to 180°.)e � 48°(Angles in a triangle add up to 180°.)

3.6 More simultaneous equationsExampleSolve the following pair of simultaneous equations.3x � 4y � 72 (1)6x � 3y � 129 (2)

Exercise� Practise multiplying both sides of these equations by the value given.

a) 3d � 4e � 7 (multiply by 4) b) 3e � 3r � 6 (multiply by 3)c) 5g � 3d � 12 (multiply by 3) d) 8h � 2r � 10 (multiply by 5)e) 3e � 2k � 7 (multiply by 4)

� Below are pairs of simultaneous equations.a) 8k � 3e � 28 (1) b) 12e � 2h � 48 (1)

2k � e � 8 (2) 6e � 4h � 6 (2)i) By what would you multiply equation (2) to get the same number of e’s?ii) Multiply equation (2) by that value and call your answer equation (3).iii) Now solve the simultaneous equations (1) and (3).

� Solve each of these pairs of simultaneous equations:a) 3a � 3e � 36 b) 3h � k � 6 c) 6r � 3t � 36

a � e � 2 4h � 4k ��8 3r � t � 13

6x � 8y � 144 (3)6x � 3y � 129

5y � 15y � 3

3x � 4y � 723x � 12 � 72

3x � 60x � 20 So x � 20 and y � 3.

To solve simultaneous equations, we need the same number of oneof the unknowns. To achieve this, we can multiply both sides ofequation (1) by 2, so it now has a 6x term. Call this equation (3).

Now subtract equation (2) from (3) in the usual way.

Having found the value of y substitute it intoequation (1) to find x.

e

f

t

72°

60°

a b

c dh j

kfe

g

70°

60°70° 80°

55°

a) b) c)

ab

c

d fe g

h

m n p

kj


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4.2 Pythagoras’ theorem 1ExampleCalculate the lengths of the missing sides in these right-angled triangles.

Exercise� Find the lengths of the missing sides in each of these right-angled triangles to

1 d.p.

� Julie says that the hypotenuse of this right-angled triangle is exactly 5.4 cm. Is she correct? Show your working.

a) a2 � b2 � c2 b) a2 � b2 � c2

a2 � 82 � 62 152 � 82 � c2

� 64 � 36 c2 � 152 � 82

� 100 c2 � 225 � 64a � �100� c � �161�a � 10 cm c � 12.7 cm (to 1 d.p.)

We need to make theunknown variable thesubject of this equation.

a is the hypotenuse (the longest side).

8 cm

a) b)

8 cm

6 cm 15 cma

c

7 cm

4 cm

4 cm

4 cm3 cm

5 cm

a) b) c)

d) e) f)

5 cm

7.5 cm

8 cm

4 cm 8.4 cm3 cm

4.6 cm

3 cm

4.3 Pythagoras’ theorem 2ExampleAre the following triangles acute-angled, obtuse-angled, or right-angled?a) A triangle with sides 7.5 cm, 12.5 cm and 10 cmb) A triangle with sides 6 cm, 9 cm and 11 cm

Exercise� Are the triangles with these sides acute-angled, obtuse-angled or right-angled?

a) 10 cm, 12 cm, 20 cm b) 3 cm, 3 cm, 4 cmc) 10 m, 8 m, 6 m d) 4 cm, 7 cm, 6 cm

� An aeroplane flies 6 km due west and then 10 km due north. How far is it fromthe point at which it took off?

� How far apart are the two points (2, 1) and (9, 3) on a coordinate grid?

� Gary says that if he places the foot of his 3 m ladder 1.4 m from the base of awall, the ladder reaches exactly 2.6 m up the wall. Is he correct?

� These are the first four Pythagorean triples.

Comment ona) the heights of the trianglesb) the relationship between the base and the hypotenuse of the trianglesc) the square of the height compared to the size of the base and hypotenuse.

a) b2 � c2 � 7.52 � 102

� 56.25 � 100 � 156.25a2 � 12.52 � 156.25

Here a2 � b2 � c2, so the triangle is right-angled.b) b2 � c2 � 62 � 92

� 36 � 81 � 117a2 � 112 � 121

Here a2 � b2 � c2, so the triangle is obtuse-angled.

Compare a2 (the longest side) with b2 � c2.

a2 � b2 � c2 means the triangle is right-angled.a2 b2 � c2 means the triangle is acute-angled.a2 � b2 � c2 means the triangle is obtuse-angled.

4 m 24 m 40 m

41 m25 m7 m

13 m5 m5 m3 m

12 m

9 m

[Let a, b and c be the sides ofeach triangle.]



3R3R4.4 Congruent trianglesExampleState whether the following triangles are congruent, giving reasons for your answer.

Exercise� Which of the following triangles are congruent? Give reasons for your answers.

� Tanya says that if all the angles of one triangle are the same as in anothertriangle, then they are congruent. Is she correct?

� Sketch each of the following shapes. Draw a diagonal on each one and statewhether the two triangles formed are congruent or not. If they are congruent,state which condition of congruency they satisfy.a) parallelogram b) trapezium c) rectangle d) kite

No: The given side joins the angles 45° and 65° in the second diagram, but joinsthe 45° angle to the third angle in the first diagram. This triangle is notisosceles, so it is not possible for both sides to be 4 m.

Triangles are congruent if they share one of thefollowing sets of information.SAS: two sides and the angle between themASA: two angles and the side joining themSSS: three sidesRHS: right-angled, hypotenuse and one other side

45°

45°65° 65°

4 m

4 m

40°

40°

40°40°

60°

60°

60°

80°

80°

80°5

35°

70°

75° 35°4 cm

a) b) c)

5

4 cm

6 cm

6 cm

5

4.5 Unique trianglesExample

Is this the only triangle that can be constructed with the information given?

Exercise� a) Construct two different triangles ABC with �A � 30° and sides

BC � 3.2 cm and AB � 4.2 cm.b) Construct a triangle with �A � 90° , AC � 3 cm, BC � 5 cm.c) Construct two different triangles with �A � 40°, �B � 90°, �C � 50°.d) Are the triangles you drew in part a) congruent?e) Are the triangles you drew in part c) unique?

� Construct the following triangles. If either triangle is not unique, draw asecond triangle that has the same given properties but is not congruent to thefirst.a) A triangle with sides of 3 cm, 4 cm, 5 cmb) A triangle with sides of 4 cm and 3.5 cm and an angle of 40°

� a) Draw two triangles that have the same angles and are congruent.b) Draw two triangles that have the same angles and are not congruent.

� Can the following triangles ABC be constructed?a) AB � 4.5 cm, AC � 3 cm, BC � 3 cmb) AB � 4 cm, AC � 5 cm, �A � 70°c) AB � 5 cm, AC � 3 cm, BC � 9 cmd) �A � 90°, AB � 4 cm, BC � 3 cm

Given one of the following sets of information, a unique triangle canbe constructed.SAS: two sides and the angle between themASA: two angles and the side joining themSSS: three sidesRHS: right-angled, hypotenuse and one other side

There is more than one triangle that can be drawn given this information.

53°

4 cm5 cm


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4.7 Symmetry of shapesExamplea) Draw sketches to show all the

planes of symmetry of a rectangular prism.

b) Draw sketches to show all the axes of symmetry of a rectangular prism.

Exercise� How many planes of symmetry do each of these 3-D objects have?

a) a rectangular pane of glass b) a sphere c) a calculator� How many axes of rotational symmetry does each of the objects in Q1 have?� What is the order of rotational symmetry

of each of the following?a) a rectangular pane of glassb) a spherec) a calculator

� Trace the following shapes and draw on the planes of symmetry, using a separate diagram for each plane.

A plane of symmetry divides a 3-D shape intotwo mirror images.

An axis of symmetry is the line about which ashape can be rotated to map onto itself.

a)

b)

4.6 Lines on a coordinate gridExamplea) What is the length of the line joining A (3, 2) to B (5, 6)?b) C is the point (3, 2). The line CD is 2.5 units long and the x-coordinate of D is 5.

What is the y-coordinate of D? Note that CD has a positive slope.

Exercise

� Find the length of the lines joining these points:a) (4, 7) and (8, 10) b) (8, 11) and (12, 4) c) (0, 6) and (7, 2)

� a) A � (5, 8). The line AB is seven units long and the x-coordinate of B is 11.What is the y-coordinate of B?

b) A � (5, 8). The line AD is eight units long and the y-coordinate of D is 5.What is the x-coordinate of D?

� What are the coordinates of the mid-points of lines joining these two points?a) (5, 9) and (11, 17) b) (4, 7) and (8, 2) c) (8, 6) and (3, 0)

� P � (3, 5) and Q � (6, 3).a) Find the distance between P and Q.b) How far is Q from the mid-point of the line?c) State the coordinates of the mid-point of the line.

The y value has gonefrom 2 to 6, so thisline is four units long.

a) AB2 � (5 � 3)2 � (6 � 2)2

� 22 � 42

� 4 � 16� 20

AB � �20�� 4.5 (to 1 d.p.)

b) 2.52 � (5 � 3)2 � height2

height2 � 2.52 � 22

� 6.25 � 4� 2.25

height � �2.25�� 1.5

So the y-coordinate is (2 � 1.5) � 3.5.

10

0

234567

A

B

(3, 2)

C

D

(5, 6)

y

1 2 3 4 5 x

10

0

234

y

1 2 3 4

2.5

5 x

?

(3, 2)

x has gone from 3to 5, so this line istwo units long.

Sketching adiagram may help.

5 � 3 � 2

The order of rotational symmetry is thenumber of times a shape can be mappedonto itself by rotation in a 360° turn.



3R3R4.9 Finding a locus using ICTExampleSketch a diagram showing what shape you would get with the following LOGOinstruction: REPEAT 5[FD 200 RT 72].

Exercise� Sketch diagrams showing the shapes you would achieve by following these

LOGO instructions.a) REPEAT 6 [FD 200 RT 60]b) REPEAT 6 [FD 200 RT 120]c) REPEAT 5 [FD 100 RT 144]

� a) Abdul draws two points, A and B, on a sheet of paper. He then moves hispen so that the pen is equidistant from A and B. Sketch the locus of the pen.

b) Abdul draws two points, C and D, on a sheet of paper. He moves his penso that the sum of the distances from his pen to C and from his pen to D isalways equal to twice the shortest distance between C and D. Sketch thelocus of the pen.

� Draw two points, F and G, 5 cm apart. Now show the locus of the points P thatare nearer to F than G but less than 3.5 cm from G.

� On a coordinate grid, plot the points A � (0, 3), B � (0, 0), C � (7, 3) andD � (7, 0).a) Write down three points equidistant from the lines AB and CD.b) Write down three points equidistant from the lines AC and BD.c) Give the point equidistant from A, B, C and D.

� Write a set of LOGO instructions to draw the following shape, where eachtriangle is equilateral with side length 200.

4.8 LociExamplea) Sketch a diagram showing the locus of a mark at the

end of a propeller blade on an aircraft moving down a runway, when viewed from the side.

b) What shape is formed by all points in space at a fixed distance from a star?

Exercise� A hoop is rolled along the ground. Show the

positions of the point marked x on the hoop through two complete revolutions.

� A rectangular box is rolled side-over-side along the ground. Draw the locus of the point x on the corner of the box during two full turns.

� Col is at football practice. The team coach has put cones out and instructs Colto dribble the ball so that it remains the same distance from the cones. Drawdiagrams to show the locus of the ball ifa) there is just one coneb) there are two cones placed 3 m apartc) there is a row of touching cones forming a

straight line 3 m longd) there are two rows of cones set out like this:

� Sally put some birdseed on the lawn, and watched three birds around theseed. For each of the birds use either words or diagrams to describe the locus.a) The first bird on the ground stayed exactly 5 m from the seed.b) The second bird on the ground stayed was always within 4 m of the seed.c) The third bird didn’t always stay on the ground, but was always within

4 m of the seed.

The path of a movingpoint is called the locus.

a)

b) The shape would be a sphere, with the star at the centre.

x

x3 cm

1.5 cm


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5.1 Planning an investigation 1ExampleMeena writes the following hypothesis:‘Pupils don’t cycle to school because they don’t think their bikes are secure during the day.’What would Meena need to think about before starting to collect evidence to support her hypothesis?

Exercise� Kendra and Alice are discussing

a) the time taken to get to schoolb) pupils’ attitudes towards smokingc) whether the weather is warmer in May than in September.For each case:i) Identify sources of data and the size of sample needed.ii) Describe how bias could be avoided.

� Would a bigger range of food on the menu at school dinners persuade morepupils to stay? Ahmed and Gerry decide to collect primary data.Ahmed says they need to take a sample from pupils who stay for dinners.Gerry says they should take a sample of four pupils in each form.a) Are they correct in saying that their data will be primary?b) Write a hypothesis for them.c) What is the flaw in Ahmed’s suggestion?d) What steps could they take to avoid bias?

Data can come from primary sources whereyou collect the data yourself throughquestionnaire, survey, experiment, etc, orsecondary sources in which the data havebeen collected for you, such asnewspapers, books, on the internet, etc.

5.2 Grouped dataExampleThe result of a survey into the ages of people living in my street is as follows:

Find an estimate for the mean age of the people living in my street.

Exercise� The ages of the passengers on a trans-Atlantic flight were as follows:

a) Find estimates for the mean age and the median age.b) Estimate the range of the ages. Can we be sure that this is correct?

� The marks of the pupils taking French and German tests are as follows:

a) Estimate the mean marks in French and in German.b) Which was the harder of the two tests?

Estimated mean �

� 35.2(7.5 � 8) � (22.5 � 9) � (37.5 � 10) � (52.5 � 6) � (67.5 � 5) � (82.5 � 1)��

39

Her first main steps would be to decide on her source of data, and how much data tocollect. In this case the data would be probably primary, collected through aquestionnaire or a survey.Meena would also need to avoid bias. She could do this by making sure that shegathered a full range of opinions rather than just from a small like-minded group ofpeople. For example, Year 7 pupils may be more worried than Year 10 pupils about thesecurity of their bikes. Also, she might need to consider whether boys’ attitudes tocycling were the same as those of girls.

Age (years) 0 � a � 15 15 � a � 30 30 � a � 45 45 � a � 60 60 � a � 75 75 � a � 90

Mid-point 7.5 22.5 37.5 52.5 67.5 82.5

Age (years) 0 � a � 15 15 � a � 30 30 � a � 45 45 � a � 60 60 � a � 75 75 � a � 90

Frequency 8 9 10 6 5 1

To estimate the mean: find the mid-point of each interval, then multiply the mid-point by thefrequency. Sum these answers and then divide by the total frequency. (To estimate the median,

find the �(n �

2

1)�th value and estimate the position within the group of this value.)

Age (years) 0 � a � 20 20 � a � 40 40 � a � 60 60 � a � 80

Frequency 12 107 97 21

Mark 0–10 11– 20 21– 30 31–40 41– 50

French 0 7 18 13 10

German 0 9 14 20 5



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Interquartile range � upper quartile � lower quartile.To find the lower quartile, construct a cumulativefrequency graph and read off the value that is �

14� of the

way into the data. The upper quartile is �34� of the way

into the data.

Time (min) 0 � t � 10 10 � t � 20 20 � t � 30 30 � t � 40 40 � t � 50

Frequency 0 4 8 9 5

The interquartile range � (38 � 23) minutes

� 15 minutes

Time (min) 0 � t � 10 10 � t � 20 20 � t � 30 30 � t � 40 40 � t � 50

Cum. Freq. 0 4 12 21 26

To find the cumulativefrequencies, add thefrequencies so far.

38 and 23 minutes are theupper and lower quartiles,read from the graph20

25

15

5

Cum

ulat

ive fr

eque

ncy

10

0100 20 30

Time (min)40 50 60

Time (min) 0 � t � 5 5 � t � 10 10 � t � 15 15 � t � 20 20 � t � 25

Frequency 6 15 21 6 2

No. of people 0 � t � 10 10 � t � 20 20 � t � 30 30 � t � 40

Frequency 7 15 6 3

5.4 Drawing diagramsExamplea) Draw a frequency diagram of the following data showing the time spent on a

maths puzzle.b) Superimpose a frequency polygon on your diagram.

Exercise� The heights of pupils in a maths group are shown in the following table.

a) Draw a frequency diagram showing these data.b) Superimpose a frequency polygon on your diagram.

� In a survey about homework, three sets of data are collected. Comment on thesuitability of the diagrams suggested for each data set.a) The number of pieces of homework given each week: frequency diagramb) How much time pupils spent on homework at the weekend: time series

graphc) Where pupils usually do their homework: pie chart

Choosing the most suitable diagramTime Series: used for data that varies over time.Pie Charts: used where categories are parts of a whole.Frequency Diagrams: show the overall distribution ofdata.

8

12

Freq

uenc

y

4

0100 20 30Time (min)

40Frequency polygons are formed by joining themid-points of each part of the frequency diagram.

Time (min) 0 � t � 10 10 � t � 20 20 � t � 30 30 � t � 40

Frequency 3 12 9 3

Height (cm) 140 � a � 145 145 � a � 150 150 � a � 155 155 � a � 160 160 � a � 165 165 � a � 170

Frequency 3 9 10 6 9 2

5.3 Cumulative frequencyExampleDraw a cumulative frequency diagram showing the following data and use it to find the interquartile range.The times spent by pupils doing last night’s History homework:

Exercise� For the following data draw a cumulative frequency diagram and find the

median and the interquartile range.a) Times taken by pupils to get to school

b) Number of people getting off the buses at the town centre


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5.5 CorrelationExampleThis table below shows the marks of eight pupils in Physics and Chemistry exams.a) Draw a scatter graph for this data.b) Calculate the mean for each set of data.c) Give the coordinates of the mean point.d) Draw the line of best fit.

Exercise� For the following data:

i) Draw a scatter graph and the line of best fit.ii) Calculate the mean for each set of data.a) Shoe sizes and collar sizes for a group of men

b) Jumpers sold on the first of each month, and the temperature those days

a) b) Physics � 27Chemistry � 27

c) (27, 27)

d) Line of best fit – see graph.

The mean point of two sets of datais given by the coordinate (meanof x, mean of y).Your line of best fit should passthrough the mean point.

5.6 Misleading statisticsExampleThe following graph in a newspaper shows the number of road accidents in atown during a five-year period.The headline says ‘Road accidents numbers dramatically down.’ Is this true?

Exercise� This data shows the sales of a new toy.

a) Draw a graph that gives an honest impression of the sales.b) Draw a graph that gives the impression that sales are rising more quickly

than they really are.

� It has been proved that most accidents occur at home. Does this mean that ifyou die in an accident, it is most likely to be at home? Explain.

� List three ways in which graphs and statistics can be misleading.

No. The graph is misleading as the y-axis doesn’t start from 0. The accidentnumbers have fallen, but only from 275 to 225 over the five-year period.

Chem

istr

y

10152025303540

010 20 30 40 500

Physics

Physics 33 11 41 25 33 27 19 26

Chemistry 38 14 31 25 32 26 21 26

Shoe size 42 44 42 46 44 50 46 40 46

Collar size 38 40 42 40 44 48 42 38 40

Temp. (°C) 1 9 7 15 13 22 16 21 11 13 8 4

Jumper sales 30 40 20 10 10 5 5 10 15 30 35 35

280

260

220

240

Nu

mb

er o

f ac

cid

ents

020000 2001 2002

Year

2003 2004

Week 1 2 3 4

Sales 145 150 160 175



3R3R6.1 Compound measuresExampleA bus travels from London to Bristol in 180 minutes. The distance between thecities is about 150 miles. The return journey takes 2�

12� hours.

a) What is the speed of the bus for the first half of the journey?b) What is its speed for the second half of the journey?c) What is the average speed for the whole journey?

Exercise� Murphy travels in a car from London to Manchester, a distance of 210 km. The

journey takes 3�12� hours. What is his average speed?

� Abigail travels by bicycle through Cornwall. She travels at 24 km/h for 3hours, rests for an hour and then travels at 21 km/h for 2 hours.a) How far did she travel? b) What was her average speed?

� To travel on the Underground on the Piccadilly Line from Heathrow toCockfosters involves 33 stops on the way. Each stop lasts 40 seconds onaverage. The journey takes 2 hours 22 minutes. If the average speed is32 km/h how far does the train travel?

� The distance from the Earth to the moon is approximately 400 000 km. It tookthe Apollo spaceship 2 �

12� days to reach the moon. What was the average

a) daily speed b) speed in km/h?

� 60 cm3 of mercury weighs 816 g. Calculatea) the density of mercury b) the mass of 75 cm3 of mercury.

a) Speed � �dis

ttiman

ece

� � �1530� � 50 mph

b) Speed � �dis

ttiman

ece

� � �125.50� � �

305

0� � 60 mph

c) Average speed ��to

ttoatladlisttiman

ece

�� 150

5�

.5150

� � �350.50

�

� �60

110

� � 54.5 mph

Convert minutes to hours

6.2 MeasurementExampleLook at the scale to the right.a) Give the readings shown at A, B, C and D.b) Write down the range of each.

Exercise� Look at the scales below.

i) Write down the readings at each letter.ii) Write down the range of each reading.a) b)

� The masses of four rugby players are 102.5 kg, 110.6 kg, 98.0 kg and 95.6 kgcorrect to 1 d.p.a) Find the upper bound of their total mass.b) Find the lower bound of their total mass.c) Use your answer in a) to find the maximum possible value of their mean

mass.d) Use your answer in b) to find the minimum possible value of their mean

mass.e) Write down the range of their mean mass.

� A box has dimensions 40 cm by 50 cm by 30 cm to the nearest cm. What is therange of the volume of the box?

� The dimensions of a floor are 4.6 m by 1.6 m.a) Calculate the range of the area of the floor.b) How many tiles 30 cm by 30 cm does one need to cover the floor?

a) A � 10, B � 25, C � 34 and D � 46.b) Range of A is 9.5 � A � 10.5. Range of B is 24.5 � B � 25.5.

Range of C is 33.5 � C � 34.5. Range of D is 45.5 � D � 46.5.

We multiply top andbottom by 2 to makethe calculation easier.

0 10

A B C D

20 30 40 50

0 10 20 30 40

B A C D E

16 17 18 19 20

D A B C


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6.4 Circles 2ExampleDraw two equilateral triangles with sides 5 cm.a) Bisect the angles of the first triangle and draw an inscribed circle.b) Construct the perpendicular bisectors of the second triangle and draw the

circumscribed circle.

Exercise� Draw a triangle ABC with AC � 6 cm and BC � BA � 5 cm. By bisecting the

angles draw the inscribed circle for triangle ABC.

� Draw triangle XYZ with XY � 5 cm, YZ � 4 cm and XZ � 3 cm. Byconstructing perpendiculars draw the circumscribed circle for triangle XYZ.

� Draw a circle with radius 4 cm. Draw another circle also with radius 4 cm whose centre is a point on the circumference of the first circle. Draw a line joining the centres of the two circles, a line joining the intersection points of the circles, and lines joining each of the centres with each of the points of intersection. Calculate the sizes of x, y and z.

� Draw a circle with a radius 4 cm. Draw the radius and extend it by 1 cm. From the end point draw another circle also with a radius of 4 cm. Join the intersections of the circles with a straight line and join the intersections of the circles with the centers as shown.

60°x yz

6.3 Circles 1ExampleDraw a circle with a radius of 3 cm, and make an accurate drawing of a regularhexagon inscribed within that circle.

Exercise� Draw a circle with a radius of 3 cm and make accurate drawings of

a) an inscribed square b) an inscribed decagon (ten sides).

� Calculate the angles marked in the diagram below.

� Calculate the angles marked w, x, y and z in the diagrams below.a) b)

� a) Calculate the length of tangent AD. b) Calculate the area of ABCD.

O

31°

xy z

wO

x

y

z

w

a

cb

B D

C

A

13 cm

5 cm



3R3R6.5 ArcsExampleFind the length AB and the arc length AB, correct to two decimal places.

Exercise� Given 2�r � 144 cm, what is the length of arc AB if is

a) 180° b) 90° c) 60°d) 45° e) 40° f) 145°?

� For each of the circles below calculate the length of the arc AB, correct to 1 d.p.a) b) c) d)

� If the circumference is 216 cm, what is the size of when arc AB isa) 54 cm b) 72 cm c) 96 cmd) 90 cm e) 81 cm f) 111 cm?

