Essential Mathematics. Book 1 (Rayner D.)

Essential Mathematics

Book I

David Rayner

Oxford

Essential Mathematics

Book I

David Rayner

Oxford University Press

Oxford University Press, Walton Street, Oxford OX2 6DP

Oxford New York
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Oxford is a trade mark of Oxford University Press

© David Rayner

All rights reserved. This publication may not be reproduced, stored
or transmitted, in any forms or by any means, except in accordance
with the terms of licences issued by the Copyright Licensing Agency,
or except for fair dealing for the purposes of research or private
study, or criticism or review, as permitted under the Copyright,
Designs and Patents Act, 1988. Enquiries concerning reproduction
outside those terms should be addressed to the Permissions
Department, Oxford University Press.

First published 1995 by Elmwood Press

Reprinted 1996 by Oxford University Press

ISBN 0 19 914661 6

Artwork by Angela Lumley
Emma Djonokusumo
Rachel Flowerday
Lisa Lawton
Lisa Lee
Bidisha Bandyopadahay
Fiona Lait
Paulina Spencer
Eleanor Galvin

Typeset and illustrated by Tech-Set, Gateshead, Tyne and Wear.
Printed in Great Britain by
Butler & Tanner Ltd, Frome and London

PREFACE

This is the first of a three book series written for pupils in the age
range 11-14 years. Many classrooms will contain children with a
range of abilities in mathematics. This book is written to cater for
this situation and also has an ample supply of questions and
activities to stretch the most able.

The author believes that children learn mathematics most effectively
by doing mathematics. Interest and enthusiasm for the subject is
engendered by working at questions which stretch the pupil
forward in his or her knowledge and understanding. The author,
who is a teacher, emphasises a thorough grounding in the
fundamentals of number and algebra when working in the lower
secondary classroom.

It is hoped that the material in the books will stimulate young minds
and encourage logical thinking. Some exercises provide the neces-sary
practice in the basic mathematical skills required for later study.
There is no set path through the book and it is anticipated that
most teachers will prefer to take sections in the order of their own
choice.

Many activities, investigations, games and puzzles are included to
provide a healthy variety of learning experiences. The author is
aware of the difficulties of teaching on 'Friday afternoons' or on
the last few days of term, when both pupils and teachers are tired,
and suitable activities are included.

The author is indebted to the many students and colleagues who
have assisted him in this work. He is particularly grateful to Julie
Anderson, David Moncur and Philip Cutts for their contributions
to the text. The author would also like to thank Michelle Hawke
for her work and sense of humour in editing and checking answers
and Micheline Rayner whose encouragement to write instead of
playing golf has been invaluable.

David Rayner



CONTENTS Page

Part 1 1
1.1 Accurate drawing 9
1.2 Three dimensional objects : nets
1.3 Coordinates 12
1.4 Prime numbers, factors, multiples
1.5 Puzzles and games 1 26
31
Crossnumbers 1, Gel).eral Knowledge, Lines,
Curves from straight lines. 35

Part 2 45
2.1 Calculating angles 52
2.2 Decimals 60
2.3 Mixed problems 68
2.4 Area
2.5 Sequences 75
2.6 Puzzles and games 2
78
Puzzles, Happy numbers.
85
Part 3 93
3.1 Using algebra
3.2 Solving equations 102
3.3 Percentages 108
3.4 Symmetry
3.5 Puzzles and games 3 110
112
Clocks, Puzzles. 114

Part 4 120
4.1 Approximating 124
4.2 Using a calculator
4.3 Travel graphs 128
4.4 Problem solving I
4.5 Puzzles and games 4 135
141
Triangles and quadrilaterals,
Crossnumbers 2, Biggest number. 154

Part 5
5.1 Metric and Imperial units
5.2 Fractions
5.3 Charts and graphs
5.4 Puzzles and games 5

Printing a book, Diagonals,
Find the hidden treasure.

Part 6 156
6.1 Probability 168
6.2 Problem solving 2
6.3 Mental arithmetic 170
6.4 Puzzles and games 6 175

Puzzles, Boxes, Break the codes, 177

Part 7 182
7.1 Multiple choice papers
7.2 Mixed exercises

Part 1

1.1 Accurate drawing

When an architect designs a
house he has to draw accurate
plans for a builder to follow.
In this section you will use a
protractor and a pair of
compasses to construct
accurate diagrams involving
triangles and quadrilaterals.
You will also construct nets
from which three dimensional
shapes can be made.

Measuring angles

When angles are measured accurately they
are usually measured in degrees.
We write degrees as a number followed
by the degree symbol o .

A full tum = 360°

A half tum = 180°
A quarter tum = 90°.

An angle of 90° is called a right angle.

Exercise 1

In the drawings below some of the angles are correct and some are

obviously wrong. State which are correct and which are wrong.

You are not expected to measure the angles . 5.\ 6. \

p;- 4.,3.~

I I~~~

2 Part 1

7. 8.; r -~~2701 9.
10. 11. 12.

~

13. 14. 15. 16. 17. 18.

Labelling angles ~/ ~ ~160°
The angle shown is ABC (or CBA).
~
A
This angle is POR (or ROP).

The 'B' must be in the middle. p

The '0' must be in the middle.

Exercise 2

Write down the size of the angles stated.

1. D 2. z 3. B

\ X (a) B~C
~A c (b) ACB
B
(c) CBA
(a) ABD (b) DBC (a) WXZ (b) ZXY

Accurate drawing 3

K 6. D
c

F L

(a) DEF (a) KLM A
(b) DFE (b) KML
(c) FDE (c) NM:K (a) DCB
(b) ABD
(c) COB

Using a protractor

The protractor is one of the most commonly misused of
mathematical instruments.

It is important to read the answer from the correct scale.

Remember: When measuring an acute angle the answer must be less
than 90°.
When measuring an obtuse angle the answer must be
more than 90°.

Exercise 3
Copy and complete the table.

