Integrated mathematics scheme: IMS T2 - National STEM Centre

INTEGRATED 1"5,12 

MATHEMATICS 

SCHEME 

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MATHEMATICS 

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Teacher's Book 2 

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INTEGRATED 

MATHEMATICS 

SCHEME 

Teacher's Book 2 

Peter Kaner 

Bell & Hyman

First published in 1982 by 

Bell & Hyman Limited 

Denmark House 

37-39 Queen Elizabeth Street 

London SE1 2QB 

Reprinted 1984 

Reprinted with amendments 1985 

© Peter Kaner 1982 

All rights reserved. No part of this publication may be reproduced, 

stored in a retrieval system, or transmitted, in any form or by any 

means, electronic, mechanical, photocopying, recording or otherwise, 

without the prior permission of Bell & Hyman Limited. 

Kaner, Peter 

Integrated mathematics scheme. 

Teacher's book 2 

1. Mathematics-1961- 

I. Title 

510 QA39.2 

CENTRE 

ISBN 0 7135 1339 X 

Typeset by Polyglot Pte Ltd, Singapore, printed and bound in 

Great Britain by William Clowes Limited, Beccles and London

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- REVISION BOOK- REVISION BOOK 

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NON ACADEMIC (30%) CSE (50%) GCE (20%)

Unit 1, page 1 

Comment 

Negative numbers has been considered a difficult topic in the 

past. The use of a calculator makes it into a topic that can be 

handled and explored by most children. The key to the 

whole thing lies in considering whether a larger number is 

being taken from a smaller. Many of the physical explanations 

confuse the issue so we use them as applications 

rather than explanations. Once children understand that 

+ 1 + -1 = 0 the rest can be developed easily. A helpful aid 

is to make red and blue cards 1+11,1-11with [Q] on the back 

of both cards, showing that a + 1 and -1 together can be 

treated as zero. This leads to the following explanations of 

addition and subtraction of negatives. 

(i) 3 + 5 ... (+1, +1, +1) + (+1, +1, +1, +1, +1)~ +8 

(ii) -3+-5 ... (-1,-1,-1)+ 

(-1, -1, -1, -1, -1)~-8 

(iii) - 3 + 5 ... -1 -1 -1 

+1 +1 +1 +1 +1~ +2 

tit 

000 

(iv) +3+-5 +1+1+1 

-1-1-1-1-1~-2 

t t t 

000 

From (iii) it is clear that adding - 3 has the same effect as 

subtracting 3, and this leads to all the required subtraction 

results. 

(v) 4-3~4+-3~1 

(vi) - 2 - - 3~ - 2 + 3~ 1 

(vii) 2 - - 3~ 2 + 3~ 5 

(viii) -4 -7~ -4 + -7~ -11 

The subtraction can be confirmed by the calculator and also 

by 'take away reasoning' as follows. 

1

We wish to take - 3 away so we introduce 3 zero pairs made 

up of +1 -1 without changing the value 

+1+1+1+1+1+1+1 

-1-1-1 

now subtract -3 and clearly this leaves 7. 

Or grouping 

the + Is 

4 - -3 ~ [4+ (+ 3 + - 3) - -3] ~ 7 

Note: It is helpful to give the number line a tilt so that 

smaller numbers are lower than larger ones. 

Multiplication by negative numbers is developed in the 

supplementary units PI and E 1 . 

Answers 

M1 1. (a) -3 (b) -8 (c) -9 (d) -7 

(e) -29 (f) -23 (g) -36 (h) -28·2 

2. Checks. 

3. (a) b is 5 more than a (b) b - a = 5 

4. (a) 0 (b) 0 (c) 0 (d) 0 

5. (a) -2 (b) -2 (c) -3 (d) -6 (e) 16 

(f) -6 (g) -1·7 (h) -2·3 

M2 1. Checking 2. Checking 

3. (a) a = b (b) a = b (c) Always true so it tells you nothing 

about a and b. 

(d) Always true. 

4. (a) -2 (b) -9 (c) -4 (d) 1 

(e) -8 (f) -18 (g) -2 (h) -17 

(i) -6 (j) -3 (k) -3 (I) -6 

(m) -5 (n) -7 (0) -2 (p) -5 

(q) 8 (r) 13 (s) 14 (t) 23 

M3 1. (a) 4 (b) 4 (c) 2 (d) 6 (e) 5 

(f) -3 (g) -7 (h) -5 (i) -8 (j) -9 

(k) -9 (I) -24 

2. (a) 2 (b) 3 (c) 4 (d) 5 (e) 7 

(f) 5 (g) 9 (h) 10 (i) -2 ( j) -5 

(k) -7 (I) -14 

2

3. (a) 11 

(f) 0 

(k) -53 

(b) 8 

(g) -5 

(I) -100 

(c) 10 

(h) -14 

(d) 8 

(i) 15 

(e) -7 

(j) 31 

M4 A. 1. All in a vertical line 2. Vertical line 

3. Horizontal line 4. Vertical line 

5. Two lines which are reflections in the y axis 

B. 1. The points all lie on a straight line. If (x, y) satisfies y = x + 1 the 

point will be on a certain straight line through (0,1), (2,3) 

2. (a) (0,0) (b) (2,0); (0,2); (-2,0); (0,-2) 

3. A parallelogram, centre (0,;) 

4. Drawing 

5. (-2, -1); (-4, -2) Yes 6. Drawing 

M5 B. 1. -16°e 2. -20 o e 3. -6°e 4. -16°e 5. -16°e 

6. 7°e 

C. 1. Ground level 2. -5 3. +2 4. -1 

5. +7 6. +1 7. -9 8. -7 9. -14 

He loses another 16000ft in landing 

P1 A. 1. (a) 1 (b) -1 (c) 3 (d) -8 (e) 1 

(f) -3 (g) -25 (h) -43 (i) 3 (j) 3 

(k) 11 (I) 8 (m) -9 (n) -30 (0) -80 

(p) -96 

2. (a) 5 (b) 5 (c) -4 (d) -13 (e) 16 

(f) 21 (g) 29 (h) 60 (i) -8 (j) -17 

(k) -26 (I) -40 (m) 2 (n) 4 (0) 0 

(p) -21 

B. 1. (a) -2 (b) -5 (c) -6 (d) -5 (e) 3 

(f) 7 (g) 3 (h) -8 (i) 20 (j) 36 

(k) -9 (I) -30 

2. (a) 4 (b) 13 (c) 8 (d) 26 (e) 54 

(f) -10 (g) -8 (h) 80 (i) -10 (j) -20 

(k) -57 (I) -6 

3. (a) 30 (b) -10 (c) 58 (d) 57 (e) 6 

(f) 16 (g) -28 (h) -20 (i) 22 (j) 12 

(k) -52 (I) -23 

3

P2 A. 1. 6·3 2. 25·7, -19·4 

3. (a) £26·88 (b) £-41·95 

4. (a) 157 m above ground 

(b) +200, -70, +60, -25, -38, +40, -50, +175, -55, -80 

(c) 157 (d) The balloon had come back to Earth 

5. 0·7°C above normal 

B. (a) £570 (b) £262 (c) £1145 (d) £900 

C. 1. (a) -20 (b) -21 (c) -10 (d) -30 (e) -60 

(f) -32 (g) -21 (h) -96 (i) -21 (j) -8 

(k) -25 (I) -54 

2. (a) 4, 8, 12, 16, 20 (b) 6,12,18,24 

3. (a) 21 (b) 30 (c) 16 (d) 5 (e) 36 

(f) 81 (g) 60 (h) 55 

E1 

A. 1. 

2. 

B. 1. 

2. 

3. 

As for P1 and (q) 0·7 (r) -2·3 (s) -4·4 (t) -1·17 

As for P1 and (q) 7·2 (r) 6·5 (s) 1·2 (t) 0·88 

As for P1 and (m) 5·6 (n) 0·6 (0) -7·3 (p) 13·6 

As for P1 and (m) -7·8 (n) +3·6 (0) 5 (p) 3·4 

As for P1 and (m) 9·2 (n) 6·6 (0) -0·5 (p) -2·2 

E2 

E3 

1. 

2. 

3. 

A. 2. 

B. 2. 

As for P2C and (m) -9·6 (n) -11·5 (0) -10·2 (p) -31·5 

As for P2C and (c) -12, -24, -36, -48 

As for P2C and (i) 4·5 (j) 5·2 (k) 22 (I) 33·6 

(a) (5 x -8) + (5 x 8) = 5 x (8 + -8) = 5 x a = a 

So 5 x -8 is the negative of 5 x 8, Le. 5 x -8 = -40 

(a) Suppose n, 

is the negative of 22 and n2 is another negative. 

n, + 22 + n2 = (n, + 22) + n2 = a + n2 = n2 also 

n, + 22 + n2 = n, + (22 + n2) = n, + a = n, 

So n, = n2, both negatives are the same, etc. 

C. 2. 16x16=16x(16+0)=?16x16=16x16+(16xO) 

=? 16 x a must be zero 

4


Decimals and percentages 

A calculator will give correct answers to decimal calculations 

without reference to any rule so that fear of decimals can 

evaporate overnight. The weaker children may remain 

scared, however, and will need careful reassurance that the 

well understood system of grouping by tens continues 

below 1. Percentages are something which many people fear 

but the grouping of the two topics proves very helpful. A 

percentage, e.g. 35% means 35/100 and so can be written 

0·35. If this is established, percentage questions become 

very easy. If the calculator has a % button this should be 

investigated by the students following some set examples. 

Answers 

M6 A. 1. (a) 2 (b) 4 (c) 7 (d) 1·1 (e) 1·7 

(f) 1·9 (g) 2·7 (h) 3·3 (i) 4·5 (j) 5·7 

(k) 6·3 (I) 9·2 (m) 12 (n) 15 (0) 35 

(p) 38 (q) 0·4 (r) 0·8 

2. (a) 30 (b) 50 (c) 80 (d) 12 (e) 18 

(f) 26 (g) 36 (h) 39 (i) 45 (j) 48 

(k) 58 (I) 60 

3. (a) 0·2 (b) 0·7 (c) 0·9 (d) 1·1 

(e) 2·2 (f) 2·7 (g) 3·1 (h) 4·0 

(i) 0·36 (j) 0·49 (k) 0·51 (I) 0·72 

4. (a) 0·05 (b) 0·08 (c) 0·09 (d) 0·1 (e) 0·14 

(f) 0·18 (g) 0·27 (h) 0·38 (i) 3 (j) 4·2 

(k) 6·5 (I) 7·2 (m) 1·85 (n) 2·66 (0) 3·71 

(p) 5·04 (q) 0·372 (r) 0·493 

B. 1. (a) 1·5 (b) 1·5 (c) 1·6 (d) 0·7 

(e) 1·11 (f) 1·27 (g) 1·42 (h) 1·29 

(i) 0·4 (j) 0·3 (k) 0·6 (I) 0 

Check the additions by adding again in reverse order. 

Check the subtractions by adding back what you have just subtracted. 

5

2. (a) 2·5 (b) 5·0 

(f) 9·53 (g) 16·01 

(k) 14·26 (I) 14·5 

(c) 6-7 

(h) 16·27 

(d) 9·2 

(i) 8·51 

(e) 5-95 

(j) 8·58 

Check on the calculator (these are 'pencil and paper' practice, so 

should not be done on the calculator first). 

C. 1. (a) 0-24 (b) 0·14 (c) 0-09 (d) 0-30 

(e) 0-02 (f) 0·016 (g) 0-018 (h) 0·021 

(i) 0·26 (j) 0·72 (k) 0-8 (I) 1·28 

Take care with decimal point. Check by rule (number of places 

after decimal point is the same before and after multiplication). 

2. (a) 2 (b) 1-5 (c) 3 (d) 0-5 

(e) 20 (f) 40 (g) 20 (h) 30 

(i) 3 (j) 12 (k) 5 (I) 7 

Multiply answer by divisor to get back first number (calculator). 

M7 

A. What is ... 1. 

3. 

5. 

B. 

1. 0.6, 60% 

4. 0·46, 46% 

7. 0·325, 32·5% 

0·2 x £40 

0·15 x £60 

0·18 x 5 kg 

2. 0.467, 46-7 % 

5. 0·675, 67·50/0 

8. 0·816, 81·6% 

2. 0·01 x £50 

4. 0·12 x 6m 

6. 0·25 x 650 gals 

3. 0_45, 45% 

6. 0·76, 76% 

9. 0·75, 75% 

C. 1. 0·8, 80% 

4. 0·06, 6% 

7. 0·736, 73·6% 

10. 0·25, 25% 

2. 0·068, 6·8% 

5. 0-57, 57% 

8. 0·091, 9·1% 

11. Increase 0·002, 

0.20/0 

3. 0·033, 3·3% 

6. 0·607, 60·7% 

9. 0·258, 25·8% 

12. 0·696, 69·6% 

M8 A. 1. Mr Jones 27·8%, Mr Green 18·2% 

2. (i) July 6·8%; August 1·2%; September 6·5% 

(ii) July 6·8%; August 8·0%; September 15·1% 

3. (i) £106 

(ij) 17% rent; 45·3% food; 22·6% entertainment, 15·1% other. 

Check that they add up to 100. 

4. (i) £160 

(ij) rent 17·5%; food 31·25%; entertainment 16·25%; other 35%. 

Smith family spend a higher percentage on food. 

6 

B. 1. (b) 38·5p per 200 g 

(c) 28·6 (29p) per tin 

(d) 81·4p per kilo

Note: (a) Rounded to nearest !p. (b) Adding 10 % can be done 

by multiplying by 1·1. 

2. Use brackets if the calculator has them, otherwise work out the 

product, subtract the original price and. ignore the minus sign. 

(b) £2·04 (c) 81 p (d) £2·30 

Note: Reducing 15% can be done by multiplying by 0·85. 

3. (a) £18 (b) £21 (c) £31·25 

(d) £46 (e) £54 (f) £68·25 

4. Percentage growths are 8%, 7.40/0,17.2 % ,17.6 % ,150/0 and 10·9%. 

Biggest growth during the 4th month. 

M9 

1. All match 

2. (i) 0·36 

(v) 0·0625 

3. (i) 0·25 

(v) 0·1296 

4. (i) 0·13 

(ii) 0·16 

(vi) 0·4096 

(ii) 0·49 

(vi) 0·1936 

(ij) 0·74 

(iii) 0·04 

(iii) 0·81 

(iii) 0·72 

(iv) 0·1225 

(iv) 0·0784 

(iv) 0·185 

P3 

A. 1. 35% 

5. 88% 

9. 2·5% 

B. 1. 0·3 

5. 0·08 

9. 0·01 

2. 40% 

6. 92% 

10. 31·5% 

2. 0·45 

6. 0·12 

10. 0·333 

3. 66% 

7. 8% 

11. 44·4% 

3. 0·5 

7. 0·99 

11. 0·425 

4. 75% 

8. 3% 

12. 52·5% 

4. 0·56 

8. 0·64 

12. 0·085 

C. 1. 60% 2. 58·3% 3. 45% 4. 80% 5. 16·5% 

6. 33% 7. 33·75% 8. 65% 9. 34·06% 

10. 80%, 65%, 60%, 58·3%, 45%, 34%, 33·75%, 33%, 16·5% 

P4 A. 1. 120 g 2. 62·5% 

4. Home Economics 5. 14000 

3. Maths 

6. 75% 

B. 1. (a) £20 (b) £60 (c) £16 (d) £10 

2. (a) £90 (b) £157·50 (c) £216 (d) £81 

3. (a) £32·40 (b) £162 (c) £194·40 (d) £388·80 

4. (a) £10400 (b) £32500 (c) £50050 (d) £60450 

C. 1. £102 2. £105·60 3. £88·80 4. £110·40 

5. £5100 6. £4536 7. £4272 

8. £5010 (can be done by multiplying present pay by 1·2) 

7

D. 1. 140 + 45 = 185, 37% 

2. (a) 7700 (b) 12600 (c) 5250 

3. (a) 10320 (b) 4334 (c) 4231 (d) 1754 

No decimal people, please. So the addition is 1 out. 

4. VAT 150/0 25% 35% 

Camera £52·33 £56·88 £61·43 

boots £16·62 £18·06 £19·51 

bike £44·74 £48·63 £52·52 

crackers £ 4·89 £ 5·31 £ 5·74 

ladder £21·28 £23·13 £24·98 

drill £18·86 £20·50 £22·14 

Multiply original price by 1·15, 1·25, 1·35 

P5 1. (a) 1 (b) 2·8 (c) 5·6 (d) 6 

(e) 7·8 (f) 10·6 (g) 1·9 (h) 4·2 

(i) 3·3 (j) 0·25 (k) 1·96 (I) 7·84 

(m) 0·75 (n) 3·36 (0) 10·64 

2. (a) 2·55 (b) 7·35 (c) 9·18 (d) 1·15 

(e) 5·95 (f) 7·78 (g) -1·6 (h) -0·61 

(i) -2·21 (j) 0·7225 (k) 6·0025 (I) 9·3636 

(m) -0·1275 (n) 3·5525 (0) 6·3036 

3. (a) (b) (c) (f) true (d) (e) not true 

4. (b), (e) true; (a), (c), (d), (f) not true 

E4 A. 1. Use constant multiplier 1·16, ten times. £22057·17 

2. Use constant multiplier 1·24, 1988 80·1 thousand 

1989 99·3 thousand 

1990 123·1 thousand 

3. Use constant multiplier 1·43, 1980 429 million 

1990 613·47 million 

2000 877·26 million 

4. (a) 4·55% (b) 5·94% 

5. (a) 7·446 million (b) 9·57 million cars, i.e. 2·124 million more 

B. 1. Multiply everything by 0·85. 

(a) £80·75 (b) £47·60 (c) £17 

(e) £46·75 (f) £27·20 (g) £23·80 

2. Divide by 0·85 each time. 

(a) £94·12 (b) £47·06 (c) £23·53 

(e) £75·29 (f) £32·94 (g) £11·76 

3. (a) 8·3% (b) 12·5% (c) 40% 

(e) 37·5% (f) 36% (g) 28·6% 

(d) £12·75 

(h) £29·75 

(d) £14·12 

(h) £37·65 

(d) 16·6% 

(h) 40% 

8

E5 A. 1. (a) £39900 (b) £8960 (c) £13440 (d) £7980 

(e) £17640 (f) £5110 (g) £5320 (h) £3360 

2. (The tip is 100/0of the bill before VAT is added.) Multiply each by 

1·25. 

(a) £5·62 (b) £8·06 (c) £13·12 

(d) £10 (e) £12·37 (f) £15·25 

3. (a) (Divide by 1·014) ... 29586 

(b) (Multiply by 1·014) ... 30420 

4. (a) 8·89 kg (b) 222·25 g (c) 0·76 g (d) 0·05 g 

5. East 31·2% West 30%. East has larger percentage. 

6. Various answers. 

B. As for Exercise P5. 

c. All true except for (d). 


Even though Euclidean geometry is no longer taught as a 

complete system in school mathematics it is still worth 

knowing something about congruent triangles. The triangle 

is a widely used shape in building, surveying and navigation 

and much depends on the fact that three sides of a given 

length define a triangle so that the angles are fixed. All this is 

another way of saying that the triangle is a rigid shape. Once 

you join three rods the construction is rigid even though the 

joints may be free. Many examples of this phenomenon may 

be observed from cranes to bicycle frames and the structure 

of the human body. 

Drawing triangles is one way (one of the best) to get their 

'feel'. If triangles are drawn with the same dimensions, then 

cut out and compared, an understanding of congruence can 

follow. The collapsing of a four-rod linkage provides a 

dramatic contrast and has many consequences from the 

collapse of buildings to the infuriating problem of keeping a 

door-frame an exact right angle when hanging a door. 

The work of the unit provides extensive practice in the use 

of drawing instruments and in careful measurement of 

9

lengths and angles. A basic metric rule of geometry follows 

from the discussion of three sides which cannot form a 

triangle. In space ... 

(distance from A to B) + (distance from B to C) 

~ (distance A to C) 

which may be stated as the theorem two sides of a triangle 

are greater than the third. Congruent triangles can be used to 

prove many geometrical properties and this aspect is interesting 

to quite a number of children. 

Answers 

M10 A. 

B. 

c. 

1. (Approximate) A = 49~0 

2. (Approximate) A = 27 0 

3. (Approximate) A = B = 57~0 

4. (Approximate) A = 62 0 



Check by fitting. 

B = 22.5 0 

B = 36 0 

B = 42 0 

B = 22~0 

B = 36~0 

= 108 0 

= 117 0 

= 65 0 

= 76 0 

= 108 0 (half Q. 1) 

e = 117 0 

Put a sheet of thin paper over the triangle in the book and mark the 

corners. Then use a pin to transfer the points to an exercise book.

M13 

Ship to Ship to Distance 

lighthouse church to line 

1. 8·49 km 8·49 km 6·00 km 

2. 23·0 km 21·0 km 19·9 km 

3. 47·1 km 44·5 km 44·3 km 

4. 27·3 km 44·4 km 27·3 km 

5. 20·2 km 8·10 km 7·58 km 

6. 9·10 km 13·5 km 8-65 km 

These are correct to 3 significant figures so drawings will be approximately 

these values_ 

M14 1. 69 m 2. 84m 3. 42m 4. 22m 

P6 A. 

