Integrated mathematics scheme: IMS T2 - National STEM Centre
Integrated mathematics scheme: IMS T2 - National STEM Centre
Integrated mathematics scheme: IMS T2 - National STEM Centre
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INTEGRATED 1"5,12<br />
MATHEMATICS<br />
SCHEME<br />
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INTEGRATED<br />
MATHEMATICS<br />
SCHEME<br />
Teacher's Book 2<br />
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N32233
INTEGRATED<br />
MATHEMATICS<br />
SCHEME<br />
Teacher's Book 2<br />
Peter Kaner<br />
Bell & Hyman
First published in 1982 by<br />
Bell & Hyman Limited<br />
Denmark House<br />
37-39 Queen Elizabeth Street<br />
London SE1 2QB<br />
Reprinted 1984<br />
Reprinted with amendments 1985<br />
© Peter Kaner 1982<br />
All rights reserved. No part of this publication may be reproduced,<br />
stored in a retrieval system, or transmitted, in any form or by any<br />
means, electronic, mechanical, photocopying, recording or otherwise,<br />
without the prior permission of Bell & Hyman Limited.<br />
Kaner, Peter<br />
<strong>Integrated</strong> <strong>mathematics</strong> <strong>scheme</strong>.<br />
Teacher's book 2<br />
1. Mathematics-1961-<br />
I. Title<br />
510 QA39.2<br />
CENTRE<br />
ISBN 0 7135 1339 X<br />
Typeset by Polyglot Pte Ltd, Singapore, printed and bound in<br />
Great Britain by William Clowes Limited, Beccles and London
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- REVISION BOOK- REVISION BOOK<br />
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NON ACADEMIC (30%) CSE (50%) GCE (20%)
Unit 1, page 1<br />
Comment<br />
Negative numbers has been considered a difficult topic in the<br />
past. The use of a calculator makes it into a topic that can be<br />
handled and explored by most children. The key to the<br />
whole thing lies in considering whether a larger number is<br />
being taken from a smaller. Many of the physical explanations<br />
confuse the issue so we use them as applications<br />
rather than explanations. Once children understand that<br />
+ 1 + -1 = 0 the rest can be developed easily. A helpful aid<br />
is to make red and blue cards 1+11,1-11with [Q] on the back<br />
of both cards, showing that a + 1 and -1 together can be<br />
treated as zero. This leads to the following explanations of<br />
addition and subtraction of negatives.<br />
(i) 3 + 5 ... (+1, +1, +1) + (+1, +1, +1, +1, +1)~ +8<br />
(ii) -3+-5 ... (-1,-1,-1)+<br />
(-1, -1, -1, -1, -1)~-8<br />
(iii) - 3 + 5 ... -1 -1 -1<br />
+1 +1 +1 +1 +1~ +2<br />
tit<br />
000<br />
(iv) +3+-5 +1+1+1<br />
-1-1-1-1-1~-2<br />
t t t<br />
000<br />
From (iii) it is clear that adding - 3 has the same effect as<br />
subtracting 3, and this leads to all the required subtraction<br />
results.<br />
(v) 4-3~4+-3~1<br />
(vi) - 2 - - 3~ - 2 + 3~ 1<br />
(vii) 2 - - 3~ 2 + 3~ 5<br />
(viii) -4 -7~ -4 + -7~ -11<br />
The subtraction can be confirmed by the calculator and also<br />
by 'take away reasoning' as follows.<br />
1
We wish to take - 3 away so we introduce 3 zero pairs made<br />
up of +1 -1 without changing the value<br />
+1+1+1+1+1+1+1<br />
-1-1-1<br />
now subtract -3 and clearly this leaves 7.<br />
Or grouping<br />
the + Is<br />
4 - -3 ~ [4+ (+ 3 + - 3) - -3] ~ 7<br />
Note: It is helpful to give the number line a tilt so that<br />
smaller numbers are lower than larger ones.<br />
Multiplication by negative numbers is developed in the<br />
supplementary units PI and E 1 .<br />
Answers<br />
M1 1. (a) -3 (b) -8 (c) -9 (d) -7<br />
(e) -29 (f) -23 (g) -36 (h) -28·2<br />
2. Checks.<br />
3. (a) b is 5 more than a (b) b - a = 5<br />
4. (a) 0 (b) 0 (c) 0 (d) 0<br />
5. (a) -2 (b) -2 (c) -3 (d) -6 (e) 16<br />
(f) -6 (g) -1·7 (h) -2·3<br />
M2 1. Checking 2. Checking<br />
3. (a) a = b (b) a = b (c) Always true so it tells you nothing<br />
about a and b.<br />
(d) Always true.<br />
4. (a) -2 (b) -9 (c) -4 (d) 1<br />
(e) -8 (f) -18 (g) -2 (h) -17<br />
(i) -6 (j) -3 (k) -3 (I) -6<br />
(m) -5 (n) -7 (0) -2 (p) -5<br />
(q) 8 (r) 13 (s) 14 (t) 23<br />
M3 1. (a) 4 (b) 4 (c) 2 (d) 6 (e) 5<br />
(f) -3 (g) -7 (h) -5 (i) -8 (j) -9<br />
(k) -9 (I) -24<br />
2. (a) 2 (b) 3 (c) 4 (d) 5 (e) 7<br />
(f) 5 (g) 9 (h) 10 (i) -2 ( j) -5<br />
(k) -7 (I) -14<br />
2
3. (a) 11<br />
(f) 0<br />
(k) -53<br />
(b) 8<br />
(g) -5<br />
(I) -100<br />
(c) 10<br />
(h) -14<br />
(d) 8<br />
(i) 15<br />
(e) -7<br />
(j) 31<br />
M4 A. 1. All in a vertical line 2. Vertical line<br />
3. Horizontal line 4. Vertical line<br />
5. Two lines which are reflections in the y axis<br />
B. 1. The points all lie on a straight line. If (x, y) satisfies y = x + 1 the<br />
point will be on a certain straight line through (0,1), (2,3)<br />
2. (a) (0,0) (b) (2,0); (0,2); (-2,0); (0,-2)<br />
3. A parallelogram, centre (0,;)<br />
4. Drawing<br />
5. (-2, -1); (-4, -2) Yes 6. Drawing<br />