� What is the difference in the lengths of arc CD and arc AB (to 1 d.p.)?

r � 12 B

A 140°

r � 15 B

A

85°r � 20

B

A

120°

r � 10

60° B

A

Arc AB � �61� � 2�r (or arc AB � �3

6600� � 2�r)

� 7.96 (2 d.p.)Length AB � 7.6 (OAB is an equilateral triangle)

6.6 SectorsExample Calculate the shaded area in the diagram.

Exercise� Given �r2 � 144 cm2, what is the area of the shaded section

if �a) 60° b) 90° c) 40° d) 85°?

� What is the area of each of these sectors (correct to 2 d.p.)?a) b) c) d)

� Calculate the shaded area (correct to 1 d.p.).

� Calculate the angle if r � 12 cm and the length of the arc � 10 cm.

110°

r � 10

120°

r � 7

72°

r � 15r � 10

60°

Area of sector AOB � �36

0°� � �r2

� �39600°°

� � �r2 � 113.1 cm2

Area of triangle � �21

� � 12 � 12 � 72 cm2

Shaded area � 41.1 cm2

60°7.6 B

A

rθ

BA

30°

A

BD

C

10

75

A

B12O

A

Bθ

55°912

θ

10

12

rθ

BA


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7.2 Numbers in standard formExample a) Write the following numbers in standard form.

i) 0.000003709 ii) 605 000 000b) Do the following calculations, presenting each result in

standard form.i) 3.2 � 104 � 5.7 � 10�6 ii) 4.8 � 107 1.6 � 10�3

Exercise� Write the following numbers in standard form.

a) 8 500 000 b) 0.0041 c) 300 500 d) 10.0005e) 0.0000002 f) 50 001 000 g) 0.333437 h) 7.000007

� Xenocrates calculated the number of syllables that could be formed from theletters of the Greek alphabet to be 1 002 000 000 000. Write this number instandard form.

� Do the following calculations, writing the results in standard form.a) 4.6 � 105 � 5.3 � 106 b) 3.25 � 108 1.3 � 103

c) 6.44 � 10�5 � 7.2 � 105 d) 4.914 � 10�7 9.1 � 104

e) 8.7 � 103 � 5.5 � 104 f) 6.3 � 105 � 5.1 � 104

� Write twenty-seven million as a number in standard form.

� A large number is engraved on the Columna Rostrata in the Roman Forum. To show the number the Roman symbol for 100 000 is repeated 23 times. Write the number in standard form.

a) i) 0.000003709 � 3.709 106 � 3.709 � 10�6

ii) 605 000 000 � 6.05 � 108

b) i) 3.2 � 104 � 5.7 � 10�6 � 3.2 � 5.7 � 104 � 10�6

� 18.24 � 104 � 10�6

� 1.824 � 10 � 104 � 10�6

� 1.824 � 10�1

ii) 4.8 � 107 1.6 � 10�3 � 4.8 1.6 � 107 10�3

� 3 � 1010

7.1 Powers of 10Example a) Complete the following calculations.

i) 3.75 � 104 ii) 21.75 106

b) Write each of the following as a single power of 10.i) 105 � 103 ii) 10�2 106

Exercise� Complete the following calculations.

a) 8.42 � 105 b) 19.37 103 c) 5.66 0.001 d) 6.1 � 10�4

e) 0.002 � 106 f) 5 � 10�3 g) 0.007 104 h) 52 � 0.0001

� Write each of the following as a single power of 10.a) 104 � 106 b) 102 105 c) 107 � 10�3 d) 103 10�4

e) 103 � 100 � 104 f) 10�3 � 0.001 g) 10�3 10�7 h) 0.1 103

� Write these numbers in order of size, smallest first:6.7 � 103 5.9 � 104 6.305 � 104 6.35 � 103 6.9 � 10�1

� Copy and complete:

a) 2.7 � 10 � 0.0027 b) 5.9 � 10 � 5900 c) � 105 � 320

d) � 10�2 � 71 e) 103 � 0.4 f) 10�4 � 60

� Copy and complete with a power of 10:

a) 5.6 � 106 � 56 � b) 7 � 104 � 700 �

c) 3.4 � 10�3 � 34 � d) 0.8 � 102 � 80 �

e) 0.4 102 � 40 � f) 0.0005 104 � 5 �

a) i) 37 500ii) 0.00002175

b) i) 108

ii) 10�8

3.75 � 104 � 3.75 � 10 00021.75 106 � 21.75 1 000 000

105 � 103 � 105 � 3

10�2 106 � 10�2 � 6 � 10�8

In Roman numeral notation,repeating a ‘number’ ntimes means that ‘number’is multiplied by n.



3R3R7.3 RoundingExamplea) There are 50 847 534 prime numbers less than 109.

Round the number 50 847 534 toi) the nearest million ii) three significant figures iii) four significant figures.

b) �9�

0

4

� is the limit of the sum of a sequence. Its value is 1.082323…

Round this number toi) one decimal place ii) two decimal places iii) three decimal places.

Exercise� In bridge the probability that you will be dealt a complete suit is 1 in

158 753 389 900. Round 158 753 389 900 toa) the nearest thousand millionb) four significant figuresc) five significant figures.

� The side length of the largest cube that can be passed through a unit cube is1.060660… Round this number toa) one decimal placeb) two decimal placesc) three decimal places.

� The number of ways of tiling a standard chessboard with dominoes is12 988 816. Round this number toa) the nearest millionb) three significant figuresc) four significant figures.

� The cube root of 2 is 1.259921049894… Round this number toa) one decimal placeb) two decimal placesc) three decimal places.

a) i) 51 000 000 ii) 50 800 000 iii) 50 850 000b) i) 1.1 ii) 1.08 iii) 1.082

7.4 Upper and lower boundsExampleFind the upper and lower bounds of each of the following:a) The number of people (P) at a demonstration was 46 000 to the nearest thousand.b) The area of a field (A) is 30 000 m2 to the nearest ten thousand square metres.

Exercise� The distance along a bird’s migration route is recorded as 2730 miles, to the

nearest ten miles. The actual distance might be more than this. What is themaximum distance it could be?

� Find the upper and lower bounds of each of the following:a) The number of apples (N) picked in an orchard was 3500 to the nearest

hundred.b) The area of a lake (L) is 87 000 m2 to the nearest thousand square metres.

� The sides of a triangle are measured to the nearest mm. They are given as3.7 cm, 5.2 cm and 7.3 cm.a) What is the lower bound for the perimeter?b) What is the upper bound for the perimeter?

� The lengths of the parallel sides of a trapezium and the perpendicular distancebetween them are measured to the nearest mm. The side lengths are recordedas 2.4 cm and 4.1 cm. The distance between them is recorded as 6.6 cm.a) Calculate the area of the trapezium using the recorded measurements.b) Calculate the lower bound for the area.c) Calculate the upper bound for the area.d) Work out the difference between the upper and lower bounds for the area

as a percentage (to 2 d.p.) of the area found using the recorded lengths.

a) lower bound � 45 500,upper bound � 46 49945 500 � P � 46 499

b) lower bound � 25 000,upper bound � 35 00025 000 � A � 35 000

46 500 would round to 47 000.

45 499 would round to 45 000.

The value is less than 35 000 and not equal to it.


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7.6 DividingExampleCalculate 24.852 3.8.

Exercise� Given that 76 9.5 � 8, write the value of

a) 760 9.5 b) 7.6 0.95 c) 76 0.95.

� Calculate:a) 47.31 5.7 b) 1.189 0.29 c) 1.836 0.034

� Use inverse operations to check your answers to Q1.

� A gardener wants to make a rectangular flowerbed of area 30 square metres.The border has to be 2.4 metres wide.How long should it be?

� An urn contains 21.72 litres of tea. How many mugs, each containing 0.3 litresof tea, can be filled from the urn?

20 4 � 5

24.852 3.8 � 248.52 38

6.5438)248.52

228 38 � 620.519.0 38 � 0.5

1.521.52 38 � 0.04

24.852 3.8 � 248.52 38� 6.54

Estimate was 5.Result is 6.54.

Multiply both original numbers by a power of 10so that you are dividing by a whole number.

Estimate the product:● Round each number to 1 sig. fig.● Divide one rounded number by the other.

Carry out the division.

Compare your result with your estimate.

7.5 MultiplyingExampleCalculate 2.53 � 6.4.

Exercise� Given that 56.3 � 3.75 � 211.125, write the value of

a) 5.63 � 0.375 b) 563 � 37.5 c) 0.563 � 0.375.� Calculate:

a) 4.13 � 3.5 b) 95.7 � 8.6 c) 0.56 � 37.9� A rectangular plot in a garden is 6.8 m by 10.35 m.

a) Calculate the area of the plot.b) The average cost of sowing a special grass seed is £3.48 per square metre.

Find, to the nearest penny, the cost of sowing this seed over the whole plot.� Some square paving stones have side length 0.83 metres. 238 of these paving

stones are placed edge-to-edge to make a path one paving-stone wide. Howlong is the path?

� Water is running out of a tap at 0.64 litres per second. How much water runsout of the tap in four minutes?

3 � 6 � 18

2.53 � 6.4 � 253 � 64 1000

253� 64

1012 �415 180 �6016 192

2.53 � 6.4 � 16 192 1000� 16.192

Estimate was 18.Result is 16.192.

Write the product to involve whole numbers, andcompensate by dividing by the appropriate power of 10.

Estimate the product:● Round each number to 1 sig. fig.● Multiply the rounded numbers.

Work out the product of the whole numbers.

Compare your result with your estimate.

Divide by the appropriate power of 10.



3R3R7.7 Recurring decimals 1ExampleConvert the following numbers to decimals.State whether each is a recurring decimal or a terminating decimal.a) �1

52� b) �

58� c) �1

31�

Exercise� Which of the following are recurring decimals?

a) 0.7.

b) 3.313131 c) 5.6.7.

� Write each of the following recurring decimals using the ‘dot’ notation.a) 0.444444… b) 3.565656… c) 0.145145145…d) 6.723232323… e) 1.03467467467…

� Convert the following numbers to decimals. State whether each is a recurringdecimal or a terminating decimal.a) �1

72� b) �

38� c) �

49� d) �

37�

� Write each set of numbers in order of size, smallest first.a) 5.7

., 5.7, 5.77 b) 3.34

., 3.34, 3.344, 3.3

.4.

c) 0.191, 0.1.9., 0.191111, 0.19

.d) 0.5675

., 0.567

.5., 0.5

.67

.

a) 0.416.: This is a recurring decimal.

b) 0.625: This is a terminating decimal.c) 0.2

.7.: This is a recurring decimal.

7.8 Recurring decimals 2ExampleConvert the recurring decimal 0.2

.73

.to a fraction.

Exercise� Convert each of the following decimals into fractions.

a) 0.6 b) 0.5.

c) 0.7 d) 0.8.

e) 0.2.

� Check your results for Q1 by converting each of your fractions into a decimalnumber.

� Convert each of the following decimals into fractions.a) 0.6

.1.

b) 0.2.5.

c) 0.3.6.

d) 0.8.1.

e) 0.2.7.

� Make a generalisation about the denominators of the fractions that you foundin Q3.

� Convert each of the following decimals into fractions.a) 0.3

.09

.b) 0.1

.08

.c) 0.1

.18

.d) 0.2

.49

.e) 0.5

.24

.

� Make a generalisation about the denominators of the fractions that you foundin Q5.

Let x � 0.2.73

.

x � 0.273273273273…

1000x � 273.273273273… (1)x � 0.273273273273… (2)

Subtracting (2) from (1):999x � 273

x � �92

973

9�

� �39313�

There are three recurringdigits so multiply by 1000.


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8.2 IndicesExamplea) 74 � 75 � 74 � 5 � 79

b) 96 � 92 � 96 � 2 � 94

c) (84)3 � 84 � 3 � 812

d) 6�5 � (�16�)

5

Exercise� If possible, simplify each product by writing it as a single power.

a) 35 � 33 b) 54 � 45 c) 3117 � 3116

d) n2 � n7 e) a4 � b4 f) a7 � b5

� If possible, simplify each division by writing it as a single power.a) 35 � 33 b) 58 � 52 c) 3117 � 3116

d) n9 � n6 e) a3 � b3 f) a10 � b7

� Write each of the following as a single power.a) (85)2 b) (33)3 c) (3119)4

d) (221)35 e) (b4)3 f) (na)2

� Write each of the following as a single power.a) 7�3 � 75 b) 34 � 3�6 c) 3117 � 311�6

d) n8 � n�7 e) a4 � a�3 f) y7 � y�7

� Write the following using negative indices.

a) �81

2� b) �2155� c) �

n1

8� d) �34

15100�

74 � 75 � 7 �7 � 7 � 7 � 7 � 7 � 7 � 7 � 7 � 79

96 � 92 � � 9 � 9 � 9 � 9 � 949 � 9 � 9 � 9 � 9 � 9��

9 � 9

8.1 Prime factorisation 2Examplea) Find the HCF and LCM of 60 and 54.b) Find the HCF and LCM of xy2z3 and x2y3z.

Exercise� Find the prime factors of

a) 96 b) 140 c) 288.

� Find the HCF and LCM of the following.a) 24 and 36 b) 70 and 72 c) 18 and 60 d) 108 and 27

� Find the HCF and LCM of the following.a) x3yz and xy2z b) p2qr3 and pq2r2

c) ab4c2 and a2b2c d) d2e2f 2 and def

� Find the LCM of the denominators first, then complete these calculations:a) �

1135� � �

1270� b) �

2410� � �

2356� c) �

1214� � �2

38� d) �

3458� � �

2620�

� Complete these calculations:

a) �a32ba

2� � �b2

2

bc

� b) �d42ff

2� � �d32

de2� c) �

p12q� � �

p3q� d) �

xxyy2z� � �

y2

2

yz

�

a)

Prime factors 22 � 3 � 5 Prime factors 2 � 33

HCF � 2 � 3 � 6

LCM � 2 � 2 � 3 � 3 � 3 � 5 � 22 � 33 � 5 � 540

b) Factors of xy2z3 are x, y2 and z3. Factors of x2y3z are x2, y3 and z.

HCF � xy2z

LCM � x2y3z3

6030

155

22

3

5427

93

23

3

The combined factorswithout duplications

The combined factors without duplications

The factors that are common to both

The factors that are common to both

(84)3 � 8 � 8 � 8 � 8 � 8 � 8 � 8 � 8 � 8 � 8� 8 � 8� 812

65 � 6�5 � 65 � �5 � 60 � 1

65 � 6�5 � (6 � 6 � 6 � 6 � 6) � � 11

��(6 � 6 � 6 � 6 � 6)



3R3R8.3 Indices and surdsExamplea) Find the value of 64�12�.b) Show that �9� � �49� � �(9 � 4�9)�.c) Write �60� as a product of a whole number and a surd.

Exercise� Find the value of the following.

a) 144 �12�b) �289� c) 12 167 �13�

d) �0.49� e) 2704�12�f) �

30.027�

� By working out both sides of the equation, decide which equations are true.a) �(16 ��36)� � �16� � �36� b) �(0.04 �� 25)� � �0.04� � �25�c) �(4 � 3�6)� � �4� � �36� d) �(81 ��64)� � �81� � �64�

� Write whether each of the following is a whole number or a surd.a) �7� b) �48� c) �49�d) �120� e) �9� f) �10�

� Write each of the following as a product of a whole number and a surd.a) �44� b) �90� c) �200�d) �96� e) �75� f) �12�

a) 64�21

�� 64� � 8

b) �9� � �49� � 3 � 7 � 21�(9 � 4�9)� � �441� � 21

c) �60� � �(4 � 15�)�� 4� � �15�� 2 � �15�� 2�15�

8.4 SubstitutionExampleFind the value of each of the following expressions when n � �3.a) 2n2 � 4n � 5b) 3n(n2 � 1)

Exercise� Find the value of each of the following expressions when b � 4.

a) 2b3 � b2 b) �b2

2

� � 5b c) b4 � 7b2 � b3

� Find the value of each of the following expressions when m � 3.4. Give thevalues to one decimal place.

a) m2 � 2m b) �m5

3

� c) m3 � 3m2 � m

� Simplify each of the following expressions as much as possible, and then findtheir values when m � �

23�.

a) 2m2 � 2m b) �3mm

4

3

� c) 6m � 3m � m

� Find the value of T when A � �14�. Give the value of T to one decimal place.

T ��5A(A

2A

2 � 1)�

� Find the value of N when p � �35�. Give the value of N to three decimal places.

N � �(5p

1�

0p2)2

�

a) 2n2 � 4n � 5 � 2 � (�3)2 � 4 � (�3) � 5� 2 � 9 � 4 � (�3) � 5� 18 � �12 � 5� 18 � 12 � 5� 25

b) 3n(n2 � 1) � 3 � �3 � ((�3)2 � 1)� �9 � (9 � 1)� �9 � 10� �90

Remember the order of operations.

When you square a negativenumber put it in brackets.


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8.6 Quadratic and cubic graphsYou will need graph paper for this homework.

ExamplePlot the graph of y � x2 � 5.

Exercise� Plot the graph of y � x2 � 5.

� Plot the graph of y � �x2

3

�.

� Decide which of the following points lie on the graph of y � 3x2 � 8.A(1, 11) B(1, �5) C(2, 4) D(0, 3) E(�2, �20) F(�2, 4)

x y�3 (�3)2 � 5 � 9 � 5 � 14�2 (�2)2 � 5 � 4 � 5 � 9�1 (�1)2 � 5 � 1 � 5 � 6

0 (0)2 � 5 � 0 � 5 � 5

1 (1)2 � 5 � 1 � 5 � 6

2 (2)2 � 5 � 4 � 5 � 9

3 (3)2 � 5 � 9 � 5 � 14

Draw up a table and choose some x-values.

8.5 Parallel and perpendicular linesExampleWrite the equation of the straight-line graph thata) crosses the y-axis at y � 2 and is parallel to y � 3x � 4

b) is perpendicular to y � �x2

� � 3 and crosses the y-axis at the same point.

Exercise� Write the equation of the straight-line graph that

a) crosses the y-axis at y � 3 and is parallel to y � 4x � 2b) is perpendicular to y � 5x � 1 and crosses the y-axis at the same point.

� Write the gradient of a line that is perpendicular to the line with equation:

a) y � 6x � 10 b) y � �x7

� � 7 c) y � 4 � 2x

� Find the equation of the graph which isa) parallel to y � �x � 15 and intercepts the y-axis at �12b) perpendicular to y � �x � 15 and intercepts the y-axis at �20

c) perpendicular to y � �1x0� and intercepts the y-axis at 10.

� These lines form three sides of a square. In each case, give the equation for themissing side.a) y � x � 2 b) y � x

y � �x � 2 y � x � 4y � �x � 2 y � �x � 8

a) y � mx � cc � 2m � 3So the required equation is: y � 3x � 2

b) y � mx � cm � �2c � �3So the required equation is: y � �2x � 3

The line y � mx � c crosses the y-axis at (0, c).

The gradient of y � 3x � 4 is 3.Parallel lines have the same gradient.The line y � mx � c has gradient m.

If two lines are perpendicular, theproduct of their gradients is �1.

�3

�15

x

y

�10

�5

5

10

15

�2 �1 10 2 3

Calculate the value of y by substituting thevalue of x into the equation.

Draw a suitable pair of axes andplot the points.Join the points with a smoothcurve.



3R3R8.7 Graphs of quadratic functionsExampleWrite the equation of a quadratic graph that crosses the y-axis at the point (0, 3).

There are lots and lots of possibilities.In fact, there are infinitely many!Here are a few of them.

y � x2 � 3y � 4x2 � 3y � x2 � 2x � 3y � x2 � 5x � 3y � 7x2 � x � 3

Exercise� The equation of a graph is y � x2 � k.

Write the value of k if the graph crosses the y-axis at the pointa) (0, 7) b) (0, �8) c) (0, 50) d) (0, �0.75).

� Write the coordinates of the point where the graph of each of the followingfunctions crosses the y-axis.a) y � x2 � 3x � 9 b) y � 3x2 � x � 11c) y � 0.5x2 � 0.6x � 1.7 d) y � 88x2 � 27x � 115

� For each point, write three different quadratic functions whose graphs crossthe y-axis at that point.a) (0, 3.5) b) (0, �17) c) (0, 100) d) (0, �0.25)

� a) What is the value of y at a point on a quadratic graph where it crosses thex-axis?

b) Write a value of x for which 0 � (x � 3)2.c) Write the coordinates of a point where the graph of the function

y � (x � 3)2 intercepts the x-axis.

� a) Write a value of x for which 0 � (x � 7)2.b) Write the coordinates of a point where the graph of the function

y � (x � 7)2 intercepts the x-axis.

At the point on the graph where it crossesthe y-axis, x � 0. So you can write anyquadratic function that gives 3 for thevalue of y when x � 0.

8.8 Solving simultaneous equations graphically

ExampleBy plotting the graphs, solve the following pair of simultaneous equations:2y � 6 � x (1) x � 2y � 30 (2)

Exercise� By plotting the graphs, solve the following pairs of simultaneous equations.

a) y � 6 � 2x (1) b) x � y � 10 (1)x � y � 9 (2) y � 8 � 2x � 0 (2)

c) 8 � 4y � x (1) d) y � x2 � 2x � 2 (1)y � x � �5 (2) y � x � 2 (2)

2y � 6 � x (1) x � 2y � 30 (2)2y � x � 6 2y � �x � 30

y � �2x

� � 3 y � ��

2x

� � 15

x y

0 �02� � 3 � 3

2 �22� � 3 � 4

4 �42� � 3 � 5

x y

0 ��02� � 15 � 15

2 ��22� � 15 � 14

4 ��42� � 15 � 13

�15

�15

x

y

�10

�5

5

10

15

�10 �5 50 10 15

Calculate three pointsthat lie on each line.

Plot the graphs on asuitable pair of axes.

Rearrange the equationsto make y the subject.

The graphs intersect at thepoint (12, 9) so the solutionto the simultaneousequations is x � 12, y � 9.


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9.1 ProbabilityExamplea) An ordinary six-sided dice is thrown. Calculate the probability of throwing an

odd number or a six.b) A box contains three red counters and one black counter. A bag contains four

red counters and two black counters. One counter is picked at random from thebox and one from the bag. What is the probability that both counters are black?

Exercise� An eight-sided dice, with the numbers 1 to 8 on its faces, is thrown. Calculate

the probability of throwing an even number or a 1.� A box contains four red counters and five blue counters. A bag contains three

red counters and six blue counters. One counter is picked at random from thebox and one from the bag. What is the probability that both counters are blue?

� A box contains 3 red, 3 white and 4 blue counters. A bag contains 2 red, 5 white and 3 blue counters. One counter is picked at random from the boxand one from the bag. Calculate the probability that the two counters area) both red b) both white c) both blue d) different colours.

� If a pupil in Lily’s class is chosen at random the probability that the pupil walkedto school is �

13�, and the probability that the pupil brought a packed lunch is �

14�.

a) Calculate the probability that the pupili) walked to school and brought a packed lunchii) brought a packed lunch but did not walk to schooliii) did not bring a packed lunch but did walk to schooliv) neither brought a packed lunch nor walked to school.

b) What is the sum of the four probabilities that you calculated in part a)?Explain why the sum is that number.

a) P(odd or 6) � P(odd) � P(6)� �2

1� � �6

1�

� �64

�

� �32

�

b) P(both counters black) � P(counter from box black) � P(counter from bag black)� �4

1� � �3

1�

� �112�

8.9 Using simultaneous equationsExampleJo has fence panels of two different lengths. One short and one long panel putend-to-end make seven metres of fencing. Two short panels are two metres longerthan one long panel.a) If a short panel is x metres long and a long panel is y metres long, write two

different equations to represent the information in the problem statement.b) Find the length of a short panel and the length of a long panel.

Exercise� Amy has stone paving-slabs of the same width, but of two different lengths.

Three short and two long slabs put end-to-end make 120 cm of path. A longslab is 10 cm longer than a short slab.a) Let the length of a short slab be x cm, and the length of a long slab be y cm.

Write two different equations to represent the information above.b) Solve the equations algebraically to find the length of a short slab and the

length of a long slab.� Alice is making a necklace by joining straight silver and gold links touching

end to end. Seven silver links joined to two gold links makes 5 cm of thenecklace. The silver links are 2 mm longer than the gold links. Find the lengthof one silver link and the length of one gold link.

� The sum of Amy’s and Alice’s ages is 44. In two years Amy will be three timesas old as Alice will be. How old are Amy and Alice?