AB BC CA A B C 

1. 4cm 5-5cm 6-1 cm 62° 78° 40° 

2. 4cm 5-1 cm 5-3cm 65° 70° 45° 

3. 70mm 48mm 37mm 40° 30° 110° 

4. 38mm 60mm 65mm 65° 80° 35° 

5. 3·2cm 6·5cm 6·7cm 72;° 79;° 28° 

B. Drawing 

C. Drawing 

P7 A_ Post 2 Post 1 

1. 18-4km 16-3 km 

2. 19-1 km 10-1 km 

3. 28-3 km 20km 

4. 26-9 km 10-6 km 

5. 10-3 km 24km 

6. 12-2 km 12-2 km 

B. 1. 11-8 km 2. 15-5 km 3_ 101-6km 4. 29-2 km 

C_ 1. 25-8m 2_ 50-8m 3_ 49-2m 4. 80-5m 

E6 

A, B,C. See P6 

11

E7 

See P7 

E8 A. 1. Draw AD and BC and then consider triangles ABO and BAC. These 

are congruent because two sides are equal and so is the angle 

between them. Thus the third sides are equal. These are the 

required diagonals. 

2. Prove that triangles ABC and BCD are congruent. 

3. Consider triangles AOB, COD. 

4. AC..l BD, both BD and AC are axes of symmetry leading to the 

following 

BD bisects Band 0, AC bisects A and C, AOB, BOC, COD, DOA all 

right angles. 


The cuboid is man's most commonly used shape for many 

reasons. These reasons can be elucidated from the class 

during general discussion. The usual procedure of defining a 

cuboid, then rushing into formulae for volume and surface 

area leaves out a very important exploration stage. Remember 

also that this will probably be the first attempt children 

will make to represent a solid by a careful two-dimensional 

drawing. This takes a lot of practice. 

For bright children measuring a 'solid diagonal' of a 

cuboid makes an intriguing problem. This has practical 

applications when solid cuboid objects such as furniture and 

refrigerators have to be moved. 

The P and E units are very close for this exercise as 

brighter children may well need a considerable amount of 

practical work on this topic. 

Answers 

M15 A. 

1. Bricks, concrete blocks, breakfast cereal boxes and books are usually 

cuboid. 

2. (a) True (b) True (c) Not true 

(d) False (e) False (f) It will need 8 

3. (a) Yes, though one dimension is very small. 

(b) Could be (c) No (d) No (e) No (f) Roughly 

12

B. 1. (c) (i) Two faces are squares (largest faces) 

(ii) Two faces are square (smallest faces) 

(iii) The cuboid is a cube 

2. Measurement 3. Measurement 

C. A good supply of isometric paper will lead to coloured designs which 

can be used to brighten up the classroom. 

M16 A. 1. Drawing 2. Drawing 

3. (a), (b) and (c) all have square faces. (c) is a cube. 

P8 A. 1. (for example) They fit together, they lie on top (very stable), easy 

to manufacture. 

2. (a) They wouldn't fit together. 

(b) The sharp edges would crumble, they would not build into 

stable shapes. 

(c) They wouldn't pack (although they are used for insulation). 

(d) They would not form stable shapes. 

3. (a) The drain pipes (b) Floors, walls (c) Pillars 

(d) The ball-cock in a water tank 

B. 

C. 

1. (a) 40 bricks (b) 66 bricks 

1. 20 2. 12, 8, none 

3. 6 with 1 yellow face; 12 with 2 yellow faces; 8 with 3 yellow faces; 

1 with no yellow faces 

P9 

Drawing 

P10 1. There are only three basic shapes where four equilateral triangles 

are joined edge to edge, two of these are shown in the introduction 

to the exercise. 

2. Two of the shapes make a tetrahedron but the ~ hexagon folds 

into a pyramid with four sloping sides and without a base (two of 

these shapes will make a regular octahedron). 

3. Add one triangle in all possible positions to each of the three basic 

three triangle shapes and work on from there. 

E9 A. 1,2,3 see P8. 

4. This question should be discussed. Suggested answers are 

(a) It is sharp at the bows to reduce water resistance and wide for 

stability. 

(b) Flat to give good pressure, long for leverage. 

13

(c) Sharp-fronted to travel fast and reduce air resistance. 

(d) Hollow to hold liquids, flat handle to make it easy to hold steady. 

(e) Narrow neck to prevent fast flowing of liquid and to present small 

area of liquid to the air. 

(f) Many features are designed for safety and efficiency. 

B. 1. 25 2. 52 3. 51 

C. 3. (a) 5 x 2 x 2, 5 x 4 x 1, 10 x 2 x 1, 20 x 1 x 1 cm 

(b) 5 x 2 x 2* has least number of squares exposed 

(c) 1 x 1 x 24, 1 x 2 x 12, 1 x 3 x 8, 1 x 4 x 6, 2 x 2 x 6 and 

2 x 3 x 4. Least squares exposed 2 x 3 x 4 = 52 

4. (a) 6 

(b) 12 with 2 yellow faces, 8 with 3 yellow faces and 1 with no 

yellow paint 

(c) There are 24 with 1 yellow face, 24 with 2 yellow faces, 8 with 

3 yellow faces and 8 with no yellow face (making a 2 x 2 x 2 

cube inside) 

E10 

Drawing 

E11 See P10, 1 and 2 and 3 

4. Treat the area outside the triangle as the base of the tetrahedron. 

It is not possible to move along all the edges without going over 

one edge a second time. 


Although vectors are considered a new topic in secondary 

mathematics they are conceptually easy when thought of as a 

combination of movements. The addition rule agrees with 

common sense (which makes it easier than fractions as a 

topic). The work on vectors can relate many problems in the 

real world to the newly introduced Cartesian co-ordinate 

system. The idea of negative as direction is reinforced and 

the geometrical notion 'translation' comes under further 

discussion. The arithmetic of vectors presents no problems 

as the numbers should be kept small while the geometry 

leads to careful measurement of length and direction. A 

* Note the least squares correspond to the cuboid where the 3 dimensions add to the 

least total. 

14

Answers 

further bonus from the topic is the discussion of 'operational 

rules' which can be related to the rules of numbers, especially 

the commutative and associative law. It may be necessary 

to explain the need for the two types of representation, 

(x,y) for the point and G) for the vector from (0,0) to 

(x, y). Although this notation is complicated it is not 

difficult. Remember that the topic will be presented again 

and that exploration is more important than memorizing at 

this stage. 

M17 A. 1. 

(~) 2. (~) 3. (:) 

4. 

(~) 5. G) 

6. 

(~) 7. m 8. (~) 9. G) 

10. 

(~) 

B. 1. 

(=~) 2. (-~) 3. (-~) 4. (-~) 5. (=~) 

e. (-~) 

7. (-~) 8. (-~) 9. (=~) 10. (=:) 

11. 

(=~) 12. (=~) 

C. Drawing 10. (F) 

(=:), (G) (=~),(H) (=:) 

M18 A. 3. (a) 

(~) (b) G) 

(c) 

(~) (d) G) 

(e) (-~) (f) (-~) (g) (-~) (h) (-~) 

B. 2. A parallelogram 4. A parallelogram 

5. Nothing at all other than move it 

6. Nothing at all other than move it 

C. 

(~) 

2. 

3. m, (~),(~) 

Ṭhe last is the sum of the first two, obtained 

by simple addition of the corresponding numbers. 

D. 3. (-~) 4. ( - ~) - (~) = ( -~) 

15

M19 A. 1. 

m 

2. 

(~) 3. C~) 

4. 

(:) 

5. 

(~) 

6. 

(:) 

7. (-~) 

8. 

(1~) 

9. (-~) 10. (=~) 

11. (=~) 

12. (-~) 

B. All four are (8) because the two vectors are equal and opposite. It is 

like asking Ihow far have I travelled after going from here to London 

and back again'. After that journey I am back where I started. 

c. 1. 

(~) 

2. (1~) 

3. (~~) 

4. 

(~) 

5. W 6. 

(~) 

D. 1. 

(:) 

2. 

G) 

3. 

(~) 

4. 

(~) 

5. (,~) 

6. (-~) 

M20 A. 1. Direction needed 2. Speed needed 3. Velocity 

4. Direction needed 5. Direction needed 6. Velocity 

B. 1. M 2. L 3. M or A 

4. 0, Hand D 5. C and G or E and G. 

c. Various answers 

P11 A. A B C D E F G H 

1. (-:) (~) (:) (:) (-:) (-~) (-~) (=~) (=:) 

2. (a) 

C) 

(b) 

(~) (c) (-~) 

(d) 

(-~) 

(e) 

(=~) 

(f) (-~) 

3. (a) 

(=~) 

(b) 

(=~) 

(c) 

(~) 

16

(d) (-~) (e) 

G) 

(f) 

(-~) 

B. 1. J 2. H 3. F 4. D 

5. B 6. J 7. H 8. F 

C. 1. FACE 2. CABBAGE 3. CHIEF 

P12 A. 1. 

(~) 2. (=~) 

3. 

(=~) 

4. 

(~) 

B. 1. L.JKL 2. L.JKL 3. L.DEF 4. L.ABC 5. L.JKL 

C. 1. L.JKL 2. L.DEF 3. L.GHI 4. L.MNO 

P13 A. 1. Drawings 2. Drawings 

B. 1. Agreed 2. Should be (2~) 

3. Should be (2~) 4. Agreed 

C. Drawings 

E12 A. 1., 2., 3.-See Exercise P11. 

4. The general results work for all values of x and y. 

5. All these vectors are parallel to (~) and simple multiples of its 

length. 

B. See Exercise P11 

C. See Exercise P11 

D. Experiment 

E13 A., B., C.-See Exercise P12. 

D. There are many translations in this diagram. If you assume the figure 

shows part of a tessellation, e.g. 

s, ~ S2 which maps S2~ S3, S4~ S5, etc. 

s, ~ S3; s, ~ S4; s, ~ S5; s, ~ S6; S2~ S4; 

S2~ s,; S3~ S4; S3~ s,; S4~ s,; S4~ S2; S4~ S3; 

S5~S,;S6~S,. 

17

E14 A. See Exercise P13 

B. 1. Agreed 

2. Agreed 

3. Should be (~b) 

4. Agreed 

5. Agreed 

6. Agreed 

c. 1. Drawing 2. All the steps in the x direction add to zero and so 

do the steps in the y direction. 

vectors) must be (8). 

Thus the total effect (the sum of the 


This unit on problem solving relates some simple mathematics 

to tools in everyday use. It is not obvious to children 

that things are as they are by design rather than by chance. 

Looking carefully at objects is the same skill as the observation 

taught in Book 1 and which is being taught in all the 

other subjects of the School curriculum. There is very good 

reference material in a book called Machines, Mechanisms 

and Mathematics, part of the Schools Council Mathematics 

for the Majority Project. This book is particularly good on 

linkages. Simple tool making has been one of the most 

significant activities of human beings. 

Answers 

M21 

Note: You may decide to replace these objects with others. A good 

class lesson could be organized as a quiz-game with pupils having·to 

bring in mysterious objects and others having to guess what their 

function is. 

M22 3. These answers may need to be discussed. 

(a) Type (i) (b) Type (ii) (c) Type (i) 

(d) Type (i) (e) Type (i) (f) Type (i) (g) Type (iii) 

M23 1. 12.5 : 1 2. 312·5 : 1 3. 10·4 : 1 

4. 10·7 : 1 5. 18 : 1 6. 20 : 1 

P14 A. 1. (a), (b), (c), (g) 2. Different answers 

18

B. 2. Load 

(a) Friction holding lid 

(b) Weight 

(c) Reaction of water 

Force 

Push down on spoon 

Pull of muscle 

Pull of arms 

Fulcrum 

Edge of tin 

Elbow joint 

Rowlock 

P15 A. 1. (a) B moves down (b) B moves up 

(c) B moves away (d) 8 moves towards you 

2. (a) A moves down (b) A moves up 

(c) A moves away (d) A moves towards you 

3. (a) B moves in a circle 

(b) B moves in a square 

(c) B moves in a triangle 

B. 1. A parallelogram 

2. Band 0 move towards each other 

3. A and 0 move towards each other 

C. 1. 8 and C move outwards 

2. Band C move towards each other 

3. No 

4. Not without moving A and 0 as well 

5. Yes 

P16 A. 1. 4 kg 2. 4 kg 3. 2 kg 

6. Many possible answers 

4. 2 kg 5. 31 kg 

B. 1. 2cm 

6. 2cm 

2. 6cm 3. 2cm 4. 5cm 5. 2·5cm 

E15 A. 

1. (a), (b), (c), (g), (i), (j) 

B. 

2. Load 

(a) Friction holding lid 

(b) Tension in tyre and friction 

(c) Weight 

(d) Reaction of water 

(e) Reaction of water 

Force 

Push down on spoon 

Force exerted by hand 

Pull of biceps muscle 

Pull of arms 

Pull of arm 

Fulcrum 

Edge of tin 

Rim of wheel 

Elbow joint 

Rowlock 

Other hand 

E16 See P15 A and B. 

4. A rhombus 

C. As for P15 

5. Approx 14 em 6. 180 0 

E17 

A., B. See P16 

C. 2. 0) 7 Oi) 3 

3. (a) 16·2 kg 

(iii) 0·5 

(b) 8·6 kg 

(iv) Many answers possible 

(c) 4·7 em (d) 48 em 

19


The circle is the most perfect of all plane shapes having 

infinite symmetry. The fascinating relationship between 

circumference and diameter has interested mathematicians 

for centuries and the extraordinary properties of the number 

1T continue to stimulate research. 

A thin strip of 1 mm 2 graph paper makes an excellent 

'curved~ ruler and seems to follow the circumference of a 

circle in a most natural way. There are other ways circular 

measurements may be explored such as putting an ink blob 

on a coin and rolling it down a slope or wheeling a bicycle 

along after making a wet paint-mark on one of its wheels. 

Practical alternatives emphasize that all circles are the same 

in the ratio of diameter to circumference and fix the number 

1T far better than calculations. (Incidentally ~ all regular 

polygons have fixed circumference/diameter relationships. 

Squares for example have perimeters which are 1·414 times 

their diameter ~while the perimeters of all regular hexagon~ 

are 3 times their diameters.) .~ 

Work on circles leads naturally to the exploration of the 

cylinder which is a far more common object than a circle 

pure and simple. 

C + D comes to approximately 3·14 every time. 

20

C. The graph should give 

(a) 12·5 em (b) 15·7 em 

(c) 37·7 em (d) 26·4 em 

D. The ratio C -7 D should come to roughly 3·1"4each time so the average 

should be very close to 3·14. 

M25 

Practical work. 

M26 A. Completed table. Answers marked* 

Radius 

1. 22cm 

2. 11 m 

3. 14mm 

4. 8·5 cm* 

5. 4·25cm* 

6. 2·2 m* 

7. 15·9cm* 

8. 7·5mm* 

9. 0·9mm* 

10. 9·25 m* 

11. 4·3cm* 

12. 72 km 

44cm* 

22m* 

28mm* 

17cm 

·8·5cm 

4-4m 

31-8cm* 

15 mm* 

1·8mm* 

18·5m 

8·6cm* 

144 km* 

Diameter 

Circumference 

138cm* 

69m* 

88mm* 

53-4cm* 

26-7 cm* 

13·8 m* 

100cm 

47mm 

5·8mm 

58m* 

27cm 

452 km* 

B. 1. 235·5 mm 

2. Approx 125 em 

3. 1mm 

4. 220 em, 220 m 

5. 471 em 

(a) 21 230 approx 

(b) 212300 approx 

P17 A. 1. 37 mm 

2. 50mm 

3. 72mm 

4. 39mm 

B. 1. 146 mm 

2. 174mm 

3. 328mm 

21

P18 A. 1. (a) 12·5 (b) 18·8 (c) 28 (d) 63 (e) 94 

(f) 188 (g) 107 (h) 182 (i) 270 

Note: To find 34 use 30 + 4. Find 30 from 3. 

2. (a) 4·8 (b) 8 (c) 32 (d) 64 (e) 5·7 (f) 22 

3. (a) 50 (b) 157 (c) 264 (d) 754 

B. 1. 6·3 m 2. 4·4 em 3. About 60 em 

4. About 31 m/sec or 113 km/hr 

C. 1. 3~ is very accurate ... 3·141593. This is easily remembered as 

355/113 (113355) 

2. (a) 16400 km (b) 16400 km 

(c) 103044km (d) 21467km/hr 

E18 A. 1. Drawing. Use as large a scale as will fit on your graph paper. 

2. 3. 1 See P18 A. 1, 2, 3 

4. 

B. 1. 6·3m,159times 

2. 4·5 em, the largest side of square is 3·2 em (the square with 

diagonal 4·4 em) 

3. About 60cm 4. 31m/sec=113kmjhr 

C. 1. 12 732 km 2. 10917 km 

3. 2419026km 4. 584·3 million miles 

5. 40 000 km, 1666 km/hr 6. 25133 km, 1047 km/hr 

E19 1. (b) 

2. (a) 16366 km (b) 102831 km (c) 21 423 km/hr 

3. Joe is right. If C is the circumference of the Earth and R is the 

radius of the Earth. C= 27TR. If the circumference is increased by 6 

metres C + 6 = 27T (R + r) where r is the increase of radius. Thus 

6 = 27Tr, so r = 6/27T~ 1. 

22 


Most people in the past have found algebra difficult to 

understand. I believe this is because the beginnings are too 

rushed and too isolated. Also algebra is built on unsatisfactory 

arithmetical foundations. Both of these faults can be

emedied by using a calculator. The problems of arithmetic 

are cleared up for almost all children and the speed of the 

calculator means that many more exercises can be performed 

giving a much wider range of initial experience in the 

use of letters. 

The use of random decimal numbers to test such relationships 

as the associative and distributive laws is more 

convincing than demonstrations with easy numbers only. 

There is a lot of conventional notation in the beginnings of 

algebra and these have to be made clear before any progress 

can be made. All these conventions are man-made and often 

led to controversy when they were proposed in the first 

place. It is important that children should not have their 

confidence undermined by a 'right and wrong' attitude or 

they will reject the whole of algebra as incomprehensible. 

Discussion on how to write a + a + a + a + a ... to save 

space should lead to the suggestion Sa from pupils and this 

can be compared with other notations proposed by members 

of the class. All this material will be presented again during 

the 3rd year, so at this stage it is more important to avoid 

anxiety and confusion than to achieve memorization. 

Answers 

M27 1. (a) ael (b) abd 

ale ... word adb 

ela 

bad ... word 

eal ... 

bda 

lea ... word dba 

lae 

dab ... word 

(c) aemn eamn maen *name 

aenm eanm *mean naem 

*amen eman *mane neam 

amne enam men a nema 

anem emna mnae nmae 

anme enma mnea nmea 

2. There are 120 possible groups of letters but the only words are 

dame, dare, dear, made, mare, mead, read and ream. 

3. Barb 4. Rubber 5. Report, Porter 

M28 A. 1. (a) 5 (b) 11 (c) 11 (d) 4 (e) 4 

(f) 9·6 (g) 10·7 (h) 35·5 

2. (a) 15 (b) 16 (c) 0 (d) 0 (e) 16·1 

(f) 9·3 (g) 0 (h) 3·24 

23

3. (a) 9 (b) 21 (c) 0 (d) 48 (e) 8·4 

(f) 10·2 (g) 1·41 (h) 1·08 

4. (a) 8 (b) 0 (c) 40 (d) 36 (e) 8·6 

(f) 0·9 (g) 1·36 (h) 15·54 

5. (a) 20 (b) 25 (c) 128 (d) 59 (e) 58·4 

(f) 33·4 (g) 14·75 (h) 11·12 

B. 1. (a) 11·3 (b) 28·4 (c) 9 (d) 14·4 (e) 30·53 

(f) 55·4 

2. (a) 6 (b) 8·5 (c) 7 (d) 0 (e) 10 

(f) 17·4 (g) 9 (h) 10 (i) 12 (j) 19·333 

(k) 90 (I) 66·666 

C. 1. (a) Both sides 8 (b) Both sides 27·48 

(c) Both sides 87·94 (d) Both sides 19·008 

2. (a) Not true (b) True (c) Not true 

(d) True (e) Not true (f) True 

3. (a) Always true (b) Only true if a = b 

(c) Always true (d) True (e) True (f) True 

M29 A. B. check. All the examples in C show that subtraction and division 

are not associative. 

M30 These examples demonstrate that multiplication is distributive over 

addition. But addition is not distributive over multiplication. Easily 

proved as follows: 

a + (b x e) = (a + b) x (a + e) => a + be = a 2 + ab + be + ea 

=> a = a 2 + ab + ae 

=> 1=a+b+e 

Thus addition is only distributive over multiplication if a + b + e = 1, 

e.g. 

a = 2, b = -1, e = 0 

a-~ - 2, b-~ - 4, e-~ - 4 

2 + (-1 x 0) = 1 x 2 True 

~+ (~x ~) - ~ +i - .l!- 

2 4 4 - 2 16 - 16 

H + a) x (! + a) = 1~ 

P19 A. 1. Me! 2. Tea 3. Tame 

24 

B. 1. 6a 2. a+4b 3. 3a + 5b 4. 3a - 2b 

5. a+2b 6. a 3 7. a 3 x b 2 8. a 3 + b 3 

9. 6a + 8b 10. a+b 11. 8a 12. m+3n 

13. 3m 14. 5x + 9y 15. 5x - 4y 16. 2x - 2y

c. 

P20 A. 

B. 

C. 

P21 A. 