M5 B. 1. -16°e 2. -20 o e 3. -6°e 4. -16°e 5. -16°e<br />
6. 7°e<br />
C. 1. Ground level 2. -5 3. +2 4. -1<br />
5. +7 6. +1 7. -9 8. -7 9. -14<br />
He loses another 16000ft in landing<br />
P1 A. 1. (a) 1 (b) -1 (c) 3 (d) -8 (e) 1<br />
(f) -3 (g) -25 (h) -43 (i) 3 (j) 3<br />
(k) 11 (I) 8 (m) -9 (n) -30 (0) -80<br />
(p) -96<br />
2. (a) 5 (b) 5 (c) -4 (d) -13 (e) 16<br />
(f) 21 (g) 29 (h) 60 (i) -8 (j) -17<br />
(k) -26 (I) -40 (m) 2 (n) 4 (0) 0<br />
(p) -21<br />
B. 1. (a) -2 (b) -5 (c) -6 (d) -5 (e) 3<br />
(f) 7 (g) 3 (h) -8 (i) 20 (j) 36<br />
(k) -9 (I) -30<br />
2. (a) 4 (b) 13 (c) 8 (d) 26 (e) 54<br />
(f) -10 (g) -8 (h) 80 (i) -10 (j) -20<br />
(k) -57 (I) -6<br />
3. (a) 30 (b) -10 (c) 58 (d) 57 (e) 6<br />
(f) 16 (g) -28 (h) -20 (i) 22 (j) 12<br />
(k) -52 (I) -23<br />
3
P2 A. 1. 6·3 2. 25·7, -19·4<br />
3. (a) £26·88 (b) £-41·95<br />
4. (a) 157 m above ground<br />
(b) +200, -70, +60, -25, -38, +40, -50, +175, -55, -80<br />
(c) 157 (d) The balloon had come back to Earth<br />
5. 0·7°C above normal<br />
B. (a) £570 (b) £262 (c) £1145 (d) £900<br />
C. 1. (a) -20 (b) -21 (c) -10 (d) -30 (e) -60<br />
(f) -32 (g) -21 (h) -96 (i) -21 (j) -8<br />
(k) -25 (I) -54<br />
2. (a) 4, 8, 12, 16, 20 (b) 6,12,18,24<br />
3. (a) 21 (b) 30 (c) 16 (d) 5 (e) 36<br />
(f) 81 (g) 60 (h) 55<br />
E1<br />
A. 1.<br />
2.<br />
B. 1.<br />
2.<br />
3.<br />
As for P1 and (q) 0·7 (r) -2·3 (s) -4·4 (t) -1·17<br />
As for P1 and (q) 7·2 (r) 6·5 (s) 1·2 (t) 0·88<br />
As for P1 and (m) 5·6 (n) 0·6 (0) -7·3 (p) 13·6<br />
As for P1 and (m) -7·8 (n) +3·6 (0) 5 (p) 3·4<br />
As for P1 and (m) 9·2 (n) 6·6 (0) -0·5 (p) -2·2<br />
E2<br />
E3<br />
1.<br />
2.<br />
3.<br />
A. 2.<br />
B. 2.<br />
As for P2C and (m) -9·6 (n) -11·5 (0) -10·2 (p) -31·5<br />
As for P2C and (c) -12, -24, -36, -48<br />
As for P2C and (i) 4·5 (j) 5·2 (k) 22 (I) 33·6<br />
(a) (5 x -8) + (5 x 8) = 5 x (8 + -8) = 5 x a = a<br />
So 5 x -8 is the negative of 5 x 8, Le. 5 x -8 = -40<br />
(a) Suppose n,<br />
is the negative of 22 and n2 is another negative.<br />
n, + 22 + n2 = (n, + 22) + n2 = a + n2 = n2 also<br />
n, + 22 + n2 = n, + (22 + n2) = n, + a = n,<br />
So n, = n2, both negatives are the same, etc.<br />
C. 2. 16x16=16x(16+0)=?16x16=16x16+(16xO)<br />
=? 16 x a must be zero<br />
4
Unit 2, page 6<br />
Decimals and percentages<br />
A calculator will give correct answers to decimal calculations<br />
without reference to any rule so that fear of decimals can<br />
evaporate overnight. The weaker children may remain<br />
scared, however, and will need careful reassurance that the<br />
well understood system of grouping by tens continues<br />
below 1. Percentages are something which many people fear<br />
but the grouping of the two topics proves very helpful. A<br />
percentage, e.g. 35% means 35/100 and so can be written<br />
0·35. If this is established, percentage questions become<br />
very easy. If the calculator has a % button this should be<br />
investigated by the students following some set examples.<br />
Answers<br />
M6 A. 1. (a) 2 (b) 4 (c) 7 (d) 1·1 (e) 1·7<br />
(f) 1·9 (g) 2·7 (h) 3·3 (i) 4·5 (j) 5·7<br />
(k) 6·3 (I) 9·2 (m) 12 (n) 15 (0) 35<br />
(p) 38 (q) 0·4 (r) 0·8<br />
2. (a) 30 (b) 50 (c) 80 (d) 12 (e) 18<br />
(f) 26 (g) 36 (h) 39 (i) 45 (j) 48<br />
(k) 58 (I) 60<br />
3. (a) 0·2 (b) 0·7 (c) 0·9 (d) 1·1<br />
(e) 2·2 (f) 2·7 (g) 3·1 (h) 4·0<br />
(i) 0·36 (j) 0·49 (k) 0·51 (I) 0·72<br />
4. (a) 0·05 (b) 0·08 (c) 0·09 (d) 0·1 (e) 0·14<br />
(f) 0·18 (g) 0·27 (h) 0·38 (i) 3 (j) 4·2<br />
(k) 6·5 (I) 7·2 (m) 1·85 (n) 2·66 (0) 3·71<br />
(p) 5·04 (q) 0·372 (r) 0·493<br />
B. 1. (a) 1·5 (b) 1·5 (c) 1·6 (d) 0·7<br />
(e) 1·11 (f) 1·27 (g) 1·42 (h) 1·29<br />
(i) 0·4 (j) 0·3 (k) 0·6 (I) 0<br />
Check the additions by adding again in reverse order.<br />
Check the subtractions by adding back what you have just subtracted.<br />
5
2. (a) 2·5 (b) 5·0<br />
(f) 9·53 (g) 16·01<br />
(k) 14·26 (I) 14·5<br />
(c) 6-7<br />
(h) 16·27<br />
(d) 9·2<br />
(i) 8·51<br />
(e) 5-95<br />
(j) 8·58<br />
Check on the calculator (these are 'pencil and paper' practice, so<br />
should not be done on the calculator first).<br />
C. 1. (a) 0-24 (b) 0·14 (c) 0-09 (d) 0-30<br />
(e) 0-02 (f) 0·016 (g) 0-018 (h) 0·021<br />
(i) 0·26 (j) 0·72 (k) 0-8 (I) 1·28<br />
Take care with decimal point. Check by rule (number of places<br />
after decimal point is the same before and after multiplication).<br />
2. (a) 2 (b) 1-5 (c) 3 (d) 0-5<br />
(e) 20 (f) 40 (g) 20 (h) 30<br />
(i) 3 (j) 12 (k) 5 (I) 7<br />
Multiply answer by divisor to get back first number (calculator).<br />
M7<br />
A. What is ... 1.<br />
3.<br />
5.<br />
B.<br />
1. 0.6, 60%<br />
4. 0·46, 46%<br />
7. 0·325, 32·5%<br />
0·2 x £40<br />
0·15 x £60<br />
0·18 x 5 kg<br />
2. 0.467, 46-7 %<br />
5. 0·675, 67·50/0<br />
8. 0·816, 81·6%<br />
2. 0·01 x £50<br />
4. 0·12 x 6m<br />
6. 0·25 x 650 gals<br />