Q
p\

HI ~""' <§l

~

G'---......_

-B
--A

(a) AQD = 20° AON= AOL = 60° AOK=
ZOT =
(b) ZOF = ZOP = ZOR =
(c) ZQI = AOV =
(d) AOQ = ZOG = AOC=
(e) ZOH = AOB =
(f) AOG = AQP = AOF =
(g) AOR = ZOD =
ZOB = zoe=
AOM=
AOH= AOI
ZOK =
ZOE = 155° ZOJ =

4 Part 1

Exercise 4 Measure the following angles

~ ~ 3. DER 4. E~B 5. DRC 6. BEA 7. SRB 8. ACB
ll.CDE 12. DSC 13. DCB 14. EDS 15. UDQ 16. ECB
1. BAC 2. RCD

9. D'TB 10. CPE

Constructing triangles

Draw the triangle ABC full size and measure the length x. c

(a) Draw a base line longer than 8·5 em

and make two marks on it exactly 8·5 em apart.

(b) Put the centre of the protractor on A and

measure an angle 64°. Draw line AP. 8.5 em B

(c) Similarly draw line BQ at an angle P

40° to AB. Q

(d) The triangle is formed .

Measure x = 5·6 em

A 8.5 em B

Accurate drawing 5

Exercise 5
Construct the triangles and measure the lengths of the sides marked x.

1. 2. 3.

6cm

4. 5. 6.

7. 8. 9.

10. 11. 12.

Scm ?em

6 Part 1
y
Constructing a triangle given three sides

Draw triangle XYZ and measure XZY.

z

X 7cm y

(a) Draw a base line longer than
7 em and mark X and Y exactly
7 em apart.

(b) Put the point of a pair of compasses
on X and draw an arc of radius 8 em.

(c) Put the point of the pair of compasses
on Y and draw an arc of radius 5 em.

(d) The arcs cross at the point Z
so the triangle is formed.

Measure XZY = 60°

X 7cm

Exercise 6
In Questions 1 to 6 use a pair of compasses and measure the angle x .
1. 2. 3.

4. 5. 6.
Scm

Accurate drawing 7

In Questions 7 and 8 you need to use the formula

t'area = base x height' to find the area of a triangle.

7. A disused airfield is to be sold at a price of £5500 per hectare.
(1 hectare = 10 000 m2).
The outline of the airfield is a quadrilateral but it is not a
rectangle. The area can be found by splitting it into two triangles
and then finding the area of each part.
Find the selling price of the airfield.
[Use a scale of l em to lOOm]

600m

1400m

IOOOm I 800m

I
I
I
I
I

8. The largest office building in the world is the Pentagon in the
U.S.A. and 29 000 people work there. The building is in the
shape of a regular pentagon with each side 920 feet long. Make
a scale drawing of the building, with a scale of 1em to 100 feet,
and work out the area in acres.
[1 acre= 4840 square yards, 1 square yard = 9 square feet] .

--*I --
/-_;:, ,

I 72° \

I\
I
I
I

920

8 Part 1

Keep the price down

A road is to be built between two towns: Brocket and Newton. The
River Lea runs across the land between the towns. The land south
of the river is very marshy and road building is more expensive
than on the dry land north of the river. Building costs are as follows:

On dry land 1km of road costs £100000;
On marshy land 1 km of road costs £170 000;
A bridge over the river costs £400 000.

One possible 'scenic route' for the road is shown by the broken line.
(a) Find the total cost of building the road along the scenic route.
(b) Find the cheapest possible route for the road. Give the total cost

by this route correct to the nearest £1000.

' ' '/' ', Dry

/
/

•/ Bridge

/ land

/
'Dry
'land /
'/
'/
'/
'/
'/ Marshy
'/
'/
land
';
' '/
/ ' '/ X
/ Newton

/

/

/

/

Scale: I em to I km

Marshy
land

Three-dimensional objects :nets 9
net
1.2 Three-dimensional objects: nets

If the cube shown was made of cube
cardboard, and you cut along
some of the edges and laid it out
flat, you would have a net of the
cube.

There is more than one net of a
cube as you will see in the exercise
below.

The net for a regular tetrahedron \
[tetra: four, hedron: faces] consists I
of four equilateral triangles.

We are going to use equilateral triangles for several different objects
so it is worth spending some time on a method for constructing a
whole series of equilateral triangles all at once.

(a) Draw the line AB. Use a pair of
compasses to draw a pattern of
circles as shown. Make the radius
of the circles 3·5 em.

(b) Draw the lines to produce a series of equilateral
triangles.

We can also use this method to produce a series of
regular hexagons.

10 Part 1

Exercise 7

1. There are several different nets for a cube.

(a) Draw the diagrams below on squared paper and cut them

out. Which of the diagrams is the net for a cube?

(i) ·································· r - - (ii) (iii)

I +\
II
r--.
! .. .. i

(b) Design a net for a cube which is different to those above.

2. The diagram shows a pyramid with a square base. The vertex is
vertically above the centre of the base.
What sort of triangles make the sloping faces?
Draw a net for the pyramid on squared paper and cut it out to
see if it works.

In Questions 3, 4, 5 objects are shown together with the net which
will produce them. Draw the net on a piece of cardboard and cut it
out, leaving 'tabs' around the net for glue.

3. Tetrahedron net 4. Octahedron net

5. Icosahedron (an object with 20 faces)

Three-dimensional objects : nets ll

6. (More difficult) Here you are going to construct the net for a
dodecahedron (12 faces). Take extra care with this one!

(a) Draw a circle of radius lOcm B
and construct a large pentagon
using the 72° angles as shown. c
Draw the lines OA, OB, OC,
OD, OE very faintly because it
is best to rub them out as soon
as we have found the positions
for A, B, C, D, E.

(b) Join A to C and A to D. B
Join B to E and B to D.
Join C to E.
Mark the points V, W, X, Y, Z.

(c) Draw a line through Z and W. B
Draw a line through V and X.
Draw a line through W and Y.
Draw a line through Z and X.
Draw a line through V andY.

This pattern forms a series of regular
pentagons shown by the thick lines.

Draw the pattern of regular pentagons
twice, as shown and join them along the
line PQ.

12 Part 1

Questions 7 and 8 are for the enthusiast!
7. Cuboctahedron

8. Truncated octahedron

1.3 Coordinates

Plotting points

We use coordinates to describe the position of an object. You may be sitting at the desk which is

'in the back row, third from the left' . This is a simple example of how positions may be described.