� Amy and Alice are going shopping. Between them they have £100. Amy has£20 less than three times as much money as Alice. How much money do theyeach have?

a) x � y � 7 (1)2x � y � 2 (2)

b) (1) � (2): x � y � 7 (1)2x � y � 2 (2)

3x � 9x � 3

Substitute x � 3 into equation (1): 3 � y � 7y � 4

The outcomes aremutually exclusive.

The events are independent. The outcome of oneevent does not affect the outcome of the other.



3R3R9.2 Possible outcomes 1ExampleA box contains three red counters and one black counter. A counter is picked atrandom, its colour noted, and then it is returned to the box. Another counter ispicked at random and its colour is noted.a) Draw a tree diagram to show all the possible outcomes.b) What is the probability that the counters are different colours?

a)

b) P(the counters are different colours) � �166� � �8

3�

R

First counter Second counter Outcome Probability

RR

red, red 116

red, redRB

R

RRRB

R

RRRB

B

RRRB

red, redred, black

red, redred, redred, redred, black

red, redred, redred, redred, black

black, redblack, redblack, redblack, black

1161

161

16

1161

161

161

16

1161

161

161

16

1161

161

161

16

These outcomes are shownin italics in the tree diagram.

9.2 Possible outcomes 1Exercise� Solve the problems stated in the Example if the box contains one blue counter

and two white counters.

� An ordinary six-sided red dice and an ordinary six-sided green dice arethrown, and the pair of scores is noted. How many equally likely outcomesare possible?

� An ordinary six-sided red dice and an ordinary six-sided green dice are thrown, and the sum of the scores is noted.a) Complete this table to show the possible

total scores.b) What is the probability that the total

score is 9?c) What is the probability that the total

score is less than 5?d) Which total score is most likely?

How do you know?

� An ordinary six-sided red dice and an ordinary six-sided green dice arethrown, and the difference between the scores is noted. What is the probabilitythat the difference is 1?

red dice

green dice

5

9


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9.3 Estimating probabilitiesExampleA box contains 10 counters: some black and some white. Alex picks one counter atrandom from the box, notes the colour, and then returns the counter to the box.She repeats this trial 100 times, and records the number of times she picked awhite counter.

a) For each number of trials, enter into the table the relative frequency of pickinga white counter.

b) For 60 trials, what was the estimated probability of picking a white counter?c) Plot the relative frequencies on a graph.d) How many white counters do you think are in the box? Justify your answer.

a)

b) 0.35c)

d) Four of the 10 counters are almost certainly white because the estimatedprobabilities are tending to a limit of 0.4 � �1

40�.

0.6

0.2

0.4

Rela

tive

freq

uenc

y

04020 60Number of trials

80 100

Number of trials 20 40 60 80 100

Frequency of white 6 18 21 33 41

Relative frequency of white



Relative frequency of white 0.3 0.45 0.35 0.4125 0.41

9.3 Estimating probabilitiesExerciseA box contains 20 beads. Some are red and some are blue. Fay takes one bead atrandom out of the box, notes the colour, and then drops the bead back into thebox. She repeats this trial many times. This table of results shows the number oftimes she picked a red bead.

� For each number of trials, calculate the relative frequency of picking a redbead. Copy the table and enter the relative frequencies.

� For 80 trials, what was the estimated probability of picking a red bead?

� Plot the relative frequencies on a graph.

� How many red beads do you think are in the box? Justify your answer.



Relative frequency of white



3R3R10.1 TransformationsExampleP is a translation 15 right. Q is a reflection in the x-axis. Are P and Q commutative?

Exercise� Draw the trapezium T on a grid in the

position shown here. V is a translation 10 units up, parallel to the y-axis. W is areflection in the y-axis.Are these transformations commutative?Draw shapes on your grid to justify your answer.

� Transformation X is a clockwise rotation of 90° about (5, �5); Z is a reflection in the y-axis.Copy the diagram.a) Draw the image (A) of P after the

transformation X followed by Z.b) Draw the image (B) of P after the

transformation Z followed by X.c) What do you notice about the order of the

transformations?d) What single transformation maps A onto B?

9.4 Comparing probabilitiesExampleAli throws two six-sided dice 72 times, and each time he records the total score.He gets a total score of six 11 times.a) Calculate the relative frequency of getting a total score of 6.b) Prepare a table showing all the possible total scores when two dice are thrown.c) Why do you think Ali decided to throw the two dice 72 times?d) Find the theoretical probability of a total score of 6 when two dice are thrown.e) Compare the estimated (experimental) probability of getting a total score of 6,

given by the relative frequency, with the theoretical probability. Which isgreater? By how much?

ExerciseJan throws two eight-sided dice 128 times, and each time he records the differencebetween the scores. He gets a difference of three 18 times.

� Calculate the relative frequency of getting a difference of 3.

� Draw a table showing all the possible total differences when two eight-sideddice are thrown.

� Why do you think Jan decided to throw the two dice 128 times?

� Calculate the theoretical probability of getting a difference of 3 when twoeight-sided dice are thrown.

� Compare the estimated (experimental) probability of getting a difference of 3,given by the relative frequency, with the theoretical probability. Which isgreater? By how much?

a) �7121� � 0.1527

.

b) 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

�15

�15

x

y

�10

�5

5

0

10

15

�10 �5 5 10 15

T

Yes. For example, in the diagram the trapezium, T,can first be translated by P to A, then reflected by Q to B. Or T can first be reflected by Q to C, then translated by P to B.

�15

�15

x

y

�10

�5

5

0

10

15

�10 �5 5 10 15

T A

C B

If two transformations give the same resultregardless of the order in which they areperformed then they are commutative.

�15

�15

x

y

�10

�5

5

0

10

15

�10 �5 5 10 15

P

c) Because 72 is a multiple of 36,which is the number of equallylikely possible outcomes.

d) �356�

e) The estimated probability was�7121�, and the theoretical

probability is �7102�. The estimated

probability is bigger. Thedifference is �7

12�.


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10.2 Changing shapesExampleImagine two sheets of acetate, on each of which is the outline of a square.The squares are the same size.Imagine that one of these sheets is placed on top of the other so that the edges of the squares are parallel.What shapes can you make with the overlapping parts of the squares?Sketch them.

Exercise� Imagine the same two sheets of acetate as in the

Example, on each of which is the outline of a square. Imagine that one of the sheets is placed on top of the other and rotated through 45°.Sketch a diagram for each of a) to f) to show how the shape can be made by the overlapping parts of the squares.a) a right-angled triangle b) an irregular quadrilateralc) a kite d) a regular octagone) an irregular pentagon f) an irregular pentagon with one

line of symmetry� Imagine two sheets of acetate, on each of which is the

outline of an equilateral triangle. Imagine that one of these sheets is placed on top of the other so that the sides of the acetate sheets are parallel.Sketch a diagram for each of a) to d) to show how the shape can be made by the overlapping parts of the equilateral triangles.a) parallelogram b) a rhombusc) an irregular hexagon d) a regular hexagon

You can make a square or a rectangle.

10.3 SimilarityExampleThese two trapeziums are similar.a) Find the ratio of corresponding sides.b) Calculate the value of x.c) Write the ratio of the two perimeters.

Exercise� These two shapes are similar.

a) What is the ratio of the heights?b) Calculate the value of d.c) Write the ratio of the two perimeters.

� The diagram shows a square fitting inside another square.a) What is the ratio of the side of the small square to

the side of the large square?b) What is the area of the small square?c) What is the area of the large square?d) What is the ratio of the areas of the squares?

� The small grey right-angled triangle is similar to the large right-angled triangle.Calculate the value of x, showing your reasoning.

6 cm

10 cm 8 cm

x

14 cm7 cm 5 cmd

a) 10 : 8 � 5 : 4

b) �6x

� � �180�. So x � �

4108� 4.8 cm.

c) 5 : 4

The perimeters will be in the sameratio as corresponding sides.

2

2

1 cm

x

2 cm



3R3R10.5 Enlargements 2Examplea) Enlarge shape ABCD by scale factor �

12� using centre of enlargement (0, 0).

Label the image A�B�C�D�.b) Enlarge shape A�B�C�D� by scale factor �


Label the image A�B�C�D �.c) Write the scale factor of the single

enlargement from ABCD to A�B�C�D �.

Exercise� a) Enlarge shape ABCD by scale factor �

12�

using centre of enlargement (0, 0).Label the image A�B�C�D�.

b) Enlarge shape A�B�C�D� by scale factor �12�

using centre of enlargement (0, 0).Label the image A�B�C�D�.

c) Write the scale factor of the single enlargement from ABCD to A�B�C�D�.

� Some pupils draw plans of their classroom. Find the scale factor for each.a) 1 cm represents 1 m b) 2 cm represents 50 cmc) 1 cm represents 2 m d) 1 cm represents 20 cm

� For each of these pairs of enlargements, write the scale factor for theequivalent single enlargement from the same centre.a) s.f. 20 followed by s.f. 3 b) s.f. 10 followed by s.f. �

15�

a), b)

c) �61�

10.4 Enlargements 1ExampleCopy this shape on a coordinate grid.Enlarge the shape by scale factor �

12� using centre of

enlargement (0, 0).

Exercise� Copy the following shapes on different coordinate grids.

a) b)

� On a coordinate grid, plot the points (15, 15), (27, 30) and (24, 0). Join thepoints to make a triangle. Enlarge the triangle by scale factor �


Enlarge the shape by scale factor�14� using centre of enlargement (15, 0).

Enlarge this shape by a scalefactor �

12� using centre of

enlargement (0, 0).

500

5

10

15

20

25

30y

x10 15 20 25 30

5005

1015

202530

y

x10 15 20 25 30

500

5

10

15

20

25

30y

x10 15 20 25 30 500

5

10

15

20

25

30y

x10 15 20 25 30

500

5

10

15

20

25

30 A B

CD

y

x10 15 20 25 30

5005

1015

202530 A

A�

A�

B

B�

B�C

C�

C�

D

D�

D�

y

x10 15 20 25 30

500

5

10

15

20

25

30

D

A

C

B

y

x10 15 20 25 30


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10.6 Further enlargementExamplea) Explain why these shapes, each made from two cuboids, are similar.b) What is the ratio of the lengths of their edges?c) What is the ratio of their surface areas?d) What is the ratio of their volumes?

Exercise� a) Explain why these shapes, each made from two cuboids, are similar.

b) What is the ratio of the lengths of their edges?c) What is the ratio of their surface areas?d) What is the ratio of their volumes?

� For these two trapeziums, what is the ratio ofa) their perimetersb) their areas?

a) Similar because each pair of corresponding edges is in the ratio 1 : 3.b) 1 : 3 c) 1 : 32 � 1 : 9 d) 1 : 33 � 1 : 27

11.1 The product of two linear expressionsExampleMultiply out and simplify (x � 3)(x � 2).

Exercise� Multiply out and simplify each of the following.

a) (x � 4)(x � 3) b) (x � 7)(x � 2)c) (2a � 5)(a � 1) d) (d � 10)(2d � 2)

� Expand and simplify:a) (p � 6)2 b) (x � 9)2

c) (2y � 5)2 d) (3d � 10)2

e) (p � q)2 f) (4p � q)2

� Multiply out and simplify:a) (x � 4)(x � 4) b) (x � 7)(x � 7)c) (a � 5)(a � 5) d) (2d � n)(2d � n)

� a) Write an expression for the area of this rectangle.

b) Multiply out and simplify the expression you wrote in a).

� Mel walks at a speed of x � 6 metres per second.a) Write an expression for the distance she walks in x � 5 seconds.b) Multiply out and simplify the expression you wrote in a).

Simplify the expression by collecting like terms.

You could construct a multiplication table.30 cm30 cm

30 cm90 cm

45 cm

90 cm90 cm

30 cm

120 cm

15 cm

10 cm

40 cm

10 cm

10 cm10 cm

30 cm

20 cm

40 cm

50 cm

20 cm

20 cm

20 cm

60 cm

100 cm

(x � 3)(x � 2) � x2 � 2x � 3x � 6� x2 � x � 6

x �2

x x2 �2x

�3 3x �6

x � 8

x � 11

3.5 cm

4 cm

5 cm

7 cm

2 cm

2.5 cm



3R3R11.3 Algebraic fractionsExample

Simplify �52nm

2

� � �23n�.

Exercise� Add or subtract the following pairs of fractions. Simplify the expressions

where possible.

a) �3xy� � �

2x

xy� b) �

ac

� � �acb2� c) �

4atb

� � �s5t�

d) �x2

2

my� � �

yn

2� e) �2qp� � �

53pqq

� f) �j3k� � �

j2

9k2�

� Multiply the following pairs of fractions. Cancel the fractions down as muchas possible before multiplying.

a) �rs

� � �ut� b) �

ca2� � �

ac3

2� c) �3aac

2b� � �

56abb2

c� d) �

2gm2n

n2� � �

mn

2�

� Simplify:

a) �rs

2

� � �ut� b) �

ca2� � �

bc3

2

� c) �2aabc

2

� � �6bac2� d) �

gmnn2� � �

gmn

2�

� Write as many different pairs of fractions as you can that multiply to give �2ab�.

11.2 FactorisingExampleFactorise the expression 8n3 � 12n.

Exercise� Factorise each of the following expressions.

a) 3x � 12 b) 10 � 5n c) 7a � 77d) 56 � 16y e) 15a � 3 f) 8 � 24b � 16c

� Copy and complete each of the following.a) 18x2 � 12x � 6x(3x � ) b) 20x3 � 12x2 � 4x2(5x � )

c) 9x � 3x2 � 3x( � ) d) 10x2 � 25x3 � 5x2( � )

e) 28x2 � 21x � (4x � ) f) 55x3 � 22x2 � ( � 2)

� Factorise each of the following expressions to simplest possible form.a) 6n2 � 30n b) 26a3 � 13a2 c) 9p4 � 33pd) 72 � 60y2 e) 15x � 9x2 � 30x3 f) 8m5 � 24m3

� The area of a triangle is x2 � 6x. The height of the triangle is six units greaterthan half the base length.a) Write an expression for the base length.b) Write an expression for the height.

Check by expanding the brackets.

8n3 � 12n � 4n( � )8n3 � 4n � 2n2

12n � 4n � 3So 8n3 � 12n � 4n(2n2 � 3)Check: 4n(2n2 � 3) � 4n � 2n2 � 4n � 3

� 8n3 � 12n

The highest common factor of 8 and 12 is 4.The highest common factor of n3 and n is n.So the highest common factor of both terms is 4n.

Now we can subtract the fractions.

�52

nm

2

� � �23n� � �

52

nm

2

�

�

33

� � �22nm�

�

23m

�

� �165

mn2

� � �46nmm�

��15n2

6�

m4nm�

��n(15n

6�

m4m)

�

The LCM of the denominators is 2m � 3.We must multiply the numerator anddenominator by the same number.

The numerator can be simplified by factorising.


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11.5 Solving inequalitiesExampleFind the solution to the inequality 4 � 2x � 8 and represent the solution graphically.

Exercise� Solve the following inequalities.

a) n � 3 � 5 b) k � 4 � 7 c) 15 � n � 5d) 0 � n � 17 e) m � 50 � 71 f) �3 � a � 8

� Solve the following inequalities.

a) 2n � 12 b) �4k

� � 2 c) 15 � 3n

d) 3 � �n8

� e) �34m� � 21 f) 12 � �

45a�

� Solve the following inequalities and represent each solution on a number line.

a) 2x � 1 � 13 b) 5x � 2 � 22 c) 4 � �x3

� � 3

d) �3 � �x4

� � 2 e) 4x � 18 � �2 f) 10 � �x2

� � 7

� Represent each of the solutions to Q3 on a graph.

Draw the graph of x � 6, and shade thearea to which the inequality applies.Since the inequality is 6 � x, draw asolid line at x � 6. If the inequality was6 � x, we would draw a dotted line.

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15

11.4 InequalitiesExampleRepresent �3 � n � 4 on the number line below:

Exercise� Represent each of the following inequalities on number lines. Draw a new

number line from �7 to 7 for each part.a) �5 � n � 3 b) �2 � n � 6 c) 0 � n � 7d) 4 � n � �6 e) �1 � n � �7 f) 5 � n � �3

� List all integers which satisfy each of the following inequalities.a) �2 � x � 2 b) �5 � a � �1 c) �3 � k � 2d) 3 � n � 0 e) �2 � n � �5 f) 6 � p � �2

� If x is a positive integer, decide whether each of the following statements is A – Always True or B – Never True.a) x � 5 � x b) 52 � 48 c) 3x � 5x

d) 34 � 35 e) �1x0� � �

1x00� f) x � 6 � x � 5

� If x is a negative integer, decide whether each of the following statements is A – Always True or B – Never True.a) x � 5 � x b) 3x � 5x

c) �1x0� � �

1x00� d) x � 6 � x � 5

To show that n cannot be �3 we draw an ‘open’ (hollow) dot.To show that n can be 4 we draw a ‘solid’ or closed dot.

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

4 � 2x � 812 � 2x6 � x

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7



3R3R12.1 Solving ratio, proportion and percentage

problemsExampleA washing machine is in a ‘15% off’ sale and costs £306. How much would thewashing machine have cost without the reduction?

Exercise� A local shoe shop increases its prices by 5%. A pair of trainers now costs

£52.50. How much did the trainers cost before the increase?

� Jasneet enjoys cycling: every day she cycles to the nearest town and backagain, a total of 8 miles. She can cycle at 20 mph. One day when she is halfwayhome her bicycle chain breaks, and she walks the rest of the way at 4 mph.a) What is her average speed that day?b) Give an example using speeds of 4 mph and

20 mph where her average speed is 16 mph.

� A supermarket offers pasta in two sizes, 500 g and 1 kg bags. The 500 g bagcosts 49p and the 1 kg bag costs 95p. The 500 g bag is offered at 25% off andthe 1 kg bag is offered at 30% extra free. Which bag is better value?

� Matthews Construction Ltd have 6 workers who can dig 10 holes in 2 days.George & Sons have 3 workers who can dig 3 holes in 1 day.a) Which company is more productive?b) Matthews Construction Ltd charge £95 a day, and George & Sons charge

£60 a day. John needs 15 holes digging: which company will offer him thecheapest deal, and how much will it cost?

� Andy and Nicola are saving for a holiday that costs £500 each. Andy saves £45a month. Nicola saves a lump sum of £400. Andy opens a savings account thatpays 0.5% interest monthly. Nicola opens a savings account that pays 1.5%interest quarterly.a) Which account pays a higher rate of compound interest?b) If Andy and Nicola want to go on their holiday together at the end of the

year, will they have enough money between them?

The answer is not 16 mph.

11.6 Solving inequalities in two variablesExampleOn a pair of axes, draw the graph of y � x � 2 and then illustrate the solution setfor y � x � 2.

Exercise� a) Draw a pair of axes showing �15 � x � 15 and �15 � y � 15.

b) Draw the graph of y � 3x.c) Shade the area of the grid in which points satisfy the inequality y � 3x.

� a) Draw a pair of axes showing �15 � x � 15 and �15 � y � 15.b) Draw the graph of y � x � 5.c) Shade the area of the grid in which points satisfy the inequality y � x � 5.

� a) Draw a pair of axes showing �15 � x � 15 and �15 � y � 15.b) Draw the graphs of y � 2 and y � �10.c) Shade the solution set for the inequality �10 � y � 2.

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15

To work out which part of the grid toshade choose a point on one side ofthe graph line. Decide whether or notthe coordinates of that point satisfythe inequality.

100% of the full price is � £306 85 100 � £360Check: 306 as a percentage of 360 is (�

33

06

60� 100)% � 85%.

The reduced priceof £306 is 85% ofthe full price.


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12.2 Solving geometrical problemsExampleThe line AC is a diameter of the circle.Find the missing angles.

Exercise� Find all the missing angles.

� Find all the missing angles.

� Look at the diagram in the Example. What is �ABC? What is �ABC if �OABis a) 33° b) 72°? Make a conjecture about �ABC and prove it.

� Look at the quadrilateral ABCD in Q2. What is �DAB � �DCB? What is�ABC � �ADC? Try different values for �PAD, �QAB and �RBC, and find�DAB � �DCB and �ABC � �ADC.Make a conjecture about the size of �A � �C and �B � �D. What type of triangle are the four triangles in the quadrilateral? Use this to prove your conjecture.

Since OA � OB, the triangle OAB is isosceles; thereforea � 53° and b � 74°. Angles on a straight line add upto 180° so c � 106°. Since OB � OC the triangle OBC isalso isosceles. So d � e � 37°.

12.3 Mixed problemsExampleTanya stands 20 m away from a building and measures the angle of elevationwhich is 74°. How tall is the building, to an appropriate degree of accuracy?

Exercise� Mr Fowler is returfing his lawn, with rolls which are 1.5 m long and 35 cm

wide. He buys 15 rolls.a) What area can he cover?b) Give your answer to a) to a sensible degree

of accuracy, supporting your answer with appropriate calculations and reasoning.

� Miranda wants to measure the distance to a wall using echoes. She fires a sirenin the direction of the wall and starts her stopwatch. She stops her stopwatchwhen she hears the echo. She does this 3 times and gets times of 3.53 s, 3.67 sand 3.46 s. Find the distance to the wall to an appropriate degree of accuracy.The speed of sound in air is roughly 340 m/s.

� What fraction of the square is shaded?Investigate the fraction as you increase the number of largest possible identical circles in the square.

Smallest height is 19.5 tan(73.5°) � 65.83and largest height is 20.5 tan(74.5°) � 73.92.So, height � 70 m, to the nearest 10 m.

Draw a diagram to help youunderstand the situation.

A53°

CO

Ba

b c e

d

42° b

f

c g

40°

e d

a

DA

C

B

O

36°

ji g

a

hd

f cbe

23°45°

Consider the sum of all theangles in the four triangles.

74°20 mO A

?

B

Here, tan � � �A

O

B

A�.

(You will learn more aboutfunctions like tan � later.)

Assume that Tanya’s distancefrom the building is accuratewithin 1 m, and the angle isaccurate to within 1 degree.

Work out the largest andsmallest possible dimensionsof a roll.

Give your answerin terms of �.

Height � 20 m tan (74°)� 69.75 m



3R3R12.5 Extended problemsExampleInvestigate the sum of the first n odd numbers. Make a conjecture about the sum,and use a diagram to explain why your conjecture is true.

Exercise� A game at a fair involves throwing three dice. You pay 10p to throw the dice.

You get your money back if you throw one six, you get 20p if you throw twosixes and you get 30p if you throw three sixes. Is the game worth playing?Design your own game with three dice; state how much profit you will make.

� Prove that the sum of three consecutive numbers is divisible by 3. Is the sumof four consecutive numbers divisible by 4? Investigate larger sums.

� The large triangle is made up of 16 smaller triangles. Some of the triangles have two edges on the outside, others have one edge or zero edges. How many of each type of triangle are there? Investigate for different-sized triangles and other shapes.

� The local newsagent sells pens in packs of 6, packs of 9 and packs of 20. You want to make packs of different sizes, by combining packs from the newsagent, without opening them. Investigate which sizes you can make and which sizes you cannot.

n 1 2 3 4 5

Sum of first n odd numbers 1 4 9 16 25

12.4 Multistep problemsExampleTo make two cuboidal pillars measuring 50 cm 50 cm 1 m requires 20 kg ofcement, 40 kg of sand, 30 kg of gravel and 25 litres of water. How much is neededto make three cylindrical pillars of diameter 50 cm and height 1 m?

Exercise� If the minute-hand and the hour-hand of a clock are touching at midday,

when will they next be touching? The answer isn’t 5 minutes past 1!

� Jeremy borrows £1200 from a bank which charges an APR of 12%. He investsthis money in his savings account which pays 0.5% interest monthly. Hewithdraws £100 at the start of every month for the first year. At the end of theyear he pays back the bank the remaining money in his savings account.a) How much does Jeremy still owe the bank?

In the second year, he starts to pay back the loan.b) At £100 every month, how long will it take him to pay back the loan?c) To pay the loan back in two years, how much

will he pay each month and in total?

� 6 is a perfect number: If we sum the proper divisors of 6 we get 1 � 2 � 3 � 6.a) Which of the integers between 20 and 30 is a

perfect number?b) Find the sum of the proper divisors of 496.

Is 496 a perfect number?

Volume of cuboidal pillars � 2 0.5 m 0.5 m 1 m � 0.5 m3

Volume of cylindrical pillars � 3 � (0.25 m)2 1 m � 0.589 m3

So, we need 20 kg 0.5 0.589 � 23.6 kg of cement; 40 kg 0.5 0.589 � 47.1 kg of sand; 30 kg 0.5 0.589 � 35.3 kg of gravel; and 25� 0.5 0.589 � 29.5� of water.