3. 0_45, 45%<br />
6. 0·76, 76%<br />
9. 0·75, 75%<br />
C. 1. 0·8, 80%<br />
4. 0·06, 6%<br />
7. 0·736, 73·6%<br />
10. 0·25, 25%<br />
2. 0·068, 6·8%<br />
5. 0-57, 57%<br />
8. 0·091, 9·1%<br />
11. Increase 0·002,<br />
0.20/0<br />
3. 0·033, 3·3%<br />
6. 0·607, 60·7%<br />
9. 0·258, 25·8%<br />
12. 0·696, 69·6%<br />
M8 A. 1. Mr Jones 27·8%, Mr Green 18·2%<br />
2. (i) July 6·8%; August 1·2%; September 6·5%<br />
(ii) July 6·8%; August 8·0%; September 15·1%<br />
3. (i) £106<br />
(ij) 17% rent; 45·3% food; 22·6% entertainment, 15·1% other.<br />
Check that they add up to 100.<br />
4. (i) £160<br />
(ij) rent 17·5%; food 31·25%; entertainment 16·25%; other 35%.<br />
Smith family spend a higher percentage on food.<br />
6<br />
B. 1. (b) 38·5p per 200 g<br />
(c) 28·6 (29p) per tin<br />
(d) 81·4p per kilo
Note: (a) Rounded to nearest !p. (b) Adding 10 % can be done<br />
by multiplying by 1·1.<br />
2. Use brackets if the calculator has them, otherwise work out the<br />
product, subtract the original price and. ignore the minus sign.<br />
(b) £2·04 (c) 81 p (d) £2·30<br />
Note: Reducing 15% can be done by multiplying by 0·85.<br />
3. (a) £18 (b) £21 (c) £31·25<br />
(d) £46 (e) £54 (f) £68·25<br />
4. Percentage growths are 8%, 7.40/0,17.2 % ,17.6 % ,150/0 and 10·9%.<br />
Biggest growth during the 4th month.<br />
M9<br />
1. All match<br />
2. (i) 0·36<br />
(v) 0·0625<br />
3. (i) 0·25<br />
(v) 0·1296<br />
4. (i) 0·13<br />
(ii) 0·16<br />
(vi) 0·4096<br />
(ii) 0·49<br />
(vi) 0·1936<br />
(ij) 0·74<br />
(iii) 0·04<br />
(iii) 0·81<br />
(iii) 0·72<br />
(iv) 0·1225<br />
(iv) 0·0784<br />
(iv) 0·185<br />
P3<br />
A. 1. 35%<br />
5. 88%<br />
9. 2·5%<br />
B. 1. 0·3<br />
5. 0·08<br />
9. 0·01<br />
2. 40%<br />
6. 92%<br />
10. 31·5%<br />
2. 0·45<br />
6. 0·12<br />
10. 0·333<br />
3. 66%<br />
7. 8%<br />
11. 44·4%<br />
3. 0·5<br />
7. 0·99<br />
11. 0·425<br />
4. 75%<br />
8. 3%<br />
12. 52·5%<br />
4. 0·56<br />
8. 0·64<br />
12. 0·085<br />
C. 1. 60% 2. 58·3% 3. 45% 4. 80% 5. 16·5%<br />
6. 33% 7. 33·75% 8. 65% 9. 34·06%<br />
10. 80%, 65%, 60%, 58·3%, 45%, 34%, 33·75%, 33%, 16·5%<br />
P4 A. 1. 120 g 2. 62·5%<br />
4. Home Economics 5. 14000<br />
3. Maths<br />
6. 75%<br />
B. 1. (a) £20 (b) £60 (c) £16 (d) £10<br />
2. (a) £90 (b) £157·50 (c) £216 (d) £81<br />
3. (a) £32·40 (b) £162 (c) £194·40 (d) £388·80<br />
4. (a) £10400 (b) £32500 (c) £50050 (d) £60450<br />
C. 1. £102 2. £105·60 3. £88·80 4. £110·40<br />
5. £5100 6. £4536 7. £4272<br />
8. £5010 (can be done by multiplying present pay by 1·2)<br />
7
D. 1. 140 + 45 = 185, 37%<br />
2. (a) 7700 (b) 12600 (c) 5250<br />
3. (a) 10320 (b) 4334 (c) 4231 (d) 1754<br />
No decimal people, please. So the addition is 1 out.<br />
4. VAT 150/0 25% 35%<br />
Camera £52·33 £56·88 £61·43<br />
boots £16·62 £18·06 £19·51<br />
bike £44·74 £48·63 £52·52<br />
crackers £ 4·89 £ 5·31 £ 5·74<br />
ladder £21·28 £23·13 £24·98<br />
drill £18·86 £20·50 £22·14<br />
Multiply original price by 1·15, 1·25, 1·35<br />
P5 1. (a) 1 (b) 2·8 (c) 5·6 (d) 6<br />
(e) 7·8 (f) 10·6 (g) 1·9 (h) 4·2<br />
(i) 3·3 (j) 0·25 (k) 1·96 (I) 7·84<br />
(m) 0·75 (n) 3·36 (0) 10·64<br />
2. (a) 2·55 (b) 7·35 (c) 9·18 (d) 1·15<br />
(e) 5·95 (f) 7·78 (g) -1·6 (h) -0·61<br />
(i) -2·21 (j) 0·7225 (k) 6·0025 (I) 9·3636<br />
(m) -0·1275 (n) 3·5525 (0) 6·3036<br />
3. (a) (b) (c) (f) true (d) (e) not true<br />
4. (b), (e) true; (a), (c), (d), (f) not true<br />
E4 A. 1. Use constant multiplier 1·16, ten times. £22057·17<br />
2. Use constant multiplier 1·24, 1988 80·1 thousand<br />
1989 99·3 thousand<br />
1990 123·1 thousand<br />
3. Use constant multiplier 1·43, 1980 429 million<br />
1990 613·47 million<br />
2000 877·26 million<br />
4. (a) 4·55% (b) 5·94%<br />
5. (a) 7·446 million (b) 9·57 million cars, i.e. 2·124 million more<br />
B. 1. Multiply everything by 0·85.<br />
(a) £80·75 (b) £47·60 (c) £17<br />
(e) £46·75 (f) £27·20 (g) £23·80<br />
2. Divide by 0·85 each time.<br />
(a) £94·12 (b) £47·06 (c) £23·53<br />
(e) £75·29 (f) £32·94 (g) £11·76<br />
3. (a) 8·3% (b) 12·5% (c) 40%<br />
(e) 37·5% (f) 36% (g) 28·6%<br />
(d) £12·75<br />
(h) £29·75<br />
(d) £14·12<br />
(h) £37·65<br />
(d) 16·6%<br />
(h) 40%<br />
8
E5 A. 1. (a) £39900 (b) £8960 (c) £13440 (d) £7980<br />
(e) £17640 (f) £5110 (g) £5320 (h) £3360<br />
2. (The tip is 100/0of the bill before VAT is added.) Multiply each by<br />
1·25.<br />
(a) £5·62 (b) £8·06 (c) £13·12<br />
(d) £10 (e) £12·37 (f) £15·25<br />
3. (a) (Divide by 1·014) ... 29586<br />
(b) (Multiply by 1·014) ... 30420<br />
4. (a) 8·89 kg (b) 222·25 g (c) 0·76 g (d) 0·05 g<br />
5. East 31·2% West 30%. East has larger percentage.<br />
6. Various answers.<br />
B. As for Exercise P5.<br />
c. All true except for (d).<br />
Unit 3, page 11<br />
Even though Euclidean geometry is no longer taught as a<br />