When more accurate descriptions are needed it is helpful to draw reference lines and to measure

distances from these lines.

In the plan of the garden shown, the tree is 2 units •2:! 5 trre~lv\
from the line on the left and 3 units from the line
at the bottom. We say it has coordinates (2, 3). The ~"c'
coordinates of the ant's nest are (4, 1).
8u 4
It is obviously very important to agree which
direction is taken first: we always go across (--.) "c0
and then up (j). Remember: 'Along the corridor
u0

~3

and up the stairs'. Notice also that the lines are 2
numbered, not the squares.

Patio ants nest

II .....
IIII
345
02 First coordinate

Coordinates 13

Exercise 8

The map below shows a remote Scottish Island used for training by the S.A.S.

~
1.910~------~---.---.---.---.--,1,--,---.---.
Hand
g l~ 9 f---+--grenade t:::--
Parachute

/area Rocket launcher--drop zone

Ble~~ 8~~~~~-+---+---+--~--~--1--r~~
\ ._Tan Hospital AT /

~ ~ ~ubmarine7~--~~r---1 . .-+---+--~---r--,_r-~~
Intenogation ...........
') -\6 r--- centre~fJ).
Tourist
Deep-\+-+---+----+--"(r---·~
7 l5 r-----+V-1-Inlforma~ion !. '--r--d..ro, p zone
bridge ( River Deep I '
-1
j H.IQ.
1
Rift~ range
B !Cliff J."'
Officers'
4
edge I'Ho;pital

~=~:~t l \ I I3
I '\.. Radar _Hospital C::: mess -1111+--- -

"'control X'

2 ......._~ r--... H.e"·licopte:r } Look out --
point
1'

2 3 4 5 6 7 8 9 10
First coordinate

1. Write down the coordinates of the following places:

(a) Rocket launcher (b) H.Q. (c) Hospital A

(d) Rifle range (e) Officers's mess (f) Radar control

2. Make a list of the places which are at the following points:

(a)(2,8) (b)(7,8) (c)(3,3) (d)(6,3t)

(e)(2,6) (t)(6,2) (g)(2,4) (h)(8t,2t)

3. The map below shows the first two Second hole
holes on a rather hazardous golf
course. ~
What is at the following points?
(a) (6, 3) (b) (4, 2) (c) (2, 6) .9"' 6

(d) (6, 4t) (e) (3 t, 3) (f) (7, 3) ~

4. (a) Write down the positions in which 0

the ball might land if you played ~5

c

0

~

r:/J4
3

the first hole. - ....!. El::J

(b) Where would you like the ball to Bupker

go if your maths teacher was

playing the hole?

0 234567
First coordinate

14 Part 1

5. Make up your own map. Mark some interesting points and
make a list, giving the coordinates of eight points.

x and y coordinates y axis
4.
We call the first coordinate of a point the x-coordinate
and the second coordinate the y-coordinate. So for the 3
point (l, 4) the x-coordinate is l and they-coordinate is 4.
2
The line across the page at the bottom is called the x
axis and the line up the page at the side is called the y axis.

0 23456
x axis

Coordinate pictures

Plot the points below and join them up in order.
(a) (2, 4), (8, 1), (6, 3), (4, 4),

(2, 6), (2, 4), (0, 3), (6, 2).

t.t ),(b) (5, 3 (4, 5), (3 4f ).

2 3 4 5 6 7 8X

Exercise 9

Plot the points given and join them up in order. Write down what
you have drawn in each question.

1. Draw axes with values from 0 to 15.
(8, 1), (10, 1), (11, 3), (11, 3!), (12, 4!), (12, 7),
(14, 9), (14, 10), (13, 10), (11, 12), (10, 14),
(10, 14! ), (8, 14), (6!, 14! ), (6!, 15), (4, 15),
(3,14), (1,13), (1,11), (2,9), (3,8!), (4,9),
(6, 9), (6, 8), (7, 6), (6!, 5), (7, 4), (8, 1).

2. Draw axes with values from 0 to 14.
(a) (6, 13), (1, 3), (2, 1), (12, 1), (8, 9), (6, 5),
(4, 5), (8, 13), (6, 13), (8, 13), (13, 3), (12, 1).
(b) (1, 3), (9, 3), (7, 7), (6, 5), (8, 5)

Colour in the shape.

Coordinates 15

3. Draw axes with values from 0 to 18.
(a) (0, 3), (I, 4), (2, 6), (4, 8), (6, 8), (8, 9), (12, 9),
(13, II), (12, 12), (12, 14), (14, 12), (15, 12),
(17, 14), (17, 12), (16, II), (17, 10), (17, 9),
(16, 9), (15, 8), (14, 9), (13, 9)
(b) (16, 9), (16, 7), (14, 5), (14, 1), (15, 1), (15, 6),
(13, 4), (13, 1), (12, 1), (12, 4), (11, 5), (9, 5),
(9, 6t ), (9, 4), (8, 3), (8, 1), (7, 1), (7, 4), (6, 6),
(6, 4), (5, 3), (5, 1), (6, 1), (6, 3), (7, 4), (6, 6),
(6, 7), (3, 2), (1, 2), (0, 3).

4. Draw axes with values from 0 to 10. (2, 7), (I, 6),
(a) (3, 2), (4, 2), (5, 3), (3, 5), (3, 6), (4, 5), (6, 4),
(1, 8), (2, 9), (3, 9), (5, 7), (4, 6),
(8, 4), (8, 5), (6, 7), (5, 7).
(b) (7, 4), (9, 2), (8, 1), (7, 3), (5, 3).
(c) (I, 6), (2, 8), (2, 9), (2, 7).
(d) Draw a dot at (3, 8).

In Questions 5 and 6 you will draw a picture and then alter it by
changing the coordinates according to different rules.