Divide by the oldvolume and multiplyby the new volume.

First work out the monthlyinterest rate of the loan.

Conjecture: the sum of the first n odd numbers is equal to n2.

Using the diagram we can see that the sum of the first five odd numbers is 52 � 25.

For example, you can make apack of 15, by combining a packof 6 and a pack of 9. You cannotmake a pack of 8, because apack of 6 is too small and packsof 9 and 20 are too big.

The proper divisors of anumber are all the divisorsexcept for the number itself.


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13.1 Planning an investigation 2ExampleAlan has been asked to plan an investigation into how accurately pupils can estimate a time of one minute. He is told toa) write a suitable hypothesisb) state what he will do nextc) state whether the data he collects will be primary or secondary.

Exercise� Plan an investigation to find whether boys or girls are better at reciting the

English alphabet backwards.a) Suggest a hypothesis.b) State how your hypothesis can be tested.c) Decide whether the data will be from a primary or a secondary source.d) Say what steps you will take to eliminate bias.

� Someone on TV said that we get more stormy weather now than we did in thepast. Plan an investigation into this, as in Q1.

� Ben is planning to investigate litter around school. He decides to use aquestionnaire.a) Write a suitable hypothesis for Ben.b) Ben decides to aim his questionnaire at Year 8 pupils. Is this a good idea?c) Does a questionnaire collect primary or secondary data?d) Plan the questionnaire.

13.2 Planning and collecting dataExampleAlan is planning an investigation into the hypothesis that‘When estimating a time of one minute, boys will underestimate but girls will overestimate.’a) Write a plan, stating what data will be collected and how.b) State the sample size.c) Design a data collection sheet.

Exercise� You are planning an investigation to find whether boys or girls are better at

reciting the alphabet backwards.a) Write a detailed plan, showing what data will be collected and how.b) What steps would you take to eliminate any bias?c) State a suitable sample-size.d) Design a data collection sheet.e) Test your data collection sheet on a small number of people.f) Make any necessary amendments to your data collection sheet.

� Philip is also planning an investigation to find whether boys or girls are betterat saying the alphabet backwards, but he is interested in which group willimprove their results most if they practise.a) Write a suitably detailed plan for Philip’s investigation, showing what

data will be collected and how.b) Design his data collection sheet.c) Test this data collection sheet on a small number of people.d) Make any necessary amendments to this data collection sheet.

Remember the planning cycle.Stage 2: Collect data from avariety of sources.

a) On the word ‘GO’ I’ll start the stop-watch. The subject has to say ‘Stop’when they estimate that one minute has passed. I stop the watch andrecord their time on a data collection sheet.

b) 25 boys and 25 girls, in equal numbers from each year group.c) Name Year Boy/Girl Their estimate Under/over estimate

a) His hypothesis: When estimating time, boys will underestimate but girls willoverestimate.

b) Alan says that he will theni) collect data he needs, ensuring that he avoids biasii) process the dataiii) analyse the data and decide whether the hypothesis is correct.

c) Primary.

Remember the planning cycle.Stage 1: Specify the problem and plan.



3R3R13.3 Processing dataExampleAlan is planning an investigation into the hypothesis that‘When estimating a time of one minute, boys will underestimate but girls will overestimate.’These were the times (in seconds) that the boys estimated.53, 37, 64, 50, 43, 78, 65, 47, 48, 59, 50, 37, 46, 62, 61, 58, 52, 63, 49, 58, 43, 68, 71, 51, 50a) How could the data be represented?b) What are the main statistical calculations that could be performed?c) Calculate the mean.

Exercise� Ben has organised his data on how well pupils can recite the alphabet

backwards into this grouped frequency table. No one had more than 16 correct.

For each of the groups of boys and girls, find a) the modal group b) theestimated mean c) which group has the larger median.

� Looking at Ben’s data in Q1, would it be appropriate for him to producea) a stem and leaf diagram b) a cumulative frequency diagramc) pie charts?

� For the data in the Example:a) Produce a stem and leaf diagram.b) Find the median.c) Produce a grouped frequency table. Your first grouping should be 35–39.

Remember the planning cycle.Stage 3: Process andrepresent data.

a) A histogram or frequency polygon could be used.b) Mean, mode, median, range. c) 54.5 seconds.

Number of letters correct Frequency Frequencybefore an error was made (boys) (girls)

0–3 6 3

4–7 7 10

8–11 6 9

12–15 5 3

16� 1 0

13.4 Representing data 1ExampleAlan has collected the following frequency table on the length of time (to the nearest second) that boys and girls estimate one minute to be.

Draw a suitable graph that may be used to compare this data.

Exercise� This frequency table shows data collected in an investigation into whether

boys or girls were better at saying the alphabet backwards.

a) Draw two frequency polygons that can be used to compare the results.b) Draw a bar chart that would allow the two sets of data to be compared.c) Name one advantage of using frequency polygons for comparing data.d) Name one advantage of using comparative bar charts for comparing data.

Remember the planning cycle.Stage 3: Process andrepresent data.

Time (sec) Boys Girls

30–39 3 1

40–49 5 5

50–59 9 10

60–69 5 8

70–79 2 1 2

0

30–39

40–49

50–59

60–69

70–7

9

4

6

Freq

uenc

y

Estimates (sec)

8

10

12BoysGirls

No. of letters correct 0–3 4–7 8–11 12–15 16�

Girls’ frequency 3 10 9 3 0

Boys’ frequency 6 7 6 5 1


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13.5 Interpreting data 1ExampleAlan has to comment on these bar charts which show the length of time that boys and girls estimate one minute to be.

Freq

uen

cy

2

4

6

8

10 Girls

0300 40 50

Estimates (sec)

60 70 80

Remember the planning cycle.Stage 4: Interpret and discussdata.

Freq

uen

cy

2

4

6

8

10

12

Boys

0300 40 50

Estimates (sec)

60 70

The modal group for boys is lower than that of the girls and the range of the girls’times shows a far greater spread than that of the boys. Girls are better atestimating a time of one minute, as boys tend to underestimate the length of time.

13.5 Interpreting data 1Exercise� In an investigation into reciting the alphabet backwards, John has calculated

the following statistics regarding the number of letters correct before an errorwas made.

a) Explain carefully what the boys and girls having the same medians tellsyou.

b) Comment on the ranges of the boys’ and girls’ results.c) Comment on the interquartile

ranges of the boys and the girls.

� Two rival supermarkets are making claims about their performance. The factsof their weekly sales are shown in this table.

a) True or false: Supashop sell twice as much as Quikmark.b) True or false: Supashop sell more than twice as much food as Quikmark.c) True or false: Supashop have a larger range of goods than Quikmark.d) What true claim could Quikmark make to sound better than Supashop?e) State another true claim that Supashop could make.f) Does this information help you decide where to shop?

Food Household goods Clothes Electrical

Supashop (£) 10 000 2000 3000 5000

Quikmark (£) 4800 2400 2400 2400

Mean Range Median Interquartile range

Boys 7.5 16 7 9

Girls 7.4 12 7 5

You must demonstrate that you understandwhat an interquartile range is showing you.


14.1 Investigating trigonometryExampleUsing results from the table (for unit radius) below, find the coordinates of thepoints P and Q in the following diagrams:

Exercise� The steepest public road in England is at Hard Knott Pass in Cumbria with the

steepest section at an angle of 30°. Find the difference in height between thestart and the end of this section after travelling 150 m horizontally.

� A Boeing-747 approaches an airport runway at an angle of 10°. If the plane isflying at an altitude of 300 m, how far does it travel horizontally beforelanding? Give your answer in kilometres to 2 decimal places.

� A window 2.6 m above the ground is being replaced. To ensure a safereplacement a 3 m long ladder has to maintain an angle of 60° with theground. How far from the vertical wall should the ladder be placed?

� A kite is flying at a height of 34 m above the ground. If the string is held tautat an angle of 70° how far away horizontally is the person from the kite?

60°(0, 0)

4

P

10°(0, 0)

50Q

Angle x y

10° 0.98 0.17

20° 0.94 0.34

30° 0.87 0.50

40° 0.77 0.64

50° 0.64 0.77

60° 0.50 0.87

70° 0.34 0.94

80° 0.17 0.98

For point P the enlargement is by a factor of 4. The coordinates of P arex-coordinate: 0.5 2 � 2.0; y-coordinate: 0.87 4 � 3.48

For point Q the enlargement is by a factor of 50. The coordinates of P arex-coordinate: 0.98 50 � 49.0; y-coordinate: 0.17 50 � 8.5

Enlarge the triangles by theappropriate scale factors.


3R3R13.6 Writing your reportExampleOnce the analysis of an investigation is complete, the information must be broughttogether as a report. The final stage of the report is an evaluation.

ExerciseBelow is a checklist of steps that should be taken in an investigation, but some ofthe words are missing. The first letter of the missing word has been given in somecases.Copy and complete the checklist in full.

Introductiona) The Titleb) An explanation of the p . . . . . . . .c) Your h . . . . . . . .

Planninga) A list of the i . . . . . . . . you need.b) The amount of data required, known as the s . . . . . . . . s . . . . . . . .c) An explanation as to why you chose that amount of data.d) How you plan to minimise b . . . . . . . .

Collection of the dataa) . . . . . . . . you collected the datab) . . . . . . . . you collected the datac) An explanation of the accuracy of your data

Processing and representing your dataa) What c . . . . . . . . you performedb) The relevant . . . . . . . . you drew, and why you chose those ones in particularc) T . . . . . . . . s of results

Interpreting and discussing the dataa) Link your . . . . . . . . back to the original . . . . . . . .b) Show what e . . . . . . . . you have to back up any statement that you make.

Evaluationa) What p . . . . . . . . you had, and how you overcame them.b) How you could e . . . . . . . . your investigationc) The i . . . . . . . . that you would need in order to extend your investigation.d) What c . . . . . . . . would you make to your original plan.


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14.2 Trigonometry 1ExampleA rectangular paving slab 2 m long rests against a vertical wall.a) What is the height of the slab above the ground?b) How far away from the wall is the slab?

Exercise� Draw a graph of both sine and cosine functions, with the x-axis between 0°

and 90° and the y-axis between 0 and 1.a) What are the values of sine and cosine when � � 0° and � � 90°?b) How would you describe the curves?c) At what angle do the two curves cross?

� A right-angled triangle has a base angle of 36° and a hypotenuse of length 15 cm. Find the length of the sides that are opposite and adjacent to this angle, to 1 d.p.

� An aeroplane is approaching a runway at an angle of 12° and is positionedsome 2.6 km horizontally away from the airport. How high is the plane at thisdistance, to the nearest metre?

� A chord of length 12 cm, when connected by two radii to the centre of a circle, forms an angle of 72° at the centre. What is the radius of the circle?

70°

2 m

� cos � sin �

10° 0.98 0.17

20° 0.94 0.34

30° 0.87 0.50

40° 0.77 0.64

50° 0.64 0.77

60° 0.50 0.87

70° 0.34 0.94

80° 0.17 0.98

a) Opposite � Hypotenuse sin �� 2 sin 70°� 1.88 m (2 d.p.)

b) Adjacent � Hypotenuse cos �� 2 cos 70°� 0.68 m (2 d.p.)

Draw and label a diagram.

A diagram here isvery important.

14.3 Trigonometry 2ExampleOne of the most famous right-angled triangles is the Pythagorean triple triangle known as the 3-4-5 triangle, shown opposite. Find the angle � to 1 d.p.

Exercise� A ladder of length 3.5 m leans against a vertical wall so

that the base of the ladder is 1.3 m away from the wall. Calculate the angle between the ladder and the horizontal to the nearest degree.

� An ocean-going yacht sails 200 km on a bearing of 236°.a) How far to the south has it travelled?b) How far west has it sailed?

� A kite flying into a strong breeze reaches a height of 45 m with the stringmaking an angle of 65° to the horizontal. What is the length of the string thatholds the kite in this position (to the nearest metre)?

� An isosceles triangle has sides of length 8 cm, 8 cm and 5 cm. Find the angle between the two equal sides.

� Look at the diagram. Find the side marked x? Give your answer to 3 sig. fig.

θ4

35

cos � � �AH

� � �54

� � 0.8

So � � cos�1 (0.8)� 36.9°

On your calculator press:0.8 inv cos �

Draw and label thediagram.

Draw a diagram and label theangle that is required.

47°35°

AB

C

D

6 cmx



3R3R14.5 Lengths, areas and volumes 1ExampleThe volume of a cone is given by the expression V � �

13��r2h, where r is the base

radius of the cone and h is the vertical height of the cone. If r � 10 cm and the baseangle is 60° find a) the vertical height h, and hence b) the total volume of the cone.

Exercise� The curved surface area of a cone is given by �rl, where r

is the base radius and l is the slant height of the cone. The total surface area S of the cone will also include the area of the base circle. Find S when r � 5 cm and h � 12 cm.

� A model rocket shown below comprises two sections – a cylindrical body oflength 12 cm and radius 4.2 cm (open at both ends), and a conical nose sectionof height 5 cm. Find a) the outer surface area, and b) the volume of the rocket(ignore the stabilising vanes).

a) tan 60° � �hr

�

h � r tan 60°h � 17.3 cm

b) V � �31

�� 100 17.3V � 1810 cm3

Draw a diagram and labelwith the information given.

Use Pythagoras to find l.

60°

h

r

h

r

l

14.4 Trigonometry 3ExampleFrom the top of a tower 30 m high, it is possible to see the main gate-house at anangle of 38° below the horizontal. How far is the gate-house from the tower?

Exercise� From a height of 2 m above the ground the angle of

elevation for a large tree in the garden is 57°. If this angle is measured 4 m away from the tree how tall is the tree to the nearest metre?

� A rocket is launched vertically for 3 km before stage two is fired. It then flies atan angle of 20° to the vertical for a further 2 km. Calculate the angle ofelevation from the point of take-off at the end of stage two.

� A local church steeple is 32 m high. Find the angle of elevation from a pointthat is 80 m from its base.

� From the top of a building a small lake can be seen at 27° below the horizontal.If the base of the building is 350 m away from the lake how high is the building?

� Find the angle of elevation of the top of a flagpole 5.6 m tall in a town squarefrom a point 12 m away from its base. Give your answer to the nearest degree.

Tower38°

38°A

30 m

House

tan 38° � �3A0�

A � �tan

3038°�

� 38.4 m

O and A are the sidelengths of the right-angled

triangle so use tan � = �O

A�.

Do not forget toadd on the 2 m.

Always consider drawing a diagram.


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14.6 Lengths, areas and volumes 2ExampleA solid brass cylinder of radius 5.6 cm and height 14.7 cm is melted down to form a solid cube. Find a) the side length of this cube, and b) the shape that has the larger surface area?

Exercise� A cylinder has a radius of 2.6 cm and a length of 10 cm.

a) Calculate the volume of the cylinder.b) Find the surface area of the cylinder.

� Find the radius of a cylinder whose volume is 45 cm3 and length is 4.5 cm.(Give your answer to 3 sig. fig.)

� A cylindrical can is used to collect rainwater. It has a height of 15 cm and aradius of 4 cm. If 600 ml of rain has collected during the week, calculate theheight of the rainwater in the cylinder to 3 sig. fig.

� A garden hose has an internal diameter of 2.2 cm. When in use water flowsthrough the pipe at 8 cm/s. If the pipe is full of water how much water isissued from the pipe every minute? (Give your answer in litres to 3 sig. fig.)

� Oil from a cylindrical can is used to top-up the engine oil in a car. If two litresof oil is used and the level in the can has fallen by 15 cm, what is the radius ofthe can? (Give your answer to 2 sig. fig.)

Now take the cube root of thevolume to find the cube side-length.

The surface area of the cylinder includesboth circular ends and the cylinder itself.

r

h

a) V � �r2h� � 5.62 14.7� 1448.2 cm3

Hencel � �

3V�

� 11.3 cmb) S(cylinder) � 2�r2 � 2�rh

� 714 cm2

S(cube) � 6l2

� 766 cm2

The cube has the larger surface area.

15.1 Planning a probability investigationExampleLook at Information Sheet 1. This shows results for tossing a drawing pin 100 times, showing whether the pin landed point up or point down. Theprobability result (for ‘Up’) is the experimental probability calculated from theseresults.

On a data collection sheet, ensure that there is room to record the data. As a guidefor your own work, look at the type of information and level of detail included inthe table in Information Sheet 1. Depending on the investigation, you might needto record results from individual trials rather than just the total for each set oftrials, perhaps using a tally chart.

Exercise

Choose one of the following investigations. Suggest a hypothesis to test (if appropriate). Include what size sample you would choose, how you might split this number into groups, where you would collect the data, and how you would beginto analyse the data.

� A red dice and a blue dice are thrown at the same time. Design a datacollection sheet to record 120 results. Use this to draw a sample space diagramfor the theoretical probabilities and find the probability of obtaining a total of4 by listing all of the possible outcomes. How does this result compare withthe experimental probability?

� A bag contains two red counters, two blue counters, three green counters andone black counter. Counters are removed from the bag one at a time and arenot replaced. Design a data collection sheet for this investigation. Find theprobability that two red counters will be removed from the bag within the firsttwo attempts at removing the counters.

� Four coins are tossed at the same time. List all of the possible outcomes in asystematic way using a data collection sheet. What is the probability ofobtaining at least one tail?

� Design a data collection sheet using a dice and a spinner of your own design.The spinner design may involve colours or numbers. Is it possible to comparethe experimental and theoretical probabilities for your investigation?

Analyse means ‘break downwhat the data is telling you’.


15.2 Experimental probabilitiesOnce you have collected sufficient data, you will need to process it. Look atInformation Sheet 1. In the table, the experimental probability was calculatedusing the expression:

experimental probability �

ExampleOften it is necessary to calculate the summary statistics for your data such as themean, mode, median and range. You will not always need to use the samestatistical techniques for all investigations. Here is an example taken from Exercise15.1 Q1, showing a summary of the first 20 trials in which a blue and a red dicewere thrown together:

For the blue dice: For the red dice:mean � �

6290� � 3.45; mode � 4 mean � �

7240� � 3.7; mode = 2, 4 and 6

Exercise� The marks of 10 students on a recent mathematics test were as follows:

35 40 44 51 59 65 66 74 80 82Find a) the mean b) the mode c) the median d) the range for this data.

� Two dice thrown together 12 times, with the total sum recorded each time,gave the following scores: 7, 2, 12, 5, 3, 11, 8, 9, 11, 4, 6, 10Carry out a summary analysis of the statistics for these results.

� Four biased dice were thrown together a total of 200 times. Find the mean number of sixes scored per throw. What bias do you notice in the results?

� Toss two ordinary dice a total of 20 times and record the results of the totalsscored. Find the mean total score. Repeat the exercise. Are the results similar?Explain your findings.

number of times an outcome happens��number of times the experiment was carried out


3R3R

Trials Blue dice Red dicenumbers numbers

1 2 3 4 5 6 1 2 3 4 5 6

20 3125641644 6365624152 4 3 2 5 3 3 2 4 3 4 3 42514265413 4361244352

Blue dicenumber frequencies

Red dicenumber frequencies

Number of sixes 4 3 2 1 0

Number of throws 1 3 9 36 151

15.3 Possible outcomes 2ExampleInformation Sheet 2 shows the results from an investigation in which three coinswere tossed simultaneously.

The experimental probabilities for the four possible outcomes can be calculatedusing the second table in Information Sheet 2.

P(3t) � �1510� � 0.220 P(2t, 1h) � �

2510� � 0.420

P(1t, 2h) � �1520� � 0.240 P(3h) � �5

60� � 0.120

You can now compare your results with the theoretical probabilities. You canobtain these using the sample space diagram in Information Sheet 2.

The theoretical probabilities are

P(3t) � �18� � 0.125 P(2t, 1h) � �

38� � 0.375

P(1t, 2h) � �38� � 0.375 P(3h) � �

18� � 0.125

ExerciseAttempt two of the following three problems.

� A bag contains 5 red balls and 3 green balls. A ball is selected at random, notreplaced, and then a second ball is selected. Design a data collection sheet forobtaining a realistic estimate of the experimental probabilities for theoutcomes. What size sample do you think is required? Now list all of thepossible outcomes using a tree diagram and calculate the theoreticalprobabilities. How do these compare with the experimental probabilities?

� Two coins are tossed at the same time as a fair dice. Prepare a data collectionsheet for this experiment and compare the results with the theoreticalprobabilities obtained through a list all of the possible outcomes. What is theprobability of obtaining at least one tail and an even number?

� A red, a white and a blue dice are thrown simultaneously and the totalsfound. Design a data collection sheet for this investigation and compare yourexperimental results with those found by using a sample space diagram tofind the theoretical probabilities. What is the probability ofa) obtaining a total of 15b) throwing the same number on each dicec) the numbers on each dice all being even numbersd) the total number being a prime number?

Check that your theoreticalprobabilities add up to 1:0.125 � 0.375 � 0.375 � 0.125 � 1


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15.4 Representing data 2Example 1The following table shows the results of a class survey on the sports pupils have participated in during the week. Draw a comparative bar chart for these results.

Example 2 Information Sheet 3 shows an example of a scatter graph.

Exercise� Using the information in Information Sheet 2, draw a comparative bar chart

for the experimental and theoretical probabilities. What would you need to doto improve the accuracy of the experimental results?

� What type of diagram would you use fora) continuous data b) discrete data (this may be grouped)c) two (or more) sets of discrete data? What would make the comparison

more valuable?

� For 10 pupils in your class, record the height and shoe size of each individualmember and draw a scatter graph. Is there any correlation between shoe sizeand height? Is your sample size large enough?

Always give a bar chart a title and includea key. Here the shaded bars represent‘boys’ and non-shaded bars the ‘girls’.

2

4

6

8

10

12

0

Sports played by pupils

Freq

uen

cy

RugbyTe

nnisHock

eyNetb

allFootb

all

Sport Rugby Tennis Hockey Netball Football

Boys 6 2 4 0 10

Girls 1 3 6 12 3

Total 7 5 10 12 13

15.5 Interpreting data 2ExampleInterpret the line graph from the drawing pin experiment (displayed inInformation Sheet 1).

The results appear to indicate that as the number of trials increases the probability of landing pin ‘Up’ approaches the value 0.61. The accuracy of this result would be improved by carrying out more trials.

The nature of the investigation does not allow a theoretical probability to beobtained, hence no comparisons are possible.

Exercise� Look at the data provided from tossing three coins simultaneously

(Information Sheet 2). Write a short report on the data and the resultsobtained. (Hence, first calculate the experimental and theoretical probabilitiesagain, and then prepare the comparative bar chart.) Comment specifically onhow the experimental data could be improved.

� Look at the following data for the mass and height of eight people. Draw ascatter diagram for this data. Describe and comment on the degree ofcorrelation, if any, between the two quantities.

� Crude oil (petroleum) is our main source of fuel. The following list is the main use of crude oil in the UK. Draw a pie chart to represent this information. Comment on the findings.

Try to explain what your results show. Ifpossible try to link your interpretationto your original hypothesis.

Person A B C D E F G H

Height (m) 1.9 1.7 2.0 1.5 1.9 1.8 1.6 1.7

Mass (kg) 63 60 76 49 69 57 52 55

Use of crude oil Percentage used

Fuel for vehicles 29

Fuel for generating electricity 22

Fuel for heating (homes, factories) 35

Use in plastics industry 4

Other uses 10



3R3R15.6 Writing a reportWhen writing up an investigation, include the following points.

● Explain what your investigation is about.● Include a detailed plan of what you did and how you did it.● Prepare relevant diagrams and tables of results.● Discuss your results and state your conclusions, linking back to the original

problem. Include evidence from your results.● Evaluate the investigation; suggest possible improvements to the original plan.

You could use the following checklist:

1. Introduction2. Planning3. Collection of data4. Processing and representing the data5. Interpreting and discussing the data6. Evaluation

ExerciseAttempt Q1 or Q2.

� Write a report on the drawing pin investigation, based on the results shown inInformation Sheet 1. Write down how you think the experimental data couldbe made more accurate. Use the checklist to help you.

� In Information Sheet 3, a scatter graph was drawn for the mean temperaturesin different capital cities in the northern hemisphere. Prepare a brief report onthe findings. How could the data be improved? What could you include toextend the investigation further? How could you extend the investigation toinclude other types of information, not just temperature? Give examples.