complete system in school <strong>mathematics</strong> it is still worth<br />
knowing something about congruent triangles. The triangle<br />
is a widely used shape in building, surveying and navigation<br />
and much depends on the fact that three sides of a given<br />
length define a triangle so that the angles are fixed. All this is<br />
another way of saying that the triangle is a rigid shape. Once<br />
you join three rods the construction is rigid even though the<br />
joints may be free. Many examples of this phenomenon may<br />
be observed from cranes to bicycle frames and the structure<br />
of the human body.<br />
Drawing triangles is one way (one of the best) to get their<br />
'feel'. If triangles are drawn with the same dimensions, then<br />
cut out and compared, an understanding of congruence can<br />
follow. The collapsing of a four-rod linkage provides a<br />
dramatic contrast and has many consequences from the<br />
collapse of buildings to the infuriating problem of keeping a<br />
door-frame an exact right angle when hanging a door.<br />
The work of the unit provides extensive practice in the use<br />
of drawing instruments and in careful measurement of<br />
9
lengths and angles. A basic metric rule of geometry follows<br />
from the discussion of three sides which cannot form a<br />
triangle. In space ...<br />
(distance from A to B) + (distance from B to C)<br />
~ (distance A to C)<br />
which may be stated as the theorem two sides of a triangle<br />
are greater than the third. Congruent triangles can be used to<br />
prove many geometrical properties and this aspect is interesting<br />
to quite a number of children.<br />
Answers<br />
M10 A.<br />
B.<br />
c.<br />
1. (Approximate) A = 49~0<br />
2. (Approximate) A = 27 0<br />
3. (Approximate) A = B = 57~0<br />
4. (Approximate) A = 62 0<br />
5. (Approximate) A = 49~0<br />
6. (Approximate) A = 26~0<br />
Check by fitting.<br />
B = 22.5 0<br />
B = 36 0<br />
B = 42 0<br />
B = 22~0<br />
B = 36~0<br />
= 108 0<br />
= 117 0<br />
= 65 0<br />
= 76 0<br />
= 108 0 (half Q. 1)<br />
e = 117 0<br />
Put a sheet of thin paper over the triangle in the book and mark the<br />
corners. Then use a pin to transfer the points to an exercise book.
M13<br />
Ship to Ship to Distance<br />
lighthouse church to line<br />
1. 8·49 km 8·49 km 6·00 km<br />
2. 23·0 km 21·0 km 19·9 km<br />
3. 47·1 km 44·5 km 44·3 km<br />
4. 27·3 km 44·4 km 27·3 km<br />
5. 20·2 km 8·10 km 7·58 km<br />
6. 9·10 km 13·5 km 8-65 km<br />
These are correct to 3 significant figures so drawings will be approximately<br />
these values_<br />
M14 1. 69 m 2. 84m 3. 42m 4. 22m<br />
P6 A.<br />
AB BC CA A B C<br />
1. 4cm 5-5cm 6-1 cm 62° 78° 40°<br />
2. 4cm 5-1 cm 5-3cm 65° 70° 45°<br />
3. 70mm 48mm 37mm 40° 30° 110°<br />
4. 38mm 60mm 65mm 65° 80° 35°<br />
5. 3·2cm 6·5cm 6·7cm 72;° 79;° 28°<br />
B. Drawing<br />
C. Drawing<br />
P7 A_ Post 2 Post 1<br />
1. 18-4km 16-3 km<br />
2. 19-1 km 10-1 km<br />
3. 28-3 km 20km<br />
4. 26-9 km 10-6 km<br />
5. 10-3 km 24km<br />
6. 12-2 km 12-2 km<br />
B. 1. 11-8 km 2. 15-5 km 3_ 101-6km 4. 29-2 km<br />
C_ 1. 25-8m 2_ 50-8m 3_ 49-2m 4. 80-5m<br />
E6<br />
A, B,C. See P6<br />
11
E7<br />
See P7<br />
E8 A. 1. Draw AD and BC and then consider triangles ABO and BAC. These<br />
are congruent because two sides are equal and so is the angle<br />
between them. Thus the third sides are equal. These are the<br />
required diagonals.<br />
2. Prove that triangles ABC and BCD are congruent.<br />
3. Consider triangles AOB, COD.<br />
4. AC..l BD, both BD and AC are axes of symmetry leading to the<br />
following<br />
BD bisects Band 0, AC bisects A and C, AOB, BOC, COD, DOA all<br />
right angles.<br />
Unit 4, page 16<br />
The cuboid is man's most commonly used shape for many<br />
reasons. These reasons can be elucidated from the class<br />
during general discussion. The usual procedure of defining a<br />
cuboid, then rushing into formulae for volume and surface<br />
area leaves out a very important exploration stage. Remember<br />
also that this will probably be the first attempt children<br />
will make to represent a solid by a careful two-dimensional<br />
drawing. This takes a lot of practice.<br />
For bright children measuring a 'solid diagonal' of a<br />
cuboid makes an intriguing problem. This has practical<br />
applications when solid cuboid objects such as furniture and<br />
refrigerators have to be moved.<br />
The P and E units are very close for this exercise as<br />
brighter children may well need a considerable amount of<br />
practical work on this topic.<br />
Answers<br />
M15 A.<br />
1. Bricks, concrete blocks, breakfast cereal boxes and books are usually<br />
cuboid.<br />
2. (a) True (b) True (c) Not true<br />
(d) False (e) False (f) It will need 8<br />
3. (a) Yes, though one dimension is very small.<br />
(b) Could be (c) No (d) No (e) No (f) Roughly<br />