1. 5a 2. 3a 3. -a 4. a+b 

5. 2a + 3b 6. -2m +4n 7. -4m-n 8. 3y 

9. x+ 11y 10. -2x+ y 

1. 7·4 2. 18·6 3. 28·6 4. 25·6 

5. 9 6. -4·7 

1. 2 2. 3·5 3. -1 4. 3·5333 

1. b = 3·7, ab = 8·51 2. b = 9·7, ab = 142·59 

3. b = -7'8, ab = -49·92 4. b = 32, ab = 672 

5. a = 6·7, ab = 35·51 6. a = 3·9, ab = 4·68 

7. a = 4·6666, ab = 21 8. a = 3·56, ab = 9·256 

The laws work for all values of a, band c 

B. Diagram 1 shows 3(a + b) = 3a + 3b and a = xxx, b = 0000, i.e. Distributive 

Law 

E20 A. See P19 A. 

Diagram 2 shows a + (b + c) = (a + b) + c, i.e. Associative Law (addition) 

Diagram 3 shows 4 x (3 x 2) = (4 x 3) x 2, i.e. Associative Law (multiplication) 

Diagram 4 shows 4 x (3 x 2) = (4 x 3) x 2 again in different form 

Diagram 5 shows ab + ac = a(b + c), i.e. Distributive Law 

Diagram 6 shows Associative Law (addition) 

B. 1. There are 

(i) 20 different pairs possible (5 x 4) (ii) 60 different threes 

(5 x 4 x 3) 

(iii) 120 different fou rs (5 x 4 x 3 x 2) (iv) 120 different fives 

(5 x 4 x 3 x 2 x 1) 

The words of 5 letters are bread, beard and bared. 

C. 

D. 

2. Letters 

1 

2 

3 

4 

5 

6 

See P19 B. 

Arrangements 

1 

2 

6 

24 

120 

720 

See P19 C. and ... 11. p-2q 12. q-6p 

14. 0 15. -5·7u 16. -4u + 6v 

13. -3r -35 

25

E21 A. See P20 (A) and ... 7. 8·65 8. 6 9. 34·72 10. -0·51 

B. See P20 (B) and ... 5. -3·3 6. 4 7. -46 8. 39·64 

C. See P20 (C) 

E22 

See P21 


The squaring of numbers is an easy process which can give 

rise to many pattern searching and problem solving experiences. 

The calculator and tables of squares should be used 

and compared. On the calculator the process of repeated 

squaring leads to the calculator limit very quickly or to zero 

if the number you start with is less than 1. Facility in 

squaring is needed in applications of Pythagoras' theorem 

and also in physics and in a general understanding 

of nature. 

The sequence 1, 4, 9, 16 ... should be as familiar as 10, 20, 

30, 40 and is much more interesting. 

The P units introduce notion of square roots and inverse 

operations. It is not necessarily true that [ill 6ZI[D will give 

you back n exactly. This will depend on the circuitry. 

Nevertheless [ill Ix 2 1 6ZIwill give n back again if n is not too 

large. Results should be compared with those obtained from 

tables. Some discussion of finding square roots without a 

calculator or without 6ZIwill lead to approximating methods 

of which 'divide and average' is simplest. 

Algebraic squares link spatial instinct with algebraic notation 

and the distributive law of numbers. There are many 

interesting patterns and puzzles relating to square numbers. 

The graph of y = x 2 will come later. 

Answers 

M31 A. 

1. 324 

6. 24·01 

2. 1849 

7. 14·44 

3. 3025 4. 12100 

8. 1789·29 9. 26·3169 

5. 27889 

10. 806·56 

26

B. 1. 484 2. 1296 3. 1936 4. 3249 5. 3721 

6. 77·44 7. 56·25 8. 4·84 9. 75·69 10. 98·01 

c. It is true because the squares end in 0, 1, 4, 5, 6 or 9. 

M32 A. 1. 0·09 2. 0·49 3. 0·81 4. 0·36 5. 0·64 

6. 0·7225 7. 0·3969 8. 0·1764 9. 0·0225 10. 0·0064 

B. 1. 0·04 2. 0·16 3. 0·25 4. 0·0484 5. 0·5625 

6. 0·6724 7. 0·8836 8. 0·0324 9. 0·0064 10. 0·0036 

c. 0·2~0·04 0·71 ~ 0·50 0·4~0·16 

0·5~ 0·25 0·63~0·40 

0·3~0·09 0·25~0·06 

M33 A. 1. 49 2. 144 3. 144 4. 324 5. 900 

6. 36 7. 22·09 8. 88·36 9. 156·25 10. 174·24 

B. If you have a calculator without I x 2 1 or brackets you will have to use 

something like ~ [K] la El ~ [±] ~ 5J ~ ~ for the decimal 

squares.* 

1. 25 (appears as 24·999999.) 

2. 74 3. 72 4. 170 5. 452 6. 18·72 

7. 11·45 8. 44·26 9. 79·93 10. 101·7 

(a + b)2 = a 2 + b 2 is only true if a = 0 or b = O. 

c. 1. II III IV V VI VII VIII 

(col lv-col 

VII) 

a b a+b (a + b)2 a 2 b 2 a 2 + b 2 (a + b)2 - (a 2 + b 2 ) 

2 3 5 25 4 9 13 12 

4 7 11 121 16 49 65 56 

5 3 8 64 25 9 34 30 

2·5 4 6·5 42·25 6·25 16 22·25 20 

1·7 2·8 4·5 20·25 2·89 7·84 10·73 9·52 

3·6 6·6 10·2 104·04 12·96 43·56 56·52 47·52 

4·2 0·8 5·0 25 17·64 0·64 18·28 6·72 

2. The numbers in col VIII are twice the product of a and b, e.g. 

(2 x 3) x 2 = 12, (1·7 x 2·8) x 2 = 9·52. This is explained in the next 

section. 

*If you press ~ [K] [;] [±] [ft] [K] [;] you get the square of (a 2 + b). 

27

M34 A. All drawings. 

B. These should be demonstrated by the teacher and related to drawings. 

The actual steps are quite simple, but important. 

C. 1. They are all examples of (a + b)2 where a + b = 10. 

2. (c) should be d + 14c +49. 

3. This exercise shows that one can get by sometimes without using 

a calculator. 

P22 A. 1. 14~ 196 7·6~57·76 33~ 49~2401 

3·7~ 13·69 2-3~ 5·29 8·8~ 77·44 

60~3600 0·83~ 0·6889 0·35 ~ 0·1225 

2. (a) 84100 (b) 8-41 (c) 0-0841 (d) 0·000841 

3. (a) 532900 (b) 53-29 (c) 0·5329 (d) 0·005329 

4. (a) 1444 (b) 144400 (c) 0·1444 (d) 0·001444 

B. 1. 16 2. 5041 3. 4624 4. 1369 

5. There isn't one unless you count 7744. 

C. 1. 44 2. 65 3. 9·3 4. 9·8 5. 22 

6. 20 7. 1·5 8. 0·43 9. 0-66 10. 0-75 

P23 This exercise introduces the new concept of a square root. If a calculator 

with a ) button is not available tables will be needed. The best type are 

Basic Mathematical Tables (Bell & Hyman) which give) together with 

other functions in a form suited to calculator use. Some care will be 

needed where, for example, square roots of 3 figure numbers are to be 

found. The tables will give the square root of 8,3, 83 and 830 but if )835 is 

28 

required we use )8·3 = 2,881, )8·4 = 2·898 so )8·35 = 2·890 (about halfway). 

Thus )835= 28·9. A more complicated method of finding square 

roots using the multiplication tables or 4 function calculator is divide and 

average, taking a rough estimate from the ) tables. E.g. wanted ) n. 

Estimate x. Now follow the sequence [Q] El [K] [±] [K] El [2J ~. This gives 

at least two more decimal places of the square root. The process is 

repeated with the latest answer used for x and so on until a constant result 

is achieved. (This may not be as easy as it sounds.) 

A. 

B. 

1. 7 2. 11 3. 22 4. 40 5. 44 

6. 85 7. 58 8. 92 9. 4·1 10. 7·5 

11. 6-1 12. 5·8 13. 1·3 14. 1·9 15. 1-2 

16. 2·1 17. 0-46 18. 0·85 19. 0·9 20. 0-49 

1. 7 2. 17 3. 21 4. 23 

5. 25 6. 32 (31-62) 7. 57 8. 75 (74-53)

C. 4, 9, 25, 64, 100, 144, 256, 361, 400, 484, 625 

P24 A. Diagrams 

B. (x + 4)2~X2 + 8x + 16 

(x + 5)2~X2 + 10x + 25 

(x + 6)2~X2 + 12x + 36 

(x + 7)2~X2 + 14x + 49 

(x + 8)2~X2 + 16x + 64 

C. 1. (x + 2)2 2. (x + 5)2 3. (x+ 9)2 4. (x + 10)2 

5. (3 + X)2 6. (4 + X)2 7. (7 + X)2 8. (6 + X)2 

E23 A. See also the discussion of the product of negative numbers in IMS 

Book C, Problem Number 17. 

B. See P22 A. 

C. See P22 B. 

D. See P22 C. 

E24 See P23 and ... C 2 

(a) True (b) Not true (c) True (d) True (e) True 

(f) True (g) Not true (h) True (i) Not true 

E25 A. See P24 A. 

B. 1. See P24 B. 2. See P24 C. 

C. 1. (a) x 2 -4x +4 (b) x 2 -8x + 16 

(c) x 2 -10x+25 (d) x 2 -14x+49 

(e) x 2 -16x + 64 (f) x 2 - 20x + 100 

2. Use the diagram for (x + 1)2, but make x the full side of the square. 

We then have x 2 = (x - 1)2 + 2x - 1. The small unit square is 

counted twice in the 2x strips and therefore 1 has to be subtracted. 

This leads to (x - 1)2 = x 2 - 2x + 1. 

D. 1. (a) 56 (b) 117 (c) 180 (d) 66 (e) 1512 (f) 3233 

2. (a + b)2 - (a - b)2 = 4ab 

Note: The table of squares can be constructed from differences 

I 3 5 7 etc. 

1 4 9 16 

So the whole process of multiplication can be converted to 

addition, subtraction and halving. 

29


This unit blends together practical work, measurement, 

geometry and calculations around the topics of right-angled 

triangles and Pythagoras' theorem. The various puzzle-type 

demonstrations of Pythagoras' theorem make very decorative 

work for the classroom and also enjoyable puzzles. The 

Pythagorean triples 3, 4, 5 ... etc., will later provide interesting 

investigations which again link geometry, algebra 

and number theory. 

The generalizations of Pythagoras' theorem are very 

interesting for the faster pupils. Geometrical proofs should 

not be presented at this stage but those children who ask for 

proof can be shown the traditional one with the areas 

moving by stages. There are quite a number of films on 

Pythagoras' theorem. 

Familiarity with right-angled triangles is a basic skill of 

secondary mathematics as many applications depend on it. 

Answers 

M35 A. 2. (a) DC = 31 mm (b) PO = 3·2 em (e) WX =40mm 

BC = 15mm PS = 2·5em WZ =33mm 

BAC = 26° 

PSO = 52° 

XWY= 39·5° 

BCA = 64° 

DAC = 64° 

SQR = 52° 

OSR = 38° 

WYX = 50·5° 

YWZ = 50·5° 

30 

3. There are 5 different sizes of right-angled triangles in this figure. 

Their hypotenuses are 15 mm, 21 mm, 36 mm, 51 mm, 72 mm 

B. 1. (a) 2em 2 (b) 2·1 em 2 (e) 212·5 mm 2 (d) 261 mm 2 

(e) 275 mm 2 (f) 3·4em 2 (g) 162mm 2 (h) 4·2 em 2 

(i) 198mm 2 (j) 536·5 mm 2 (k) 666 mm 2 

2. (a) 4·32em 2 (b) 546 mm 2 (e) 15.62 m 2 (d) 0.3645 m 2 

(e) 1400 em 2 (f) 53·2 em 2 (g) 4860 m 2 (h) 5·7575 km 2 

c. 1. (a) 50° (b) 58° (e) 62° (d) 46° (e) 49° 

(f) 30° (g) 39° (h) 58° 0) 60° ( j) 47° 

2. (a) Yes (b) No (e) Yes (d) No 

(e) Yes (f) No (g) Yes (h) Yes 

3. (a) BAD = 53° DAC = 37° 

(b) BAD = 51° BOA = 90° BCD = 39° 

(e) BAD = 18° BOA = 90° DAC = 18° 

(d) BCD = 45° BOC = 90° DCA = 40°

M36 A. 1. Drawing to confirm Pythagoras' theorem. 

2. Checking drawings 

3. Note that (a) If A < 90°, BC 2 < AC 2 + AB 2 

(b) If A> 90°, BC 2 > AC 2 + AB 2 

B. 1. (a) 4·47 em (b) 5·83 em (c) 7·21 em (d) 4·95 em 

2. Use b 2 = c 2 - a 2 or a 2 = c 2 - b 2 

(a) a = 9·75 em (b) b = 8·66 em (c) a = 18·3 em 

(d) b=93·67cm (e) b=27·2mm (f) 16km 

M37 A. 1. 5·83 m 

2. Yes, the base will be 4·5 m from the wall 

3. 5·6m 

B. 1. 4·7 m, 4·3 m 2. 21·6 m 

3. Roughly 88 m 

C. 1. 5·83 units 2. 4·24 units 3. 3·6 units 

5. 3·6 units 6. 6·4 units 7. 5 units 

4. 1·41 units 

8. 6·3(25) units 

P25 A. 1. c = 3·97 em so area = ~ac = 8·93 cm 2 

2. C = 48° 3. No, the third angle is 80° 

B. 1. DAX = 55° 

ADX = 35° ABY = 55° 

XDC = 55° YBC = 35° 

XCD = 35° BCY = 55° 

2. The angles are either 90°, 65° or 25° 

3. BAD = 30° DAC = 20° ADC = 90° 

4. ABD = 45° DCA = 43° 

4. 9·22 em 

C. 1. True 

2. Not true except in special cases where the angles are 45°, 45° and 

90° 

3. True 4. True 

P26 A., B. and C. practical work 

D. 1. (a) 8·06cm 

2. 4·39 m 

3. (a) 5·385 units 

(b) 

(b) 

38·01cm 

2·83 units 

(c) 6·74 em 

(c) 3·61 units (d) 5·83 units 

E26 A. See P25 A. 

31

B. See P25 B. and 5. CDA = 90° 

CAD = 50° 

DAB = 40° 

ABD = 50° 

6. DCA = 54° 

DAC = 36° 

DBC = 54° 

C. 1. (a) They are congruent (b) Rectangle (c) All equal 

(d) Equal (diagonals of a rectangle) 

2. Following question 1, the middle point of AC is the same distance 

from A, Band C. So the centre of the circle through A, Band C is 

the mid-point of AC. 

3. ~ABX and ~BXC are 45°, 45°, 90° triangles, so ~ABC also has two 

angles of 45° and B = 90° 

E27 A., B., C. all practical work 

D. 1. Yes. Heights of the triangles are 

J3 J3 J3 

-xa -xb -Xc 

2 ' 2 ' 2 

. J3 J3 J3 

SO their areas are 4a2, 4b2, and 4C2 

Since the triangle is right-angled a 2 + b 2 = c 2 

Thus J3 a2 + J3 b2 = J3 c2 

444 

2. Not so simple, depending on the size of right-angled triangle you 

start with (3, 4, 5 is easy) 

3. Pythagoras' theorem extends to both hexagons and semi-circles 

for the same reasons as given in question 1 

32 


This unit approaches the trial and error method of solving 

problems. The salient feature of this method is that the error 

is used to arrive at a better guess. This will be developed 

later into numerical methods of finding solutions that use 

iteration. Further exploration of this is to be found in IMS 

Book C, Problem Number 28 (Iteration). Children should 

not be discouraged from 'guessing' but instead, encouraged 

to scrutinize their guesses and use them to approach an 

answer. They should also be encouraged to be interested in 

the processes by which they themselves attempt to solve 

problems.

Answers 

M38 The answers are less important than the process. The first number is 

gradually built up to the second, noting what has been added. 

A. 1. 128 + 300 + 40 + 7 Check 128 + 347 = 475 

write 300 40 7 

Total 428 468 475 

2. 23 + 100 + 70 + 4 

3. 245 + 800 = 1045 (too much by 4) 

245 + 796 = 1041 

4. 390 + 10 + 328 (tidy up the hundreds first) 

390 + 338 = 728 

5. 3·4 + 3 + 0·7 --) 

6. 5·9 + 0·1 + 6,--) 

7. 8·3 + 3 + 0·3 --) 

8. 14·4 + 0·6 + 85--) 

9.5·85+0·15+11--) 

10. 60·47 + 0·53 + 9 + 50--) 

11. 3852 + ,8+ 40 + 100 + 1283J 

3·4 + 3,7= 7·1 

5·9 + 6·1 = 12 

8·3 + 3·3 = 11·6 

14·4 + 85,6 = 100 

5·85 + 11·15 = 17 

60·47 + 59·53 = 120 

3852 + 1431 (adding by calculator) = 5283 

12. 10·45 + 18·9 = 29·35 

(add 19 and subtract 0·1) 

B. Similar methods to A (or direct subtraction) 

1.40-10-3} 

40 -13 = 27 

or40-27=13 

2. 47 3. 3·05 4. 5·7 5. 62·5 

7. 371 8. 1·15 9. 26·9 10. 2·132 

12. 2·228 

C. 1. m is somewhat less than 20, try 18 

2. 12 x 10 will be too large; try 12 x 9 

6. 58 

11. 2229 

3. 8 x 100 will be too large, try 8 x 90, then 8 x 89 

4. 22 x 40 = 880, this leaves 44 so try 22 x 42 

5. 42 x 100 is too big, 42 x 90 is too big (subtract 84) 42 x 88 

6. 1·7 x 10 is much too big. Try 1·7 x 7 too big by 0·34. Try 1·7 x 6·8 

7. We know 42 -;-7 = 6 so try 0·77 x 60. Too big, subtract 42·35, rem. 

3·85. Try 0·77 x 54 = 41·58, add 0·77. Try 0·77 x 55 

8. 9 x 6 = 54 so try 9·1 x 5·9. Well done! 

9. 741. We know 25 x 3 = 75 so try 241 x 2·9. This gives 698·9 

(1·1 too small). Try 241 x 2·905 ... near enough 

10. 10 is too small so try 3·65 x 11. Too small by 1·85, so try 

3·65 x 11·5,error is 0·025 so try 3·65 x 11·508.Too large by 0·0042 

so try 3·65 x 11·507. Error 0·00055 so try 3·65 x 11·5068, etc. 

33

11. Try 7·0 error 0·021, about 0·05 x 0·4. Try 0·403x6·95= 

12. 0·1 will be too large: try 0·09 etc. 

D. Can be solved by straight division (see subtraction) 

1. 9 2. 8 3. 18 4. 9 

5. 55 6. 71 7. 28 8. 28 

9. 8·5 10. 0·47 11. 9·6 12. 96 

M39 A. 1. x = 1 2. x = 4·5 3. x = 3 4. x = 0·75 

5. x = 2 6. x = 5 7. x = 6 8. x = 1·5 

9. x=2 10. x=0·5 11. x=3 12. x=1·9 

Of course ax + b = c may be solved by the calculator sequence [Q] B 

[hJ El [ill EJ but the object of the exercise is to improve guessing. 

B Solved by [Q] [±] [hJ El [ill EJ 

1. x = 6 2. 7·5 3. 9 4. 8 

5. 

5 

3 = 1·6666 6. 

16 

'3 = 5·3333 7. 

5·5 

3 = 1·8333 8. 

4 

3 = 1·3333 

9. 2 10. 2·6 11. 3·4 12. 1·3 

C. 1. 1·2857 2. 2 3. 4·25 4. 1·5 5. 2·125 

6. 2·057 7. 0·1333 8. 0·3 9. 5 10. 12 

11. 22·857 12. 27·5 

M40 A. 1. 5 cm x 15 cm 2. 8 cm x 12 cm 3. 3 cm x 17 cm 

4. 3 cm x 17 cm gives an area of 51 cm 2 while 4, 16 gives an area of 

64 cm 2 • It is a good idea at this point to plot a graph of shortest 

side against area. You can then read off the side length which gives 

an area of 60 cm 2 • Otherwise use a trial and error method. 

3·6cm x 16·4cm has area 59·04cm 2 while 3·7 x 16·3 has area 

60·31 cm 2 • Try 16·32 x 3·68 

5. 5·53cmx14·47cm 6. 10cmx10cm 

7. 3·3cm x 16·7cm 8. 2·2cm x 17·8cm 

34 

B. 1. 25cmx25cm 

2. (a) They lie on a curve shaped like a circle inside out. This curve is 

ca IIed a Hyperbola. 

(b) The square is the largest rectangle. 

C. 1. B = 60° 2. 23·65 cm x 6·35 cm 

3. This can be done by folding as follows. 

(i) Estimate one-third of the strip and make a fold F,. Fold the 

other end of the strip into F, and make a fold F 2 • Fold the first

end into F 2 and make a fold F 3 , etc. The folds F 1 , F 2 , F 3 ••• will get 

nearer and nearer to Exactly One-Third of the strip. (The error 

is halved each time.) This method can be used to fold a strip 

into exactly 7 parts or 13 parts or any other prime number. 

4. 17·32 

5. 7 is a bit too big. Try 6·5, too small (274·625), try 6·7 ... just too 

large (300·763), try 6·69 ... too small (299·418).6·7 is a satisfactory 

answer. 

6. Examples can be found in a dictionary, e.g. abacus, abdicate 

... etc. 

P27 A. 1. 42 mm 

2. There will be a number of lines which reduce AX + BY + CZ but 

the best one will be the line passing through A and C. 