Information Sheet 1The ups and downs of drawing pins(For Homeworks 15.1, 15.2)Tossing a drawing pin in 10 groups of 10 throws gives a sample size of 100 trials.The total numbers of ‘Up’ and ‘Down’ are recorded after every 10 trials and thetotal (accumulated) number of ‘Up’ results is also recorded. The experimentalprobability for throwing the drawing pin ‘Up’ is then calculated after each groupof 10 trials, and recorded in the final column.

(For Homeworks 15.5, 15.6)The experimental probability calculated from the drawing pin investigation allowsa line graph to be drawn for each set of trials. As the number of trials increases theexperimental probability should begin to approach the theoretical value.

In this line graph the y-axisdoes not have to begin at0. Give the graph a title toshow what it represents.

0.58

10 20 30 40 50 60 70 80 90 100

0.60

0.62

0.64

0.66

0

Number of trials

Line graph showing the probability of pin landing ‘Up’ asthe number of trials increases

Pro

bab

ilit

y

Trials ‘Up’ ‘Down’ Total Probability‘Up’ of ‘Up’

1–10 6 4 �160� 0.60

11–20 7 3 �12

30� 0.65

21–30 4 6 �13

70� 0.57

31–40 8 2 �24

50� 0.63

41–50 6 4 �35

10� 0.62

51–60 5 5 �36

60� 0.60

61–70 5 5 �47

10� 0.59

71–80 8 2 �48

90� 0.61

81–90 7 3 �59

60� 0.62

91–100 5 5 �16010� 0.61

The probability of‘Up’ results is theexperimentalprobability.

It is a good idea toshow your working.


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Information Sheet 2Coin tossing(For Homeworks 15.3, 15.4, 15.5)Three coins are tossed at the same time and the data recorded using the followingsheet. The coins are tossed in five groups of 10 throws each, giving a sample sizeof 50 trials. Each throw is recorded as either tails (t) or heads (h).

The experimental frequencies of the four possible outcomes are shown below.

Total sum of all frequencies � 11 � 21 � 12 � 6 � 50 (correct number of trials)

Sample space diagram for a three-coin toss

Trial 1 2 3 4 5 6 7 8 9 10

1–10 ttt tth thh thh ttt tth tth thh tth thh

11–20 tth thh ttt thh tth hhh tth hhh tth ttt

21–30 ttt thh tth tth tth hhh ttt ttt ttt thh

31–40 tth tth tth tth tth tth ttt hhh ttt tth

41–50 ttt thh thh hhh thh tth tth tth hhh thh

Trial ttt tth thh hhh

1–10 2 4 4 0

11–20 2 4 2 2

21–30 4 3 2 1

31–40 2 7 0 1

41–50 1 3 4 2

Total 11 21 12 6

h t

hh hhh hht

th thh tht

ht hth htt

hh hhh hht

Outcomefrom 2 coins

Outcomefrom 1 coin

Information Sheet 3Mean temperatures in the northern hemisphere(For Homeworks 15.4, 15.6)The following table shows the results of a survey of the mean (average)temperature in eight different cities in the northern hemisphere together with theirposition in terms of latitude (given to the nearest degree). A scatter graph showsthe degree and type of correlation between latitude and mean temperature.

City Latitude Mean annual temperature(degrees) (°C)

Bombay 19 30

Calcutta 22 25

Dublin 53 13

Istanbul 41 18

London 51 12

Oslo 60 10

Paris 49 16

St. Petersburg 60 7

10

10 20 30 40 50 60 70

20

30

40

0

Latitude (degrees)

Mean temperatures in cities

Tem

per

atu

re (°

C)

y

x

This scatter graph showsa good example of strongnegative correlation. Theline of best fit has alsobeen drawn. As in allgraphs, it has a title toshow what it represents.



3RUnit 1 Homework answers1.1 Quadratic sequences� a) 4, 13, 28, 49 b) 7, 16, 31, 52

c) 4, 16, 36, 64, 100 d) �2, 10, 30, 58, 94

� 148, 301, 433

� a) Multiply each term by 5.b) Divide each term by 2.c) Double each term and then add 4.

�

� a) 3, 5, 7, … b) 3, 5, 7, … c) 3, 5, 7, …

1.2 The general term� T(n) � n2 � 13

� a) Linear b) Quadratic c) Linear

� a) T(n) � 3n � 1 b) T(n) � n2 � 3c) T(n) � 2n � 2

� a) T(n) � n2 b) T(n) � n2 � 1c) T(n) � 2n2 d) T(n) � 2n2 � 1

� Sequence First Seconddifferences differences

a) 4, 7, 12, 19, 28 3, 5, 7, 9 2, 2, 2b) 6, 12, 22, 36, 54 6, 10, 14, 18 4, 4, 4c) 3, 12, 27, 48, 75 9, 15, 21, 27 6, 6, 6d) 5, 17, 37, 65, 101 12, 20, 28, 36 8, 8, 8

� The second differences are always double thecoefficient of n2. (Accept similar statements.)

1.3 Special sequences� a) T(n) � �

n �

n2

� b) T(n) � �2nn2� � �

21n�

c) T(n) � �n2

2� 2�

� a) �11

24� � �

67� b) �2

1828� � �2

14� c) �1

246� � �7

13�

� a)

b) n more; 4 � n

� a) T(n) � �n �

n1

�

b) i) �45� ii) �

78� iii) �1

90� c) i) 0.8 ii) 0.875 iii) 0.9

d) No. The numerator will always be one less thanthe denominator.

1.4 Formulae� a) C � 0.4m � 2 b) m � �

C0�

.42

�

c) £8.40 d) 7.5 miles

� x � �y �

312

�

� a) T � 35K � 25 b) 2 hours 27�12� minutes

c) i) 4 kg ii) 4000 g

� a) Equilateral. The angle at the centre of thecircle � 360° � 6 � 60°, and the other angles areequal because the sides are radii of the circle.

b) 6r

1.5 The inverse of a linear function� a) x → → y b) x → → → y

c) x → → → y d) x → → → y

� a) x � y � 5 b) x � �y �

37

�

c) x � 4(y � 1) d) x � �y �

23

�

� a) Divide the amount by 1.78.

b) US$ → → £

c) US$4.50

� a) T(°C) → → → T(°F)

b) 69.8°F c) T(°C) ��5(T(°F

9) � 32)�

d) 27.8°C

1.6 Graphs� a)

b)

c)

�

� a) y � 3x � 2 b) y � �2x

3� 7�

c) y � 4(�3x � 2) or y � �12x � 8

� a) Gradient � 3; y-intercept � �2b) Gradient � �

23�; y-intercept � ��

73�

c) Gradient � �12; y-intercept � 8

� (2, 2) and (�1, 5)

8

2

4

6

y

x�5 0 5

�4

�2

�6

�8

�10

2

y

x�5 0 5

�2

�4

�6

�8

10

12

14

2

4

6

8

y

x�5 0 5

�32�5�9

�1.78

�3�2�1�4

�7�3�5

Term 1 2 3 4

T(n) � n2 1 4 9 16

T(n) � 2n2 � n 3 10 21 36

T(n) � 3n2 � 3 0 9 24 45

x 36 0 6 45

y 0 12 10 �3

E HWS Ans 114-128.qxd 20/4/05 11:57 am Page 114

Unit 2 Homework answers2.1 Adding and subtracting fractions� a) 1�

12� b) �2

51� c) �

23

95�

� a) 12 b) ah c) 4dd) 4cd e) 12cde f) 24y2

� a) �(a �

bd)

� b) �(ad

b�

dc)

� c) �6c

�

� a) �(4xy

6

2

y�

2

3x)� b) �

(4x3

6�

x2

9yxy2)

��(4x2

6�

xy9y2)

�

2.2 Multiplying fractions� a) �

12

58� b) �

16� c) 3�

12� d) 1

� a) �59� b) 2�

29� c) �

3a

�

d) �2xpy

� e) �23x�

� a) �6parc

� b) �6ht2

� c) �35x� d) 5p2

� a) 12�47� cm b) �

272r� cm c) �

67t� cm

2.3 Reciprocals� a) �

43� b) �

18� c) �

45� d) 1

� a) �140� � 2.5 b) �

11

03� c) �

12

08

07� d) �

130�

� a) �170� � 1�

37� b) 1

� a) �32� b) �

cad� c) �1

43� d) �

25�

� a)

b) As the value of y increases, the value of �1y

� decreases.

2.4 Dividing fractions� a) �1

85� b) �

12

61� c) 1�

18� d) 3

� a) �34

54� b) 1�

12

78� c) 1�

15� d) �

12�

� a) �acbtd

� b) �c1f� c) a2 d) �

ba2

2�

� �ae2

2

fhc

�

� a) fk b) 7y3

2.5 Mental methods� a) 0.23 b) 0.069 c) 0.0384 d) 0.432

� a) 40 b) 10.6 c) 34 d) 30

� a) 0.5 � 0.6 � 0.3 b) 0.7 � 0.4 � 0.28c) 0.4 � 0.8 � 0.5 d) 2.4 � 0.4 � 6

� a) 221.2 b) 2.212 c) 0.2212 d) 2212

� a) 420 b) 42 c) 0.0042 d) 0.3

2.6 Percentages� £95 000

� a) £374.50 b) £3.36 c) £77.08 d) £83.25

�

� £5955.08

� The larger one

2.7 Direct proportion� a) €1.45

b) y � 1.45xc) Yes. As one amount increases, the other increases

proportionally.

� a) Yesb) k � 0.035

� a) Nob) Ashok. 4 km would be represented by 1 cm, not

0.5 cm.c) 26 km

2.8 Inverse proportion� a) 4 days

b) 2 days (insufficient for three days)c) Yes (exactly)

� 7�15� hours � 7 hr 12 min

�

� a) 2�23� hours � 2 h 40 min

b) 64 km/hc) 160

2.9 Ratio� a) 1 : 1.5 or 24 : 36

b) 1 : (1.5)2 � 1 : 2.25

� a) 1 : 0.67 b) 1 : 1.25 c) 1 : 0.2 d) 1 : 1.21

� a) 4 : 25b) 8 : 125

� a) 4.93 : 3.91 b) 24.3 : 15.29 c) 1.26

Unit 3 Homework answers3.1 Constructing and solving equations 1� a) c � £2.50 b) 5c � £5 c) i) £2.50 ii) £5

� a) 3b) 10 and 8c) 80 units2

� £160

3.2 Solving more complex equations� 3.4

� a) 2.7b) 2.71

� Pupils have choice of methods here, but only allowalgebraic in part a) and trial and improvement inpart d).

� a) 0.7 b) 5 c) 0.6 d) 1.3

� 8w3 � 40; w � 1.7 cm


3R

y 0.2 0.4 0.6 0.8 1

�1y

� �120� � 5 �

140� � 2.5 �

160� � 1.67 �

180� � 1.25 1

% 30% 20% 12.5% 30%change Decrease Increase Decrease Increase

Length 5 cm 2 cm 8 cm 1 cm

Width 4 cm 10 cm 2.5 cm 20 cm


3.3 Constructing and solving equations 2� 25, 26 and 27

� Length � 7 cm; width � 4 cm

� 5

� 27.5

� a) x � 2 b) 24 and 26

3.4 More than one unknown� a) a � 3, e � 9 b) f � 4, g � 9

c) h � 1, k � 3

� a) a � 4, e � 1 b) m � 5, n � 3c) m � 4, n � 2

� a) m � 5, p � 15 b) r � 5, t � 2c) m � 4, e � 3 d) r � 5, e � 3

� a) a � 0.5, r � 4 b) e � �1, t � 3

3.5 Eliminating one unknown� a) p � 2, r � 5 b) a � 2, b � 4

c) r � 5, t � 6

� a) h � 8, r � 5 b) x � 8, t � 3c) q � 1, r � 4

� a) e � 5, f � �23� b) h � 1.5, k � 2.5

c) e � 4, h � �2

� a) 2c � 3y � 48 b) 2c � y � 32c) i) c � 12 ii) y � 8p

3.6 More simultaneous equations� a) 12d � 16e � 28 b) 9e � 9r � 18

c) 15g � 9d � 36 d) 40h � 10r � 50e) 12e � 8k � 28

� a) i) 3ii) 6k � 3e � 24iii) k � 2, e � 4

b) i) 2ii) 12e � 8h � 12iii) h � 6, e � 5

� a) a � 7, e � 5b) h � 1, k � 3c) t � 2, r � 5

Unit 4 Homework answers4.1 Conventions� a) b � 50° (Angles in a triangle sum to 180°.)

a � 130° (Angles on a straight line sum to 180°.)b) e � 55° (alternate angles)

g � 125° (angles on straight line)f � 125° (alternate to g)c � 125° (angle on straight line with e) or(vertically opposite to f )d � 55° (angle on straight line with f ), etc.

c) k � 110° (angle on straight line with 70°)h � 110° (alternate angle to k)j � 100° (sum of angles in quadrilateral)

� Answer b): a practical demonstration

� a) Sketch showing at least six tessellating hexagonsb) Sketch showing octagons and squares tessellating

� a � e, c � f, n � h � k, m � j � p

4.2 Pythagoras’ theorem 1� a) 8.1 cm b) 5.7 cm c) 5.8 cm

d) 5.6 cm e) 6.9 cm f) 7.8 cm

� No (The length of the hypotenuse would be 5.5 to1 d.p.)

4.3 Pythagoras’ theorem 2� a) obtuse b) acute

c) right-angled d) acute

� 11.7 km (to 1 d.p.)

� 7.3 units

� False

� a) Heights are consecutive odd numbers startingfrom 3.

b) Hypotenuse � base � 1c) Height squared � sum of base and hypotenuse

4.4 Congruent triangles� a) Yes (SAS)

b) No: The 4 cm line joins different angles.c) Yes: Both triangles have angles 35°, 70°, 75°,

satisfying ASA.

� No. They are similar. The sides could all be longer orall be shorter.

� a) Yes (SAS)b) Noc) Yes (SAS)d) It depends on which diagonal is selected.

4.5 Unique triangles� a) Correct construction of two different triangles,

ABC, with �A � 30° and sides BC � 3.2 cm andAB � 4.2 cm

b) Correct construction of triangle ABC where�A � 90°, AC � 3 cm, BC � 5 cm

c) Correct construction of two different triangleswith �A � 40°, �B � 90°, �C � 50°

d) Noe) No

� a) Single drawing of a unique triangle with sides of3 cm, 4 cm, 5 cm

b) Correct construction of two different triangleswith sides of 4 cm and 3.5 cm and an angle of 40°

� a) Drawing of two congruent trianglesb) Drawing of two similar, but non-congruent

triangles

� a) Yes b) Yes c) No d) No

4.6 Lines on a coordinate grid� a) 5

b) 8.1 (to 1 d.p.)c) 8.1 (to 1 d.p.)

� a) 11.6 or 4.4b) 12.4 or �2.4

� a) (8, 13)b) (6, 4.5)c) (5.5, 3)

� a) 3.6 units (to 1 d.p.)b) 1.8 units (to 1 d.p.)c) (4.5, 4)


3R


4.7 Symmetry of shapes� a) 3 b) An infinite number c) 0

� a) 3 b) An infinite number c) 0

� a) 4 b) Infinite c) 1

� Two drawings of the ‘T’ shape, showing the twoplanes of symmetry.Two drawings of the ‘C’ shape, showing the twoplanes of symmetry.

4.8 Loci� �

� a) b)

c) d)

� a) a line forming a circle 5 m round the seedb) a circular area, 4 m round the seedc) a hemisphere of radius 4 m

4.9 Finding a locus using ICT� a) Drawn diagram of a hexagon

b) Drawn diagram of an equilateral trianglec) Drawn diagram of a five pointed star

� a) Sketch of a line passing mid-way between thepoints A and B, perpendicular to line AB

b) Sketch of a straight line joining C and D

�

� Drawing of rectangle with vertices A � (0, 3) B � (0, 0)C � (7, 3) D � (7, 0)a) Any three points listed satisfying x � 3.5b) Any three points satisfying y � 1.5c) (3.5, 1.5)

� One possible solution is as follows:RT 90, FD 1000, LT 120, REPEAT 5[FD 200, LT 120,FD 200, RT 120]

Unit 5 Homework answers5.1 Planning an investigation 1� a) Primary data; accept any correct suggestion as to

how this could be collected, i.e. questionnaire,survey, etc. Range of ages; equal numbers fromeach gender

b) Primary data; accept any correct suggestion ashow this could be collected, i.e questionnaire,survey, etc. Range of ages; equal numbers fromeach gender

c) Secondary data would be needed, from referencebooks, internet, etc.Data would need to cover a number of years ratherthan, for example, just what happened last year.

� a) Yesb) ‘A larger range of food on the menu would

persuade more pupils to stay to school dinners’(or other relevant hypothesis).

c) He is not seeking the opinions of the people whodon’t stay to dinners and so will have biasedresults.

d) To avoid bias, we need a full range of ages. Also,equal numbers of boys and girls, etc.

5.2 Grouped data� a) 40.7 years

b) Can’t say for certain; for example, the people inthe upper age group could all be 61.

� a) French 30.9; German 29.9b) German

5.3 Cumulative frequencyIn all questions allow a reasonable margin of error (i.e.within 2 mm) when reading values from graphs.

� a)

median � 11; interquartile range � 7

b)

median � 17; interquartile range � 10

5.4 Drawing diagrams� Correct drawing of the frequency diagram with

frequency polygon superimposed.

� a) Yes b) No c) Yes

8

Freq

uen

cy

4

2

6

10

0 140 145 150

Height (cm)

155 160 165 170

20

30

Cu

mu

lati

ve f

req

uen

cy

10

0100 20 30

Number of people

40

40

50

30

10

Cu

mu

lati

ve f

req

uen

cy

20

050 10 15

Time (min)20 25

FG


3R


5.5 Correlation� a)

Mean shoe size � 44.4 (to 1 d.p.)Mean collar size � 41.3 (to 1 d.p.)b)

Mean temperature � 12°CMean number of jumpers sold � 20.4

5.6 Misleading statistics� a) Pupils should have drawn a graph starting at 0

on the y-axis.b) Pupils should have drawn a graph that starts at a

figure higher than 0 on the y-axis.

� No. Whilst most accidents occur at home, these areoften not serious accidents resulting in death.

� Scales not starting at zero; incomplete diagrams;small sample sizes.

Unit 6 Homework answers6.1 Compound measures� 60 km/h

� a) 114 km b) 19 km/h

� 64 km

� a) 160 000 km/day b) 5000 km/h

� a) 13.6 g/cm3 b) 1020 g

6.2 Measurement� a) i) A � 10, B � 2, C � 18, D � 26 and E � 32.

ii) Range of A is 9 � A � 11.Range of B is 1 � B � 3.Range of C is 17 � C � 19.Range of D is 25 � D � 27.Range of E is 31 � E � 33.

b) i) A � 18.0, B � 18.6, C � 19.2 and D � 16.4.ii) Range of A is 17.9 � A � 18.1.

Range of B is 18.5 � B � 18.7.Range of C is 19.1 � C � 19.3.Range of D is 16.3 � D � 16.5.

� a) 406.9 kg b) 406.5 kg c) 101.725 kgd) 101.625 kg e) 101.625 � W � 101.725 kg

� 57 679.0 � V � 62 380.1 cm3 (1 d.p.)

� a) 7.0525 � A � 7.6725 m2

b) At least 90, but no more than 96

6.3 Circles 1� a) b)

� a � 60°, b � 60°, c � 60°

� a) w � 90°, x � y � 45° and z � 90°b) w � 62°, x � 31°, y � 59° and z � 90°

� a) 12 cm by Pythagorasb) 60 cm2

6.4 Circles 2� x � y � 30°, z � 90°

6.5 Arcs� a) 72 cm b) 36 cm c) 24 cm

d) 18 cm e) 16 cm f) 58 cm

� a) 10.5 b) 41.9 c) 22.3 d) 29.3

� a) 90° b) 120° c) 160°d) 150° e) 135° f) 185°

� Arc AB � 39.3, arc CD � 44.5, difference � 5.2

6.6 Sectors� a) 24 cm2 b) 36 cm2 c) 16 cm2

d) 34 cm2

� a) 52.36 b) 141.37 c) 51.31 d) 95.99

� 30.2 cm2

� 47.75°

Unit 7 Homework answers7.1 Powers of 10� a) 842 000 b) 0.01937 c) 5660

d) 0.00061 e) 2000 f) 0.005g) 0.0000007 h) 0.0052

� a) 1010 b) 10�3 c) 104

d) 107 e) 109 f) 10�6

g) 104 h) 10�2

� 6.9 � 10�1 6.35 � 103 6.7 � 103

5.9 � 104 6.305 � 104

� a) 2.7 � 10�3 � 0.0027 b) 5.9 � 103 � 5900c) 0.0032 � 105 � 320 d) 7100 � 10�2 � 71e) 400 � 103 � 0.4 f) 0.006 � 10�4 � 60

� a) 5.6 � 106 � 56 � 105

b) 7 � 104 � 700 � 102

c) 3.4 � 10�3 � 34 � 10�4

d) 0.8 � 102 � 80 � 100

e) 0.4 � 102 � 40 � 10�4

f) 0.0005 � 104 � 5 � 10�8

00

5

10

15

Tem

per

atu

re (°

C)

20

25

5 10 15 20Jumpers sold

25 30 35 40

44

46

48

50

42

38

Col

lar

size

40

363836 40 42

Shoe size44 46 48 50


3R



3R7. 2 Numbers in standard form� a) 8.5 � 106 b) 4.1 � 10�3

c) 3.005 � 105 d) 1.00005 � 101

e) 2 � 10�7 f) 5.0001 � 107

g) 3.33437 � 10�1 h) 7.000007 � 100

� 1.002 � 1012

� a) 2.438 � 1012 b) 2.5 � 105

c) 4.6368 � 101 d) 5.4 � 10�12

e) 6.37 � 104 f) 5.79 � 105

� 2.7 � 107

� 2.3 � 106

7. 3 Rounding� a) 159 000 000 000 b) 158 800 000 000

c) 158 750 000 000

� a) 1.1 b) 1.06 c) 1.061

� a) 13 000 000 b) 13 000 000 c) 12 990 000

� a) 1.3 b) 1.26 c) 1.260

7. 4 Upper and lower bounds� 2735 miles

� a) lower bound � 3450, upper bound � 35493450 � N � 3549

b) lower bound � 86 500, upper bound � 87 50086 500 � L � 87 500

� a) 16.05 cm b) 16.35 cm

� a) 21.45 cm2 b) 20.96 cm2

c) 21.945 cm2 d) 4.59%

7. 5 Multiplying� a) 2.11125 b) 21 112.5 c) 0.211125

� a) 14.455 b) 823.02 c) 21.224

� a) 70.38 m2 b) £244.92

� 197.54 metres

� 153.6 litres

7. 6 D i v i ding� a) 80 b) 8 c) 80

� a) 8.3 b) 4.1 c) 54

� Pupils’ own checks

� 12.5 metres

� 72.4

7. 7 Recurring decimals 1� a) and c)

� a) 0.4.

b) 3.5.6.

c) 0.1.45

.

d) 6.72.3.

e) 1.034.67

.

� a) 0.583.: This is a recurring decimal.

b) 0.375: This is a terminating decimal.c) 0.4

.: This is a recurring decimal.

d) 0.4.28571

.: This is a recurring decimal.

� a) 5.7, 5.77, 5.7.

b) 3.34, 3.3.4., 3.344, 3.34

.

c) 0.191, 0.191111, 0.1.9., 0.19

.

d) 0.5675., 0.5

.67

., 0.567

.5.

7. 8 Recurring decimals 2� a) �1

60� � �

35� b) �

59� c) �1

70�

d) �89� e) �

29�

� Pupils’ own checks

� a) �69

19� b) �

29

59� c) �

39

69� � �1

41�

d) �89

19� � �1

91� e) �

29

79� � �1

31�

� Each denominator is either 99 or 11.

� a) �39

09

99� � �

13

03

33� b) �

19

09

89� � �1

1121� c) �

19

19

89�

d) �29

49

99� � �3

8333� e) �

59

29

49�

� Each denominator is either 999, 333 or 111.