12
B. 1. (c) (i) Two faces are squares (largest faces)<br />
(ii) Two faces are square (smallest faces)<br />
(iii) The cuboid is a cube<br />
2. Measurement 3. Measurement<br />
C. A good supply of isometric paper will lead to coloured designs which<br />
can be used to brighten up the classroom.<br />
M16 A. 1. Drawing 2. Drawing<br />
3. (a), (b) and (c) all have square faces. (c) is a cube.<br />
P8 A. 1. (for example) They fit together, they lie on top (very stable), easy<br />
to manufacture.<br />
2. (a) They wouldn't fit together.<br />
(b) The sharp edges would crumble, they would not build into<br />
stable shapes.<br />
(c) They wouldn't pack (although they are used for insulation).<br />
(d) They would not form stable shapes.<br />
3. (a) The drain pipes (b) Floors, walls (c) Pillars<br />
(d) The ball-cock in a water tank<br />
B.<br />
C.<br />
1. (a) 40 bricks (b) 66 bricks<br />
1. 20 2. 12, 8, none<br />
3. 6 with 1 yellow face; 12 with 2 yellow faces; 8 with 3 yellow faces;<br />
1 with no yellow faces<br />
P9<br />
Drawing<br />
P10 1. There are only three basic shapes where four equilateral triangles<br />
are joined edge to edge, two of these are shown in the introduction<br />
to the exercise.<br />
2. Two of the shapes make a tetrahedron but the ~ hexagon folds<br />
into a pyramid with four sloping sides and without a base (two of<br />
these shapes will make a regular octahedron).<br />
3. Add one triangle in all possible positions to each of the three basic<br />
three triangle shapes and work on from there.<br />
E9 A. 1,2,3 see P8.<br />
4. This question should be discussed. Suggested answers are<br />
(a) It is sharp at the bows to reduce water resistance and wide for<br />
stability.<br />
(b) Flat to give good pressure, long for leverage.<br />
13
(c) Sharp-fronted to travel fast and reduce air resistance.<br />
(d) Hollow to hold liquids, flat handle to make it easy to hold steady.<br />
(e) Narrow neck to prevent fast flowing of liquid and to present small<br />
area of liquid to the air.<br />
(f) Many features are designed for safety and efficiency.<br />
B. 1. 25 2. 52 3. 51<br />
C. 3. (a) 5 x 2 x 2, 5 x 4 x 1, 10 x 2 x 1, 20 x 1 x 1 cm<br />
(b) 5 x 2 x 2* has least number of squares exposed<br />
(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<br />
2 x 3 x 4. Least squares exposed 2 x 3 x 4 = 52<br />
4. (a) 6<br />
(b) 12 with 2 yellow faces, 8 with 3 yellow faces and 1 with no<br />
yellow paint<br />
(c) There are 24 with 1 yellow face, 24 with 2 yellow faces, 8 with<br />
3 yellow faces and 8 with no yellow face (making a 2 x 2 x 2<br />
cube inside)<br />
E10<br />
Drawing<br />
E11 See P10, 1 and 2 and 3<br />
4. Treat the area outside the triangle as the base of the tetrahedron.<br />
It is not possible to move along all the edges without going over<br />
one edge a second time.<br />
Unit 5, page 21<br />
Although vectors are considered a new topic in secondary<br />
<strong>mathematics</strong> they are conceptually easy when thought of as a<br />
combination of movements. The addition rule agrees with<br />
common sense (which makes it easier than fractions as a<br />
topic). The work on vectors can relate many problems in the<br />
real world to the newly introduced Cartesian co-ordinate<br />
system. The idea of negative as direction is reinforced and<br />
the geometrical notion 'translation' comes under further<br />
discussion. The arithmetic of vectors presents no problems<br />
as the numbers should be kept small while the geometry<br />
leads to careful measurement of length and direction. A<br />
* Note the least squares correspond to the cuboid where the 3 dimensions add to the<br />
least total.<br />
14
Answers<br />
further bonus from the topic is the discussion of 'operational<br />
rules' which can be related to the rules of numbers, especially<br />
the commutative and associative law. It may be necessary<br />
to explain the need for the two types of representation,<br />
(x,y) for the point and G) for the vector from (0,0) to<br />
(x, y). Although this notation is complicated it is not<br />
difficult. Remember that the topic will be presented again<br />
and that exploration is more important than memorizing at<br />
this stage.<br />
M17 A. 1.<br />
(~) 2. (~) 3. (:)<br />
4.<br />
(~) 5. G)<br />
6.<br />
(~) 7. m 8. (~) 9. G)<br />
10.<br />
(~)<br />
B. 1.<br />
(=~) 2. (-~) 3. (-~) 4. (-~) 5. (=~)<br />
e. (-~)<br />
7. (-~) 8. (-~) 9. (=~) 10. (=:)<br />
11.<br />
(=~) 12. (=~)<br />
C. Drawing 10. (F)<br />
(=:), (G) (=~),(H) (=:)<br />
M18 A. 3. (a)<br />
(~) (b) G)<br />
(c)<br />
(~) (d) G)<br />
(e) (-~) (f) (-~) (g) (-~) (h) (-~)<br />
B. 2. A parallelogram 4. A parallelogram<br />
5. Nothing at all other than move it<br />
6. Nothing at all other than move it<br />
C.<br />
(~)<br />
2.<br />
3. m, (~),(~)<br />
Ṭhe last is the sum of the first two, obtained<br />