3. For this line AX + BY + CZ is about 30 mm 

B. Drawing exercise 

C. Exact to 3 decimal places 3·302 

1. (a) 2·714 (b) 3·684 (c) 10 

2. Yes 

D. 62500m 2 

E. The shortest route is found by using reflections of X and Y in the river. 

x 

X' 

The line from X' to Y is the shortest possible one. Therefore X~ P~ Y 

is the shortest route. 

F. Compare results to see who can swap the counters in the smallest 

number of moves. 

E28 A. See P27 A. 

B. 1. The line should pass through the two points which are farthest 

apart. (Can you see why this will give the shortest distance from 

the third point to the line?) 

35

2. Not usually true 

3. Not true 

C. 1. The diagonals 

2. Can only be found by trial and error. Mark the 4 points in ink and 

explore with pencil lines, rule out the line each time you find a 

better fitting one. 

E2Sa A. 1, 2, 3 see P27 C. 

4. The suggestion is true for all numbers 

B. 1. 250 m x 250 m = 62 500 m 2 • 

2. Your drawings should approximate to (a) 72 168 m 2 

(b) 75 445 m 2 (c) 79 577 m 2 

C. 1. See P27 E. 

2. (i) and (iii) can be drawn, (ii) and (iv) cannot 

3. The shortest route varies according to his starting point; starting 

from A or E he can visit all 5 towns by travelling only 31 km, 

although he will not get back to his starting point. 

36 


Powers are a simple concept when a calculator is available. 

They lead to important generalizations through experience. 

It has been shown that the laws of indices can be used and 

understood by 8-year-olds, hence the topic should not be too 

difficult at this stage. The first problem is to clarify the 

language and symbolism. It is not hard to understand why 

children get confused between x 2 and x2 or 3 2 an

Answers 

M41 A. 1. 

5. 

9. 

13. 

Some of the facts which should appear from exploration 

are: 

(i) xn~ 00 as n~ 00 if x> 1 

(ii) x n ~ 0 as n ~ 00 if x < 1 

(iii) XO = 1 

(iv) x m • x n = x m + n 

(v) (xmt = x mxn = (xn)m 

4 2 2. 7 2 3. 52 4. 6 2 

(2·5)2 6. 9x9 7. 10x10 8. 12 x 12 

8 3 10. 11x11x11 11. 53 12. 6 3 

(4'5)3 14. (2·8) cubed 15. 31 x 31 x 31 

B. 1. 

6. 

11. 

C. 1. 

2. 

3. 

8 2. 9 3. 125 4. 64 5. 1000 

2·25 7. 484 8. 4913 9. 18·49 10. 1936 

175·616 12. 0·64 13. 0·343 14. 0·04 15. 0·027 

6 2 7 2 8 2 9 2 10 2 

36 49 64 81 100 

11 13 15 17 19 ~ differences 

20 2 21 2 22 2 23 2 . .. etc. 

400 441 484 529 

41 43 45 

1.1 2 1.2 2 1.3 2 1.4 2 . .. etc. 

1·21 1·44 1·69 1·96 

0·23 0·25 0·27 ... etc. 

4. (a) Cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 

(b) 7 19 37 61 91 ~ differences 

12 18 24 30. . . etc. 

666 

(c) The 6 times table appears. Differences of this are all 6s. 

M42 A. 1. (a) 7 7 = 823543 (b) Also 7 7 in fact 7 8 -T 7 (c) Also gives 7 7 

2. (a) 3 8 (b) 3 16 (c) 3 8 = 6561 

3. (a) (1·6)10 (b) (1·6)18 (c) (1·6)10 

4. (a) (0·9)12= 0·2824295 (b) (0·9)12 (c) (0·9)16 

B. Some calculators will set in the constant multiplier automatically. 

Others will require a double press on the [K] button. There are also 

other systems. This should be checked before starting the exercise. 

1. See above 

2. (a) 20·4, 27,2, 37·4, 54·4, 27·88 

(b) 2401 (c) 32 768 (d) 15·625 (e) 0·03125 

37

(f) 0·43046721 (9) 131621 7 (h) 2-0121964 

3. Same questions as 2 

C. 1. True 2. True 3. Not true 4. True 

5. Not true 6. Not true 7. Not true 8. True 

M43 A. 1. 7 5 2. 2 8 3. 4 10 4. 59 

5. 11 4 6. 16 11 7. 23 9 8. 99 18 

9. (0·4)5 10. (0·8)10 11. (0.36) 10 12. (0-47)21 

B. 1. Yes 2. Yes 3. Yes 4. Yes 

5. Yes 6. Yes 7. Yes 8. Yes 

C. 1. True 2. Not true, (a 2 )3 = a 6 3. True 

4. Not true (a 3 )3 = a 9 5. True 6. True 

P28 A. 1. 27 2. 256 3. 3125 4. 0·25 5. 0·512 

6. 2·197 7. 2·8561 8. 5·832 9. 2·0736 10. 1-61051 

B. 1. 10 % of 50000 is 5000 so population after 1 year is 

55 000 = 50 000 x 1-1 

2. 10 % of 55000 is 5500, so population after 2 years is 

60 500 = 50 000 x (1.1 )2 

3. 10% of 60500 is 6050, so population after 3 years is 

66 550 = 50 000 x (1.1 )3 

C. 1. Use constant multiplier 

(1·1)3 = 1·331 (1·1) 7 = 1·949 

(1.1 )4 = 1.464 (1·1)8 = 2.143 

(1-1)5=1'610 (1·1)9 = 2.358 

(1·1)6=1·771 (1·1 )10= 2.593 

2. (1·1F 3_ Between 7 and 8 years. 

D. Use constant multiplier 

1. 4 2. 5 3. 6 4. 6 5. 4 

6. 6 

P29 A. All true 

B. 1. Not True 2. True 3. Not True 4. True 

5. True 6. True 7. Not true 8. Not true 

C. 1. The sequence is [m [KJ ~ ~ [KJ~. Your sequence may be 

a a 2 a 3 a 6 

different if your calculator has a different mechanism for constant 

multiply. 

(a) 4096 (b) 1838·2656 (c) 3·635215 

(d) 0·007256 (e) 0-001838 (f) 0·94148 

38

2. Use [ill [K] ~ ~ [K] ~ [K] ~ but see question 1. 

a a 2 a 3 a 6 a 12 

(a) 5·350 (b) 

3. (a) 2·1828746 

(d) Correct 

4. (a) True 

(e) Not true 

2·252 (c) 1·601 

(b) 23·2 (not 232) 

(e) Correct 

(b) True (c) True 

(f) True 

(d) 8·064 

(c) 0·0387595 

(f) Correct 

(d) .Not true 

E29 A. 1. 

2. 

3. 

B. 1. 

3. 

C. 1. 

3. 

4. 

5. 

6. 

7. 

(b) They form the x 6 table (c) They are all 6 

The patterns continue 

No, the 4th differences are all 24, the 3rd differences are the x 24 

table 

See P28 C. 1 

See P28 B. 1,2 and 3 

30 (= 1 2 + 2 2 + 3 2 + 4 2 ) 

36 (= 1 + 8 + 27) 

2. (1.1 )7 

4. Between 7 and 8 years 

2. 55,91,140,204 

100 (=1 3 +2 3 +3 3 +4 3 ) (Note: This also equals (1 +2+3+4)2, 

see front cover of the Book A2 or B2.) 

The numbers are (2 3 )2 = 64, (3 3 )2 = 729, (4 3 )2 = 4096 

x Y is a power function. [ill IxyllliJ ~ gives a b on the display 

(a) 1024 (b) 17·576 (c) 0·4096 (d) 3·583 

E30 A. See P28 D. 

B. See P29 A. 

C. See P29 B. 

D. See P29 C. 1,2,3 

E. The result is am -;- an = a m - n (a) 32 (b) 52 (c) (1·7)4 (d) (2·3)4 


This is a straight unit on careful use of calculator. There are 

many possibilities of error and these can usually be avoided 

by improving technique. The work is not unrelated to the 

beginnings of computer programming which will interest 

many children. Once again various calculators will have 

different methods of working and this should be discussed 

with the class. If someone has a programmable calculator in 

class this will have a more subtle way of doing the calcula- 

39

tions. A calculator with a statistics circuit will add a collection 

of numbers in the memory without the (±] key. Just press 

in each number and M + or 'x'. The total will be recovered 

on pressing LX. It will also count how many numbers have 

been added. This can be obtained by pressing ill], 

There are no P or E exercises with this Unil. 

Answers 

M44 A. 1. 331·93 2. 1444·86 3. 4107 4. 3·290 

B. 1. £55·70 2. 33·0875 cm 3. 550·4444 g 4. 672·7 ml 

C. 1. Correct 2. Correct 3. Correct 4. Should be 21·48 

D. 1. 330 2. 4·3 3. 0·3 4. 53·4 

M45 A. 1. 6·8660865 2. 344·13984 

3. 1·0267781. This may come out to 1·0267776 in reverse because of 

rounding off errors 

4. 0·2835144. This may come out to 0·28275 in reverse 

B. 1. 4, (14 -7- 7) x 2 

2. 2·0681818, (2·6 x 3·5) -7- 4·4 

3. 616, (44 -7- 0.8) x 11·2 

4. 4·6882978, (5,65 x 3·9) -7- 4·7 

C. The calculation can be treated as 'straight through' if you can ignore 

the brackets and get the same answer as with the brackets. For 

example (2 x 3) x 6 is the same as [2] ~ ~ ~ [ill. 

1. 14 -7- (2 -7- 3) = 14 -7- (0·666666) = 21 [I] @] B [2] B ~ ~ gives 

2,333333, so this is not a straight through calculation 

2. 5·733333, not straight through. 

3. 0,0009, straight through 

4. 1067·7333, not straight through 

M46 A. 1. 

2. 

3. 

4. 

5. 

6. 

2464 @] [I] [±] I]] [ill ~ ~ [lJ. The change of order is possible 

because a x b = b x a 

76, straight through 

18·395, straight through 

46·64, change order 

71, change order 

0·0105, change order 

40

B. 

c. 

7. 26·75, straight through 

8. 1·0231481, straight through 

If you have a calculator with brackets you can use those instead of 

memory. 

1. 176 2. 28·53 3. 93·6 4. 6·9384 

5. 8·1374092 6. 1·475 7. 3·0425531 8. 0·7430703 

1. 5·92 2. 0·0175 3. 23355 4. 3·6644736 

5. 482 

Answers 

M47 A. 


The topic of area is made very easy by the use of the 

calculator. The child must be sure that moving boundaries 

will not change the area they enclose and must have the 

concept of conservation of area before he can tackle 

trapezia, etc. There are many applications in physics, 

geography and mathematics itself. In the non-school world, 

area is, of course, of vital importance in the measurement of 

land, floor coverings, etc. though these matters may not 

light too much flame of interest in the pupils. They may be 

more interested in air resistance (getting your head down 

when cycling or even running), sailing and windmills, space 

per child in the classroom and the four-colour map problem. 

There are ancient mathematical puzzles which involve 

areas ... the proof of Pythagoras' theorem, the fact that the 

side of a square whose area is 2 cm 2 cannot be measured with 

any ruler, the fact that a circle and a square cannot have the 

same area. Finally, areas will be used to explain other things 

in mathematics so that it is most important that the fundamental 

ideas are grasped at this stage. 

1. 253 mm 2 (2·53 cm 2 ) 2. 810 mm 2 3. 320 mm 2 

4. 400 mm 2 5. 640 mm 2 

6. 12 cm 2 (can be calculated as a 5 x 5 square with a 2 x 2 square and 

a 3 x 3 square cut away). 

B. 1. 10cm 2. 12·5cm 3. 6~ cm 4. 16~ cm 

6. 26·3 cm 7. 45·45 cm 8. 2·86 cm 9. 21·7 cm 

11. 2·3cm 12. 250cm 

5. 22·2cm 

10. 9·3 cm 

41

C. 1. 112 cm 2 2. 6 cm 2 3. 1·62cm 2 4. 6 cm 5. 6·66 

6. 4·88 m 7. 7·5 m 8. 2·5 cm 9. 0·11 cm 10. 72 cm 

(Note: These answers could be written in terms of different units, e.g. 

1. 1120 mm or 1·12 m.) 

3. 2cm 2 4. 1·875cm 2 

M48 A. 1. 3cm 2 2. 2cm 2 

5. 2 cm 2 and 6 cm 2 6. 450 mm 2 

5. 3·75 cm 2 6. 3·75 cm 2 7. 3cm 2 8. 4 cm 2 

B. 1. 17·5cm 2 2. 26·25mm 2 3. 10·4m 2 4. 210 mm 2 

5. 0·14 m 2 or 1400 cm 2 

6. 8'8cm 2 

C. 1. 50cm 2. 4m 

3. 2·94cm 4. 4·59cm 

2. 4·5cm 2 3. 6cm 2 4. 2·5cm 2 

M49 A. 1. 4cm 2 

C. 1. 7cm 2 2. 12 cm 2 3. 20cm 2 4. 12cm 2 

5. 3cm 2 6. 6cm 2 

B. 1. 1·5cm 2 2. 3·5cm 2 3. 3cm 2 4. 3·375cm 2 

5. 3·125cm 2 6. 3·2cm 2 7. 6'9cm 2 

P30 A. 

B. 

1. 6·9cm 2 

1. (a) 8·46 cm 2 

2. (a) 3·03 cm 

3. (a) 12·1cm 

2. 12cm 2 

(b) 9 cm 2 

(b) 1·7cm 

(b) 2 cm 

3. 11·3cm 2 

(c) 20·46 cm 2 

(c) 16·66 cm 

(c) 0·34cm 

4. 7·5cm 2 

(d) 14·21 cm 2 

(d) 5·7 cm 

(d) 6·57 mm 

P31 A. 1. 533 cm 2 , 4467 cm 2 2. 5·3 cm 

3. Each equilateral triangle has area 10·825 cm 2 • So estimated area of 

circle is 6 x 10·825 = 64·95 cm 2 

Error 13·59 cm 2 

4. They should all be correct. The rule is very accurate. 

42 

B. 1. (15 x 4) x 3 plus 2 x 4 x 4 x 0·433 (see question 4 of Exercise 

A) = 193·8 cm 2 

2. 15·5 cm 2 • This is quite a difficult trapezium to draw. Start off with 

the isosceles triangle 4 cm, 4 cm and 2 cm. The rest is easy 

3. Areas: rectangle 588 mm 2 , bows 63 mm 2 , stern 45 mm 2 . 

Total 696 mm 2 • Since 1mm ~ 1 metre, 1mm 2 ~ 1m 2 • 

Total deck area is 696 m 2 

C. 1. Areas 6ABC = 999 mm 2 , .6ADE = 426 mm 2 , .6AFG = 952 mm 2 

~BCED = 327 mm 2 , ~DEGF = 526 mm 2

2. Find the area by dividing into rectangles and right angled 

triangles. The double dimension enlargements will have four 

times the area of the figure you started with in each case. 

(a) Areas (i) 1430 mm 2 (ii) 680 mm 2 

E31 See P31 and C.2 (iii) 10·30 mm 2 

E32 A. 1. (a) 17'32cm 2 (b) 24cm 2 (c) 17·1 cm 2 (d) 27·7cm 2 

2. All the results should agree to 1 decimal place. 

B. 1. The largest will have one corner in a 

corner of the square with a diagonal of 

the square as an axis of symmetry. 

Area = 46·4 cm 2 

2. The largest square in an equilateral triangle is hard to find. Draw 

some of the rectangles which sit on the base and calculate their 

area. This will have a maximum at the square. Another method is 

to draw as large a rectangle as you can. Multiply the sides and 

take the square root of the product. This will give the approximate 

length of the side of the square. A large square can be drawn with 

3 corners on the three sides and the fourth corner on the axis of 

symmetry, but it will not be as large as the largest square which 

sits on the base. 

C. This is a fascinating puzzle. The secret is that the rectangle is not 

exact. There is a thin space along the diagonal which accounts for the 

extra cm 2 • 


This unit explores the area of a triangle, developing from 

!base x height to !(length of side) x (altitude standing on 

the side). People learning mathematics tend to be too 

43

inflexible about configurations and this work emphasizes 

that the notion of 'base' is only a practical arrangement for 

identifying one of the sides. Thus a more dynamic view of a 

triangle is encouraged with the children feeling they have 

more control. 

Altitudes are easy to draw accurately and the concurrence 

of the altitudes is an interesting and curious property of 

triangles that is quite unexpected. Why should they meet at 

a point? It is especially striking where the triangle contains 

an obtuse angle and the altitudes meet outside the triangle. 

This is much more subtle than symmetry and can arouse a lot 

of mathematical interest in children. It can also improve the 

drawing and measuring skills. 

Answers 

M50 A. Side Length Altitude Length Product 

from 

1. AB 43mm C 29mm 1247 

BC 40mm A 31 mm 1240 

CA 34mm B 38mm 1292 

2. AB 45mm C 59mm 2655 

BC 61 mm A 44mm 2684 

CA 68mm B 39mm 2652 

3. AB 62mm C 46mm 2852 

BC 53mm A 55mm 2915 

CA 60mm B 48mm 2880 

The products are roughly the same for 

the given triangle. The product is the 

area of a rectangle in which the triangle 

would fit. 

B. 1. 8cm 2 2. 7·875cm 2 3. 9cm 2 4. 10·5cm 2 

5. 4cm 2 6. 5·5cm 2 

M51 A. 1. When C is above the mid-point of AB 

2. Always the same length, simply the distance between the two 

parallel lines 

3. The area is always the same (it is always (base x height) -;- 2) 

44 

B. 1. 6s AQB, ARB, XQY, XRY and XSY are all equal to 6APB

M52 1. 2cm 2 

2. 6.PAO = 6.PBO = 6.PXO, 6.PXR = LPAR = 6.PBR, 

6.OBR = L OXR = 6.OYR = 6.OAR, 

6. RXS = 6.RYS, 

6. BOX = L BRX = 6. BPX, 6.OXS = 6. OYS 

6. 2cm 2 2. 3 cm 2 3. 2 cm 2 4. 5 cm 2 

7. 2·25cm 2 8. 3·75cm 2 9. 4cm 2 5. 5cm 2 

P32 

A. 1. 

B. 1. 

2. 

3. 

Drawing 2. Drawing 

Drawing 

Yes, the altitudes should meet at a point 

Yes, the altitudes will meet at a point (outside the triangle) 

P33 

A. 1. 

3. 

B. 

231 km 2. 2·8 m 

You construct each triangle with compasses. Start with 6.ABC. 

Areas are LABC 770 m 2 6.ACD 680 m 2 LADE 774 m 2 

Total area = 2224 m 2 

1. True 

2. (a) True 

3. Not true 

(b) True 

4. 

(c) True 

Not true 

5. True 

C. This is the formula Area = abc/4R and it works for all triangles. This is 

especially easy if you want to find the areas of a family of triangles all 

drawn with their vertices on the same circle. Area = 800 mm 2 • 

E33 A. 2. The figure made up of P and the three arc intersections '" '2 and x 

is a rhombus. It has 4 sides equal because that is how you have 

drawn them. Therefore PX and AB are diagonals of the same 

rhombus and must be perpendicular. 

3. Drawing 

B. 2. (b) is true 

3. The altitudes still meet at a point but outside the triangle 

C. Find points " and '2 on the line, an equal distance from P. Now draw 

two circles with " and '2 as centres and with the same radius. Choose 

the radius large enough for the two circles to meet above and below 

the line. The points '" '2 and the two points of intersection of the 

circles will form a rhombus. 

E34 A. B., C. see P33 

D. 1. Measurement 2. All th ree suggestions are true 

45


Many problems can be simplified by some sort of listing 

process and this unit considers a number of such problems. 

The listing either solves the problem completely (as in 

probability questions) or helps towards a solution by identifying 

the nature of the operations involved in the problem. 

This work is an introduction to programming by means of 

flow charts and so the type of numbering utilized in BASIC 

is used here. Children enjoy this type of work very much and 

it can be developed to involve problems devised by the 

pupils themselves. There are bound to be some computer 

enthusiasts present who will already know the fundamentals 

of programming and these can be brought into the discussion. 

The idea of a loop (and decision box) is introduced in the 

second half of the unit but is kept fairly simple. You may 

wish to introduce analogous programme language at this 

point in which case you should refer to the BASIC book 

being used in your school. (Or use Eagle: An Introduction to 

Basic, published by Bell & Hyman.) 

Flow charting is a very helpful method of organizing 

thought and will have applications in many of the traditional 

activities of mathematics such as proving theorems and 

solving equations. In addition flow charting problems on 

numbers can give valuable insight into many properties 

which are part of number theory. 

Answers 

M53 A. 

B. 

1. Run, jump, fall 2. Get on, buy, get off 

3. Homework, breakfast, school (7) 

4. Study, learn, pilot 5. Choose, insert, play 

6. Add fat, heat, add egg, eat 

7. Cut bread, butter, marmalade, eat 

8. Think, double, subtract 

9. Think, double, add 7, subtract 4 (others possible) 

10. Draw line, choose A, compass at A, mark off length 

Just as A but number lines, 1~, 2~, 3~, etc. 

2. (a) 05 (b) 15 or 25 (c) 35 

(e) 25 or 35 (f) 25 (g) 25 

(i) 45 (j) 45 

(d) Could be 15 or 35 

(h) 25 or 35 

46

C. 1. Should give 5 more than the number you started with 

2. Should give the perpendicular bisector of AS 

3. Should give a rhombus 

4. 6 lines 

M54 A. 1. (a) Depends on the letters, e.g. E, N, O~ ONE, CONE 

M55 A. 1. 

x 2 -1 -'-10 

(b) For example 9~ 81~80 ~ 8 (Answer should be one 

less than the number you started with) 

(c) For example 1~~difference 198 -T 9 = 22 remainder O. 