5. Draw axes with x from 0 to 18 andy from 0 to 9.
(a) Plot the following points and join them up in order.
(1, 2), (3, 3), (6, 3), (6, 2), (4, 1), (1, 1), (1, 2),
(4, 2), (4, 1), (4, 2), (6, 3).
(b) Multiply all the coordinates by 3, plot the new points and
join them up to draw a new picture.
(c) For each of the original points add 10 to the x coordinate
and keep the same y coordinate.
So, for example, (1, 2) becomes (11, 2) and (3, 3) becomes
(13, 3). Plot the new points and join them up.

6. Draw axes with both x andy from 0 to 12.
(a) Plot the following points and join them up in order.
(5, 11), (5, 10), (6, 8), (7, 9), (7, 10), (6, 12), (4, 12),
(3, 11), (0, 10), (1, 9), (3, 9), (1, 9), (4, 8), (5, 7),
(8, 8), (7, 9).
Draw a dot at (3t, lOt).
(b) Plot a new picture by swapping the x and y coordinates.
So: instead of (5, 11) plot (11, 5);
instead of (5, 10) plot (10, 5) and so on.

Describe what has happened to the first picture.

16 Part 1

Complete the shape

Two sides of a rectangle are drawn.
Find (a) the coordinates of the fourth vertex of the

rectangle
(b) the coordinates of the centre of the rectangle.

2 3 4 5 6X

The complete rectangle is shown. y , . ..........;. ......... +·········
(a) Fourth vertex is at (6, 3) 5 -+ ··· ····· > ;... . ··!

(b) Centre of rectangle is at (3 t, 3) 4~····· ····· ····· ·· ! / +············: ~:,· ···· ·······i+··········

3
2

2 3 4 5 6X

Exercise 10 y ¥ · · ··········i +·· · ·········i i··· ·· ···· ··· i+····· ·· ·· ··· i ····~ · ·· · · ·· · ···· ··· i+

1. The graph shows several 12~· ····· ·· · · ·· ··· ··i
incomplete quadrilaterals. Copy
the diagram and complete the 10 -+ ············!· /
shapes.
9 + ·· ···!·········"""'!!<
(a) Write down the coordinates 8 ~ -· ············! ··
of the fourth vertex of each
shape. 7

(b) Write down the coordinates 6
of the centre of each shape.

4

3 ~ ······+···············

2 -+ ;...~...:.-/ ·····

2 3 4 5 6 7 8 9 10 II 12 X

Coordinates 17

2. You are given the vertices but not
the sides of two parallelograms P
and Q.

For each parallelogram find three
possible positions for the fourth
vertex.

3. y + ············: • +············ 2 3 4 5 6 7 8 9 10 II 12 X

7~ ·················: The crosses mark two vertices of an isosceles triangle A.
Find four possible points, with whole number
6 j~ ········· !············: +······· coordinates, for the third vertex of the triangle.

5~· ·········· · · · ···!+

2 3 4 5 6x

4. The diagram shows one side of an isosceles
triangle B.
(a) Find six possible points, with whole number
coordinates, for the third vertex of the
triangle.
(b) Explain how you could find the coordinates
of several more positions for the third vertex.

2 3 4 5 6 7 8X

18 Part 1

Negative coordinates L

The x axis can be extended to the left and the y axis X
can be extended downwards to include the negative
numbers -1, -2, -3 etc.

The name 'FALDO' can be found using the letters in
the following order:
(2, 3), (-2, -1), (-1, 2), (2, -2), (-2, -3).

Similarly the coordinates of the points which spell out
the word 'LOAD' are (-1, 2), (-2, -3),
(-2, -1), (2, -2)

Exercise 11 j -3t +·•••••••••••••+•m •••m +•••••• •• •

o:...........~ ...............~

The letters from A to Z are shown on the grid.
Coded messages can be sent using coordinates.

For example (-4, -2) (-4, 2) (-4, 2) (4, 2) reads 'FOOD'.

Decipher the following messages

1. (5, 5) (4, 0) (1, 3) (2, 5) # (4, 2) (-4, 2) # (-5, -3)
(-4, 2) (-2, 5) # (-2, -2) (1, 3) (-5, -5) (-5, - 5) #
(1, 3) # (4, -4) (1, 3) (0, 1) # (5, 5) (-2, 1) (2, 5) (4, 0) #
(1, 3) # (5, -2) (-5, 4) (1, 3) (4, 2) (-2, 2) # (-2, 1)

(0, 1) # (4, 0) (-2, 1) (5, -2) # (4, 0) (-2, 2) (1, 3)
(4, 2) ? # (4, 2) (-4, 2) (-2, 5) (4, 4) !

Coordinates 19

2. Change the seventh word to: (5, 5) (-2, 1) (2, 5) (4, 0)
(-4, 2) (-2, 5) (2, 5).
Change the last word to: (4, 2) (-4, 2) (-2, 5) (4, 4)
(-5, -5) (1, 3) (5, -2).

3. (5, 5) (4, 0) (1, 3) (2, 5) # (4, 2) (-4, 2) # (-5, -3)
(-4, 2) (-2, 5) # (-2, -2) (1, 3) (-5, -5) (-5, -5) #
(1, 3) # (4, 2) (-2, 2) (1, 3) (4, 2) # (-5, 4) (1, 3) (-3, -4)
(-3, -4) (-4, 2) (2, 5) ? # (-5, 4) (-4, 2) (-5, -5) (-5, -3)
(4, 4) (-4, 2) (0, 1) !

4. (5, 5) (-2, 1) (2, 5) (4, 0) # (5, 5) (4, 0) (1, 3) (2, 5) #
(4, 2) (-4, 2) # (-5, -3) (-4, 2) (-2, 5) # (5, -2) (2, 5)
(-2, 5) (-4, -2) (-4, -2) # (1, 3) # (4, 2) (-2, 2) (1, 3)
(4, 2) # (-5, 4) (1, 3) (-3, -4) (-3, -4) (-4, 2)
(2, 5)? # (-5, 4) (-4, 2) (-5, -5) (-5, -3) (-4, -2) (-2, 1)

(-5, -5) (-5 , -5) (1, 3) !