by simple addition of the corresponding numbers.<br />
D. 3. (-~) 4. ( - ~) - (~) = ( -~)<br />
15
M19 A. 1.<br />
m<br />
2.<br />
(~) 3. C~)<br />
4.<br />
(:)<br />
5.<br />
(~)<br />
6.<br />
(:)<br />
7. (-~)<br />
8.<br />
(1~)<br />
9. (-~) 10. (=~)<br />
11. (=~)<br />
12. (-~)<br />
B. All four are (8) because the two vectors are equal and opposite. It is<br />
like asking Ihow far have I travelled after going from here to London<br />
and back again'. After that journey I am back where I started.<br />
c. 1.<br />
(~)<br />
2. (1~)<br />
3. (~~)<br />
4.<br />
(~)<br />
5. W 6.<br />
(~)<br />
D. 1.<br />
(:)<br />
2.<br />
G)<br />
3.<br />
(~)<br />
4.<br />
(~)<br />
5. (,~)<br />
6. (-~)<br />
M20 A. 1. Direction needed 2. Speed needed 3. Velocity<br />
4. Direction needed 5. Direction needed 6. Velocity<br />
B. 1. M 2. L 3. M or A<br />
4. 0, Hand D 5. C and G or E and G.<br />
c. Various answers<br />
P11 A. A B C D E F G H<br />
1. (-:) (~) (:) (:) (-:) (-~) (-~) (=~) (=:)<br />
2. (a)<br />
C)<br />
(b)<br />
(~) (c) (-~)<br />
(d)<br />
(-~)<br />
(e)<br />
(=~)<br />
(f) (-~)<br />
3. (a)<br />
(=~)<br />
(b)<br />
(=~)<br />
(c)<br />
(~)<br />
16
(d) (-~) (e)<br />
G)<br />
(f)<br />
(-~)<br />
B. 1. J 2. H 3. F 4. D<br />
5. B 6. J 7. H 8. F<br />
C. 1. FACE 2. CABBAGE 3. CHIEF<br />
P12 A. 1.<br />
(~) 2. (=~)<br />
3.<br />
(=~)<br />
4.<br />
(~)<br />
B. 1. L.JKL 2. L.JKL 3. L.DEF 4. L.ABC 5. L.JKL<br />
C. 1. L.JKL 2. L.DEF 3. L.GHI 4. L.MNO<br />
P13 A. 1. Drawings 2. Drawings<br />
B. 1. Agreed 2. Should be (2~)<br />
3. Should be (2~) 4. Agreed<br />
C. Drawings<br />
E12 A. 1., 2., 3.-See Exercise P11.<br />
4. The general results work for all values of x and y.<br />
5. All these vectors are parallel to (~) and simple multiples of its<br />
length.<br />
B. See Exercise P11<br />
C. See Exercise P11<br />
D. Experiment<br />
E13 A., B., C.-See Exercise P12.<br />
D. There are many translations in this diagram. If you assume the figure<br />
shows part of a tessellation, e.g.<br />
s, ~ S2 which maps S2~ S3, S4~ S5, etc.<br />
s, ~ S3; s, ~ S4; s, ~ S5; s, ~ S6; S2~ S4;<br />
S2~ s,; S3~ S4; S3~ s,; S4~ s,; S4~ S2; S4~ S3;<br />
S5~S,;S6~S,.<br />
17
E14 A. See Exercise P13<br />
B. 1. Agreed<br />
2. Agreed<br />
3. Should be (~b)<br />
4. Agreed<br />
5. Agreed<br />
6. Agreed<br />
c. 1. Drawing 2. All the steps in the x direction add to zero and so<br />
do the steps in the y direction.<br />
vectors) must be (8).<br />
Thus the total effect (the sum of the<br />
Unit 6, page 28<br />
This unit on problem solving relates some simple <strong>mathematics</strong><br />
to tools in everyday use. It is not obvious to children<br />
that things are as they are by design rather than by chance.<br />
Looking carefully at objects is the same skill as the observation<br />
taught in Book 1 and which is being taught in all the<br />
other subjects of the School curriculum. There is very good<br />
reference material in a book called Machines, Mechanisms<br />
and Mathematics, part of the Schools Council Mathematics<br />
for the Majority Project. This book is particularly good on<br />
linkages. Simple tool making has been one of the most<br />
significant activities of human beings.<br />
Answers<br />
M21<br />
Note: You may decide to replace these objects with others. A good<br />
class lesson could be organized as a quiz-game with pupils having·to<br />
bring in mysterious objects and others having to guess what their<br />
function is.<br />
M22 3. These answers may need to be discussed.<br />
(a) Type (i) (b) Type (ii) (c) Type (i)<br />
(d) Type (i) (e) Type (i) (f) Type (i) (g) Type (iii)<br />
M23 1. 12.5 : 1 2. 312·5 : 1 3. 10·4 : 1<br />
4. 10·7 : 1 5. 18 : 1 6. 20 : 1<br />
P14 A. 1. (a), (b), (c), (g) 2. Different answers<br />
18
B. 2. Load<br />
(a) Friction holding lid<br />
(b) Weight<br />
(c) Reaction of water<br />
Force<br />
Push down on spoon<br />
Pull of muscle<br />
Pull of arms<br />
Fulcrum<br />
Edge of tin<br />
Elbow joint<br />
Rowlock<br />
P15 A. 1. (a) B moves down (b) B moves up<br />
(c) B moves away (d) 8 moves towards you<br />
2. (a) A moves down (b) A moves up<br />
(c) A moves away (d) A moves towards you<br />
3. (a) B moves in a circle<br />
(b) B moves in a square<br />
(c) B moves in a triangle<br />
B. 1. A parallelogram<br />
2. Band 0 move towards each other<br />
3. A and 0 move towards each other<br />
C. 1. 8 and C move outwards<br />
2. Band C move towards each other<br />
3. No<br />
4. Not without moving A and 0 as well<br />
5. Yes<br />
P16 A. 1. 4 kg 2. 4 kg 3. 2 kg<br />
6. Many possible answers<br />
4. 2 kg 5. 31 kg<br />
B. 1. 2cm<br />
6. 2cm<br />
2. 6cm 3. 2cm 4. 5cm 5. 2·5cm<br />
E15 A.<br />
1. (a), (b), (c), (g), (i), (j)<br />
B.<br />
2. Load<br />
(a) Friction holding lid<br />
(b) Tension in tyre and friction<br />
(c) Weight<br />