The remainder is always zero 

B. Different answers are possible. 

C. Different answers are possible though the order is strict, for example 

stir with a spoon could not come first 

No 

Catch bus 

2. 

3. 

Join school 

team 

47

4. 

Borrow from 

friend 

B. 1. 

Put in sugar 

and stir 

2. Could be quite a long process. Don't forget to tryon the second 

shoe of a pair. 

3. You have to avoid over-watering the plant 

4. Remember that the 27 s have to be counted as well as subtracted 

5. In this chart you will have to find a way of getting from one 

number to the next 

C. 1. Pumping up a bicycle tyre 

2. Bathing a baby. Always put the cold water in first to avoid the risk 

of scalding the baby. 

3. This is a programme for using a calculator and checking as you go 

along 

48

P34 A. 1. (a) The years are counted from 0, the birth of Christ. Different 

countries and faiths number the years differently. Thus 1984 is 

1404 in the Muslim calendar and 5744 in the Jewish calendar. 

(b) The minutes are counted, grouped into 60s. Each 60 is 1 hour. 

This system of counting time goes back to the Babylonians 

(many thousands of years ago) 

(c) Homes are counted by numbering houses or flats 

(d) Clothes are numbered in sizes. They are also numbered by the 

manufacturer. 

2. (a) All the names with the same first letter are grouped together. 

These groups are ordered alphabetically. Then the second 

letter is used to order the names within a group. Then the third 

letter is used. People with the same name are grouped by 

initials ... See telephone directory or voters' list. 

(b) Similar to names but without initials 

(c) Letters of the alphabet have been used in the registration 

numbers of cars. Usually the first three letters are a code for 

the district in which the car was first registered. The last letter 

can tell the year of manufacture, e.g. X~ 1982. 

(d) Letters are used to describe how good or bad a pupil is on a 

report. 

3. (a) So that each user has his own telephone number 

(b) So that you can find your place 

(c) So that no-one else can take your savings 

(d) So that one aircraft does not get mixed up with others (and 

other reasons) 

(e) So that notes can be checked 

(f) So that you can tell where they are going 

B. 1. 30 2. 28 3. 19 4. 36 5. 30 

6. 19 7. 50 8. 15 

C. 1. No remainder, the number is divisible by 11 

2. Different remainders, the number is not divisible by 11 

3. No remainder because n + 1 is a factor of n 2 - 1 (The other factor 

is n - 1) 

P35 A. If the flow charts are satisfactory they will work on a 'test run'. 

B. 1. Different answers 

2. (a) and (b) are examples of the equation ax + b = c where a, band 

c are numbers. The steps are 

1fll calculate c - b = m 

2fll calculate m -T a = n 

49

3~ write down x = n 

(c) is an example of x 2 + b = c where band c are numbers. The 

steps are 

1~ calculate c-b=m 

2~ is m>O 

3~ Yes: x=)m 

4~ No: x 2 + b = c cannot be solved 

The following flow chart would solve the equations by trial and error 

(a) 

(b) Start with x = 1·5, d = 0 and change x to x - d/2, and find 

(3x - 2) - 1·8 = d 

(c) You need a flow chart for )3, see question C2, or use tables or 

calculator [2] button. 

C. 1. (a) The flow chart needs to try all the primes in turn up to 11. 

These are 2, 3, 5, 7 and 11. When 11 has been tried, and shown not 

to be a factor of 149 the flow chart should come to an end. 

(b) 36 is square and also a triangle number 

(c) Different answers 

50

E35 A. See P34 A. 

2. (a) (b) The flow charts are stupid ways of coming to decisions. 

There is no guide to the choice based on the results of n -;-X, or 

reading first line, so it is not worth following these procedures. 

B. 1. (a) 30 (b) 28 (c) 19 (d) 36 (e) 30 

(f) 19 (g) 50 (h) 15 (i) 32 (j) - 7 

2. (a) 50, 100 (b) 48, 98 (c) 31,61 (d) 60,120 

(e) 46, 86 (f) 49, 194 (g) 56, 1 

(h) -1 5, -160 (i) 512, 524288 (j) -37, -182 

C. See P34 C and 

4. n 2 - 1 has two factors (n + 1) and (n - 1) 

E36 A. There will be various answers to these. The process for 5 is to multiply 

both sides by 100 ... 

x = 3·747474 ... 

1OOx = 374·747474 and then subtracting to get 

99x = 371 

371 

so x=- 99 

6. Use tables or [2J to find In, otherwise use a divide and average 

method.* 

7. Use a gradual approximation process. Start somewhere between 

2 and 3, since 2 3 = 8 and 3 3 = 27. Perhaps 2·2 is a good starting 

point 

(2·2)3 = 10·648 (too small) (2·35)3 = 12·977 (too large) 

(2·3)3 = 12·167 (too small) (2·34)3 = 12·813 (too large) 

(2·4)3 = 13·824 (too small) (2·33)3 = 12·649 (too small) 

2·338 ... etc. 

(Answer: 2·3386049 to 7 decimal places) 

B. See P35 B, also (d) use a flow chart that considers x(x + 1) each time 

C. Check flow charts by giving them a Itest run' (see P34 C. 2) 

* See Basic Mathematical Tables. 

51


Answers 

M56 A. 

B. 

Recurring decimals should be a familiar phenomenon to 

most of the children by now. In this unit we make use of 

them to strengthen concepts of fractions as well as to 

investigate the recurring patterns themselves. Mathematically 

they are a very useful tool and one can demonstrate 

the existence of irrational numbers by simply constructing 

decimals that will never recur. Some examples are 

0·121122111222. .. or 0·1234567891011121314.... Later 

this technique can be used with binary numbers. Most 

important is that this investigatory unit removes fear of 

division; once the patterns are understood there is nothing 

to fear. There are also beneficial side-effects such as the 

discussion of the difference between 2 and 1·9999 ... or the 

problems that arise from approximating 0·16666 to 0·16 or to 

0·17. There is some encouragement for the use of fractions 

in their ratio forms, the advantages can be seen; division by 

9 is seen as an interesting rather than difficult process, while 

division by 99 can be found both amusing and interesting. 

This unit also highlights some of the limitations of the 

calculator, for instance the repeated pattern of 1/13 comes 

at the end of the display. Multiplication tables show up the 

remainder better than the calculator and use of these should 

be encouraged. 

This unit is more about numbers than their applications 

but this does not mean the work is less important. Numbers 

are the foundation of mathematics and any understanding 

gleaned in this unit will be of great value later on. There are 

also simpler concepts which are reinforced here such as the 

process of division, remainder and place value. Remember it 

is not always helpful to view concepts only from below. A 

feeling for place value comes more from this sort of investigation 

than from endless concentration on the 'basics'. 

1. 0·6 2. 2·6 3. 3·3 4. 5·6 5. 0·8 

6. 1·3 7. 1·6 ~. 2·4 9. 0·4 10. 0·7 

11. 0·16 12. 0·83 13. 1·6 14. 1·16 15. 3·6 

16. 5·83 

1. 

2 7 

9 

1 3 

- 2. - 3. -=1 4. -=- 

9 9 9 3 9 

52

1 1 1 5 2 1 

5. -+10=- 6. - 7. - 8. -=- 

3 30 6 6 30 15 

5 1 5 

5 1 

9. -=- 10. - 11. 1~ 12. -=- 

60 12 9 90 18 

C. These work out by converting the fractions to decimals, then adding 

or subtracting. 

M57 A. All proofs 

B. All proofs. Note: Some of these fractions can be simplified. 

45 5 63 7 6 2 36 4 

99 = 11; 99 = 11; 99 = 33 ; 99 = 11 

C. 

1. (a) 0·090909 = 0'9~ (b) 0·181818 = 0·18 

(c) 0·454545 = O·~~ (d) 0·3181818 = 0·318 

(e) 0·242424 = 0·24 (f) 0·155555 = 0·15 

(g) 0-36 

(h) 0·2 

2. (a) 31 + 99 (b) 40 + 99 

(c) 53 + 90 (0·088888 ... = 8/90 

0·5 = 45/90 Total 53/90) 

(d) 68 (e) 5 + 6 (f) 8 + 99 (g) 33 + 90 

90 

(h) 37 + 90 

3. (a) This can be checked by using decimal forms on the calculator 

1 

- = 0·111111 

9 

3 

- = 0·03030303 

99 

14 1 11 11 3 14 

and this is 99' Also "9 = 99 so 99 + 99 = 99 

Total = 0·14141414 

(b), (c), (d) all similar 

P36 A. 1. (a) 0·5 (b) 0·3 (c) 0·25 (d) 0·2 (e) 0·16 

.....----- 

(f) 0·142857 (g) 0·125 (h) 0·; 

2. (a) 0·6 . (b) 0·75 (c) 0·4 (d) 0·6 (e) 0·6 

.....-----.... 

(f) 0·714285 (g) 0·75 

53

3. Proofs. (b), (c) and (d) these can be proved by using equivalent 

fractions 

B. 

c. 

1 1 2 1 3 

-+-~-+-~- etc. 

2 4 4 4 4 

1. (a) 4·9 (b) 2·9 (c) 6·9 (d) 11·9 

2. (a) 0·0000001 (b) 0·5 

3. (a) 2·499999... (b) 0.39 (c) 0·249 

4. (a) 10·39 (b) 0·249 (c) 5·49 

1. (a) 0·5:7 (b) o·§{) (c) O·fa (d) 0·3f5 

4 5 20 

2. (a) 11 (b) 11 (c) 33 

(e) 9·9 

(d) 0·679 

(d) 0·339 

r---.. 

(e) 0·527 

P37 

A. 2. (a) 285714 (b) 428571 (c) 571428 

(e) 857142 

all are the number 142857 rotated 

3. 999999 (though 1x 7 should be 1) 

B. 

(d) 714285 

1. 0·111111, O·22222, etc. 

43 35 17 

2. (a) 90 (b) 90 (c) 30 

5 

31 23 

(d) 55 

(e) 90 (f) 90 

90 

c. 1. General investigation following the work already done 

2. ~ = 0·422222 23 = 0·696969. .. !...- = 0·07070707 

45 33 99 

t 

7 

/' 

'8 

4 

2 

59 = 0.6555555 ~ = 0.777777 47 = 0.522222 

90 9 90 

E37 A. See P36 and 4. (a) Use calculator 

54 

B. See P36 B. 

1 

(b) Yes 13= 0·0769230769230 ... 

1 

17 = 0·05882352941176470588 ... etc. 

The pattern repeats after a particular remainder comes up a 

second time in the division process.

C. See P36 C and 

3. You must get to a point at which the remainder has occurred 

before. From then on the pattern of division will repeat. An exact 

division will produce recurring zeros. ; = 0·5000000 ... 

4. Since every rational number must have some sort of repeating 

pattern, a decimal that does not repeat cannot have come from a 

rational number. 

5. The pattern is changing all the time. 

6. This is a very ancient discovery (2000 years). If a length is to be 

measured it must be measured with a ruler and in units. The 

smaller divisions are formed by dividing the units into an equal 

number of parts, i.e. rational numbers fit the marks on the ruler, 

1~, 11>1etc. A length which cannot be fitted to a rational number, 

e.g. the diagonal of a square side 1 unit, cannot therefore be 

exactly measured by an ordinary ruler, no matter how finely it is 

graduated! 

E38 A, B. 

4. 

C. 1. 

2. 

3. 

4. 

See P37 and some additional investigations 

Can be proved by simply listing all possible values for a and b. 

There are no regular patterns in 7T. 

There are 8 ones, 12 twos, 11 threes, 11 fours, 8 fives, 8 sixes, 

9 sevens, 12 eights, 14 nines, 8 zeros. 

No! If 7T could be represented by a fraction there would be a 

pattern in the decimal form. 

Investigation 


Multibase arithmetic is more than a rather unimportant 

mathematical game. It is a way of investigating the important 

rules which govern ordinary arithmetic and the system 

of relationships which control all arithmetical operations. 

For those children who have learned their arithmetic facts 

blindly this work can be very illuminating. Simple binary 

arithmetic is important in understanding electronic calculators 

and storage systems and is best seen as another counting 

system. The relationships between numbers in base 2, 4 and 

8 are also revealing. Children find the addition and multiplication 

tables in different bases easy to understand and 

55

amusing because everything looks wrong. Even though the 

numbers change, the properties of commutativity and associativity 

are clearly seen in all base systems. So are primes, 

number patterns, and fractional systems. 

It is interesting to realize tha.t for generations we have 

asked children to work in multi base systems in their work 

with fractions, even before the age of 8! 1+ -! is the addition 

of two numbers from different base systems, base 3 and base 

4, yet we have expressed surprise that many children become 

confused and frightened when asked to manipulate 

fractions in primary school. 

Answers 

M58 A. 1. dots: 178 10 =262 8 =454 6 

3. dots: 89 10 =131 8 =225 6 

B. 

c. 

1. 11 2. 13 

6. 33 7. 23 

11. 50 12. 44 

1. (a) (i) 18 (ii) 36 

(v) 324 (vi) 612 

(b) (i) 1 box + 4 packets 

(ii) 

(iii) 

(iv) 

(v) 

3. 29 

8. 42 

13. 88 

(iii) 144 

(vii) 744 

4. 36 

9. 63 

14. 157 

(iv) 216 

(viii) 1104 

2 boxes + 4 packets + 4 eggs 

5 boxes + 3 packets + 2 eggs 

4 boxes + 1 packet + 5 eggs 

2 crates + 2 boxes + 3 packets + 3 eggs 

2. (a) (i ) 16 (ii) 48 (ii i) 44 

(iv) 64 (v) 160 (vi) 204 

(b) (i) 1 carton + 2 boxes (ii) 2 cartons 

(iii) 3 cartons + 2 glasses 

(iv) 1 case + 2 cartons + 1 box + 3 glasses 

5. 28 

10. 69 

15. 229 

M59 A. 1. (a) 113 6 (b) 201 6 (c) 244 6 (d) 532 6 (e) 1103 6 (f) 1443 6 

Note: The easy calculator check. 1443 6 

56 

[I]~[ID[±]@]~[ID[±]@]~[ID[±]~E] 

iii 

i 

1 443 

2. (a) 115 8 (b) 137 8 (c) 240 8 

(d) 321 8 (e) 674 8 (f) 1365 8

Check 1365 8 

[I]~lID[±][JJ~lID[±][[]~lID[±][]J~ 

Note: Some calculators will require ~ after each operation. 

3. (a) 13 5 (b) 32 5 (c) 124 5 

(d) 214 5 (e) 1212 5 (f) 2132 5 

B. 1. 133 6 is wrong 2. 2112 4 is wrong 3. 2445 6 is wrong 

c. 1. (a) 7 (b) 14 (c) 24 

(d) 44 (e) 60 (f) 237 

2. (a) 19 (b) 29 (c) 39 

(d) 77 (e) 122 (f) 407 

3. (a) 23 (b) 63 (c) 64 

(d) 168 (e) 219 (f) 1253 

M60 A. 1. 13 8 2. 12 8 3. 5 4. 5 

5. 36 8 6. 61 8 7. 43 8 8. 6 8 

B. The 5 times table in base 6 is 5, 14, 23, 32, 41, 50. Similar to the 

ordinary 9 times table. There is a similar pattern for x7 in base 8 and 

x9 in base 10. In general the pattern is the same for x (n - 1) in base n. 

c. 1. 1,4, 13,24,41, 100. Note the pattern 1 434 1 

2. Base 4 1 10 21 100 (121) 

pattern of endings 1 0 1 0 1 0 ... 

Base 5 1 4 14 31 100 (121) 

pattern of endings 1 4 4 1 0 ... 

Base 7 1 4 12 22 34 51 100 

pattern of endings 1 4 2 2 4 1 

Base 8 1 4 11 20 31 44 61 100 

pattern of endings 1 4 1 0 1 4 1 0 

3. The numbers 9, 16,25 will answer this question whatever the base 

(a) 12 2234 (b) 11 20 31 (c) 10 17 27 

P38 A. 1. 40 6 , 20 9 , 20 8 , 30 5 2. 200 8 , 400 6 

3. 120 4 , 110 3 

B. 1. 243 8 2. 1302 5 3. 11031 4 4. 628 9 

C. 1. 160, no remainder when divided by 8 

2. 56, remainder 2 when divided by 6 

3. 69, remainder 4 when divided by 5 

57

4. (a) 4 (b) 0 (c) 2 (d) 2 

The number tells you the remainder by the last digit 

P39 A. 1. + 

1 

2 

3 

4 

5 

6 

7 

8 

1 2 345 678 

2 3 4 5 6 7 8 10 

3 4 5 6 7 8 10 11 

4 5 6 7 8 10 11 12 

5 6 7 8 10 11 12 13 

6 7 8 10 11 12 13 14 

7 8 10 11 12 13 14 15 

8 10 11 12 13 14 15 16 

10 11 12 13 14 15 16 17 

2. (a) 16 

(b) 10 

(c) 13 

(d) 4 

(e) 1 

(f) 7 

3. (a) 6 

(b) 6 

(c) 5 

(d) 4 

(e) 6 

(f) 5 

B. 1. x 

1 234 5 678 

1 

2 

3 

4 

5 

6 

7 

8 

1 234 5 678 

2 4 6 8 11 13 15 17 

3 6 10 13 16 20 23 26 

4 8 13 17 22 26 31 35 

5 11 16 22 27 33 38 44 

6 13 20 26 33 40 46 53 

7 15 23 31 38 46 54 62 

8 17 26 35 44 53 62 71 

All numbers in the table are base 9 

2. (a) 44 (b) 40 (c) 38 (d) 7 (e) 7 

(f) 6 (g) 6 (h) 4 (i) 13 

(40g -:- 6 so 80g -:- 6 = 6 x 2 = 13g) 

3. (a) Is like the 9 times table. 9, 18, 27, .... 

going down by 1 in the units and up by 1 in the 10s. 

(b) The last figures of the squares are 1 4 0 717 0 4 1 in base 9. In 

base 10 they are 1 4 9 6 $ 6 9 4 1. Both patterns are 

symmetrical. 

(c) The 3 times table, base 9 is like the 5 times table in base 10 

with a repeating simple pattern. 

58

E39 A. 1,2, 3, see P38 

4. (a) 2438 (b) 3022 4 (c) 2313 5 (d) 628 9 (e) 1031 7 

(f) 112211 3 (g) 100100112 (h) 2034 6 

B. 1. 161'0; zero 2. 56 10 ; 2 

3. (a) 4 (b) 5 (c) 0 (d) 3 (e) 9 (f) 2 (g) 2 

(h) 1 (i) 4 

Explanations: 568 9 -7- 9 leaves a remainder of 8 so 568 9 -7- 3 will 

leave the same remainder as 8 -7- 3, i.e. 2 

C. 1. 265 is 324 9 and 100211 3 

2. 3 9 2 9 4 9 

10 3 02 3 11 3 

3. 363 is 11223 4 and (0)1011010112, Each digit of 11223 4 is replaced 

by its base 2 form 

4. 363 is 553 to base 8 

101 ........., 101 ........., 011 ........., each base 8 digit corresponds to a 

5 5 3 group of 3 binary digits 

5. This is a quick method of converting to base 2 (especially for large 

numbers). 

(a) 100~ 1448~ 11001002 

Note: The first group of 3 binary digits is 001 but we can 

leave the zeros off the front 

(b) 260 ~ 4048 ~ 1000001002 

(c) 359 ~5478 ~1011001112 

(d) 826 ~14728 ~11001110102 

(e) 6160~ 140208~ 11000000100002 

(f) 7665~ 167618~ 11101111100012 

E40 A. See P39 for base 9 tables 

1. (a) 16 9 (b) 57 9 (c) 124 9 (d) 6 (e) 6 (f) 31 

2. (a) 6 (b) 15 9 (c) 16 9 (d) 15 9 (e) 28 9 (f) 45 9 

In these questions it is easier to 'build up' the smaller number than 

to subtract by the usual rules 

3. (a) 44 9 (b) 40 9 (c) 80 9 (d) 6 (e) 7 (f) 13 9 

4. (a) They are similar to the 9 times table in base 10 

(b) The last digits follow the pattern 1 4 0 7 7 0 4 1 

(c) The 3 times table, base 9, is like the 5 times table in base 10 

with a simple repeating pattern 

B. Base 12 tables 

59

c. 

2. (a) 14'2 

(f) 30'2 

(k) =9: 

(b) 13'2 

(g) 13'2 

(I) 9 R 3 

(c) 13'2 

(h) 71'2 

(e) 42'2 

(j) 4 R 1 

2 1 1 7 1 5 13 

1. (a) "8 = 4 = 0.2510 (b) 2 (c) "8 (d) "8 + 64 = 64 

1 3 3 21 7 5 4 39 3 

(e) "2 (f) "6 + 36 = 36 = 12 (g) "7 + 49 = 49 (h) 25 

2. (a) 0.4 8 (b) 0-6 8 (c) 0-146314631 8 (d) 0-46314631 8 

(e) 0-525252... (f) 0·252525 ... 

Note: There are repeating and recurring octimals. 

; is 0.111111 8 so 0·22222 ... is ~ and so on. 

3. j is a base 3 number, ~ is a base 5 number. We can, however, 

convert them both to base 15 by using equivalent fractions. 

60


The purpose of this unit is to convince children that the 

mathematics they are learning is indeed useful. Usually, 

everything that is learned has to be 'stored' for later, but in 

this section mathematical skills are applied immediately to 

other areas of school work. 