Unit 8 Homework answers8.1 Prime factorisation� a) 2 � 2 � 2 � 2 � 2 � 3 � 25 � 3

b) 2 � 2 � 5 � 7 � 22 � 5 � 7c) 2 � 2 � 2 � 2 � 2 � 3 � 3 � 25 � 32

� a) HCF � 12, LCM � 72b) HCF � 2, LCM � 2520c) HCF � 6, LCM � 180d) HCF � 27, LCM � 108

� a) HCF � xyz, LCM � x3y2zb) HCF � pqr2, LCM � p2q2r3

c) HCF � ab2c, LCM � a2b4c2

d) HCF � def, LCM � d2e2f 2

� a) �16003

� b) �43

36

90� � 1 and �3

7690�

c) �15698� d) �

22

64

30� � 1 and �2

2430�

� a) �(3c

a�

b2

2cab)

� b) �(4e2

d�2e2

3fdf)

�

c) �(1 �

p2q3p)� d) �

y3z�

8.2 Indices� a) 38 b) can’t be simplified

c) 31113 d) n9

e) (ab)4 f) can’t be simplified

� a) 32 b) 56

c) 311 d) n3

e) ��ba

��3

f) can’t be simplified

� a) 810 b) 39 c) 31136 d) 2735

e) b12 f) n2a

� a) 72 b) 3�2 c) 311 d) ne) a7 f) y14

� a) 8�2 b) 2�55 c) n�8 d) 345�100

8.3 Indices and surds� a) 12 b) 17 c) 23 d) 0.7

e) 52 f) 0.3

� a) True b) True c) False d) False

� a) Surd b) Surdc) Whole number d) Surde) Whole number f) Surd

� a) 2�11� b) 3�10�c) 10�2� or 5�8� or 2�50� d) 4�6� or 2�24�e) 5�3� f) 2�3�



3R8.4 Substitution� a) 144 b) �12 c) 304

� a) 18.4 b) 7.9 c) 1.2

� a) �32

27� � 1 and �2

57� b) �

92� � 4 and �

12� c) �

136� � 5 and �

13�

� 2.7 � 0.167

8.5 Parallel and perpendicular lines� a) y � 4x � 3 b) y � 1 � �

x5

�

� a) ��16� b) �7 c) �

12�

� a) y � �x � 12 b) y � x � 20c) y � �10x � 10

� a) y � x � 2 or y � x � 6b) y � �x � 4 or y � 2x � 12

8.6 Quadratic and cubic graphs�

�

� B, C and F

8.7 Graphs of quadratic functions� a) 7 b) �8 c) 50 d) �0.75

� a) (0, 9) b) (0, �11) c) (0, �1.7) d) (0, 115)

� a) Any three equations of the form: ax2 � bx � 3.5,where a and b can be any numbers

b) Any three equations of the form: ax2 � bx � 17,where a and b can be any numbers

c) Any three equations of the form: ax2 � bx � 100,where a and b can be any numbers

d) Any three equations of the form: ax2 � bx � 0.25,where a and b can be any numbers

� a) 0 b) 3 c) (3, 0)

� a) �7 b) (�7, 0)

8.8 Solving simultaneous equationsgraphically

� a) x � 5, y � 4 b) x � �18, y � 28c) x � 4, y � �1d) x � �1, y � 1, and x � 4, y � 6

8.9 Using simultaneous equations� a) 3x � 2y � 120

y � x � 10b) The length of a short slab is 20 cm.

The length of a long slab is 30 cm.

� The length of a silver link is 6 mm.The length of a gold link is 4 mm.

� Alice is 10.Amy is 34.

� Alice has £30.Amy has £70.

Unit 9 Homework answers9.1 Probability� �

12� � �

18� � �

58� � �

59� � �

69� � �

12

07�

� a) �130� � �1

20� � �5

30� b) �1

30� � �1

50� � �2

30�

c) �140� � �1

30� � �2

35� d) 1 � (�5

30� � �2

30� � �2

35�) � �1

6070�

� a) i) �13� � �

14� � �1

12� ii) �

23� � �

14� � �

16�

iii) �13� � �

34� � �

14� iv) �

23� � �

34� � �

12�

b) The sum is 1 because it is certain that one andonly one of the four possibilities will be true: theoutcomes are exhaustive.

9.2 Possible outcomes 1� a)

b) P(the counters are different colours) � �49�

� 36

� a)

b) �346� � �

19� c) �3

66� � �

16�

d) 7. Six of the 36 equally likely possible outcomesare 7, more than any other total score.

� �13

06� � �1

58�

red dice

green dice

2

3

4

5

6

7

3

4

5

6

7

8

4

5

6

7

8

9

5

6

7

8

9

10

6

7

8

9

10

11

7

8

9

10

11

12

B

First counter Second counter Outcome Probability

B

W

blue, blue 19

blue, white

W

W

B

W

W

W

B

W

W

blue, white

white, blue

white, white

white, white

white, blue

white, white

white, white

1919

191919

191919

�3

�15

x

y

�10

�5

5

10

15

�2 �1 10 2 3

�3

�15

x

y

�10

�5

5

10

15

�2 �1 10 2 3


9.3 Estimating probabilities�

� 0.6625

�

� 12 out of the 20 beads are almost certainly redbecause the estimated probabilities are tending to alimit of 0.6 � �1

60� � �

12

20�.

9.4 Comparing probabilities� �1

1288� � 0.140625

�

� Because 128 is a multiple of 64, which is the numberof equally likely possible outcomes.

� �16

04�

� Estimated probability � �11288� and theoretical

probability � �12208�. The theoretical probability is bigger

by �1228� � �6

14�.

Unit 10 Homework answers10.1 Transformations� Yes, they are commutative.

� a) and b)

c) The transformations are not commutative.d) A rotation of 180° about (2.5, 0).

10.2 Changing shapes� a) a right-angled triangle

b) an irregular quadrilateral

c) a kite

d) a regular octagon

e) an irregular pentagon

f) an irregular pentagon with one line of symmetry

� a) a parallelogram

b) a rhombus

c) an irregular hexagon

d) a regular hexagon

10.3 Similarity� a) 7 : 5 b) d � 19.6 cm c) 7 : 5

� a) 2 : 2�2� � 1 : �2� b) 4 square unitsc) 8 square units d) 1 : 2

� The ratio of the sides of the smaller triangle is 1 : 2.So 1 � x � 4.Therefore x � 3.

�15

�15

x

y

�10

�5

5

0

10

15

�10 �5 5 10 15

PA

B

�15

�15

x

y

�10

�5

5

0

10

15

�10 �5 5 10 15

T

60400

0.2

0.4

Rel

ativ

e fr

equ

ency

0.6

0.8

80Number of trials

100 120


3RNumber of trials 40 60 80 100 120


Relative frequency 0.525 0.53.

0.6625 0.64 0.5916.

of white

1 2 3 4 5 6 7 8

1 0 1 2 3 4 5 6 7

2 1 0 1 2 3 4 5 6

3 2 1 0 1 2 3 4 5

4 3 2 1 0 1 2 3 4

5 4 3 2 1 0 1 2 3

6 5 4 3 2 1 0 1 2

7 6 5 4 3 2 1 0 1

8 7 6 5 4 3 2 1 0



3R10.4 Enlargements 1� a)

� b)

�

10.5 Enlargements 2� a), b)

c) �14�

� a) �1100�

b) �215�

c) �2100�

d) �210�

� a) 60b) 2

10.6 Further enlargements� a) They are similar because every pair of

corresponding edge lengths are in the ratio 1 : 2.b) 1 : 2c) 1 : 22 � 1 : 4d) 1 : 23 � 1 : 8

� a) 1 : 2b) 1 : 22 � 1 : 4

Unit 11 Homework answers11.1 The product of two linear

expressions� a) x2 � x � 12 b) x2 � 9x � 14

c) 2a2 � 7a � 5 d) 2d2 � 18d � 20

� a) p2 � 12p � 36 b) x2 � 18x � 81c) 4y2 � 20y � 25 d) 9d2 � 60d � 100d) p2 � 2pq � q2 f) 16p2 � 8pq � q2

� a) x2 � 16 b) x2 � 49c) a2 � 26 d) 4d2 � n2

� a) (x � 11)(x � 8) b) x2 � 3x � 88

� a) (x � 6)(x � 5) b) x2 � x � 30

11.2 Factorising� a) 3(x � 4) b) 5(2 � n)

c) 7(a � 11) d) 8(7 � 2y)e) 3(5a � 1) f) 8(1 � 3b � 2c)

� a) 18x2 � 12x � 6x(3x � 2)b) 20x3 � 12x2 � 4x2(5x � 3)c) 9x � 3x2 � 3x(3 � x2)d) 10x2 � 25x3 � 5x2(2 � 5x)e) 28x2 � 21x � 7x(4x � 3)f) 55x3 � 22x2 � 11x2(5x � 2)

� a) 6n(n � 5) b) 13a2(2a � 1)c) 3p(3p3 � 11) d) 12(6 � 5y2)e) 3x(5 � 3x � 10x2) f) 8m3(m2 � 3)

� a) 2x b) x � 6

11.3 Algebraic fractions

� a) �x

3�

y6

� b) �a(c

c�2

b)�

c) �4ab

sst� 5� d) �

2myx2

�

y2

nx2

�

e) �3q �

6p10p2

� f) �3(jk

j2k�

2

3)�

� a) �srut� b) �

ac

�

c) �52a2

� d) �g2

2m�

� a) �rs

2

tu� b) �

abc2�

c) �3ca

2

2b� d) �

mn2

3

�

� a) Pupils’ own pairs of fractions.

11.4 Inequalities� a)

b)

c)

d)

500

5

10

15

20

25

30

35 A

A

A

B

B

B

C

C

C

D

D

D

y

x10 15 20 25 30

500

5

10

15

20

25

30

y

x10 15 20 25 30

500

5

10

15

20

25

30

y

x10 15 20 25 30

500

5

10

15

20

25

30

y

x10 15 20 25 30

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7


e)

f)

� a) �1, 0, 1, 2b) �4, �3, �2c) �3, �2, �1, 0, 1d) 1, 2, 3e) �5, �4, �3f) �1, 0, 1, 2, 3, 4, 5, 6

� a) Bb) Ac) Ad) Be) Bf) B

� A – Always True or B – Never Truea) Bb) Bc) Ad) B

11.5 Solving inequalities� a) n � 2 b) k � 11 c) n � 10

d) n � 17 e) m � 21 f) a � 5

� a) n � 6 b) k � 8 c) n � 5d) n � 24 e) m � 28 f) a � 15

� a) x � 6

b) x � 4

c) x � 3

d) x � �4

e) x � 4

f) x � 6

� a)

b)

c)

d)

e)

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15


3R

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7

�7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7



3Rf)

11.6 Solving inequalities in two variables�

�

�

Unit 12 Homework answers12.1 Solving ratio, proportion and

percentage problems� £50

� a) 10 mphb) 7�

12� miles at 20 mph, �

12� mile at 4 mph

� The 1 kg bag

� a) George & Sonsb) Matthews Construction Ltd, £285

� a) Andy’sb) Yes

12.2 Solving geometrical problems� a � 42°, b � 69°, c � 69°, d � 42°, e � 69°,

f � 42°, g � 153°.

� a � 45°, b � 45°, c � 45°, d � 90°, e � 54°, f � 72°, g � 67°, h � 46°, i � 67°, j � 34°

� a) 90°b) 90°

�ABC is always 90°. If �OAB � then �OBA � and �AOB � 180° � 2 .So �BOC � 2 and �OBC � 90° � .� �ABC � �OBA + �OBC � � (90° � ) � 90°

� Angle A � angle C � 180°, angle B � angle D � 180°.

12.3 Mixed problems� a) 7.875 m2

b) 8 m2 to the nearest m; if measurements areaccurate to within 1 cm then the smallest area is7.737 m2 and the largest is 8.014 m2.

� 600 m

� ��

4�

12.4 Multistep problems� 5�1

51� minutes past 1.

� a) £1309.71b) 14 monthsc) £60.70; £1456.85

� a)

28 is a perfect number.b) 1 � 2 � 4 � 8 � 16 � 31 � 62 � 124 � 248 � 496.

Yes, 496 is perfect.

12.5 Extended problems� No, you expect to lose. Over the 216 possible

outcomes your overall loss is £1.70.

� n � (n � 1) � (n � 2) � 3n � 3 � 3(n � 1) so divisibleby 3.No; e.g., 1 � 2 � 3 � 4 � 10 which is not divisible by 4.Sum of k consecutive integers is n � (n � 1) � …� (n � k � 1) � kn � �

12�k(k � 1). For k odd this is

divisible by k, for k even it is not divisible by k.

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15

�15

�15

x

y

�10

�5

5

10

15

�10 �5 5 100 15

Number 20 21 22 23 24 25 26 27 28 29 30

Sum 22 11 14 1 36 6 16 13 28 1 42


� Three triangles have two edges on the outside, sixtriangles have one edge and seven triangles havezero edges.

� Cannot make packs of the following sizes:1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28,31, 34, 37, 43.Can make packs of all other sizes.

Unit 13 Homework answers13.1 Planning an investigation 2� a) Any reasonable hypothesis that states either that

‘Girls are better at saying the alphabet backwardsthan boys are’, or boys are better, or that they areboth equally good.

b) Test hypothesis by experiment.c) primaryd) Similar numbers of girls and boys, different ages

to be tested, etc.

� a) Any reasonable hypothesis that states either that‘The weather is stormier now than in the past’, orjust the same, or less stormy.

b) By gathering information from references, etc.c) secondaryd) Would need to look back over a number of years

to eliminate the possibility that the last couple ofyears were unusual, possibly using differentsources.

� a) Pupils’ answers will vary. One example: ‘Olderpupils use litter bins less than younger pupils’.

b) No, he should look at the full range of opinion(students, staff).

c) primaryd) Pupils’ answers will vary. Check that their

questions are relevant to the hypothesis.

13.2 Planning and collecting data� a) Answers will vary. One suggestion: ‘I will ask

someone to say the alphabet backwards, andcount how many letters they get right beforemaking a mistake’. All responses should mentionusing a data collection sheet.

b) Accept any reasonable answer that would helpeliminate bias.

c) Accept any suitable sample-size.d) Any suitable data collection sheet, possibly:

e) Pupils’ data collection sheet tested on a smallnumber of people.

f) Amendments made to their data collection sheetif necessary.

� a) Answers will vary. One suggestion: ‘I will asksomeone to say the alphabet backwards, andcount how many letters they get right beforemaking a mistake. Then I will given them tenminutes to practice and repeat the process’.

b)

c) Their data collection sheet tested on a smallnumber of people.

d) Amendments made to their data collection sheetif necessary.

13.3 Processing data� a) 4–7 for both girls and boys

b) Boys 7.5, girls 7.4c) They are the same.

� a) No. Stem-and-leaf diagrams are not used forgrouped data.

b) Yes, though cumulative frequency diagrams areusually used for continuous data.

c) Yes – one for boys and one for girls.

� a)

b) 52c)

13.4 Representing data 1� a) Correctly drawn frequency polygons.

b)

c) Gives a clear indication of the shape of thedistribution.

d) Shows where one set of data is higher/lowerthan the other set of data.

13.5 Interpreting data 1� a) The median is the middle value of the ordered

data. The medians are the same, which suggeststhe boys and girls achieved similarly.

b) The boys’ results are more spread out than thegirls.

c) The interquartile range is the difference betweenthe top and bottom of the middle half of the data,with the extreme values excluded. These resultsshow a greater spread amongst the boys thanamongst the girls.

� a) Falseb) True

5

00–

34–

78–

1112

–15

16�

10

15

Freq

uen

cy

Letters correct

Boys

Girls


3R

Name Year Boy/Girl Number correct

Name Year Boy/ Number Number ImprovementGirl correct correct in

first second performancetime time (�/�)

35– 40– 45– 50– 55– 60– 65– 70– 75–39 44 49 54 59 64 69 74 79

2 2 4 6 3 4 2 1 1

3 7 74 3 3 6 7 8 95 0 0 0 1 2 3 8 8 96 1 2 3 4 5 87 1 8



3Rc) We don’t know whether this is true or false.d) Various answers. For example, Quikmark sell

more household goods than Supashop.e) Various answers. For example, Supashop sell

more than double the value of electrical goodsthan Quikmark does.

f) No

13.6 Writing your reportIntroduction

b) plan or problemc) hypothesis

Planninga) informationb) sample sized) bias

Collection of dataa) and b) where and how (either order)

Processing and representing your dataa) calculationsb) diagramsc) Tables

Interpreting the dataa) conclusions, hypothesisb) evidence

Evaluationa) problemsb) extendc) informationd) changes

Unit 14 Homework answers14.1 Investigating trigonometry� 29 m

� 1.7 km

� 1.5 m

� 12.4 m

14.2 Trigonometry 1�

a) � 0°: sin � 0, cos � 1 � 90°: sin � 1, cos � 0

b) The two curves are continuous and smooth andcross over each other. Both have a maximumvalue of 1 and a minimum value of 0 in thisangular range. Sine is increasing and cosine isdecreasing in this angular range.

c) The two curves cross over when � 45°.

� O � 8.8 cm and A � 12.1 cm

� 553 m

� 10.2 cm

14.3 Trigonometry 2� 68°

� a) 112 kmb) 166 km

� 50 m

� 36.4°

� 10.0 cm

14.4 Trigonometry 3� 8 m

� 82.0°

� 21.8°

� 178 m

� 25°

14.5 Lengths, areas and volumes 1� l � 13 cm

S(cone) � 204.2 cm2

S(total) � 282.7 cm2

� a) S(cylinder) � 316.67 cm2

S(cone) � 86.16 cm2

S(total) � 402.8 cm2

b) V(cylinder) � 665.01 cm3

V(cone) � 92.36 cm3

V(total) � 757.4 cm3

14.6 Lengths, areas and volumes 2� a) 212.4 cm3

b) 205.8 cm2

� r � 1.78 cm

� h � 11.9 cm

� V � 1.82 �

� r � 6.5 cm

Unit 15 Homework answers15.1 Planning a probability investigation� Data collection sheet for the two-dice problem. The

dice are thrown in six groups of 20 giving a samplesize of 120 trials. Each time the numbers on both diceare recorded and the totals for each number are alsoreduced. The first 20 trials are shown as an example.

10

�1

1

020 30

Angle (degrees)

cosine sine

40 50 60 70 80 90

1 2 3 4 5 6 1 2 3 4 5 6

203125641644 6365624152

4 3 2 5 3 3 2 4 3 4 3 42514265413 4361244352

40

60

80

100

120

Trials Bluedice

numbers

Reddice

numbers

Bluedice

totals

Reddice

totals


Below is the sample space diagram for the theoreticalprobabilities.

The probability for throwing a total of 4 is P(Total � 4) � �3

36� � �1

12�

� P(Red, Red) � �218�

� The possible outcomes from tossing four coins

P(at least one tail) � �11

56�

15.2 Experimental probabilities� mean � 59.6

mode � all equal, hence no mode availablemedian � 62range � 47

� mean � 7.3mode � 11median � 7.5range � 10

� 0.335; about half the expected number of sixes

� Own results

15.3 Possible outcomes 2� Own design of data collection sheet.

A sample size of between 50 and 100 trials should besufficient to obtain reasonably accurate results.The tree diagram for this looks like the following:

The theoretical probabilities can therefore betabulated to give the following probabilities:

A comparison should be made between yourexperimental and theoretical probabilities.

� The sample space diagram should look like thefollowing:

Total number of possible outcomes � 24Probability of obtaining a tail and an even number � �2

94� � �

38�

� The sample space diagram for throwing three dicelooks like the following:

a) P(total: 15) � �1508�

b) P(same number) � �316�

c) P(even) � �18�

d) P(prime) � �27136�


3R1 2 3 4 5 6

1 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

4 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6

5 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6

6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6

tt th ht hh

tt tttt ttth ttht tthh

th thtt thtth thht thhh

ht httt htth htht hthh

hh hhtt hhth hhht hhhh

Outcomes from two coins

Outcomesfromtwo coins

R

1st draw 2nd draw Outcome Probability

R �

�

�

�

�

�

�

�

G

G

58

R

G

R, R

R, G

G, R

G, G

58

5838

38

38

27

27

57

57

37

37

47

47

636

1556

1556

2056

Outcome R, R G, G R, G or G, R

Theoretical�2506� �

566� �1

556� � �1

556� � �3

506�

probability

1 2 3 4 5 6

hh hh1 hh2 hh3 hh4 hh5 hh6

ht ht1 ht2 ht3 ht4 ht5 ht6

th th1 th2 th3 th4 th5 th6

tt tt1 tt2 tt3 tt4 tt5 tt6

Two coins

Dice

1 2 3 4 5 6

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

7 8 9 10 11 12 13

8 9 10 11 12 13 14

9 10 11 12 13 14 15

10 11 12 13 14 15 16

11 12 13 14 15 16 17

12 13 14 15 16 17 18

Two dice(total)

One dice

1 2 3 4 5 6

1 � 1 3 4 5 6 7 8

1 � 2, 2 � 1 4, 4 5, 5 6, 6 7, 7 8, 8 9, 9

1 � 3, 2 � 2, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10,3 � 1 5 6 7 8 9 10

1 � 4, 2 � 3, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11,3 � 2, 4 � 1 6, 6 7, 7 8, 8 9, 9 10, 10 11, 11

1 � 5, 2 � 4, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12,3 � 3, 4 � 2, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12,5 � 1 7 8 9 10 11 12

1 � 6, 2 � 5, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13,3 � 4, 4 � 3, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13,5 � 2, 6 � 1 8, 8 9, 9 10, 10 11, 11 12, 12 13, 13

2 � 6, 3 � 5, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14,4 � 4, 5 � 3, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14,6 � 2 9 10 11 12 13 14

3 + 6, 4 + 5, 10, 10, 11, 11, 12, 12 13, 13, 14, 14, 15, 15,5 + 4, 6 + 3 10, 10 11, 11 12, 12 13, 13 14, 14 15, 15

4 � 6, 5 � 5, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16,6 � 4 11 12 13 14 15 16

5 � 6, 6 � 5 12, 12 13, 13 14, 14 15, 15 16, 16 17, 17

6 � 6 13 14 15 16 17 18

Blue dice

Red dice


15.4 Representing data 2� A comparative bar chart for tossing three coins. The

shaded bars represent the experimental probabilitiesand the clear bars the theoretical probabilities.

The accuracy of the experimental results could beimproved by increasing the number of trialsundertaken.

� a) Frequency diagrams, graphsb) Bar charts, line graphs, pie charts, scatter graphsc) Compound or comparative bar charts. These are

best drawn using the same scales on both axes.

� Pupils’ own data

15.5 Interpreting data 2� Pupils’ own response.

� Scatter graph showing the relationship between massand height of ten people.

The correlation is strong and positive. A line of bestfit has also been drawn.

� The uses of crude oil in the UK

Other

Fueling forheaing

Plastics

Fuel forgeneratingelectricity

Fuel forvehicles

1.6

1.545 50 55 60

Mass (kg)

Hei

ght (

m)

65 70 75 80

1.7

1.8

1.9

2.0

0.1

0.2

0.3

0.4

0

Outcomes

Pro

bab

ilit

y

P(ttt) P(tth) P(thh) P(hhh)


3R



3RYear 9 Unit test

Unit 1 Sequences, functions and graphs� Given that T(n) � 3n � 5, find

a) the first three terms

b) the 50th term

c) the position in the sequence of the term 70.

d) Write down the inverse function of x → 3x � 5.

� Write down the first three terms of the sequence T(n) � �2n

5�

n1

�.

Leave your answers as fractions.

� Find the general term of each sequence:

a) 6 11 16 21 … b) 50 45 40 35 … c) 3 3.5 4 4.5 …

� a) What is the second row of differences for the sequence T(n) � 2n2?

b) The second row of differences for a sequence is 16. The sequence can be written as an2,where a is a constant (always the same number).Write down the value of a.

� Write down the general term of the sequence found by adding the square numbers to thesequence T(n) � 3n � 4.

� The nth term of a sequence can be represented by the area of a diagram. This diagram shows the sequence whose nth term is 4n � 3.

a) Write down the general term of the sequence represented by the diagram below.

b) Find the 5th term.

� Find the inverse of each of the following functions.

a) y � 3x � 4 b) y � 3(x � 4)

� a) Draw the graphs

3y � 2x � 12 and y � 1.5x � 2

for values of x from �8 to �8.

Draw both lines on the same grid. Label your lines clearly.

b) Write down the relationship between the two lines you have drawn.Show that this relationship is true.