(d) Reaction of water<br />
(e) Reaction of water<br />
Force<br />
Push down on spoon<br />
Force exerted by hand<br />
Pull of biceps muscle<br />
Pull of arms<br />
Pull of arm<br />
Fulcrum<br />
Edge of tin<br />
Rim of wheel<br />
Elbow joint<br />
Rowlock<br />
Other hand<br />
E16 See P15 A and B.<br />
4. A rhombus<br />
C. As for P15<br />
5. Approx 14 em 6. 180 0<br />
E17<br />
A., B. See P16<br />
C. 2. 0) 7 Oi) 3<br />
3. (a) 16·2 kg<br />
(iii) 0·5<br />
(b) 8·6 kg<br />
(iv) Many answers possible<br />
(c) 4·7 em (d) 48 em<br />
19
Unit 7, page 31<br />
The circle is the most perfect of all plane shapes having<br />
infinite symmetry. The fascinating relationship between<br />
circumference and diameter has interested mathematicians<br />
for centuries and the extraordinary properties of the number<br />
1T continue to stimulate research.<br />
A thin strip of 1 mm 2 graph paper makes an excellent<br />
'curved~ ruler and seems to follow the circumference of a<br />
circle in a most natural way. There are other ways circular<br />
measurements may be explored such as putting an ink blob<br />
on a coin and rolling it down a slope or wheeling a bicycle<br />
along after making a wet paint-mark on one of its wheels.<br />
Practical alternatives emphasize that all circles are the same<br />
in the ratio of diameter to circumference and fix the number<br />
1T far better than calculations. (Incidentally ~ all regular<br />
polygons have fixed circumference/diameter relationships.<br />
Squares for example have perimeters which are 1·414 times<br />
their diameter ~while the perimeters of all regular hexagon~<br />
are 3 times their diameters.) .~<br />
Work on circles leads naturally to the exploration of the<br />
cylinder which is a far more common object than a circle<br />
pure and simple.<br />
C + D comes to approximately 3·14 every time.<br />
20
C. The graph should give<br />
(a) 12·5 em (b) 15·7 em<br />
(c) 37·7 em (d) 26·4 em<br />
D. The ratio C -7 D should come to roughly 3·1"4each time so the average<br />
should be very close to 3·14.<br />
M25<br />
Practical work.<br />
M26 A. Completed table. Answers marked*<br />
Radius<br />
1. 22cm<br />
2. 11 m<br />
3. 14mm<br />
4. 8·5 cm*<br />
5. 4·25cm*<br />
6. 2·2 m*<br />
7. 15·9cm*<br />
8. 7·5mm*<br />
9. 0·9mm*<br />
10. 9·25 m*<br />
11. 4·3cm*<br />
12. 72 km<br />
44cm*<br />
22m*<br />
28mm*<br />
17cm<br />
·8·5cm<br />
4-4m<br />
31-8cm*<br />
15 mm*<br />
1·8mm*<br />
18·5m<br />
8·6cm*<br />
144 km*<br />
Diameter<br />
Circumference<br />
138cm*<br />
69m*<br />
88mm*<br />
53-4cm*<br />
26-7 cm*<br />
13·8 m*<br />
100cm<br />
47mm<br />
5·8mm<br />
58m*<br />
27cm<br />
452 km*<br />
B. 1. 235·5 mm<br />
2. Approx 125 em<br />
3. 1mm<br />
4. 220 em, 220 m<br />
5. 471 em<br />
(a) 21 230 approx<br />
(b) 212300 approx<br />
P17 A. 1. 37 mm<br />
2. 50mm<br />
3. 72mm<br />
4. 39mm<br />
B. 1. 146 mm<br />
2. 174mm<br />
3. 328mm<br />
21
P18 A. 1. (a) 12·5 (b) 18·8 (c) 28 (d) 63 (e) 94<br />
(f) 188 (g) 107 (h) 182 (i) 270<br />
Note: To find 34 use 30 + 4. Find 30 from 3.<br />
2. (a) 4·8 (b) 8 (c) 32 (d) 64 (e) 5·7 (f) 22<br />
3. (a) 50 (b) 157 (c) 264 (d) 754<br />
B. 1. 6·3 m 2. 4·4 em 3. About 60 em<br />
4. About 31 m/sec or 113 km/hr<br />
C. 1. 3~ is very accurate ... 3·141593. This is easily remembered as<br />
355/113 (113355)<br />
2. (a) 16400 km (b) 16400 km<br />
(c) 103044km (d) 21467km/hr<br />
E18 A. 1. Drawing. Use as large a scale as will fit on your graph paper.<br />
2. 3. 1 See P18 A. 1, 2, 3<br />
4.<br />
B. 1. 6·3m,159times<br />
2. 4·5 em, the largest side of square is 3·2 em (the square with<br />
diagonal 4·4 em)<br />
3. About 60cm 4. 31m/sec=113kmjhr<br />
C. 1. 12 732 km 2. 10917 km<br />
3. 2419026km 4. 584·3 million miles<br />
5. 40 000 km, 1666 km/hr 6. 25133 km, 1047 km/hr<br />
E19 1. (b)<br />
2. (a) 16366 km (b) 102831 km (c) 21 423 km/hr<br />
3. Joe is right. If C is the circumference of the Earth and R is the<br />
radius of the Earth. C= 27TR. If the circumference is increased by 6<br />
metres C + 6 = 27T (R + r) where r is the increase of radius. Thus<br />
6 = 27Tr, so r = 6/27T~ 1.<br />
22<br />
Unit 8, page 36<br />
Most people in the past have found algebra difficult to<br />
understand. I believe this is because the beginnings are too<br />
rushed and too isolated. Also algebra is built on unsatisfactory<br />
arithmetical foundations. Both of these faults can be
emedied by using a calculator. The problems of arithmetic<br />
are cleared up for almost all children and the speed of the<br />
calculator means that many more exercises can be performed<br />
giving a much wider range of initial experience in the<br />
use of letters.<br />
The use of random decimal numbers to test such relationships<br />
as the associative and distributive laws is more<br />
convincing than demonstrations with easy numbers only.<br />