5. (5, 5) (4, 0) (1, 3) (2, 5) # (4, 2) (-4, 2) # (3, -5) (-2, 2)
(4, 4) (-2, 2) (2, 5) (1, 3) (-3, -4) (-2, 1) (1, 3) (0, 1) #
(4, -4) (-4, 2) (0, 1) (5, -2) (2, 5) (-2, 2) (-3, -4)
(5, -2) # (-2, 2) (1, 3) (2, 5) ? (5, -2) (5, 5) (-2, 2) (4, 2)
(-2, 2) (5, -2).

6. Write a message or joke of your own using coordinates. Ask a
friend to decipher your words.

Decimal coordinates

When greater accuracy is required points may be plotted using
decimal coordinates. Be careful with the scales!

In graph 1 the tip of the church spire has In graph 2 point A has coordinates (1·2, 2·6) and
coordinates (0·9, 2·6) point B has coordinates (3·8, 4·4).

20 Part 1

Exercise 12
1. This is a game for two players A and B.
Player A rolls a dice two times. He might roll a '3 ' and then a
'5'. This means that the x coordinate is 3·5.
Player A then rolls the dice two more times. He might roll a '4'
and then a '2'. This means that they coordinate is 4·2.
The point (3·5, 4·2) represents a 'shot' at the target below.
If the shot lands inside the 'I ' the player scores 1 point.
If the shot lands inside the '2' the player scores 2 points.
For example the shot (3·5, 4·2) would score 4 points.
If the shot misses all the numbers or lands in a shaded area the
player scores nothing. If the shot lands exactly on a line the
shot bounces off the target and there is no score.
Then player B rolls the dice for his shot and the game continues.

Target

(a) Take turns to roll the dice and record the points scored. You
can play 'first to 20' or 'five goes each'.

(b) Some of the numbers are very difficult to hit. Say which

numbers and explain why they are difficult to hit.

Coordinates 21

2. Copy and complete the table for
all the remaining points

Point Position

A (1·8, 1·2)
B (2·3, 3·3)
()
c ()

D
E

Plotting lines y p
4
• The points P, Q, R and S have coordinates 3 Q
2
(4, 4), (4, 3), (4, 2) and (4, 1) and they all lie R
on a straight line. Since the x-coordinate of all 0
the points is 4, we say the equation of the line s
is x = 4. y
4 2 3 4 5 6X
• The points A, B, C and D have coordinates -3 I
(1, 3), (2, 3), (3, 3) and (4, 3) and they all lie 2
on a straight line. Since the y-coordinate of all ABcD
the points is 3, we say the equation of the line is
y = 3. y=3

0 2 3 4X

22 Part 1

Exercise 13 2. Write down the equations for the lines
1. Write down the equations for the lines marked P, Q and R .
marked A, B and C.
,--i4 13 -12
!. . . . . . .y
········-~ ............ ...
8 .1. ............1......

j

7
6
5
4

0 2 3 4 5 6 7 8 9x

In Questions 3 and 4 there is a line of dots A, a line of crosses B and
a line of circles C.
Write down the equations of the lines in each question.

3. 4.

-14 ---1 3

:!i
$........ . ! ~2--+ + ·············+··-··

·· ···~· ~3-+ i··············+ +·········~···· · · i

5. On squared paper
(a) Draw the lines y = 2 and x = 3. Where do they meet?
(b) Draw the lines y = 5 and x = 1. Where do they meet?
(c) Draw the lines x = 7 andy= 3. Where do they meet?

6. Name two lines which pass through the following points.

(a) (5, 2) (b) (3, 7) (c) (8, 0) (d) (8, 8) (e) (5, 21)

Coordinates 23

7. In the diagram, E and N lie on the line with

equation y = 1. B and K lie on the line x = 5.

In parts (a) to (h) find the equation of the line

passing through the points given:

(a) A and D (e) L and E

(b) A, Band I (f) D, K and G

(c) M and P (g) C, M , L and H

(d) I and H (h) P and F

pF
2 3 4 5 6 7 8 9x

Relating x and y

• The sloping line passes through the following points: ly ~~- ~.."
(1, 1), (2, 2), (3, 3), (4, 4), (5, 5).
For each point, the y-coordinate is equal to the 5
x-coordinate.
The equation of the line is y = x (or x = y). 4- ............... ............. ..... . ........!.. !1 / "
if
• This line passes through: 3 I' ,"' f
(0, 1), (1, 2), (2, 3), (3 , 4), (4, 5).
For each point the y-coordinate is one more than the 2 ""

x-coordinate. The equation of the line is y = x + 1. ."

We could also say that the x coordinate is always one less " 2 3 4 5X
than the y coordinate. The equation of the line could then "
be written as x = y- 1. ""
[Most mathematicians use the equation beginning 'y ='].
y
5

4

3

2

I 2 3 4 5X

""""

• This line slopes the other way and passes through: y
(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0).
'5'~···············!-+············t+············tm
The sum of the x coordinate and they coordinate is always 5.

The equation of the line is x + y = 5.

234

24 Part 1

Exercise 14
For each question write down the coordinates of the points marked.
Find the equation of the line through the points.

1. y

8
7
6

4-f ············+················

3---l·················>··············· +
2---l················,················+·

2 3 4 5 6X 2 3 4 5 6X

3. y 4. y

8 7
7 6
6 5
5 4
4
3 3
2 2

2 3 4 5 6 7X

2 3 4 5 6X

5. i y 6. y

·S-+·········+ +············: ! ············ : : ··········· 6---!················;················

LA-+ ···········+ +············ ! :··········· ! :············ 5

'J

i. 2

-I 2 3 4X

1

Coordinates 25

In Questions 7 and 8 there is a line of dots A and a line of crosses B.
Find the equation of each line.

7. y 8. y

9 9

88

77

66

55

44

33

22

2 3 4 5 6 7 8x 2 3 4 5 6 7 8x

In Questions 9 to 20 you are given the coordinates of several points

on a line. Find the equation of each line. 10.: II 2 3 4 5 6
6 7 8 9 10 11
9 23456
.: I 4 5 6 7 8 9

u.: 3 5 7 12.: II 2 4 6 8
I 0246
8 10 12 14

13. X 10 12 14 16 14.: I 1 2 3 4 5
y 4 6 8 10 2 4 6 8 10

15. X 2 4 5 6 16.: I 8 7 6 5 4 3
y 6 12 15 18 0 2345

17. :II 5 4 3 2 0 18.: I 2 3 4 5
3 5 7 9 11
0 2345

19.: I 2 3 4 5 20.: I 2 3 4
5 7 9 11 13 7 10 13 16

26 Part 1

1.4 Prime numbers, factors, multiples

A prime number is divisible by just two different numbers: by itself
and by one.
Notice that one is not a prime number.
The first five prime numbers are 2, 3, 5, 7, 11.
Mathematicians have been fascinated by the study of prime numbers
for hundreds of years. Nowadays computers are used to test for
prime numbers. Many prime numbers have been found which are
thousands of digits long.