3

1

n

4

21

n

n

3

F Unit Tests 129-148.qxd 20/4/05 12:04 pm Page 129


3RYear 9 Unit test

Unit 2 Fractions, decimals and ratio

� Work out

a) 5�17� � 3�

45� b) �

xy

� � �3y

� c) �xy

� � �y3

�.

� Calculate �12016� � �

5633� by cancelling down first.

� The diagram shows a rectangle which has been split into six pieces of equal size.

a) Write down length r in terms of q.

b) Write down length s in terms of p.

c) Multiply your answers to parts a) and b)above to find the area of one of the small rectangles.

d) Write down the ratio rs : pq in its simplest form.

e) Calculate mentally the area of one small rectangle, in m2, for p � 0.8 m and q � 4.2 m by

filling in the empty boxes: �0.8

�� 0.8 � � � �.

Write the ratio p : q in its simplest form for this rectangle.

� Write down the reciprocal of

a) �n3

� b) n.

� Work out:

a) �34� � �

57� b) �

3p

� � �2qr�.

� Find the total length of each line labelled x, giving your answer as a single fraction in itssimplest form.

a) b)

c)

� Describe each of the following as direct proportion, inverse proportion or neither.

a) The length and width of a rectangle whose length and width may change but whose arearemains fixed

b) The distance between two towns on a map and in real life

c) The number of workers required to complete a task and the length of time needed tocomplete the task

x

4a

4a

4a

x a4

116

x xab

q

r

p

s



3Rd) The relationship between x and y in y � 4x � 3

e) The relationship between x and y in y � 4x

f) The relationship between x and y in y � x2

g) The relationship between x and y in xy � 18

� A fridge cost £156.25 after a 25% increase.Write down the cost before the increase using the fact that 1252 � 15 625.

The hourly rate for a casual job increases by 10% from £x to £7.70 per hour.

a) Form an equation involving x.

b) Solve your equation mentally.


Year 9 Unit test

Unit 3 Equations and formulae� The two shapes below have equal areas:

a) Form an equation in x to represent the information above.

b) An equal area, shown shaded, is deleted from both shapes.

Form a new equation.

c) Solve your equation to find x.

� Solve each equation to find the value of x.

a) 2(x � 1) � 3(x � 3) � 37

b) 5x � 2 � 10 � 3x

c) 3x2 � 4 � 112

� The triangle is isosceles.

a) Form an equation in x.

b) Solve your equation to find x.

c) Use your solution to work out the perimeter of the triangle.

� A quadrilateral has sides of length 5x � 2, 3x � 6, 5x � 3, 7x � 5.The perimeter of this quadrilateral is 82 cm.

a) Form an equation in x.

b) Use your solution to calculate the side lengths.

c) Write down the possible quadrilaterals that this shape could be.

� T(n) � 2n � 5. One of the terms of this sequence is 441.

a) Form an equation.

b) Solve your equation to find n.


3R

4

x

13

x

7 5

12

4

x

13

x

7 5

12

4

6x � 2

10� 3x

not drawn accurately


All lengths are in cm




3R�

a) Fill in the closest pair of numbers from those listed in the left column of the table tocomplete the sentence correctly:

A solution to the equation x3 � 3 � 30 lies between ________ and ________.

b) Write down the solution to the equation x3 � 3 � 30 to one decimal place.

� Solve the simultaneous equations:

2x � 3y � 115x � 6y � 14

X X 3

3.0 27

3.05 28.373

3.1 29.791

3.15 31.256

3.2 32.768

3.25 34.328

3.3 35.937


Year 9 Unit test

Unit 4 Geometrical reasoning and loci� Fill in the gaps:

a) Angle d � angle ____

These are _____________ angles.

b) Angle e � angle ____

These are _____________ angles.

� The diagram shows a rhombus inside a rectangle.

a) Calculate angle p.

b) Work out angles q and r in the grey triangle, showing your calculations clearly.

c) The rhombus has a perimeter of 40 cm.Use Pythagoras’ theorem to calculate the base of the shaded triangle.

�

Two of these three triangles are congruent.Which two? Show how you know.

� Triangle ABC has AB � 4 cm, BC � 6 cm and BCA � 32°.Sketch two possible triangles to show that triangle ABC is not uniquely defined.Do not construct your diagrams accurately.

� A triangle ABC has coordinates A(3, 4), B(15, 4) and C(15, 9).

a) Use Pythagoras’ theorem to calculate the hypotenuse of the triangle.

b) Triangle APQ is similar to triangle ABC. AB : AP = 2 : 3.Find coordinates P and Q.

� The diagram shows a triangular prism, whose cross-section is an isosceles triangle.Make a separate sketch of the triangular prism to show each plane of symmetry.

� The base of the triangle remains fixed.Vertex A is allowed to slide.Sketch the triangle and the locus of positions of Asuch that the area of the triangle remains the same.


3R

a bd

e

c

106°

12 cm

rq

p

P Q R

1.8

2.4

3 331.8

Anot drawn accurately


Year 9 Unit test

Unit 5 Handling dataThis test is all based on the same problem in order to follow through the data handling cycle.

Zoe’s year group have just received their Key Stage 3 National Curriculum Test results formaths, English and science. Zoe’s hypothesis is that less than a third of the pupils have a highermaths level than their English level.

Zoe is in the top set for maths.

a) Give one reason why the comparison between English and maths results for the pupils inZoe’s maths class will not be representative of the year as a whole. In what way would youexpect the results to be biased?

Zoe decides to take a sample of three pupils from each of four different maths classes. Their English and maths NCT results were as follows:

b) Draw a scatter diagram to show this data.

c) Describe the correlation between the maths and English results.

d) Is Zoe’s hypothesis true or false for this data? You must show some of your steps and justifyyour answer.

e) What would Zoe need to do to get more reliable results?

f) Complete the back-to-back frequency diagram:

g) Make a further comparison between the English results and the maths results based on yourcompleted frequency diagram in f).

h) The grouped frequency table shows the maths results for the whole year group.

Use the table to help you calculate an estimate of the mean maths level.

i) The cumulative frequency curve shows the marks obtained on the mathematics (levels 5–7) paper by 60 pupils.i) Use the graph to find the

median score on this paper.ii) Use the graph to find the

quartiles and the interquartile range.

Pupils needed 55 marks for a level 6 and 89 marks for a level 7.iii) Work out the number of pupils

scoring a level 5 or below.iv) Work out the percentage of

pupils entered for the levels 5–7 paper who achieved level 7.


3R

Maths 8 7 7 6 6 6 6 5 5 4 4 3

English 7 7 6 6 5 6 7 6 5 4 5 4

Maths level 3–4 5–6 7–8

Frequency 45 120 35

Level345678

Maths English

10

20

30

Raw score on 5-7 test

40

Cu

mu

lati

ve f

req

uen

cy 50

60y

x10

020 30 40 50 60 70 80 90 10

011

012

013

014

015

0


Year 9 Unit test

Unit 6 Measures and circles� A turnstile allows people to walk through at a rate of 11 per minute. How long would it take

for 704 people to pass through the turnstile?

� Biscuits are coated at a constant speed of 30 every 12 minutes.Work out the coating rate in

a) biscuits per minute b) biscuits per hour.

� The length of a room is 4 m to the nearest metre.The width of the room is 3.2 m to the nearest 0.1 metre.

a) Write down the minimum length and width of the room.

b) Work out the minimum area of the room.

c) Work out the maximum area of the room.

� a) Write down the names relating to the circle fori) the line BCii) the line TViii) the part of the circle between the circumference

and the line BC.

b) Calculatei) �AOBii) �OBAiii) �ABT.

� a) Draw a line AB 5 cm long.Draw a circle, centre A, radius 4 cm.Draw a circle, centre B, radius 4 cm.Draw the rhombus formed by these overlapping circles, whose vertices are the points A,B and the two points of intersection of the circles.

b) Measure the vertical diagonal of your rhombus.

c) Use Pythagoras’ theorem to calculate the vertical diagonal of the rhombus in cm.

� Calculate x.


3R

O

A

B

T

C V

x


�

The diagram shows the cross-section of an ice-cream cone. The top is a semicircle withdiameter 6 cm.

Calculate the area of this cross-section. Use 3.14 as an approximation of �.

� a) Calculate the shaded area, leaving your answer as a multiple of �.

b) Calculate the perimeter of the shaded region, leaving your answer in terms of �.


3R6 cm

9 cm

Diagram not to scale

3 cm

120°



Year 9 Unit test

Unit 7 Place value and calculations� a) Complete the table.

b) Which number in the left column is not written in standard form?

� Write down the missing powers of ten, represented by question marks:

a) 3.4 � 100 � 3.4 � 10? b) 5.4 � 0.001 � 5.4 � 10?

� A � 5.0 � 1023 B � 2.5 � 107

a) Calculate AB, working in standard form and converting your final answer to standardform.

b) Calculate A � B, working in standard form.

c) Write down the answer to A � B to 2 sig. fig. in standard form.

d) Calculate A2 converting your answer to standard form.

e) Write �B� in standard form.

� a) Write down the smallest number which rounds up, to 1 d.p., to 7.3.

b) Write down the largest number with exactly three decimal places which rounds down to7.3 to 1 d.p.

� The number of people attending Magdalen Road church on 14 November was 80 to thenearest 10. What is the maximum number of people that were there?

� A square has side length 3.5 cm to 2 sig. fig.

a) Copy and complete, filling in the lower and upper bounds for the length:

cm � length cm

b) Calculate the minimum area of the square.

� Convert �37� to a decimal, giving your answer to 2 d.p.

� a) Round 0.2.5.

to 3 decimal places.

b) Convert to a fraction.

Complete this multiplication grid:


3R

Number using powers of 10 Number without using powers

31 � 104 310 000

3.2 � 104

8.6 � 10�3

0.425

� 45 451

23 1035 782

2.3 10.373


Year 9 Unit test

Unit 8 Factors, indices and graphs� Calculate each of the following when x � 6 and y � �2.

a) 3(2x � 5y)

b) 3x(2x � 5y)

c) �3x3(2x

4� 5y)�

� a) Complete the table of values for the graph of y � x2 � 6x � 9 for values of x from 0 to 7.

b) Between which two values of x in the table above is the gradient of the graph the highest?

c) Draw the graph of y � x2 � 6x � 9.

d) Write down the equation of the line of symmetry of your graph.

e) Draw this line of symmetry on your graph.

� The diagram shows the graphical solution of simultaneous equations.

a) Use the graph to write down these simultaneous equations.

b) Work out whether the two lines on the graph are perpendicular. You must show some working.

c) Solve the simultaneous equations algebraically.

� a) Write 1225 as a product of prime factors.

b) Use your answer to part a) to find the square root of 1225.

� a) Find the highest common factor of a2b and ab2c.

b) Find the lowest common multiple of a2b and ab2c.

c) Multiply your answers to parts a) and b) above.

d) Explain how to find the lowest common multiple of two numbers given their product andtheir highest common factor.

� a) Given that 27 � 128, calculate 28.

b) Given that 210 � 1024, write down the answer to 1024 � 128 as an ordinary number. Show your method using indices.

� Find the value of p for which �128� � p�2�.


3R

Use your answer to part a).

Use your answer to part b).

x 0 1 2 3 4 5 6 7

y 4

y

y � 5 �

x

x2

1

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2 3�1�1�2�3�4�5

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3RYear 9 Unit test

Unit 9 Probability� A dice is biased. The probability of it

landing on each of the numbers 1 to 5 is shown opposite:Work out the probability of the dice

a) landing on 6 b) not landing on 4 c) landing on an odd number.

The dice is thrown 200 times.

d) Work out the number of times you would expect it to land on an odd number.

� On any given day, the probability that the school bus is late is 0.3.

a) Draw a tree diagram for Monday and Tuesday showing the possible outcomes. Label each branch with its probability.

b) Calculate the probability that the bus is on time both days.

c) Calculate the probability that the bus is late on exactly one of the days.

� A spinner with 20 numbers, a dice and a coin are thrown. Work out the total number ofpossible outcomes.

� Groups of pupils threw three coins and recorded the number of times all three coins showedheads. There were five groups, each of which threw the coins a different number of times.

a) Write down the missing numbers p, q and r.

b) Altogether the coins were thrown 1000 times. Work out the estimated probability of 3 heads from the combined results. Show your working clearly.

c) Work out the theoretical probability of three heads.

d) Work out the theoretical probability of at least one tail.

e) Explain whether you think that the coins used by the pupils were biased.

Score 1 2 3 4 5 6

Probability 0.2 0.15 0.1 0.12 0.25

Group A B C D E

Number of throws 100 150 200 250 300

Frequency of 3 heads p 21 22 r 39

Estimated probability 0.15 0.14 q 0.12 0.13


Year 9 Unit test

Unit 10 Transformations� AB and BC are two fixed sides of a 2-D shape. Any proportion of the dotted line from the

point A may be used for a top side to the shape.

Sketch and name every different type of shape which can be made by joining a single linefrom C to any point on the dotted line, ignoring any ‘unwanted’ part of the dotted linewhich falls outside the shape. For example:

Do not redraw the above but you do need to include an ordinary trapezium.

� a) Describe a single transformation which maps triangle a onto triangle b.

b) Describe a combination of two separate transformations which maps triangle a onto triangle b without using a translation.

c) Triangle a is translated 6 right and 4 up. It is then reflected in the line x � 3. Describe a single transformation which would map this image onto triangle b.


3R

B

120°

C

A

B

120° 120°

Isosceles trapeziumC

A

y

x

1a

b

2

3

4

5

6

�1�1�2�3�4�5 1 2 3 4 5

�2�3�4�5�6

0


�

Write down the scale factor and centre of enlargement for each trapezium A, B, C which is anenlargement of the dark grey trapezium. State any which are not enlargements of the darkgrey trapezium.

�

a) The triangles are similar. Find p and q.

b) Write down the ratio in its simplest form of the perimeters of each of the following:i) A to B ii) B to C iii) A to C

c) Write down one aspect of all three triangles which remains the same.

d) Find the ratio of the area of triangle A to the area of triangle B.

e) Write down the ratio of the area of triangle B to the area of triangle C.

� The volumes of two cubes are in the ratio 1 : 27.

a) Write down the ratio of the side lengths.

b) Find the ratio of the surface areas.

c) The volume of the smaller cube can be written as a3.Write an expression for the volume of the larger cubei) using brackets ii) without brackets.

d) The side lengths of both cubes are whole numbers between 2 and 8 inclusive. Work out the side lengths, the surface areas and the volumes of the two cubes.

AC

B

q

7.5

Diagrams not to scale

20

p

4

3

y

x

123456789

�1�2�3�4�5

�10

�9

�8

�7

�6

1 2 3 4 5 6 7 8 9 10 11 12 130�1�2�3�4�5�6�7�8�9

B A

C


3R


Year 9 Unit test

Unit 11 Formulae and graphs� a) Work out the total area:

b) Expand and simplify (2x � 3)(x � 4).

� a) Work out the missing lengths:

b) Factorise fully the expression 6x3y2 � 9xy4.

� a) Write down a formula in its simplest form for the perimeter, P, of a regular hexagon with sides of length t � 4 units

i) using brackets ii) without using brackets.

b) Calculate the perimeter when t � 5.

c) A different regular hexagon has a total perimeter of 12x � 30. Work out the length of its sides.

� Write each of the following as a single fraction in its lowest terms:

a) �ba

� � �2a

� b) �ba

� � �2a

� c) �ba

� � �2a

�

� The diagram shows the inequality x 5.

Write down the inequalities shown by the following diagrams:

a)

b)

� a) Solve the inequality 7x � 3 � 18.

b) Katie has £14. She needs £3 for the bus and wants to buy four identical pens. Represent the cost of one pen by x. Write an inequality to represent the fact that Katie’s bus fare and pens must not cost morethan £14. Solve your inequality to find the maximum amount that Katie can spend on each pen.


3R

1

x

x 4

2721x

p q

3 45x � 505

r

t � 4

0 1 2 3 4 5 6 7 8 9 10 11�10 �9 �8 �7 �6 �5 �4 �3 �2 �1

0 1 2 3 4 5 6 7 8 9 10 11�10 �9 �8 �7 �6 �5 �4 �3 �2 �1

0 1 2 3 4 5 6 7 8 9 10 11�10 �9 �8 �7 �6 �5 �4 �3 �2 �1


�

a) Give the equation of the line.

b) Write down the inequality represented by the shaded area.

y

x

123456

�6

�5

�4

�3

�2

�11 2 3 4 5 6�6 �5 �4 �3 �2 �1 0


3R


Year 9 Unit test

Unit 12 Problem solving� The ratio of girls to boys in a class of 28 changed from 9 : 5 to 4 : 3 between 2003 and 2004.

The total number in the class was always 28. Calculate the percentage increase in the numberof boys.

� The diagram shows two similar rhombi.The vertices of the small rhombus bisect each of the sides of the rectangle.The vertices of the rectangle bisect each of the sides of the large rhombus.The diagonals of both rhombi cross at G.

a) Determine the centre of enlargement and the scale factor of enlargement of the small rhombus onto the large rhombus.

b) Work out the ratio of the area of the shaded triangle to the area of the rectangle.

c) Work out the ratio of the area of the shaded triangle to the area of the large rhombus.

d) Work out the ratio, in its simplest form, of the areas of the small rhombus to rectangle to large rhombus.

e) Use your answer to part d) above to write down the ratio of the perimeter of the smallrhombus to rectangle to large rhombus.

� The graph shows the number of dots in each term of a sequence.

a) How many dots are added each time?

b) Write down the nth term, T(n), of the sequence.

c) Calculate the number of dots in the 50th term.

d) The equation of the dotted line of another sequence is parallel to this one. Write down one thing that the two sequences have in common. You must describe the sequences, not the lines.


3R

A

B DC

GFE H I

10

8

6

4

2

20

18

16

14

12

0 3 41 2Term number

Nu

mb

erof

dot

s

5

y

x


� The graph shows the number of dots in each term of a sequence.Each term in the sequence is made up of the area of two squares, represented in dots. One of these squares is the same size for each term.The size of the other square is different for each term.

a) Sketch the first three terms of the sequence.

b) Write down the equation of the dotted graph.

� The diagrams are made up of squares with whole number side-lengths less than or equal to 10 cm.

a) The total area is 36 cm2. The four smallest squares are all the same size.Given that x y z, find x, y and z.Now find a second solution for x, y and z.

b)

The two unshaded areas are equal.Find two possible solutions for x, y and z, the side lengths of the squares.Sketch a triangle with side lengths x, y and z, labelling the sides clearly.Describe your triangle.

� Write down a counter-example to each of the following.

a) A square number greater than 1 cannot be a cube number.

b) All numbers greater than 1 have an even number of factors.

zy

x


3R

10

8

6

4

2

20

18

16

14

12

0 3 41 2Term number

Nu

mb

erof

dot

s

5

y

x

Write down the first 10 square numbers.

x

y

z


Year 9 Unit test

Unit 14 Trigonometry, areas and volumes� A map has a scale of 1 : 250 000.

a) Work out the actual distance in km represented by 1 cm on the map.

b) Broadway is 60 km from Summertown. Calculate the distance on the map.

c) The distance on the map between Summertown and Woodstock is 5 cm. What is the actual distance between them in km?

� The two triangles are similar.All lengths are in cm.

a) Write down cos a.

b) Find �a.

c) The two triangles are similar. Find length x and angle b.

d) Fill in the missing number: cos a � � x

� Find the missing length or angle in each triangle.All lengths are assumed to be in cm.

� The diagram shows an isosceles triangle.Calculate length y.


3R

x

6

a b

1

0.75

8

35°

a

b

4

340°

c

8

y

10 m

80°Split into two right-angledtriangles.


� Calculate the volume of the triangular prism.

� The diagram shows a cylinder enclosed tightly by a box.What percentage of the volume of the box is taken up by the cylinder?You must show your working, stage by stage, very clearly.


3R10 cm

13 cm

5 cm

5 cm

5 cm

10 cm


Year 9

Unit testMark schemes

Unit 1� a) �2, 1, 4 (1)

b) 3 � 50 � 5 � 145 (1)c) 75 � 3 � 25 (1)

d) x → �x �

35

� (2)

� �35�, �1

50� (or �

12�), �1

75� (2)

� a) 5n � 1 (1)b) 55 � 5n (2)c) 2.5 � 0.5n (2); accept any equivalent expression

� a) 4 (1)b) 8 (1)

� n2 � 3n � 4 (2)� a) n2 � 3n � 2 (2)

b) 38 (1)

� a) y � �x �

34

� (2) b) y � �x3

� � 4 (2)

� a) Clear axes (1)line 3y � 2x � 12 drawn (2)line y � 1.5x � 2 drawn (2)

b) perpendicular (1)�

�23� � 1.5 � �1 (1)

Unit 2� a) 1�

13

25� (2)

b) �x �

y3

� (1)

c) �3x

3�

yy2

� (2)

� a) �12016� � �

56

33� � �

12� � �

13� � �

16� (2)

� a) �3q

� (1)

b) �p2

� (1)

c) �p6q� (1)

d) 1 : 6 (1)

e) �0.8 �

64.2

� (1) � 0.8 � 0.7 (1) � 0.56 (1)

4 : 21 (1)

� a) �n3

� (1)

b) �n1

� (1)

� a) 1�210� (2) b) �

p6qr� (1)

� a) �2ab� (1)

b) �14

1�

23a

� (2)

c) �1a2� (1)

� a) inverseb) directc) inversed) neithere) directf) neitherg) inverse. (3) 1 mark for 4 correct; 2 marks for 6

� £125 (1) a) x � 1.1 � 7.7 (1)

b) x � 7 (1)

Unit 3� a) 7x � 8 � 5x � 13 (1)

b) 2x � 8 � 13 (1)c) x � 2.5 (2)

� a) x � 6 (2)b) x � 1 (2)c) x � �6 (3)

� a) 6x � 2 � 10 � 3x (1)b) x � 4 (2)c) 48 cm (2); 1 mark for correct method

� a) 5x � 2 � 3x � 6 � 5x � 3 � 7x � 5 � 82 orequivalent (1)

b) x � 4 (1); side lengths 18 cm, 18 cm, 23 cm, 23 cm (2)c) rectangle, parallelogram, kite, arrowhead (delta)

(2) Full marks for 3 correct, 1 mark for 2 correct.� a) 2n � 5 � 441 (1)

b) n � 223 (1)� a) 3.2 (1) 3.25 (1)

b) 3.2 (1)� x � 4, y � 1 (3); 2 marks for first value, 1 mark for

substituting to obtain second value (allow followthrough)

Unit 4� a) Angle d � angle c (1)

alternate (1)b) Angle e � angle a (1)

corresponding (1)� a) 74° (1)

b) q � (180° � 74°) � 2 � 53° (2)r � 90° � 53° � 37° (1)

c) �102 ��62� � �64� � 8 (2)� P and R (1) 32 � 1.82 � 5.76 � 2.42 (1)

Mentions SSS or SAS or RHS (1)�

(2)

� a) AB � 12, BC � 5 (1); AC � �144 �� 25� � 13 (2)

b) Px � 3 � �32� � 12 � 21 (2) P � (21, 4) (1)

Qy � 4 � �32� � 5 � 11.5 (2) Q � (21, 11.5) (1)

32°

4 cm

4 cm

6 cm

A

B C

y3y � 2x � 12

y � 1.5x � 2

x

�8

�6

�4

�2

2468

�8 �6 �4 �2 20 4 6 8


3R

G Mark Schemes 149-152.qxd 20/4/05 12:05 pm Page 149

�

(2)

(2)

(2)

�

Unit 5a) Zoe’s class have all been selected for their ability in

maths, not English. They are more likely to be betterat maths than at English compared with the yeargroup as a whole. (2)

b)

axes (1) labels (1) points (2)c) positive correlation (1)d) Three pupils have a higher maths level than English

level. (1) �132� (1) is less than �

13� (1) The hypothesis is

true for this sample. (1)e) Take a larger sample. (1)f)

(2)

g) The range of levels is higher for maths (5) than forEnglish (3). (1)

h) 3.5 � 45 � 5.5 � 120 � 7.5 � 35 � 1080 (3)45 � 120 � 35 � 200 (1)mean � �

1200800

� � 5.4 (2)i) i) median: answer between 64 and 66 (1)

ii) Q1: answer between 60 and 62 (1)Q3: answer between 74 and 76 (1)Interquartile range: answer between 12 and 16 (1)

iii) 9 (1)iv) �6

80� � 13% (3)

Unit 6� 64 min or 1 hr 4 min (1)� a) 2.5 biscuits/min (1)

b) 2.5 � 60 � 150 (2)� a) length: 3.5 m, width: 3.15 m (1)

b) 3.5 � 3.15 � 11.025 11 m2 (2)c) 4.5 � 3.25 � 14.625 14.6 m2 (2)

� a) i) chord (1) ii) tangent (1) iii) segment (1)b) i) 72° (1)

ii) �(180°

2� 72°)�� 54° (1)

iii) 90° � 54° � 36° (1)� a)

(3)b) answer between 6.1 cm and 6.4 cm (1)

c) 2�42 � 2�.52� � 6.24 (2)� 30° (1)� semicircle: 9 � 3.14 � 2 � 28.26 � 2 � 14.13 (2)

triangle: 3 � 9 � 27 (1)Total area � 14.13 cm2 � 27 cm2 � 41.13 cm2 (1) (ft)

� a) 3� cm2 (2)b) 2� � 6 cm (2) Do not deduct marks for missing

units.