In some ways other subjects are more 'real' to the pupils 

than the external world of commerce, etc. The problems 

they are asked to solve are not rates and taxes, or HP, but 

problems presented to them by other teachers in the course 

of learning subjects such as geography and science. Geography 

uses a considerable amount of modern mathematics 

for example, set theory, networks in the study of communications, 

transformation in theories of rock formation, etc. 

Science uses the more traditional mathematics of graphs, 

equations and metric units. All applications can make use of 

the calculator and it is in the mathematics department that 

children will learn the' effective use of this instrument. 

Further applications can be collected from the departments 

concerned (they will have their favourite mathematical skill 

which the children have failed to learn) and from the 

children who may have interesting applications of their own 

outside of school. (Hobbies, jobs, etc.) 

Answers 

M61 A. 

1. 12·75 million 


3. 103·53 million 


B. 1. 13·53 million, 47·27 million, 108·259 million, 507·754 million (use 

constant multiplier) 

2. Populations grow by 1·03 million, 5·27 million, 6·259 million, 

47·754 million 

C. 1. The populations are 12·36,12·73,13·11,13·51,13·91,14·33,14·76, 

15·2, 15·66, 16·13. Not double. 

2. After 20 years it will be 21·7 million at this rate of growth. Not 

double. 

61

3. 

Country 

1980 (est.) 

(millions) 

1990 (est.) 

(millions) 

Algeria 

Nigeria 

Turkey 

India 

Japan 

Denmark 

USA 

Gt. Britain 

19·37 

71·09 

45·68 

710·95 

111·56 

5·26 

228·70 

59·18 

26·80 

91·90 

59·63 

918·99 

124·46 

5·52 

255·14 

62·21 

Use constant multiplier or XV button if you have one. 

M62 There are different ways of tabulating the data in these questions. The 

tables should be clear and tidy with careful labels so that the information 

can be seen at a glance. 

M63 1. The angles at the centre of the pie charts are given. 10 % = 36° 

(a) 216° bat ber, 144° home haircut 

(b) 108° hairdresser, 252° home haircut 

2. Angles: Nitrogen 280-8°, Oxygen 75'6°, other gases 3-6° 

3. Ang les: Developed countries Developing countries 

protein 38° protein 34° 

fat 142° fat 34° 

sugar 57° sugar 17° 

starch 123° starch 275° 

4. Angles: land 105°, water 255° 

M64 Draw these graphs with care. They should be drawn in pencil with a 

smooth curve and labelled along the axes. 

Note: Exercises in which questions are asked about data are to be found 

in the P and E units 

P40 A. 1. (a) 77250 (b) 81 954 (c) 86945 

2. (a) Yes (b) No (c) No 

3. (a) 5250, 5512, 6381 (b) 15 years 

62 

B. 1. Heights (cm) Weights (kg) 

121-130 10 21-30 1 

131-140 7 31-40 10 

141-150 6 41-50 16 

151-160 7 51-60 3

2. Boys Girls Boys Girls 

Heights (em) only only Weights (kg) only only 

121-130 5 5 21-30 0 1 

131-140 4 3 31-40 5 5 

141-150 2 4 41-50 6 10 

151-160 3 4 51-60 3 0 

3. (a) The average weig ht of the girls is 41-5625, average weight of 

boys is 44-0714 so the statement does not agree with the data. 

(b) Agrees with the data 

(c) Does not agree with the data 

(d) Does agree with the data 

(e) Does not agree 

(f) Agrees (g) Does not agree (h) Agrees 

c. 1. The data can be tabulated giving diameter and number of acorns 

with that diameter. 

2. This can be treated in various ways, e_g. dry, a little rain, wet, very 

wet. 

P41 1. (a) 33-8 % (b) 54·g o /0 (c) 11·3% 

2. 

4. Yes, the large (41-50) classes have almost disappeared on the pie 

chart. 

E41 A. 1. 3-9735 x 10 13 km 2. 45360 years 

3. 226800000 km per hour 

63

4. There would be enormous problems of control as the rocket 

would be so far away. There would be enormous energy problems 

in escaping from the Sun's gravitational field and many 

more besides. 

B. 1. 14080 m. Lighter because of no human, no breathing apparatus, 

and no need to keep a human warm; 0·008 of earth's radius. 

2. 1·6 x 10 8 gallons per day, 5·84 x 10'0 gallons per year 

= 2·774 x 10" litres per year 

= 2·774 x 10 8 m 3 per year 

Thus a tank 28 m deep, 100 m wide and 100 km long would be 

needed to store all this water. 

3. 3 x 10 9 litres, enough for about 4 days. 

4. 1140. Not realistic as it would produce an oyster explosion! 

C. See P40 A. for 1 and 2, and 3 is just for interest. Of course such 

growth could never happen because the flies would overcrowd their 

habitat within the first 3 generations and start to die off. 

E42 A. 1. 

2. 

64

3. 

Med.3° 

4. Starts with (a) above and divides each section in proportion for 

the continents and oceans. 

B. Drawings 


Although the use of logarithm tables for multiplication and 

division will give way to the use of calculators, simple 

multiplication tables will always be important. * They demonstrate 

important principles which the hidden mechanism 

* This unit makes use of Basic Mathematical Tables (Bell & Hyman). 

65

of the calculator never reveals. They also provide a simple 

and quick calculating aid for use when two-figure accuracy 

only is required. A further valuable feature is that any child 

could construct these tables for himself. He will also use his 

mental arithmetic. 

In learning to use multiplication tables to tackle more 

extended calculations, the pupil comes face to face with the 

basic laws of arithmetic which make calculation techniques 

feasible. Thus the distributive and associative laws of arithmetic 

will be met, fractions can be converted to decimal 

form, algebraic patterns such as (a + b? can be confirmed 

and other patterns such as (n + 1)(n - 1) = n 2 - 1 will be 

noticed. Searching for primes is a revealing activity. First, if 

n is being tested, an area up to J n is cordoned off and the 

search restricted to that area. If n does not appear then it is 

definitely a prime number. 

The tables give a useful check on the calculator (whereas 

pencil and paper checking is beyond the powers of many of 

the pupils) and the manipulations needed to deal with 

decimals bring the pupil closer to understanding the numbers 

less than 1, something very difficult to glean from 

calculator usage alone. 

Answers 

M65 A. 1. 350 2. 1504 3. 5005 4. 948 5. 3969 

6. 3168 7. 2499 8. 6596 9. 9 10. 11·48 

11. 30-03 12. 25·92 13. 1-665 14. 1·2 15. 8·64 

16. 0·2844 

B. 1. 2415 2. 930 3. 430-2 4. 16650 

5. 28820 6. 2553-6 7. 16-8078 8. 148-608 

C. 1. 11 978710 2. 11425674 3. 30080424 4. 40344854 

5. 449·2345 6. 499·6986 7. 279244·8 8. 48230·7 

M66 A. 1. 9 2. 6 3. 8 4. 35 5. 35 

6. 88 7. 67 8. 157 9. 56 10. 45 

11. 35 12. 42 13. 56 14. 63 15. 54 

16. 88 

66 

B. 1. 54 R6 2. 48 R3 3. 64 R4 4. 55 R7 5. 49 R9 

6. 38 7. 82 R9 8. 90 R15 9. 87 R44 10. 60 

11. 74 R63 12. 75 R60 13. 85 R55 14. 83 R39 15. 85 R25 

16. 60 R17

C. 1. 45·88 2. 47·22 3. 72·08 

5. 20·83 6. 34·76 7. 7·16 

9. 168·47 (subtract 100 x 17 first) 

11. 82·07 (not 82·7) 12. 75·41 

14. 129·86 15. 93·73 16. 90·91 

4. 27 

8. 52·33 

10. 99 

13. 180·91 

M67 A. 1. The first two numbers go up 1 at a time while the last number 

goes down 1 at a time, e.g. 279288. The sum of the digits is 9 or 18 

for each number. 

2. The numbers are 300 + (15 x 1), (15 x 2) ... etc. so the 15 times 

table repeats. The sum of the digits is a multiple of 3 and each 

number ends in 0 or 5. 

3. Contains the numbers 222, 444, 666, etc. because 74 is 2 x 37. 

Endings have pattern 4 8 2 6 0 4 8 2 6 0 ... (like 4 itself). 

4. With the exception of 99, 209, 308 and 319, the outer two digits add 

to the middle one. 

B. Digital sums Digital roots 

1. 6, 12, 18, 6, 12, 18... 1. 6, 3, 9, 6, 3, 9 ... 

2. 18, 18, 18, 18... 2. 9, 9, 9, 9, 9 ... 

3. 8, 7, 6, 14, 13, 3, 11, 10, 9, 8. . . 3. 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, ... 

4. 9, 9, 9, 9, 9, 9, 9, 9, 9, 18, 9, 4. 9, 9, 9, 9, 9, ... 

18, ... 

5. 9, 9, 9, 9, 9, 18, 18, 18, 18, 9, 5. 9, 9, 9, 9, 9, ... 

18, 18, 9 

6. 10, 11, 12, 13, 5, 15, 16, 8, 18, 6. 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, ... 

10, 11, 21 ... 

Digital roots are often used as spot checks for errors in computing. 

(Ask a bank!) 

M68 A. 1. 81 

6. 2025 

B. 1. 49284 

5. 533~61 

2. 196 3. 484 4. 1089 5. 900 

7. 5184 8. 8649 9. 6·76 10. 18·49 

2. 119025 3. 271 441 4. 407 044 

6. 1840·41 7. 3203·56 8. 3956·41 

Note: The calculator is the easiest tool to use to answer these 

questions but it is important to know that the answers can be found 

with just a little more trouble using ordinary multiplication tables. 

M69 A. 1. (a) 13 

(f) 36 

(k) 64 

(b) 17 

(g) 42 

(I) 72 

(c) 22 

(h) 48 

(m) 75 

(d) 29 

(i) 50 

(n) 81 

(e) 33 

(j) 56 

(0) 88 

67

2. (a) 1·4 

(f) 7·2 

3. (a) 1·183 

(f) 7·720 

(b) 3·4 

(b) 1·924 

(g) 8·173 

(c) 4·3 (d) 5·2 

(c) 3·578 (d) 5·477 

(h) 8·660 

(e) 6·3 

(e) 6·870 

P42 A. 1. 374 2. 3060 3. 15200 4. 25470 5. 304·2 

6. 7·84 7. 2·96 8. 9·12 

B. 1. 5 RO 2. 5 R10 3. 44 4. 25 R19 5. 26 R26 

6. 13 R29 7. 14 R19 8 7 R57 9. 40 10. 70 R41 

11. 118 R1 12. 156 R9 

C. 1. 1·9 2. 140 3. 69 4. 1·2 5. 21 

6. 1600 

D. These can be checked by noting that the remainder stays the same: 

e.g. 7079 = 7 R7, i.e. the 70 continues. 

70079 = 77 R7 this leads to 0·70079 which again gives the same 

remainder, so the division keeps on repeating. This is what happens 

with all the questions. 

E. 1. 1764 2. 3481 3. 18496 4. 67·24 5. 44·89 

6. 15·21 7. 0·2304 8. 0·8649 9. 0·7225 10. 0·0016 

11. 0·0081 12. 0·0049 

F. 1. 14 2. 22 3. 27 4. 35 5. 47 

6. 54 7. 65 8. 73 9. 6·3 10. 7·1 

P43 A. 1. £12 2. £41·16 3. 6·29 kg 4. 10·11 kg 5. 2774m 2 

6. 0·4125m 2 7. (a) 696 km (b) 725 km (c) 5220 km 

8. 498cm 2 

68 

B. 1. £2·20 per kg 2. 11 P 3. 91·75 kg 4. 44 

5. 47em 6. 36km 7. 33 metres 8. 3'6kg 

C. 1. (a) 11·56 cm 2 (b) 7.29 m 2 (c) 0·4225 m 2 

(d) 729 mm 2 (e) 4356 mm 2 (f) 0·5625 mm 2 

2. (a) 18cm (b) 28 em (c) 3·3 em 

(d) 4·2 m (e) 68·67 m (f) 18·4m 

3. The pattern continues 7, 7, 9, 4,1,9,1,4,9,7,7, 9oo.etc. 

4. 7, 5, 3, 1, 8, 6, 4, 2, 7, 9, 5, 3, 1 ... etc. 

5. Different answers

E43 A. 1. 374 2. 15200 3. 304-2 4. 2·96 5. 5 

6. 25·6786 7. 14·5278 8. 70·5325 9. 1·8947 10. 20·506 

11. 68·387755 (Interesting? What are the next decimal places?) 

12. 1620·91 13. 1764 14. 67·24 15. 0·2304 16. 0·0016 

17. 14 18. 27 19. 47 20. 6·3 

B. 1. (a) 11·56cm 2 (b) 7·29m 2 (c) 0·4225m 2 

(d) 729 mm 2 (e) 4356 mm 2 (f) 0·5625 mm 2 

2. (a) 18 cm (b) 28 cm (c) 3·3 cm 

(d) 4·2 m (e) 68·67 m (f) 18·44 m 

3. (a) The pattern continues. This can tell you when not to expect an 

exact square root. 

(b) 7, 5, 3, 1, 8, 6, 4, 2, 7, 9, 5, 3, 1 ... etc. 

(c) Your own choice. 

4. a, b, c are all equal to )10, thus a 2 , b 2 and c 2 are all equal to 10. 

E44 A. 1. This is not necessarily true. For example a = 2, b = -4 so a> b. 

But b 2 = 16, a 2 = 4, so b 2 > a 2 . 

2. Again not necessarily true. Same example. Both of the statements 

1 and 2 are true if we restrict a and b to be positive numbers. 

3. True. 

4. Not necessarily true. Example. Suppose a = 9, n = 27. a 2 = 81 so n 

is a factor of 81 but not of 9. 

5. All are true. This is easily proved by using algebra as follows. If n 

is a factor of a then a = xn where x is a whole number and 

similarly b = yn. Thus a + b = (x + y)n. 

So a + b is a multiple of n. Also a 2 + b 2 = x 2 n 2 + y2n2 = (x 2 + y2)n 2 

and (a + b)2 = (x + y)2n2 so both a 2 + b 2 and (a + b)2 are multiples 

of n. 

6, 7, 8 are all true and can be verified by using any numbers you like 

for a and b. 

B. 1. Not true 

6. True 

2. True 3. True 4. True 5. True 

69


This unit reinforces work on Pythagoras' theorem. It considers 

applications, solution of triangles and (in the E Unit) 

proof. The work brings together algebra, arithmetic, 

geometry and real world applications. It is therefore ideal 

for fitting the various parts of mathematics together. The 

application on using Pythagoras' theorem with coordinates 

links with geography and navigation. Other aspects such as 

Pythagorean triples will be considered later in books M 1 and 

N1. All work on Greek mathematics gives opportunity to 

indicate the thread of mathematical discovery which has run 

through human history. There are many sources of information 

and legends in books such as Men of Mathematics by 

E. T. Bell or The World of Mathematics by James R. 

Newman though these will have to be edited for children's 

use. 

Answers 

M70 A. 

B. 

Because of the limitations of measurement it is only possible to 

confirm Pythagoras' theorem roughly by these examples. 

1. 20 mm, 2-0mm, 28 mm" ... squaring ... 400 + 400 = 800 = 784 

(=28 2 ) 

2. 30 mm, 20 mm, 36 mm 900 + 400 = 1300 = 1296 (= 36 2 ) 

3. 40 mm, 15 mm, 43 mm 1600 + 225 = 1825 = 1849 (=43 2 ) 

4. 20 mm, 22 mm, 30 mm 400 + 484 = 884 ~ 900 (= 30 2 ) 

5. 25 mm, 15 mm, 29 mm 625 + 225 = 850 = 841 (=29 2 ) 

6. 35 mm, 23 mm, 42 mm 1225 + 529 = 1754 = 1764 (=42 2 ) 

7. 28 mm, 28 mm, 40 mm 784 + 784 = 1568 = 1600 (=40 2 ) 

1. Rectangle 30 mm x 20 mm diagonal 36 mm 

900 + 400 = 1300 = 1296 (=36 2 ) 


441 + 1225 = 1666 = 1681 (=41 2 ) 


225 + 2500 = 2725 = 2704 (= 52 2 ) 


625 + 900 = 1525= 1521 (=39 2 ) 

70

5. Rectangle 26 mm x 18 mm diagonal 31·5 mm 

676 + 324 = 1000 = 992·25 (=31'5 2 ) 


2025 + 484 = 2509 = 2500 (= 50 2 ) 

M71 A. 

B. 

c. 

1. 38·9mm 2. 31·8mm 3. 48·8mm 4. 51 mm 5. 51·2mm 

6. 44·8mm 

1. 43·7 mm 2. 47·7 mm 3. 48·7 mm 4. 40 mm, 25 mm, 47 mm 

5. 56·6mm 6. 34·4mm 

1. 213·6mm 2. 18km 3. 71·2m 4. 73·3m 

P44 A. There are many different right angled triangles in the figure. 

Pythagoras' theorem works for all of them, of course. 

B. AD=8·5cm, CO=~AD=4·24cm, DX=6·7cm, DY=6·2cm 

C. 1. Draw a triangle with sides 6cm, 8cm and 10cm with careful 

construction. Any other lengths can be used as long as a 2 + b 2 = c 2 

2. (a) Not, AC should be 108 mm (b) Yes (c) Yes 

(d) Not, AC should be 10·8 (e) Yes 

P45 A. 1. 8'6cm, BC, EG, FH 

2. 6·4 cm, BG, AF and CE 

3. CG = 8·06, GH = 5 

CH = )90 = 9·49 cm 

90 = 7 2 + 4 2 + 52 

H 

c 

4. The diagonals are AG, CH, ED and SF. All four have the same 

length. 

B. Different answers 

71

E45 A. 1. (a) Use a triangle whose sides are 5 em and 3 em 

(b) Use a triangle whose sides are 8 em and 5 em 

(c) Use a triangle whose sides are 5 em and 6 em 

2. Drawing. For this drawing accurate right angles should be drawn. 

B. 1. (a) ABD = 40°, DBC = 50° 

2. (a) BAD = 25°, (b) DAC = 65°, (c) A~D = 25° 

3. (a) POR = 59°, (b) QPS = 31°, (c) RPS = 59° 

4. If B = x o , BAD = (90 - x)O, DAC = X O and DCA = (90 - x)O 

Thus all three triangles have angles 90°, X O and (90 - x)O 

AB 

AD 

C. AC = AB => AC x AD = AB 2 by cross-multiplication 

CD = BC => AC x CD = BC 2 

BC AC 

Now add (AC x AD) + (AC x CD) = AB 2 + BC 2 

But AC(AD + CD) = AC(AC) = AC 2 

So AB 2 + BC 2 = AC 2 ... Pythagoras' theorem 

E46 A. 1. AD = 8·6 em also the length of BC, FH and EG 

DH = 6·4 em also the length of BG, AF and CE 

2. CH = )90 = 9·49 em. AG, ED and BF are also )90 em 

B. Different answers 

72 


Percentages are widely used in everyday communication 

from wage deals to unemployment figures. They are also 

frequently quoted in science, geography, economics, home 

economics and so on. In spite of this many people would say 

that they did not understand 'percentages'. It is hard to 

understand why not as the topic is extremely simple. It must 

be surmised that some sort of confusion lingers on from 

failure to understand fractions and that this is the barrier.

Answers 

Revision Exercises 

M72 A. 1. 70p 

5. 10·08 km 

9. 42·75 mg 

B. 1. 50% 2. 

6. 5% 7. 

C. 1. 1 

(a) - 

2 

In this unit a number of different applications are explored 

and the percentage is presented in fraction and decimal 

form. The decimal form is easier to use in problems especially 

as the calculator is available. The percentage button 

on the calculator is best ignored, or perhaps explored after 

the concept has been grasped. It is not necessary at this 

stage that the applications should be learned, the purpose 

is to experience percentages being used and for the pupils 

to feel they can cope. 

The work can be made more interesting if it is extended to 

include problems produced by members of the class. These 

may concern such varying topics as how to invest their 

pocket money to their chance of surviving in a war with 

inhabitants of outer space. Probabilities are traditionally 

expressed as percentages and make sense if they are related 

to a lottery with 100 tickets. The chance of winning is 50% if 

you buy 50 tickets. This emphasizes the 'out of' aspect of a 

statement in percentage terms (see Unit 23). 

2. £6 

6. 18·75 litres 

3. 33·33% 

8. 27% 

3. £16·50 

7. 5·28 kg 

1 

4. 10 % 

9. 58·666% 

(d) ~ 5 

4. 1·7kg 

8. £7·04 

2 

(e) - 5 

(f) 

7 

- 10 

(b) ~ 4 

4 

(g) 5 

(c) - 

4 

66 

(h) 100 

2. (a) 50% 

(f) 12·5% 

M73 A. 1. £30 2. 

6. £147·50 7. 