Exercise 15

1. Find the two numbers in each line which are prime.
(a) 14, 17, 21, 27, 29, 39
(b) 41, 45, 49, 51, 63, 67
(c) 2, 57, 71, 81, 91, 93

2. The prime numbers up to 100 or 200 can be found as
follows :

• Write the numbers in 8 columns (leave space
underneath to go up to 200 later).

• Cross out 1 and draw circles around 2, 3, 5 and 7.
• Draw 4 lines to cross out the even numbers (apart

from 2).
• Draw 6 lines to cross out the multiples of 3.
• Draw 2 lines to cross out the multiples of 7.
• Cross out any numbers ending in 5.
• Draw circles around all the numbers which have not

been crossed out. These are the prime numbers.
Check that you have 25 prime numbers up to 100.

A B

3. Selmin looked at her circled prime numbers and she thought she
noticed a pattern. She thought that all the prime numbers in
columns A and B could be written as the sum of two square
numbers.
For example 17 = 12 +42
41 = 42 +52

Was Selrnin right? Can all the prime numbers in columns A and

B be written like this?

Prime numbers, factors, multiples 27

4. Extend the table up to 200 and draw in more lines to cross out
multiples of 2, 3 and 7. You will also have to cross out any
multiples of 11 and 13 which would otherwise be missed. (Can
you see why?)
Does the pattern which Selmin noticed still work?

Testing for prime numbers

(a) Is 307 a prime number?

You might think that we need to test whether 307 is divisible by
2, 3, 4, 5, 6, 7, 8, ...... 306. This would be both tedious and
unnecessary.
In fact, if 307 is divisible by any number at all it will certainly be

divisible by a prime number less than v'307. [The 'square root of
307'. E.g. JT6 = 4, VlOO = 10].
Since v'307 is about 17·5, we only need to test whether 307 is

divisible by 2, 3, 5, 7, 11, 13, 17.

Using a calculator, we find that 307 is not divisible by any of
these, so we know that 307 is a prime number.

(b) Is 689 a prime number?

Since V689:::::::: 26·2, we only need to test whether 689 is divisible

by 2, 3, 5, 7, 11, 13, 17, 19, 23.
Using a calculator we find that 689 is divisible by j13. We do not
need to go any further than this.
We now know that 689 is not a prime number.

Exercise 16

1. Use your calculator to find which of the following are prime

numbers.

(a) 293 (b) 407 (c) 799 (d) 335

(e) 709 (f) 1261 (g) 923 (h) 1009

2. One very large prime number is 286243 - 1.
The number has 25 962 digits.
(a) How long would it take to write out this
number, assuming that you could
maintain a rate of 1 digit every second?
Give your answer in hours, minutes and
seconds.
(b) How many pages would you need to
write out this number if you could write
50 digits on a line and 30 lines on a
page?

28 Part 1

3. For a long time mathematicians have tried to find a formula
which always gives a prime number.

Consider the formula p = n2 + n + 17

If n=1, p=12 +1+17

p = 19, which is prime.

n = 2, p = 22 + 2 + 17

p = 23, which is prime.

n=3, p=32 +3+17

p = 29, which is prime.

(a) Try several more values of n. Do you always get a prime
number?

(b) (Harder) Can you think of a value of n which cannot
possibly give you a prime number for p?
Explain why this is so.

Factors and multiples

At a Factor Factory clients bring their
raw numbers to 'INPUT', pay a large fee,
and then collect from 'OUTPUT' a list of
the factors of their original numbers.

The factors of a number are the numbers
which divide into it exactly.
So if the number 20 went into INPUT the
numbers 1, 2, 4, 5, 10, 20 would come out
at OUTPUT.
These are the factors of 20 as they divide
into 20 exactly, leaving no remainder.

Exercise 17

l. What numbers would be delivered from OUTPUT when the

following numbers were put into INPUT?

(a) 24 (b) 21 (c) 36 (d) 100

2. An optional extra, offered at Fingal's Factor Factory, is called
'common factors'.
The factory takes two numbers into INPUT at a time and
produces a list giving numbers which are factors of both
numbers. So the common factors of 24 and 36 would be found
like this:

number factors common factors

}24 1, 2, 3, 4, 6, 8, 12, 24 1, 2, 3, 4, 6, 12

36 1, 2, 3, 4, 6, 9, 12, 18, 36

Find the common factors of the following pairs of numbers

(a) 12 and 20 (b) 18 and 24 (c) 36 and 45

Prime numbers, factors, multiples 29

3. A final 'Executive, de-luxe, gold star' service will provide the
highest common factor (known in the trade as the H.C.F.) of
any two, three or more numbers.
For instance the H.C.F. of 12 and 20 would be found as follows:

number factors common factors H.C.F.
4
12 1' 2, 3, 4, 6, 12 } I, 2, 4
20 1, 2, 4, 5, 10, 20

Find the H.C.F. of

(a) 12 and 18 (b) 22 and 55 (c) 45 and 72
(f) 20, 40 and 50
(d) 12, 18 and 30 (e) 36, 60 and 72

4. A cut-price mail order company provides three services:
multiples; common multiples; lowest common multiple (L.C.M.)
Take, for example, the numbers 4 and 12

number multiples common L.C.M.
multiples

}4 4, 8, 12, 16, 20, 24, 28, 32, 36 12, 24, 36 12

12 12, 24, 36, 48, 60, 72

Find the L.C.M. of

(a) 6 and 9 (b) 8 and 12 (c) 14 and 35
(f) 4, 7 and 9
(d) 2, 4 and 6 (e) 3, 5 and 10

5. Don't confuse you L.C.M.'s with your H.C.F.'s!
(a) Find the H.C.F. of 12 and 30.
(b) Find the L.C.M. of 8 and 20.
(c) Write down two numbers whose H.C.F. is 11.
(d) Write down two numbers whose L.C.M. is 10.