Unit 7� a)

Entries appearing in the question are shown initalic.

b) 31 � 104 (1)� a) �2 (1)

b) �3 (1)� a) 12.5 � 1030 � 1.25 � 1031 (2)

b) 2.0 � 1016 (1)c) 5.0 � 1023 (1)d) 25 � 1046 � 2.5 � 1047 (2)e) 5.0 � 103 (2)

� a) 7.25 (1)b) 7.349 (1)

A B

Level345678

Maths English

12345678

10 2 3 4 5Maths

En

glis

h

6 7 8

y

x


3R

Number using Number withoutpowers of 10 using powers

31 � 104 310 000

3.2 � 104 32 000 (1)

8.6 � 10�3 0.0086 (1)

4.25 � 10�1 (1) 0.425


� 84 (1)� a) 3.45 cm length 3.55 cm (2)

b) 3.452 � 11.9 (2)� 0.43 (2)� a) 0.253 (1) b) �

29

59� (2)

Total marks: 4.

10 373 (1) 34 (1) as shown aboveThird column (1)Bottom row fully correct (1) (follow through � 10from row above).

Unit 8� a) 6 (1) b) 36 (1) c) 324 (1)� a)

(2) 1 mark for any 4 correct.b) 6 and 7 (1)c) Suitable axes (1) At least six points plotted

accurately (1) and joined with a smooth curve (1)f) x � 3 (1)b) line x � 3 drawn on graph (1)

� a) y � 4x � 2, 2y � 10 � x (1)

b) 4 � ��12� � �2, therefore not perpendicular (2)

c) x � �194�, y � 4�

29� (3)

� a) 52 � 72 (2) b) 35 (1)� a) ab (1)

b) a2b2c (1)c) a3b3c (2)d) product � highest common factor (1)

� a) 2 � 128 � 256 (1) b) 210 � 27 � 23 � 8 (2)� �128� � �64� � �2�, so p � 8 (2)

Unit 9� a) 1 � (0.2 � 0.15 � 0.1 � 0.12 � 0.25) � 1 � 0.82

� 0.18 (2)b) 0.88 (1)c) 0.2 � 0.1 � 0.25 � 0.55 (2)d) 0.55 � 200 � 110 (2) follow through answer from

part c)� a)

(3)

b) 0.7 � 0.7 � 0.49 (2)c) 0.3 � 0.7 � 0.7 � 0.3 � 2 � 0.21 � 0.42 (3)

� 20 � 6 � 2 � 240 (2)� a)

p � 15 (1), q � 0.11 (1), r � 30 (2)

b) � �1102070

� � 0.127 (3)

c) 0.53 � 0.125 (or equivalent) (2)d) 1 � 0.125 � 0.875 (or equivalent) (2) follow

through answer from part c)e) Not biased, because the experimental probability

based on the combined results is very close to thetheoretical probability. Accept any sensiblecomparison. (2)

Unit 10�

(2)

(2)

(2)C joins the dotted line in any position other than theones above and the example in the question.

In each case 1 mark for name, 1 mark for sketch.� a) Rotation 180° about the origin (1)

b) Reflection in the y-axis (1) followed by reflectionin the x-axis (1) Either order. Also accept twosuccessive 90° rotations about the origin, eitherdirection.

c) Reflection in the line y � 2 (1)� A: enlargement scale factor 2 (1), centre (�13, �10) (1)

B: enlargement scale factor �12� (1), centre (�5.5, 8) (1)

C is not an enlargement of the shaded trapezium. (1)

� a) p � 3 � 5 � 15 (1)q � 20 � 2 � 10 (1)

b) i) 1 : 5 (1) ii) 2 : 1 (1) iii) 2 : 5 (1)c) The angles (1)d) 1 : 25 (1)e) 4 : 1 (1)

� a) 1 : 3 (1)b) 1 : 9 (1)c) i) (3a)3 (1) ii) 27a3 (1)d) side lengths 2 and 6, surface areas 24 and 216,

volumes 8 and 216 (3)

Unit 11� a) x2 � 5x � 4 (2) b) 2x2 � 5x � 12 (2)� a) p � 7x (1) q � 9 (1) r � 9x � 10 (1)

b) 3xy2(2x2 � 3y2) (2)

B

120°

Trapezium

C

A

B

120°

Triangle

C

A

B

120°

Parallelogram

C

A

(15 � 21 � 22 � 30 � 39)��

1000

0.7 on time

late0.3

0.7 on time

late

TuesdayMonday

0.3

0.7 on time

late0.3


3R

� 45 451 4.51 34 (1)

23 1035 10373 (1) 103.73 782

2.3 103.5 1037.3 10.373 78.2

x 0 1 2 3 4 5 6 7

y 9 4 1 0 1 4 9 16

Group A B C D E

Number of throws 100 150 200 250 300

Frequency of 3 heads 15 21 22 30 39

Estimated probability 0.15 0.14 0.11 0.12 0.13



3R� a) i) P � 6(t � 4) (2) ii) P � 6t � 24 (2)

b) 54 (1)c) 2x � 5 (1)

� a) �a2 �

ab2b

� (2)

b) �2b

� (2)

c) �2ab

2

� (2)

� a) x �4 (1)b) �7 x 9 (1) or x � �7 and x 9

� a) x 3 (2)b) 3 � 4x 14 or equivalent (1) £2.75 (2)

� a) y � 2x � 1 (1)

b) y � 2x � 1 (1)

Unit 12� 18 girls, 10 boys → 16 girls, 12 boys; �1

20� � 20% (3)

� a) centre of enlargement G (1) scale factor 2 (1)b) 1 : 8 (1)c) 1 : 16 (1)d) 1 : 2 : 4 (1)e) 1 : �2� : 2 (2)

� a) 3 (1)b) T(n) � 3n � 2 (2)c) 152 (1)d) same differences (1) Accept reference to same

times-table or multiples.� a)

(2)b) y � x2 � 4 (1)

� a) First solution:62 � 22 � 4 � 12 � 36 � 4 � 4 � 36x � 1, y � 2, z � 6 (2)Second solution:92 � 72 � 4 � 12 � 81� 49 � 4 � 32 � 4 � 36.x � 1, y � 7, z � 9 (2)Third solution: x � 2, y � 4, z � 6Accept any two of these solutions. No othersolutions

b) x � 3, y � 5, z � 4 (2)x � 6, y � 10, z � 8 (2)

(1)The description includes ‘right-angled’. (1)

� a) e.g. 64 (1) is both a square number and a cubenumber. Also n6 for any integer n

b) Pupil writes down any square number (1)

Unit 14� a) 250 000 cm � 2.5 km (1)

b) 24 cm (2); 1 mark for 60 � 2.5 or 120 � 5c) 5 � 2.5 km � 12.5 km (1)

� a) 0.75 (1) b) 41.4° (1)c) 0.75 � 6 � 4.5 (2) d) 6 (1)

� a � sin 35° � 8 � 4.59 (2)b � tan�1 2 � 63.4° (2)c � tan 40° � 3 � 2.52 (2)

� y � 2 � sin 40° � 10 m = 12.86 m (3)� Height � �132 ��52� cm � 12 cm (2)

Volume � �12� � 5 cm � 12 cm � 10 cm � 300 cm3 (3)

� Volume of cylinder � 2.52 � � � 10 cm3

� 196.35 cm3 (3)Volume of cuboid � 5 � 5 � 10 cm3 = 250 cm3 (2)Percentage of box taken up by cylinder � �

1925

6035

�%� 78.54% (2)

z

xy



3RTest A (Levels 6–8)

Calculator NOT allowed

Do not turn the page until you are told to do so.Write your details in the spaces below.

Name

Class

Teacher

Date

Remember● The test is 50 minutes long.

● You must not use a calculator.

● You will need: pen, pencil, rubber, ruler, protractor or angle measurer, compass.

● Try to answer all the questions.

● Write all your answers and working on the test paper.

● Check your work carefully.

H E-Y-T 153-172.qxd 20/4/05 12:07 pm Page 153


3R Test A

1 (a) I’m thinking of a number; I multiply it by 2 then add 7. The answer is 15.

Write this as an equation in terms of n, where n is the number.

..................Solve your equation.

.................. ...........2 marks

(b) 3(n � 2) � 15. Express this equation as a ‘thinking of a number’-type problem by fillingin the gaps below.

I’m thinking of a number; I ...............................................

then .............................................................................. ...........

and the answer is 15. 2 marks

2 (a)

Circle the statement that most closely describes the relationship between A and B.

Positive correlation Negative correlation No relationship ...........1 mark

A

B

H E-Y-T 153-172.qxd 20/4/05 12:07 pm Page 154


3RTest A

(b) On the axes below, draw 8 points that suggest no relationship between A and C.

...........1 mark

(c) What relationship, if any, is there between C and B?

......................................................................................

...................................................................................... ...........1 mark

3 Four pupils hold up cards with algebraic expressions:

Carol: 5b � 3 John: 4b � 6 Matthew: 20 � 2b Hilary: b2 � 4

(a) Find the value of b that makes Carol and John’s cards worth the same.

.................. ...........2 marks

(b) i) Factorise the expression that Hilary is showing.

............2 marks

ii) The expressions represent age.Matthew is a teenager. List all the possible integer values of b.

............3 marks

(c) The expression for Hilary’s age must satisfy b2 � 4 � 0.

To solve this inequality Tom says:

b2 � 4 � 0 if b is greater than 2.

He does not get full marks for this answer: explain why.

......................................................................................

...................................................................................... ...........1 mark

A

C

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4 Evaluate:

(a) 32 � 23 .................. ...........1 mark

(b) 82/3 .................. ...........1 mark

(c) (�12�)�2 .................. ...........

1 mark

5 (a) I am a quadrilateral with both pairs of opposite sides equal and parallel, and I have noright angles: What sort of quadrilateral am I?

.............................. ...........1 mark

(b) I am a different quadrilateral. All of my sides have different lengths; my base is parallelto the side opposite and has two acute angles leading from it. What sort of quadrilateralam I?

.............................. ...........1 mark

6 The table below shows the results of three classes in their mock KS3 exams:

(a) What fraction of Class 9A achieved a level 7?

.................. ...........1 mark


3R Test A

Class Level 4 Level 5 Level 6 Level 7

9A 0 2 16 12

9B 2 8 15 7

9C 10 16 4 0

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(b) What percentage of Class 9B achieved a level 5?

.................. ...........2 marks

(c) Of all the pupils that achieved level 5, what fraction came from 9B?

.................. ...........2 marks

7 Tim has worked out the answer to 6 ÷ �58� – will his answer be …

Tick the box that corresponds to your answer.

Explain why:

......................................................................................

...................................................................................... ...........1 mark

8 What is the area of this rectangle in terms of x?

.................. ...........2 marks

If the area of this rectangle is 6x2 � 3x and the width is 3x, what is its length?

.................. ...........2 marks

6x 2 � 3x 3x

3x � 4

2x � 1


3RTest A

More than 6 Less than 6

H E-Y-T 153-172.qxd 20/4/05 12:07 pm Page 157

9

Calculate the length marked a. .................. ...........2 marks

10 Titus is doing a survey on homework. He wishes to find out whether pupils do theirhomework on the day it is set and about how long it takes them to complete it.

These are the questions he decides to use:

(a) Do you do your homework the day that it is set? YES: NO:

(b) How long do you spend on it?

Suggest a more suitable question for (b).

......................................................................................

...................................................................................... ...........1 mark

11 Calculate:

(a) �23

� � �45

� .................. ...........2 marks

(b) �3a

� � �5b

� .................. ...........1 mark

(c) �(x �

21)

� � �(x �

33)

� .................. ...........2 marks

5 cm

a 13 cm


3R Test A

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12 The distance–time graph shows Emma’s journey to school.She leaves home at 8.20 and stops at a sweet shop on the way.

(a) How long did Emma stay in the sweet shop?

.................. ...........1 mark

(b) Look at the graph. How can you tell that Emma walked faster after leaving the shop?

......................................................................................

...................................................................................... ...........2 marks

(c) What is Emma’s average speed on her journey to school?

.................. ...........2 marks

13 A dog is tethered to a post next to a shed at the point marked ‘X’. The chain is 3 m long. Thediagram is drawn to a scale of 1 cm represents 1 m. Draw an accurate diagram to show thearea that can be reached by the dog.

...........2 marks

2

3

1

Time

Dis

tan

ce (

km)

8.20 8.30 8.40 8.50


3RTest A

2 m

Shed3 m

Scale 1 cm � 1 m

1 m

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14 These two shapes are similar.

(a) Calculate the length of EF.

.................. ...........2 marks

(b) Angle EFG is 34°. Find angle ABC.

.................. ...........1 mark

15 When a drawing pin is tossed it can land or

Here are the results of tossing this drawing pin:

(a) What is the experimental probability of landing after 20 throws?

.................. ...........1 mark

(b) Estimate the probability of landing

.................. ...........1 mark

(c) Another drawing pin is used. This has a larger base.

How might this change the probability of landing ?

Tick the box that best describes your answer:


......................................................................................

...................................................................................... ...........1 mark

A D HE

F

GC

B

2 cm

5 cm 12.5 cm


3R Test A

Position landed 20 throws 50 throws 100 throws 250 throws

6 18 38 99

Increase Decrease No Change

H E-Y-T 153-172.qxd 20/4/05 12:07 pm Page 160


3RTest A

Mark scheme: Test AQuestion Correct answer Additional guidance Level Attainment

target

1 (a) 2n � 7 � 15 → n � 4 1 mark for each term 6 AAllow 1 ft mark for correctly solving an incorrect equation

(b) … add 2 then multiply the result by 3 �1 mark for any omissionAccept multiply by 3 then add 6

2 (a) b � 3 1 mark for 5b � 3 � 4b � 6 6 A(b) i) (b � 2)(b � 2) 1 mark for each bracket 8(b) ii) 1, 2, 3 �1 mark for any omitted, �1 if left as

range(c) b also � �2 Or equivalent explanation

3 (a) Positive correlation circled 6 HD(b) Points scattered HD(c) No relationship likely U & A

4 (a) 72 7 N(b) 4 8(c) 4

5 (a) Parallelogram, rhombus 6 U & A(b) Trapezium

6 (a) �25� Accept �

13

20� 6 N

(b) 25% 1 mark for �382� or similar fraction

(c) �143� Accept �2

86�

7 ‘More than’ ticked and explanation: 7 U & Adividing by less than 1 makes the answer bigger or equivalent

8 6x2 � 5x � 4 (2x � 1)(3x � 4) → 1 mark, �1 mark for 7 Aany errors

2x � 1 1 mark for evidence of suitable method

9 12 132 � 52 → 1 mark 7 SSM

10 Question including ranges in answer 7 HDe.g. �15 min, 15–30 min, etc.

11 (a) 1�175� Evidence of �

11

05� � �

11

25� � 1 mark 7 N

(b) (5a � 3b)/15 A(c) (5x � 3)/6 (3x � 3)/6; (2x � 6)/6 or equivalent → 8

1 mark each

12 (a) 5 min 7 SSM(b) Steeper gradient Or equivalent U & A(c) 6 km/hr Evidence of 3 km in 30 min → 1 mark SSM

13 7 SSM

NB compasses MUST be used (look for evidence of holes!)�1 for not changing radius at either point

2 m

Shed3 m

1 m

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3R Test A

14 (a) 5 cm Evidence of scale factor being 8 SSM2.5 → 1 mark

(b) 34° U & A

15 (a) �12

40� or 0.7 8 HD

(b) 0.4 �25� or equivalent

(c) Increase ticked and explanation Explanation including increased weight U & Aof base relative to total weight

Test A thresholds

N A SSM HD U & A

10 18 9 5 8

By level 5, 3, 2 6, 5, 7 0, 7, 2 2, 1, 2 3, 2, 3

Level 6 7 8

Marks 11–22 23–34 35–50

Balance of marks

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Test B (Levels 6–8)Calculator allowed

Do not turn the page until you are told to do so.Write your details in the spaces below.

Name

Class

Teacher

Date

Remember● The test is 50 minutes long.

● You may use a calculator.

● You will need: pen, pencil, rubber, ruler, protractor or angle measurer, compass.

● Try to answer all the questions.

● Write all your answers and working on the test paper.

● Check your work carefully.


3R

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1 Here are the first five terms of a sequence: 11, 20, 29, 38, 47, …Write down an expression, in terms of n, for the nth term of this sequence.

.................. ...........2 marks

This sequence represents the cost, in pounds, of ordering CDs from an internet website.Explain each term of your expression for the nth term.

......................................................................................

...................................................................................... ...........1 mark

2

Use this diagram and angle facts associated with parallel lines and straight lines to showthat the angles of triangle PQR add up to 180° (i.e. that b � e � f � 180°).

...........2 marks

3

(a) Fully describe the transformations of A onto B.

......................................................................................

...................................................................................... ...........1 mark

1

2

3

�2

�1

y

x

B A C

10�1�2�3 2 3 4 5 6

R Q

P

ed gf

cab


3R Test B

H E-Y-T 153-172.qxd 20/4/05 12:07 pm Page 164

(b) Fully describe the transformation of A onto C.

......................................................................................

...................................................................................... ...........1 mark

(c) A is translated 2 in the x-direction and �3 in the y direction onto D.Draw triangle D on the graph above.

...........1 mark

(d) Angus says that the area of C is double that of A. Is he right?Explain your answer.

......................................................................................

...................................................................................... ...........1 mark

4 Solve the following equations:

(a) 3a � 5 � 20

.................. ...........1 mark

(b) �b �

35

� � 3

.................. ...........1 mark

(c) 3c � 4 � 6c � 2

.................. ...........2 marks


3RTest B

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5 Points A, B and C are plotted on the axes below.

(a) What is the equation of the line AB?

.................. ...........2 marks

(b) On the axes above, shade the region where y � 3x.

...........2 marks

6 429 � 78 � 33 465.

Without calculating the answer, how do you know that this is incorrect?

......................................................................................

...................................................................................... ...........1 mark

7 Look at this table:

Explain how this shows that 343 � 2401 � 823 543.

......................................................................................

...................................................................................... ...........2 marks

2

4

6

�6

�4

�2

y

x

B

20�2�4�6 4 6

A


3R Test B

71 7

72 49

73 343

74 2401

75 16 807

76 117 649

77 823 543

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8 Fiona was earning £32 000. Her pay is increased to £33 250.Calculate the percentage increase in her pay.

.................. ...........2 marks

She is promised that over the next 3 years she will receive a 4% pay rise each year.What will she then be earning?

.................. ...........

2 marks

9 The radius of the earth is approximately 6350 km.

(a) If an airplane flew around the world at a height of 500 m above sea level, approximatelyhow far would it travel?

.................. ...........2 marks

(b) The value for the radius of the earth is 6350 km to the nearest 10 km.What is the maximum value that the radius could be?

.................. ...........1 mark

10 A photocopier enlarges an original image in the ratio 5 : 7.By what percentage is the original image enlarged?

.................. ...........2 marks


3RTest B

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11 Matthew was carrying out a survey of the heights of Year 9 pupils. His results are recordedin the table below. The boys’ results have been plotted on the frequency polygon.

(a) Why are the boys’ results plotted in the middle of the class interval?

................................................................................. ...........1 mark

(b) Using the axes above, plot the frequency polygon for the girls’ data.

...........2 marks

(c) An estimated mean for the boys has been calculated to be 161 cm.Why can Matthew only calculate an estimated mean?

......................................................................................

...................................................................................... ...........1 mark

(d) Calculate an estimate for the mean height of the girls.

.................. ...........3 marks


3R Test B

Height (cm) No. of boys No. of girls

130 � h � 140 2 3

140 � h � 150 6 9

150 � h � 160 10 15

160 � h � 170 18 16

170 � h � 180 8 2

180 � h � 190 1 0

2468

1012

Fre

qu

ency

Height (cm)

14161820

y

x130140150160170180190200

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The cumulative frequency curve of the combined data is shown below.

(e) Use the graph to find the median height of this class.

................................................................................. ...........1 mark

(f) Use your graph to estimate how many people are more than 165 cm tall.

................................................................................. ...........2 marks

12 Solve the simultaneous equations:

x � 2y � 5.5

x � y � 5

.................................. ...........2 marks

102030405060

Cu

mu

lati

ve F

req

uen

cy

Height (cm)

708090

y

x130140150160170180190200


3RTest B

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13 Martin spends his money as follows:�25� goes in income tax, �

56� of the remainder is spent on food, bills, etc.

Of the remainder, half goes into a savings account.

What fraction of his income is saved?

.................. ...........3 marks

14 The volume of a cylinder is calculated using the formula

V � r2h, where r is the radius and h is the height of the cylinder.

(a) Rearrange this formula to make r the subject.

.................. ...........2 marks

(b) The radius of the cylinder is 5 cm and the height is 8 cm.Find the volume of the cylinder.

.................. ...........2 marks

(c) Give another possible radius and height for a cylinder with the same volume as in part (b).

.................................. ...........2 marks


3R Test B

r

h

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3RTest B

Mark scheme: Test BQuestion Correct answer Additional guidance Level Attainment

target

1 9n � 2 1 mark for each term 6 A£2 post & package (or other plausible 6 U & Aexplanation);£9 per CD

2 (a) a � b � c � 180° (angles on a straight 7 SSMline); a � e and c � f (alternate angles)→ b � e � f � 180°. 1 mark for 2 of these

3 (a) Rotation 90° ACW centre (0, 0) Must be fully described 6 SSM(b) Enlargement SF2 centre (0, 1)(c) D at (3, �2), (3.5, �2), (3.5, �1)(d) No �4, �2 on each dimension (accept �4) 7

4 (a) a � 5 6 A(b) b � 4(c) c � 2 3c � 6 → 1 mark

5 (a) y � 2x � 3 1 mark for either term 7 A(b) Area below y � 3x shaded 1 mark for correct line; if incorrect line, 8

follow-through correct shading

6 9 � 8 � 72, so result must end in a 2 6 U & A

7 Explanation suggesting 73 � 74 � 77 8 N

8 3.9% 6 N£37 402 �3

16205000� � 1 mark 7

1.043 or equivalent method → 1 mark

9 (a) 39 901 km � 12 701 or 2 � 6350.5 → 1 mark 6 SSM(b) 6355 km 7 N

10 40% �25�. 1 mark 8 N

11 (a) Mean for each group 6 U & A(b) Plotted at (135, 3), (145, 9), (155, 15), HD

(165, 16), (175, 2) → �1 for 2 errors(c) Don’t know any exact values – or U & A

equivalent(d) 156 cm: 3 � 135 � 9 � 145, etc. or HD

7025 � 1 mark; �704

25

5� � 1 mark

(e) 160 8 HD(f) 27 pupils (accept 28) – 62 or HD

63 → 1 mark

12 x � 4.5, y � 0.5 1 mark for each 7 A

13 �210� 1 mark (max 2) for evidence of: 8 N

Tax � �25�, bills � �

56� � �

35� � �

12�, left �1

10�

14 (a) r � �(�Vh�) �

Vh�. 1 mark 8 N

(b) 628 cm3 (628.3) � 52 � 8. 1 mark SSM(c) r � 10, h � 2 Or any combination that they prove to U & A

equal 620 or r2h � 200

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3R Test B

Test B thresholds

N A SSM HD U & A

8 16 10 8 8

By level 2, 3, 5 6, 4, 4 5, 3, 2 2, 3, 3 3, 1, 4

Level 6 7 8

Marks 11–23 24–33 34–50

Balance of marks

H E-Y-T 153-172.qxd 20/4/05 12:07 pm Page 172

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