B. Interest 

1. £150 

2. £490 

3. £1760 

4. £395·20 

(b) 25% 

(g) 80% 

£88 

£135 

(c) 75% 

3. £104·50 4. £135 

8. £243 

Amount repaid altogether 

£650 

£1190 

£2560 

£1155·20 

(e) 70% 

5. £72·45 

73

M74 A. Profit (cash) 

1. £ 4 

2. £100 

3. £ 12 

4. £ 34 

B. Profit 

1. £16 

2. £50 

3. £8750 

4. £15·40 

C. 1. £1384·61 

Profit % 

Selling price 

(4 -;-75) x 100 = 5·33% 

23·8 

36·36 

20·36 

£80 + £16 = £96 

£175 

£43750 

£43·40 

2. £113·33 3. £55 4. £140 

M75 A. 1. North 7·9, South 8·10/0 2. North 55·7%, South 49·1 

3. North 3.6 % , South 4·6% 

4. Yes 

B. 1. 8% 2. 7·9% 3. No, very close 

M76 3. 19·7% 4. Yes 

P46 A. 1. (a) 0·25 (b) 0·36 (c) 0·08 (d) 0·12 (e) 0·18 (f) 0·165 

2. (a) 33% (b) 15% (c) 20% (d) 4% (e) 26% (f) 37;% 

3. (a) 100 % per month = 1200% per year 

(b) 25% (c) 20% per week = 1040% per year 

(d) 20% per year 

B. 1. Profit 50p, percentage profit 33·3% 

2. Loss £15, percentage loss 20% (15/75) 

3. £162 4. £62 (simple interest) 

P47 A. 1. 77·77% of rivers were unpolluted in 1975. 

2. Of people hurt in road accidents 1·9% were killed, 23·6% were 

seriously injured and 74·59% were slightly injured. 

3. In 1978 the quantity of oil was 58% of the 1973 quantity but it cost 

269% of the 1973 cost: the price had increased by 360%. 

4. 1981 children aged 5-14 form 14·3% of all people 

2001 children aged 5-14 estimated 14·8% of all people 

B. 1. 3·375 kg 

2. Gold was worth £3·2 per g. Value of gold in the ring £24. 

3. 144g 4. 24g 5. 48ml pure alcohol 

74

E47 A. 1. £13 2. £22·50 3. £58·65 4. £945 5. £92·80 

6. £198·55 7. £924 8. £649·08 

B. 1. £103·70 (= 85x 1·22) 2. £136·80 3. £492·80 

4. £4602 5. £5368 6. £7735 

C. Interest Principal + interest (0.5) . 

1. £ 54 £ 129 

2. £ 288 £ 608 

3. £ 968 £1408 

4. £5292 £8892 

E48 A. 1,2. See P47 A, 1,2 

3. The figures for 1968 and 1977 are very similar in all respects. 

4,5. See P47 A, 3,4 

B. 1. 337·5 g 2. £24 3. 144 g 4. 12 g 

C. 1. The answers depend on the choice of rough estimate. 

2. (a) 0·079% (tanker), 0·6% (car). The second is the greater percentage 

error. 

(b) 2·78% (London-New York), 2·5% (London-Paris). The first 

percentage error is the greater. 

3. Error (a) is 1/1600 Error (b) is 1/3600 so (b) has smaller percentage 

error and is therefore the better estimate. 

4. 31 x 11 x 2·5 = 852·5m 3 

30 x 10 x 3 = 900 m 3 (error 5.6 % ) 

30 x 12 x 2 = 720 m 3 (error 15.6 % ) 

So 30 x 10 x 3 is the better estimate 

5. (a) Percentage error 0·0666% (b) Percentage error 0·8333% 

So it looks as if the first scientist is more accurate even though he 

is working to the nearest 100000 kilometres!! 


Probability is one of the most important applications of 

arithmetic in that it is so often the basis for decisions. The 

use of probability is much wider than in mere gambling 

situations and ranges from 'shall I call the doctor?', 'when is 

the best time to take a holiday?', 'what are my chances of 

75

promotion?', 'will my bike be "nicked"?, to vital questions 

of health and family planning. 

The concepts have been proved to be difficult to understand 

and it is not certain that understanding in one situation 

such as tossing coins can be transferred to other situations 

such as the chance of inheriting a particular genetic weakness. 

This unit is essentially one of discussion and experiment 

rather than obtaining answers to calculations and reinforcement 

of percentages. The probability as a percentage is to be 

found in common language in such terms as 'fifty-fifty' or 'I 

am 100% sure that ... ' and these expressions should be 

related to the work of the unit. A great deal of the 

conceptual difficulty of this topic is due to careless language, 

but in tightening up the language it is very easy to lose the 

thread of meaning. Later on IMS will use the model of a 

raffle with 100 tickets to relate probability and percentage. 

Answers 

M77 The question refers to the probability of future events. 

A. 1. Likely 

4. Likely (7) 

7. 7 

10. Definitely not 

2. Unlikely 

5. ? 

8. ? 

3. Unlikely 

6. Unlikely 

9. Definitely not 

B. Different answers. Not really right or wrong here as the figures are a 

matter of opinion (except 8 which will not happen) 

c. Different answers 

M78 A. 1. 50% 

6. 23·1% 

2. 7·7% 

7. 30·80/0 

3. 7·7% 

8. 3·8% 

4. 7·7% 5. Under 1·9% 

B. All experimental resu Its. 

P48 A. 1. They are all certain or have happened already 

2. You could argue some of these. For example you could (a) forget 

to clear the calculator; (b) if you went by boat you would walk on 

the boat, etc. 

(c) Impossible because we do not allow it to happen. 

(f) Impossible because you must score at least 2 with two dice. 

76 

B. 1. (a) This is to be expected, say 70% chance, if your class is not all 

boys

(b) This is unlikely but not impossible (especially in Wales where 

so many people are Jones or Davies ... ) 

(c) Almost 100% certain. (They are only half-brothers if, for 

example, they have different fathers) 

(d) This is more likely than you would expect. At least 50% 

(e) There is no probability to this event. It either was a fine day or 

was not. 

(f) This is about the chance of having two fine days, one after the 

other. The chance is about 40% in Britain, though this will be 

different for different parts of the country. 

2. (a) More likely to be an 8 because this can happen as 1 + 7,6 + 2, 

3 + 5 or 4 + 4 

(b) More likely to be a spade (13 of them, only 4 aces) 

(c) 1 head, 1 tail is more likely 

P49 A. Experimental results 

(d) A mixture is more likely 

(e) More likely to just win the 100 m race 

B. Experimental results 

C. Experimental results 

E49 A., B., C., D. Experimental results 

ESO A. 1, 2, 3 Drawing. Simple tree diagrams 

4. (a) Around 14% chance 

(b) More likely to pick up a left and right (16-12) chance, i.e. 4n) 

(c) 3·6% chance (1/28). This problem can be simulated using a 

pack of cards (use two cards from each suit) 

B. 1. (a) No (b) The probability is ,1. Suppose her friend chooses a 

shop, then Jenny has 12 shops to choose from but only one will 

be right. 

(c) iJ 

2. 9 different ways. There are 3 different choices of the second part 

of the journey for each choice of New York to Chicago 

3. 80 

4. 125, probability 0·8% (small chance!) 

C. Different answers 

77


Mathematics is all about proof and all children following a 

mathematics course should have some experience of this 

central idea. Perhaps in the past, the formalities of proof 

were over-emphasized in geometry but in discarding formal 

geometry from today's syllabuses we are in danger of leaving 

out the main concept forming experience. Proof itself is 

about argument. There is an argument, to win it something 

stronger than a point of view is needed. All children 

appreciate this (though they may resort to physical means of 

winning an argument) and they are instinctively appreciative 

of an irrefutable argument. At this stage the language of 

proof is introduced, especially the symbol =} which is 

interpreted as a shorthand for 'if ... then'. The word 'implies' 

follows rather than leads as it is a new word whose 

meaning has to be felt before it can be understood. 

Discussion is of particular importance in this unit. Not 

only is implication being discussed but also the problems will 

usually contain profound mathematical concepts. Do not 

rush this work, the ideas will be presented again during the 

main courses. Do not over-emphasize the formal structure. 

Answers 

M79 A. 

1. n > 100 =;> n > 99 2. n < 6 =;> n < 10 

3. n = 10 + 5 =;> n = 20 - 5 

4. n = k x 6 =;> n = I x 3 where k and I are whole numbers. 

5. (n is a multiple of 2) and (n is a multiple of 5) 

=;> n is a multiple of 10. 

6. ;~~ }=;>X

5. If a number is more than 6 then its negative is less than -6. 

-6 

I I 

-n (less 

than -6) 

-5 -4 -3 -2 -1 0 2 3 4 5 6 

I I I I I I I I I I I I 

n (more 

than 6) 

6. If £x is more than £y then £(x - y) is positive. 

If I have £x, I can buy something for £y and still have something 

left over. 

C. Different answers. 

M80 A. 1. 10 0 5. True 

6. True if SAC is a straight line 

C. 1. It depends how you cook it. 

2. It depends on the line. 

3. This is only true if you start with a cylinder and cut perpendicular 

to the axis. 

4. What about 12·5, i.e. only true if you are thinking of whole 

numbers. 

5. Depends on how worn the coin is. 

M81 Remembering that anything already proved can be used in a new proof ... 

A. 1. 2n + 1 is odd (proved), adding 2 to an odd number produces 

another odd number, 2n + 3 is odd. 

2. 2n + 1 is odd so 2n - 1 is odd as well, being two less. 

3. Since 2n + 1 is odd and 2n - 1 is odd, (2n - 1) + (2n + 1) will be 

even (since odd + odd = even). 

4. 2n - 1 + 2n + 1 = 4n which is a multiple of 4 (verify by example, if 

n = 5, 2n - 1 = 9, 2n + 1 = 11,9 + 11 = 20 which is a multiple of 4). 

5. If n is even, n + 1 is odd, (n + 1)2 is odd (since odd x odd = odd). If 

n is odd, n + 1 is even, (n + 1)2 is even (since even x even = even). 

B. 1. If two angles are 60° the third must be 180°-(60° + 

60°) = 180° - 120° = 60°. Thus the triangle is equilateral. 

2. The third angle is 180°-(45° + 45°) = 180° - 90° = 90° 

3. The other angles add up to 180° - 20° = 160°. Since the angles are 

both equal, each one is 80°, 

79

4. Since a rhombus has opposite angles equal, the angle opposite 

90° is also 90°. This leaves the other pair of angles with 360°-180° 

to share. So both of these are 90° as well. 

P50 A. 1. Logical 2. Should be sister 3. Logical 

4. Logical 5. Logical 

6. Not true. For example, I have £10, you have £4 and Jan has £6. 

One example which breaks the implication is enough to make it 

false. 

B. 1. True 

2. The statement should be !n is even? n is even. 

3. Not true, something else may be wrong. Should be the bulb is 

broken? the light does not come on. 

4. Not true. Even the reverse is not always true as the bus may stop 

to let people off! 

5. (a) True (b) True (c) Not true 

(d) Not true. The score could go 0-1, 0-2, 0-3, 0-4, 0-5, 1- 5 , 

1-6, ... etc. 

(e) Must be true. 

P51 A. 1. Since odd + odd = even and even + even = even, two numbers 

adding to 21 must be one odd and one even. 

2. Since two consecutive numbers must be odd and even or even 

and odd, their sum will be odd. 

3. Four consecutive numbers are made up of two odd pairs 

e.g. 3 + 4 + 5 + 6 and odd + odd = even 

odd 

odd 

4. The numbers are n, n + 1 and n + 2. Their sum is 3n + 3 but 

3n + 3 = 3(n + 1) so their sum is a multiple of 3. 

B. 1. Put 8 = X O BAD = 90° - X O DCA = 90° - X O DAc = X O 

So 6ABD, ACD and ABC have angles x O , 90° - X O and 90° 

2. Put BDC = X O and ADB = yO then DBC = yO and DBA = X O 

A = C = 180° - (XO + yO), so both triangles have angles x o , yO and 

1800 - (X O + yO) 

3. DAB + DCB = X O + yO + ZO + W O = ADC + ABC 

All four angles add up to 360° = 2x + 2y + 2z + 2w 

Therefore DAB + DCB = 180° 

E51 A. 1. Logical 2. Should be sister 3. Logical 

4. Logical 5. Logical 6. Wrong 

7. Not true, e.g. I have £10, you have £4 and John has £6. Only one 

counter-example is needed to break an implication. 

8. Logical 9. Logical 10. Logical 

80

B. 1., 2., 3., 4. see P50 B. 

5. (a) Not true (b) True 

(c) Somewhere during the set. 

(d) Not true, e.g. the score could be 0-1, 0-2, 0-3, 0-4, 1-5, 1-6 

C. Different answers from different people. 

E52 A. See P5i A. 

B. See P51 Band 4. Simple proof 


The concept of set is fundamental to mathematics but that 

does not mean it can be taught in the abstract at an 

elementary stage. Classification, belonging to, not belonging 

to and their consequences have a much wider application 

than mathematics. The basic idea is that if a set is defined to 

have certain properties then an element will have those 

properties if it can be shown to be a member of the set. A 

good example is the classification of verbs in French. If a 

verb is an 're' verb then we know how it will form its past 

and future tenses. Belonging to the family carries a lot of 

information. Social sciences make use of the set concept in 

classifying society, chemistry classifies elements by the 

periodic tables and then discusses properties of the whole 

group. Mathematics itself does the same type of classification, 

for example, rational and irrational numbers; prime 

and composite numbers; transformations of various types, 

equations of various types, shapes of various types. The key 

question in a problem is often does ... belong to this set. 

In this unit and the next the two ideas of union and 

intersection are explored. The only symbols needed are n, 

U, E, $. and::} and these should be examined carefully and 

committed to memory after discussion. During the exploration 

of the uses of these symbols the work will touch upon 

numerous important mathematical ideas and in this way the 

work on sets does unify the mathematics syllabus though not 

in a formal, theoretical way. 

81

Answers 

M82 A. 1. The sets are the different states of Russia and Siberia. The 

elements are people who live in the states. 

2. The sets are different engineering trades unions which have been 

joined to form one big union. The sets are the small unions, the 

elements are workers. 

4. 

5. 

B. 1. 

2. 

3. 

C. 1. 

2. 

3. 

3. This is a trade union of people who work in local government. The 

elements are workers. Here union does not mean the union of 

4. 

sets. It is another name for a set of people. 

As above, but the workers are teachers. 

As above, but the workers are school-children. 

This is not a union because only parts of the sets were joined. 

The shoes they threw out formed a set which vvas the union of 

Mary's unwanted shoes with Liz's unwanted shoes. 

Not a true union since the new set is not made up of all the parts of 

the old sets. 

The set of all squares and circles. 

The set of all rectangles (including squares) 

The set of numbers 2 n where n is a whole number, positive or 

negative. 

All words beginning with A or B. The first two sections of the 

dictionary. 

M83 A. 

B. 

1. U 2. E 

6. x, x 7. y, Y 

1. U 2. E, E 

5. Cubes, primes 

3. P 

8. Y 

4. E 5. =? 

3. W, V 4. x 

6. Odds U evens 

M84 A. 1. 

_~_A~ 

3. Same as question 2. 

5. 

A 

4. Same as question 1. 

x• 

B 

82

B. 

• Ahmed 

• Fred 

B 

• Jenny 

C. 1. (a), (b), (c), (e) belong to AU B 

2. (a) (b) (c, possibly), (e, possibly), (f, possibly) 

M85 A. 1. {Dave, Pete} 2. {Britain, Netherlands} 

3. {Red, Blue} 4. {The point D} 

B. 1. (a) A teacher with a child of his/her own in the school 

(b) Nobody 

(c) A parent that works in the school 

(d) Nobody 

2. (a) People with brown eyes and fair hair 

(b) Sea animals such as whales which suckle their young 

(c) Letters that have a vertical and a horizontal axis of symmetry, 

e.g. X, 0 

(d) Flats with 3 bedrooms and gardens 

(e) Team sports in which a ball is used 

3. (a) (D·1, 0·2, 0·5, 1·0) (b) All prime numbers except 2 

(c) All even square numbers 

(d) (1, 2, 3, 4 .... ) since all these numbers appear in both sets. 

)1 = 1, )4=2, )9=3, ... and so on 

P52 A. 1. 

2. 

3. 

4. 

5. 

B. 1. 

2. 

{ABCDE 12345}. 

Teams "in 1st divisi.on together with teams in 2nd division 

(a), (b) only 

Different choice of answers, 6 even numbers would do or 6 

multiples of 1D or a mixture 

Circle, parallelogram, rhombus ... etc. 

Different answers. 

(a) Mary belongs to the Girl Guides 

(b) Roger does not belong to the Naval Cadets 

(c) Patrick is a Catholic 

(d) Elsa is not human (she was a famous lion), 

(e) Blanco is not a drink 

(f) Deadly nightshade is a poisonous plant 

83

3. (a) Is true (b) Is not true. Examples will be different. 

4. (a) Jan is not at nursery school 

(b) Peter is under five 

(c) Francis is not under 5 

(d) Gary is under 5 or at nursery school 

(e) Joanne is not under five or at nursery school 

(f) If a child is at nursery school then he or she is under five years 

old. 

C. 1. (a), (b), (e), (f), (g) 

2. (a) Countries which are in Africa or contain desert. 

(b) No, e.g. Arabia. (c) No. 

(d) China, USA, Nigeria ... China and USA have desert land. 

Nigeria is in Africa. 

3. (a) Is true (b) Is true (c) Not always true, he uses cigarettes 

or alcoholic drinks but maybe both. 

(d) Not true (e) True (f) Not true 

P53 A. 1. (a) A (b) A (c) x, y (d) z 

(e) No points in A and in B 

2. (a) Agrees (b) Does not agree (c) Agrees (d) Agrees 

3. (a) Yes (b) Yes (c) Yes (d) Yes (e) No 

(f) Yes 

B. 1. (a) (iii) and (iv) (b) (i), (ji), (iv) 

(c) 5 triangles (d) Triangle (iv) is counted twice 

2. 

c. 

1. A set made up of the members which belong to both sets 

2. (a) A n B is squares 

(b) No 

(c) Yes 

(d) Rectangles which are not squares 

3. A 

A 

8 8 

(a) 

(b) 

A 

(e) 

(d) 

4. (a) True. If x E A n a then x E A and therefore x E A U a 

(b) A n B = (10, 20, 30, 40 ... ) 

Au B (2, 4, 5,6,8, 10, 12, 14, 15 ... ) every member of An B 

belongs to A U B. We can describe this situation as 

xEAnB=?XEAUa 

(c) A U is all the people of Canada who speak French or English 

A n a is all the people of Canada who speak both French and 

English 

x E A n B =? x E A, people who speak French and English 

speak 

English 

x E A n B =? x E a, people who speak French and English 

speak French 

8 

E53 

A., B., see P52 

C. 1. (a) All even numbers are whole numbers 

(b) Isosceles triangles are 'triangles' 

(c) Potatoes are vegetables 

(d) Pupils in the class belcung to the school 

Not that the reverse of each statement is not true 

2. (a) Not true (b) Not true (because of 2) 

(c) True 

(d) True 

3. (a) A is a subset of A u a because if you belong to A you must 

belong to the union of A with B 

85

(b) If you belong to B you must belong to the union of B with A 

(c) If you belong to A n B you must belong to A (and also to B) 

(d) If you belong to A you must belong to A, i.e. A c A (this seems 

odd!) 

E54 

1. The next perfect number is 28, the next is 496. After that they take 

quite a long time to find as the next two are 8128 and 33550336. 

2. (a) It happens because 26 is 25 + 1 so 48 x (25 + 1) gives 

1200 + 48 and the 48 reappears 

(b) 76 has the same property, so has 51 

3. (a) 37 is a factor of 111 so it is a factor of all numbers such as 222, 

333, etc. 

(b) Multiples of 37 are either of the form nnn or the sum of their 

digits is the same as n + n + n. Multiples of 37 such as 2664 

have first and last digits adding to the same as the central 

digits of the number 

4. (a) The number is just rearranged but the numbers remain in the 

same order 

(b) Another number is 0·076923 which comes from ,1 or better 

still 0·0588235294117647 which comes from ,j 

5. Investigation 


The ideas of set algebra are taken a little further in this unit 

to consider the universal set and the empty set. Most of the 

exercises concern the language. In many ways set algebra is 

easier than traditional algebra because the symbols are quite 

distinctive and it is not so heavily concerned with manipulation. 

Some writers say that this type of exercise is not relevant 

to non-academic children. Careful thinking 'develops' the 

mind and if work on sets is omitted at this stage children's 

potential for analytical thinking could well remain undeveloped. 

Remember this work does not have to be 'learned'. 

It is experience of a simple symbolic language. 

Answers 

M86 A. 

1. n 

6. F 

2. n 

7. xEQ 

3. and 4. E 

8. n, A, Q 

5. M 

86

B. 1. If x does not belong to A then x does not belong to A n B. True 

2. If x belongs to A then x belongs to A n B. Not true 

3. If x does not belong to B then x does not belong to A n B. True 

4. If x belongs to A n B then x belongs to A. True 

5. If x does not belong to A n B then x does not belong to A. Not true 

6. If x does not belong to A n B then x does not belong to B. Not true 

7. If x does not belong to A n B then x does not belong to A and also 

x does not belong to B. Not true 

8. If x does not belong to A and x does not belong to B then x does 

not belong to A n B. True 

C. 1. Yes,0 2. Not 0 3. 0 4. Not 0 5. 0 

6. 0 7. Not 0 (squares) 8. Not 0 

M87 A. 1. 

4. y could be in Q instead 

5. g does not belong to the football 

team or the hockey team 

6. xn Y squares 

B. 1. (a) y 

(f) x, k 

2. (a) 6131 

(b) x (c) z (d) x, y, z (e) k 

(g) z, k 

(b) 8270 (c) 1875 (d) 6395 (e) 4256 (f) 10651 

87

3. people with both 

A 

7 

13 

4 P 

C. 1. 13 children 

2. (a) Numbers which are multiples of 3 and 4, i.e. the multiples of 

12. 