6. Just one number between 30 and 40 (inclusive) cannot be
written as a multiple of 4, or as a multiple of 5, or as the sum
of a multiple of 4 and a multiple of 5. Which number is that?

Factor trees 3 52 2

The number 60 can be split into its \1 \1
prime factors by drawing a factor tree. 15 4
~/
60

30 Part 1

You can draw a trunk around the 60 and branches to
give an authentic 'tree shape'. Some people like to
draw the prime factors inside apples, pears, bananas
and so on.

Exercise 18

Draw a factor tree to find the pnme factors of the following
numbers.

1. 36 2. 294 3. 108 4. 600
5. 1500 6. 924 7. 2464 8. 4620
9. 4000 10. 7650 11. 22 540 12. 98 175

13. Given that 1386 = 2 X 3 X 3 X 7 X 11 and 858 = 2 X 3 X 11 X 13,

find the highest common factor of 1386 and 858. [i.e. The largest
number that goes into 1386 and 858].

14. If 1170 = 2 X 3 X 3 X 5 X 13 and 10 725 = 3 X 5 X 5 X 11 X 13,
find the highest common factor of 1170 and 10 725.

15. Use your answers to Questions 2 and 3 above to find the highest
common factor of 294 and 108.

16. The number 324 is a square number [324 = 18 x 18].

In its prime factors, 324 = 2 x 2 x 3 x 3 x 3 x 3.
If we take half of the twos and half of the threes we have
2 x 3 x 3, which is 18.
We say that 18 is the square root of 324.
Find the square roots of these numbers (without a calculator!)

(a) 196 (= 2 X 2 X 7 X 7) (b) 256 (= 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2)
(c) 576 (d) 1936
(e) 38 025

17. In its prime factors, 588 = 2 x 2 x 3 x 7 x 7.
What is the smallest number by which you can multiply 588 so
that the answer is a square number?

18. Write 8820 in its prime factors. What is the smallest number by
which you can multiply 8820 so that the answer is a square
number?

19. What is the smallest whole number which is exactly divisible by

all the numbers from 1 to 10 inclusive?

Puzzles and games 1 31

1.5 Puzzles and games 1

Crossnumbers

Make four copies of the pattern below and complete the puzzles
using the clues given. To avoid confusion it is better not to write
the small reference numbers 1, 2-18 on your patterns.

I 23 4

,, ,; , ·:

5 ,\

:'; '{

67 8

','

9 nN· f / :;; 10 f'';,·:~~,;:d{;~,tr'(

15 16 II r 12

', .',,.·', 18 :

: :,~ .,,

,/
:.:,·: ;:

13 14 ; ' , ~:

·;;

; :,

17

'f,

,>'"

(I~'l:.:~:,~

Part A [No calculators] Down
Across

1. 499 + 43 1. 1% of 5700

3. 216 X 7 2. 600-365
5. 504-;- 9 4. 63

6. 8214-3643 7. 4488-;- 6
8. Half of 192
8. 302 + 3 X 6

9. 20% of 365 9. 10000-2003

10. Prime number between 30 11. 4 X 4 X 4 X 4

and 36 12. 58·93 X (67 + 33)

11. 213 + 62 + 9 14. 1136 - 315
16. 112 - 102
13. 406-;- 7

15.316 X 23

17. 1000-731

18. Next prime number after 200

32 Part 1

In parts B, C and D a calculator may be used [where absolutely
necessary!) Write any decimal points on the lines between squares.

Part B Down
Across

1. 9 X 10 X 11 1. 5 X 6 X 7 X 8 _ 11 X 68
2

3. Ninety less than ten 2. 26% as a decimal
thousand 4. 0 ·12

5. (7 t )2 to the nearest whole 7. Next in the sequence

number 102t. 205, 410,

6. 140·52 7 0·03 8. 1- 0 ·97

8. Last two digits of 992 9. 52% of £158·50

9. 32 + 42 + 52 + 62 11. 0·0854 7 (7 - 6 ·99)

10. Angle between the hands of 12. 103 + 113
14.3 X 5 X 72
a clock at 2 .00 pm

11. Eight pounds and eight 16. Half of a third of 222

pence

13. Next prime number after 89

15. 11% of 213

17. 3·1 m plus 43cm, in em

18. Area of a square of side

15cm.

Part C Down
Across
t1. 1 as a decimal
1. Next square number after
144 2. 122 + 352
1·4 + 0 ·2
3. 5·2 m written in mm
4. 66% as a decimal
5. Total of the numbers on a 7. Days in a year minus 3
8. Number of minutes between
dice
6. 0·1234 7 0·012 1322 and 1512
9. Seconds in an hour
8. Ounces in a pound 11. Double 225 plus treble 101
12. A quarter to midnight on the
9. Inches in a yard
24h clock
10. 3 4 + 56 ·78 X 0 14. 23 X 3 X 52
16. (5i )2 to the nearest whole
11. Next in the sequence
number
1, 2,6, 24,120

13. One foot four inches,

in inches

15. 234 m written in km

17. 1 as a decimal
25

18. [Number of letters in

'ridiculous') 2

Puzzles and games I 33

Part D Down
Across 1. Next in the sequence

1. 20% of 15% of £276 25,36,49,64
3. 81 ·23 X 9·79 X 11 ·2,
2. 900 - ( 17 ; 12 )
to the nearest thousand
5. Three dozen

6. 1·21 min mm 4. 0·2 X 0·2
8. Solve 2x - 96 = 72 7. Solve x3 = 1 million

9. Inches in two feet 8. 9- 0·36

i - i10. 6·6 -7- 0·1 9. 83 + 93 + 103

11. as a decimal 11. 99% as a decimal

13. Volume of a cube of 12. 20% of 2222

side 4 units 14. 0·2055 -7- 0·0005
16. 4 score plus ten
15. 555 + 666 + 777

17. A gross
18. (19i )2 to the nearest whole

number

General Knowledge
You will probably need to use the library to answer the 20

questions below. Answer as many as you can. The answer to each
question is a whole number. For example, the answer to 'Battle of
Hastings' would be 1066.