(b) (i) Yes (ii) Yes (iii) Yes (iv) No (v) Yes 

Maa A. 1. Fruits 

3. Whole numbers 

6. Animals 

B. 1. True 

6. True 

11. True 

2. True 

7. False 

12. True 

4. Words 

7. Fish 

2. 

3. True 

8. False 

Numbers and fractions 

5. Names 

a. Motor cars 

4. True 

9. True 

5. True 

10. True 

C. ·1. All the numbers except 1, 2, 3, 4 

2. The odd numbers 

3. All numbers which do not end in 2 

4. All numbers which do not begin with 2 

P54 A. 1. A U B = things that are red or edible, or both. A n B = red food. 

88 

2. (a) Yes (b) Yes (c) Yes 

(d) Could be (depends whose blood). 

3. (a) No (b) No (c) No 

4. More in A U B 

5. (a) True (b) Not true (c) Not true 

(d) True (e) Not true (f) True 

B. All different answers. 

C. 1. (a) 6 E X n Y (b) 10 E X n Y 

(c) 7 $. X n Y (d) 8 E X n Y 

2. True (a), (d); not true (b), (c)

3. (a) 10 belongs to the set X implies 10 belongs to the union of X 

with any other set 

(b) If 19 does not belong to X it does not belong to the intersection 

of X with any other set 

(c) If 9 does not belong to Y then 9 does not belong to the 

intersection of X and Y 

(d) 25 belongs to the union of X with Y implies that 25 belongs to 

X or to Y (or perhaps to both) 

4. (a) 5 ~ Y::} 5 ~ X n Y 

(b) {1a E X and 1a E Y}::} 1a E X n Y 

(c) xn Y= YnX 

P55 A. 1. 

x 

y 

2. 

3. You cannot like tea best and 

like beer best at the same 

time. 

00 

4°CltersQ ) 

5. 

B. 2. P n Q is shaded horizontally and vertically, 

P U Q is anywhere that is shaded at all 

(h) 

E55 A., B. See P54. 

3. (b) Horizontally and vertically 

(c) Yes (d) A nB=A (e) A nB=0 

(f) A n B = B (g) A n B = A ... if you think squares are a subset 

of the set of rectangles 

An B = B 

C. 1. (a) 0' = {trees other than oaks} 

(b) B' = {trees other than beeches} 

(c) 0' u B = 0' (d) S' U 0 = S' 

89

2. (a), (b) 0 u 0' = e by definition: every tree either is an oak or 

is not an oak 

(c), (d) 0 n 0' = 0 since no tree is both an oak (E 0) and not an 

oak (E 0') 

3. Different universal sets are possible, giving different complementary 

sets. 

4. The lopposite' picks out a special set which has a special opposite 

quality. The complement consists of everything else besides the 

set in question. For example the complement of the set Itall 

people' is the set Ipeople who are not tall'. This is not the same as 

the set of short people. 

5. (a) (b) (c)

2. (a) 7 (b) 9 (c) 21 

3. Drawings 

B. 1. (a) 

0 

0 2 3 4 

(b) 

0 

-1 0 2 3 

(e) 

0 

-4 -3 -2 -1 0 2 

! 

• 

(d) 

0 

-3 -2 -1 0 

(e) 

~ 0 

2 3 4 5 6 

(f) 

0 0 

-4 -3 -2 -1 0 2 3 

2. (a) The part of the number line between 3 and 5, not including 3 

or 5 

(b) the whole number line 

(c) x : x < 3, and 3 itself 

(d) x: x > 5, and 5 itself. 

C. 1. (a) (b) (e) 

~ ~ ~ 

91

(d) (e) (f) 

2. When the diagrams are shaded it is easy to see that the sets are 

equal 

3. (a) Diagram (b) (i) 3 (ii) 7 (iii) 5 (iv) 0 (v) 42 


The first step in statistics is to learn to look at collections of 

data without feeling overwhelmed. The terms data and 

frequency are important and the idea of a variable should be 

discussed at this stage. It is not necessary to define these 

terms but children will get used to using them in context. 

Data which is of particular topical interest (e.g. Olympic 

Games) or which is proposed by the class is of great value in 

giving the feeling of importance to this work. There are 

several useful collections of data some of which will be 

known to the children. A small library of data might include 

the Guinness Book of Records, Whitaker's Almanack, the 

Penguin Book of Facts, World Statistics in Brief (United 

Nations) and Annual Statistics of the EEC. Interesting 

alternative approaches may be found in the problems of the 

Schools Council Statistics in Education Project. All data 

should be presented in a very clear form. (Remember the 

school micro will have ways of presenting data which will be 

of interest to your pupils.) 

Answers 

M89 A. 

1. 4 2. 7 3. 4 4. 11 5. 19 

6. (a) 204 (b) 235 (c) 235 (d) 228 (e) 194 (f) 172 

7. 13 8. 1974 9. 1980 

10. 1970-74; 1154 students 1976-80; 1010 students 

92

B. 

c. 

1. 17 

4. 13 

7. 1st quarter 

10. 320/0 

1. 

5. 

6. 

34 2. 76 

3 to 4 hours 

More than 5 hours 

2. 22 

5. April 

8. 4th quarter 

3. 13 

6. November 

9. The total number of children 

3. 40 4. 27 

M90. A. 1. Physics 2. Biology 3. 253 4. 74 

5. 185 6. 119 

7. Chemistry 104, physics 117, biology 174, Rural science 30, 

Domestic science 68, Engineering science 44, 

General Science 51 

8. Biology 86 difference 9. Rural science 

10. Boys and girls expect to have different jobs 

B. 1. 77 2. 108 3. 175 

4. A higher percentage of boys can swim than girls 

5. A higher percentage of girls can swim over 5 lengths 

6. Once a swimmer can swim more than 3 lengths he or she is likely 

to swim longer distances 

M91 1. (a) 73p (b) 47p 

(c) 22p (d) 16p 

2. 73p, January 

3. (a) March and November 

(b) January, February 

(c) August, September 

(d) April, October 

4. October 

5. Tomatoes from Spain and the Canary Islands are ripe 

6. 57p per Ib 

7. Not quite the same as crops are harvested at different times. Also 

some crops keep better than tomatoes so they can be sold over a 

longer period of time. This avoids a glut (very low prices) followed 

by a sharp rise in price. 

P56 A. 1. Dogs 2. 73 3. 275 4. 121 5. Rabbits 

B. 1. (a) 170 (b) 243 (c) 142 

2. Bad signals 3. Alcohol/drugs 4. 1000 

93

C. 1. Chocolate 

2. Sweet 

3. (a) About 100 (b) About 70 (c) About 22 

4. About 425 

5. Chocolate, savoury, ginger, shortbread, custard, cream, cheese, 

sweet 

P57 A. 1. (a) 500 000 (b) 1 450 000 (c) 300 000 

2. (a) 1 260 000 (b) 450 000 (c) 3 150 000 

3. (a) France, Italy, Spain, W. Germany, other countries 

(b) Austria, Ireland, Switzerland 

4. Spain and France 

5. Still Spain 

6. 1968; apart from other countries, Spain, France, Ireland and Italy, 

Austria, Switzerland and W. Germany. 

1978; Spain, France, Italy, Ireland, W. Germany, Austria and 

Switzerland 

France has moved well ahead of Italy and Ireland. 

W. Germany is more popular than Switzerland and Austria (shorter 

journey, lower cost of living). 

B. 1. (a) 71 000 (b) 17000 (c) 3000 

2. (a) 4000 (b) 48000 (c) 47 000 

3. The number of farm workers dropped from 75000 to 43 000 

4. (a) True (b) True 

5. Horses were replaced by tractors as tractors were more efficient. 

(Though there is some doubt about this because 

(i) petrol is expensive; 

(ii) tractors wear out; 

(iii) horses produce new horses; 

(iv) horses produce fertilizer) 

6. Yes, roughly 

7. 43 000 8. 28000 

9. Farms became larger so that one tractor (and ploughman) served 

a much greater area. 

E57 A. See P56 A and 6. 30% 

7. Hamsters, guinea pigs, fish, birds ... etc. 

8. Different answers 

B. See P56 Band 



94

C. 1. Chocolate 

2. Sweet and cheese 

3. (a) 100 (b) 70 (c) 20 

4. About 425 

5. £138·74 

6. Chocolate, savoury, ginger, shortbread, custard, cream, cheese 

and sweet 

7. 3 cheese, 10 savoury, 3 sweet, 7 shortbread, 7 creaJ'!l, 

14 chocolate, 7 custard and 10 ginger 

D. Reasonable 

E58 A. 1, 2, 3, 4 see P57 A 

5. The answer to this question depends on how you think the 

changes will happen. For example, Austria reduced from 35 to 18, 

Le. a reduction factor of 18/35. If you apply this reduction to 18000 

you get 18000 x 18/35 = 9257. So 9300 would be a reasonable 

prediction. If you say the numbers fell by 17000 and this would 

happen again the result would be only 1000 holidays in Austria in 

1988. This method seems less reasonable. 

Applying the first method of prediction gives estimates as 

follows for 1968: Austria 93000, France 3 170 000, Ireland 

405 000, Italy 1 040 000, Spain 5 030 000, Switzerland 108 000, W. 

Germany 432 000. 

In practice, it is unlikely that the factors producing the change in 

the period 1968-1978 would continue to operate in the same way 

for the next decade. 

6. Order of popularity: 

1968 Spain, France, Italy and Ireland, Austria, Switzerland and 

W. Germany 

1978 Spain, France, Italy, Ireland, W. Germany, Austria and 

Switzerland 

1988 Spain, France, Italy, W. Germany, Ireland, Switzerland, 

Austria 

B. 1. (a) 71 000 (b) 17 000 (c) 3000 

2. (a) 4000 (b) 48 000 (c) 47 000 

3. (a) The number dropped from 75 000 to 43 000 

(b) Farms become more capital intensive 

4. True 

5. Tractors are more efficient 

6. Yes, roughly 

7. (a) 43 000 (b) 28 000 8. Different answers 

C. Drawing and different answers 

95


The idea of average is widely used for comparisons and to 

summarize data. However it is not always well understood in 

the context of problems. Many people would say that 

'nobody should be paid less than the average wage' without 

realizing that this means that no-one should be paid more 

than the average wage either. The concepts of average 

weight and height are related to health, yet a person can be 

far from average and be perfectly healthy. The idea of 

average contains the notion of acceptable range and deviation. 

We, as mathematics teachers, sometimes misapply the 

notion of average child, mixing up the idea of what an 

average child should be able to do (in which case 50% will 

not be able to do it) with the expectation of every child (in 

which case it will probably be such a low level as to be 

worthless) . 

The exercises of this unit emphasize the idea of average as 

a single number representing a set of data, the relationship 

between average and total and also the use of averages to 

make comparisons. Further interesting examples can arise 

from class discussion. Calculators with statistic circuits will 

have an 'average' button marked x. 

Answers 

M92 A. 1. 7·8 minutes 2. 7·2 minutes 3. 2·25 

4. 4·25 5. 10·49 seconds 6. 7·1 

B. 1. £5·30 2. 41·77 em 3. 3·2 kg 

4. 37·18°C 5. 26·07 km 

C. 1. 14 2. £66 3. £84·30 

4. 252 hours 5. 4320 6. Yes, by 46 kg 

M93 1. (a) 215·7, corrected to 216 as 0·7 of a child does not mean 

anything 

(b) 1974, 1975, 1976, 1977 (c) 1978, 1979, 1980. 

2. (a) 12·5 (b) 12·5 (c) 12·5 (d) Jan, Feb, May, Nov, Dec. 

(e) Dark 14·8; light 10·2 

(f) The dark months are more dangerous 

3. (a) 58·6p 

(b) Because the fruit does not ripen till October 

(c) Eggs, milk, bread ... 

96

4. (a) 10·1 (b) 1976, less rain 

(c) 1978 after two dry years the reservoirs would be low 

P58 A. 1. £659·17 2. £776·33 3. The second half 

4. March, June, July, August, September 

B. 1. 61·6 

2. 60, worse than last term so far (but better than last term's average 

on the first three marks). 

c. 1. Paul 101·7 minutes; Joanne, 106·8 minutes. Joanne's average was 

higher but things might have been different ifshe had timed her 

viewing on Thursday. 

2. Cod 266·33; haddock 142; plaice 35·2; herring 128-8; lobster 0·97. 

All the catches are declining. This is because of overfishing. 

P59 1. x f xf 2. x f xf 

0 3 0 0 42 0 

1 7 7 1 36 36 

2 14 28 2 33 66 

3 12 36 3 41 123 

4 8 32 4 6 24 

5 3 15 5 22 110 

6 3 18 

- - 

6 20 

- 

120 

- 

50 136 200 479 

Average 2-72 Average 2-395 

3. x f xf 

0 300000 0 

1 5600000 5600000 

2 9400000 18800000 

3 1300000 3900000 

4 265000 1 060000 

16865000 29360000 

The average is 1·74 radios per household. This seems reasonable 

because many people have radio alarm clocks, etc., but accurate data 

would be difficult to collect. 

97

E59 A. 1. See P58 A. 2. See P58 B. 

3. See P58 C. and (b) A reasonable guess would be a little below 

average, i.e. 90-100. 

4. See P58 C. 2 and (c) 85·4 million, about 1~ fish per 

person (d) Overfishing. 

B. See P59 

EGO A. 1. Different answers 2. Different answers 

B. 1. £106·92 2. £89·16 

3. £124·67 since (124·67 x 6) + (89·16 x 6) = (106·92 x 12). She has 

to make the same total if she wants the same monthly average. 

4. Green £75·05; Jones £71·14. They also need to know what the 

money was spent on and to compare the number of miles they 

had travelled. 

C. 3. Assuming that after 1000 throws the scores settled down you 

would expect the average score to be 

(1 + 2 + 3 + 4 + 5 + 6) 7 6 = 3·5 

5. The average score wou Id not be (1 + 2 + 3 + ... 12) 7 12 because 

the scores are not equally likely. The average score should be 7 

and not 6·5. Out of 36 throws the exact expectation is shown in 

this table. 

score 

f 

2 3 4 5 6 7 8 9 10 11 12 

1 234 5 6 543 2 1 

(7 for example can occur as 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2 or 6 + 1) 


Drawing and using straight line graphs is a basic skill of 

elementary mathematics and of everyday life. Of these, the 

family y = kx is the simplest and most important. We only 

need to know one point and the whole line may be drawn. 

The point that a straight line is a mathematical ideal can be 

discussed. This gives a greater reality to children who 

98

understand that the world of commerce allows discounts for 

large purchases. The main purpose of this unit is to recognize 

the form y = kx in problems, draw and use the graph, 

and to see the effect of adding a constant to form y = kx + c. 

Answers 

M94 A. 

B. 

1. Drawing 

2. (3,4); (2,5); (1, 1·5); (2·2,3·2); (0,4) are above the line 

while the rest are below the line 

If the y-co-ordinate is greater than the x-eo-ordinate the point 

will be above the line. 

If the x-eo-ordinate is larger the point will be below the line 

1. Drawing 

2. (1·5,3); (2·1,4·2); Because the y-co-ordinate is exactly twice the 

x-eo-ordinate, i.e. y = 2x. 

3. (1,3); (0,4) are above the line. The rest are below the line. The 

points are above the line if the y-co-ordinate is more than twice 

the x-eo-ordinate 

4. x 0 1 2 3 4 5 

y 0 1·5 3 4·5 6 7·5 

(a) Different answers are possible 

(b) (6,9); (7,10·5); (1·5, 2·75), etc. 

(c) Only one 

(d) Only one 

C. Drawing 

M95 A. 1. x = number of kilos, y = total cost, k = 10 pence 

2. x = number of kilos, y = total cost, k = 80 pence 

3. x = number of metres, y = total cost, k = £1·80 

4. x = number of miles, y = total cost, k = 75p 

5. x = number of hours, y = total distance, k = 8 km 

6. x = number of hours, y = total distance, k = 3 km 

B. 1. 120p 2. 20miles 3. 96g 4. £15 

C. 1. (a) 17p (b) 25p (c) 42p (d) 84p 

2. (a) 2 (b) 3 (c) 5 (d) 12 (e) 30 (f) 42 

M96 

Drawing 

99

P60 A. 1. Drawing 

2. (a) Yes 

(b) £26·40 

(c) a = 5 

(d) b=8·4 

3. (a) Different points are possible 

(b) Yes 

(c) Different points are possible 

(d) This is not certainly true for all points. For example (2,2·2) 

would come below the line but the y-co-ordinate is greater 

than the x-eo-ordinate. 

B. 1. Drawing 

2. (a) Only (0,0) (b) Different points 

(c) Depends on (b) (d) Should be true 

3. (a) Different points 

(b) Because the y-co-ordinate is more than double but less than 3 

times the x-eo-ordinate, e.g. (2,5) will be between the lines 

y= 2x and y= 3x 

C. 1. (0,0); (2,3); (4,6) are allan the line y = 1~x (3,5) is the other point 

asked for in the question 

2. (6,12) 3. (0,0) 4. (7,6) 

P61 A. 1. Drawing 

2. (a) £6·40 (b) £9·80 (c) £16·20 

(d) £6·96 (e) £11·76 (f) £38·40 

100 

B. 1. (a) £97·75 (b) £103·50 (c) £7·48 

(d) £20·70 (e) £3·91 (f) £7·82 

2. (a) The graph would be y = 1·25x instead of y = 1·15x, i.e. the 

slope would be greater. 

(b) The line still passes through (0,0) 

C. 1. (a) y = 4·5x + 14 

(b) £68, £122, £176 

(c) 19 months 

2. The graph required is y = 0·12x + 78. This must be drawn on 

1 mm 2 graph paper if an accurate answer is to be obtained. 

100 miles cost £90 



1000 miles cost £198

E61 A. See P60 A. 

B. 1. It is the reflection in the x-axis (and also in the y-axis) 

2. (a) Drawing (b) They are perpendicular (c) (0,0) 

3. They have the same value but opposite signs (a and -a) 

C. 1. (3,3) 2. (0,0) 3. (1, -3) 4. (3,0) 

E62 A. 1. Different answers 

2. The cost of cement decreases as you buy larger quantities so the 

graph will not be a straight line 

B. 1. (a), (c) and (d) 

2. £6·40, £6·96, £9·80, £11·76, £16·20, £38·40 

3. (a) £6210 (b) £7425 (c) £6534 (d) £8694 

(e) £9315 (f) £11475 (g) £17280 

C. See P61 C. 


Ratio is one of the least successfully taught topics in 

mathematics. This is because one is asking the children to 

hold a relationship in their head while they juggle with it. It 

is, however, a fundamental concept and failure will undermine 

many of the ideas that will be taught in science, 

mathematics, home economics and geography. At least, use 

of the calculator removes fear of arithmetic which is one of 

the contributory factors to failure in this topic. In all the 

examples the relationship between m: nand m/m + n is 

explored in the context of problems. These problems should 

be considered as material for discussion and children should 

be encouraged to persist until they fully understand what is 

going on. 

Applications in geometry provide an alternative approach 

to the concept of ratio and incidentally revise some of the 

more important geometrical relationships. 

The ratio sign is just another division sign and it is worth 

noting that both the fractional 'over' sign and the ratio sign 

are present in our division sign -:-, 1: 2, t, 1 -:-2 

101

Answers 

M97 A. Ratios of boys: girls (not girls: boys): 

1. 17: 19 2. 25: 40 (simplifies to 5 : 8 or 1 : 1·6) 

3. 6: 8 (=3 : 4) 4. 232: 250 (=0·928 : 1) 

5. 22: 18 (= 11 : 9) 

B. 1. 42: 65 2. 360: 240 (=3 : 2) 3. 480: 74 (=240 : 37) 

4. 15000: 31 000 (=15 : 31) 5. 194: 6 (=32·3 : 1) 

6. 75: 25 (=3 : 1) 

Note that a given ratio can be expressed in a number of ways. 

C. 1. (a) 1: 1·2857 (b) 1·6: 1 (c) 1: 1·2 

(d) 1:2·222 (e) 1:1·428 (f) 1:1·67 

(g) 1: 1·67 (h) 1: 4 (i) 1: 2 

2. (a) 1: 8 (weight of oxygen 0·120 g) (b) 1: 3 

(c) 5: 100 = 1: 20 

M98 1. (a) (i) 21:39 (ij) 9:34 (iii) 23:33 (iv) 27:21 

(b) 600 m 

(c) 3: 2·4 This is the only ratio in which you go up more than 

1 unit for 1 along 

2. (a) 200 (b) 45 

3. NO.4: 5 is a better ratio than 6: 8 

4. 78·75 mm 

5. (a) £296. Note: It will be larger than £185 by a scale factor 

. 

4: 2·5. Thus the fare IS 

185 x 4 

2.5 

305 x 2·5 

(b) £190·63... 4 

M99 Usually the best known constant ratio is 7T, the ratio of circumference to 

diameter of circles. 

102 

A. Drawing 

B. If the rectangles are the same proportion the ratios will be constant, 

otherwise not 

C. 1,2 Drawing.AB:AD=1·55:1. 

3. Yes, the ratio altitude: side is constant for equilateral triangles 

D. All drawing and measuring 

E. All drawing and measuring 

There are no P or E exercises with this Unit.

· .•··i •• ;;::{;i;......................... .·····..·..··.·.·...i.·? 

INTEGRATED 

M&JIiEMI\TICS 

SCHEME 

1"1 T2 

Bell& Hyman limited 

ISBN 0 7135 1339 X

Integrated mathematics scheme: IMS T2 - National STEM Centre

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