1 Feet in a Fathom. 11 Hundertdreiundvierzig.
2 Moons of Mars. 12 Battle of Trafalgar.
3 Road number of the 13 Emergency number
14 Eagle.
Great North Road. 15 Little girls from school.
4 Calling birds. 16 Legs on a spider.
5 Faces on an Icosahedron. 17 Labours of Hercules.
6 Pillars of Islam. 18 Colours of the rainbow.
7 Baker's dozen. 19 Soixante neuf.
8 Lustrum. 20 Tails on a Manx cat.
9 British T.V. standard (lines).
10 Beethoven symphonies.

'Lines': a game for two players • • •
• •
• Mark several points on a piece of paper (say
13 points). •

• Players take turns to join two of the points •
with a straight line. •

• It is not allowed to draw a line which crosses •
another line or to draw two lines from one
point.

• The winner is the last player to draw a line.

••

34 Part 1

Curves from straight lines

• On a sheet of unlined paper draw a circle of radius 8·5 em and
mark 36 equally spaced points on the circle. Use a protrac-
tor inside the circle and move around 10° for each point.

• Mark an extra 36 points in between the original 36 so that final-
ly you have 72 points equally spaced around the circle.

• Number the points 0, 1, 2, 3, .. ... . 72 and after one circuit con-
tinue from 73 to 144.

\4\ 142 143 72
69 70 71 0

• You can obtain three different patterns as follows:
A (i), Join 0---+ 10, 1-+ 11, 2-+ 12 etc
(ii), Join 0 -+ 20, 1 -+ 21, 2 -+ 22 etc
(iii), Join 0 -+ 30, 1 -+ 31, 2 -+ 32 etc.

B Join each number to double that number.
ie. 1 -+ 2, 2 -+ 4, 3 -+ 6, . .. .. .

C Join each number to treble that number.
ie. 1 -+ 3, 2 -+ 6, 3 -+ 9, .. .. . .
For C you need to continue numbering points around the circle

from 145 to 216.

Part 2

2.1 Calculating angles

Angles on a straight line

The angles on a straight line add up to 180°. ~

{

Find the angles marked with letters. (b) D
(a)

A

ABC is a straight line

X+ 42 = 180

X= 138°

ABC is a straight line

a + a = 100 = 180

a= 40°

Exercise 1 3. 4.
Find the angles marked with letters. 8.

1. 2.

c

5. 7.

6.v p
nSOO

36 Part 2

9. 10. 11.

13. 14. 15.

Angles at a point

a + b + c + d = 360°

Exercise 2 3. -~c 4.
Find the angles marked with letters.
;x;,t-
1.~ 2.~
7. 8.
~oo

5. 6.

9.'* 10. ~2"11.u

220° r

a

Calculating angles 37

Angles in triangles When the angles a, b and c are placed
together they form a straight line.
Draw a triangle of any shape on a piece
of card and cut it out accurately. Now
tear off the three corners as shown.

We see that:

J The angles in a triangle add~p~to 180:

Isosceles and equilateral triangles A

An isosceles triangle has two equal sides and two equal angles.

The sides AB and AC are equal (marked with a dash) so angles B
and C are also equal.

An equilateral triangle has three equal sides and three equal angles
(all 60°).

Intersecting lines

When two lines intersect, the opposite angles are equal.

In the diagram, a = 36° and b = 144o

Find the angles marked with letters (b)
(a)

x = 60° (angles on a straight line) a = 71 o (isosceles triangle)

y = 64° (angles on a straight line) b + 71 + 71 = 180°

z + 60 + 64 = 180 b = 38°

z = 56°.

38 Part 2

Exercise 3 4.
Find the angles marked with letters.
1. 2.

5. 6. 7. 8.

9. 12.

13. 14. 16.

17. 18.

Calculating angles 39

Angles and parallel lines y
X
Two straight lines are parallel if they never meet.
They are always the same distance apart.

In the diagram, lines AB and CD are parallel.
Lines which are parallel are marked with arrows.
The line XY cuts AB and CD .

All the angles marked a are equal.
All the angles marked b are equal.
Remember:

All the acute angles are equal and all the obtuse angles are equal.

Many people prefer to think about 'Z' angles, and 'F' angles

Find the angles marked with letters. (b) a= 72°
(a) b = 108°
c = 79°
X= 50°
y = 130° d = 101 °

Exercise 4 3. 4.
Find the angles marked with letters.
h
1. 2.

65°

40 Part 2
5. 6. 7. 8.

10. 12.

f

Angles in quadrilaterals Arrange the four angles about
a point.
Draw a quadrilateral of any shape on
a piece of paper or card and cut it
out. Mark the four angles a, b, c
and d and tear them off.

We see that:

The angles in a quadrilateral add up to 360°

Mixed questions

Exercise 5
This exercise contains a mixture of questions which reqmre a
knowledge of all parts of this section. Parts B and C are more
difficult than part A.
Part A. Find the angles marked with letters.

1. 2. 3.

Calculating angles 5. 41
4.
6.

7. 8. 9.

10. 11. 12.

d

Part B
Find the angles marked with letters. Give reasons at each stage of
your working.

13. 14.

110° 18.

17.

42 Part 2
19. 20.

Part C
Find the angle x in each diagram. Give reasons at each stage of the
working.

23.

24. 26.
27. 28.

Calculating angles 43

Exercise 6
Begin each question by drawing a diagram.

1. Calculate the acute angle between the
diagonals of the parallelogram shown.

2. In the parallelogram PQRS line QA bisects
(cuts in half) angle PQR.
Calculate the size of angle RAQ.

p

3. The diagram shows two equal squares and a
triangle. Find the size of angle a.

4. In the diagram KL is parallel to NM and
LJ = LM.
Calculate the size of angle JLM.

5. The diagram shows a
series of isosceles
triangles drawn between
two lines.
Find the value of x.