There is a lot of conventional notation in the beginnings of<br />
algebra and these have to be made clear before any progress<br />
can be made. All these conventions are man-made and often<br />
led to controversy when they were proposed in the first<br />
place. It is important that children should not have their<br />
confidence undermined by a 'right and wrong' attitude or<br />
they will reject the whole of algebra as incomprehensible.<br />
Discussion on how to write a + a + a + a + a ... to save<br />
space should lead to the suggestion Sa from pupils and this<br />
can be compared with other notations proposed by members<br />
of the class. All this material will be presented again during<br />
the 3rd year, so at this stage it is more important to avoid<br />
anxiety and confusion than to achieve memorization.<br />
Answers<br />
M27 1. (a) ael (b) abd<br />
ale ... word adb<br />
ela<br />
bad ... word<br />
eal ...<br />
bda<br />
lea ... word dba<br />
lae<br />
dab ... word<br />
(c) aemn eamn maen *name<br />
aenm eanm *mean naem<br />
*amen eman *mane neam<br />
amne enam men a nema<br />
anem emna mnae nmae<br />
anme enma mnea nmea<br />
2. There are 120 possible groups of letters but the only words are<br />
dame, dare, dear, made, mare, mead, read and ream.<br />
3. Barb 4. Rubber 5. Report, Porter<br />
M28 A. 1. (a) 5 (b) 11 (c) 11 (d) 4 (e) 4<br />
(f) 9·6 (g) 10·7 (h) 35·5<br />
2. (a) 15 (b) 16 (c) 0 (d) 0 (e) 16·1<br />
(f) 9·3 (g) 0 (h) 3·24<br />
23
3. (a) 9 (b) 21 (c) 0 (d) 48 (e) 8·4<br />
(f) 10·2 (g) 1·41 (h) 1·08<br />
4. (a) 8 (b) 0 (c) 40 (d) 36 (e) 8·6<br />
(f) 0·9 (g) 1·36 (h) 15·54<br />
5. (a) 20 (b) 25 (c) 128 (d) 59 (e) 58·4<br />
(f) 33·4 (g) 14·75 (h) 11·12<br />
B. 1. (a) 11·3 (b) 28·4 (c) 9 (d) 14·4 (e) 30·53<br />
(f) 55·4<br />
2. (a) 6 (b) 8·5 (c) 7 (d) 0 (e) 10<br />
(f) 17·4 (g) 9 (h) 10 (i) 12 (j) 19·333<br />
(k) 90 (I) 66·666<br />
C. 1. (a) Both sides 8 (b) Both sides 27·48<br />
(c) Both sides 87·94 (d) Both sides 19·008<br />
2. (a) Not true (b) True (c) Not true<br />
(d) True (e) Not true (f) True<br />
3. (a) Always true (b) Only true if a = b<br />
(c) Always true (d) True (e) True (f) True<br />
M29 A. B. check. All the examples in C show that subtraction and division<br />
are not associative.<br />
M30 These examples demonstrate that multiplication is distributive over<br />
addition. But addition is not distributive over multiplication. Easily<br />
proved as follows:<br />
a + (b x e) = (a + b) x (a + e) => a + be = a 2 + ab + be + ea<br />
=> a = a 2 + ab + ae<br />
=> 1=a+b+e<br />
Thus addition is only distributive over multiplication if a + b + e = 1,<br />
e.g.<br />
a = 2, b = -1, e = 0<br />
a-~ - 2, b-~ - 4, e-~ - 4<br />
2 + (-1 x 0) = 1 x 2 True<br />
~+ (~x ~) - ~ +i - .l!-<br />
2 4 4 - 2 16 - 16<br />
H + a) x (! + a) = 1~<br />
P19 A. 1. Me! 2. Tea 3. Tame<br />
24<br />
B. 1. 6a 2. a+4b 3. 3a + 5b 4. 3a - 2b<br />
5. a+2b 6. a 3 7. a 3 x b 2 8. a 3 + b 3<br />
9. 6a + 8b 10. a+b 11. 8a 12. m+3n<br />
13. 3m 14. 5x + 9y 15. 5x - 4y 16. 2x - 2y
c.<br />
P20 A.<br />
B.<br />
C.<br />
P21 A.<br />
1. 5a 2. 3a 3. -a 4. a+b<br />
5. 2a + 3b 6. -2m +4n 7. -4m-n 8. 3y<br />
9. x+ 11y 10. -2x+ y<br />
1. 7·4 2. 18·6 3. 28·6 4. 25·6<br />
5. 9 6. -4·7<br />
1. 2 2. 3·5 3. -1 4. 3·5333<br />
1. b = 3·7, ab = 8·51 2. b = 9·7, ab = 142·59<br />
3. b = -7'8, ab = -49·92 4. b = 32, ab = 672<br />
5. a = 6·7, ab = 35·51 6. a = 3·9, ab = 4·68<br />
7. a = 4·6666, ab = 21 8. a = 3·56, ab = 9·256<br />
The laws work for all values of a, band c<br />
B. Diagram 1 shows 3(a + b) = 3a + 3b and a = xxx, b = 0000, i.e. Distributive<br />
Law<br />
E20 A. See P19 A.<br />
Diagram 2 shows a + (b + c) = (a + b) + c, i.e. Associative Law (addition)<br />
Diagram 3 shows 4 x (3 x 2) = (4 x 3) x 2, i.e. Associative Law (multiplication)<br />
Diagram 4 shows 4 x (3 x 2) = (4 x 3) x 2 again in different form<br />
Diagram 5 shows ab + ac = a(b + c), i.e. Distributive Law<br />
Diagram 6 shows Associative Law (addition)<br />
B. 1. There are<br />
(i) 20 different pairs possible (5 x 4) (ii) 60 different threes<br />
(5 x 4 x 3)<br />
(iii) 120 different fou rs (5 x 4 x 3 x 2) (iv) 120 different fives<br />
(5 x 4 x 3 x 2 x 1)<br />
The words of 5 letters are bread, beard and bared.<br />
C.<br />
D.<br />
2. Letters<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
See P19 B.<br />
Arrangements<br />
1<br />
2<br />
6<br />
24<br />
120<br />
720<br />
See P19 C. and ... 11. p-2q 12. q-6p<br />
14. 0 15. -5·7u 16. -4u + 6v<br />
13. -3r -35<br />
25
E21 A. See P20 (A) and ... 7. 8·65 8. 6 9. 34·72 10. -0·51<br />
B. See P20 (B) and ... 5. -3·3 6. 4 7. -46 8. 39·64<br />
C. See P20 (C)<br />
E22<br />
See P21<br />
Unit 9, page 41<br />
The squaring of numbers is an easy process which can give<br />
rise to many pattern searching and problem solving experiences.<br />
The calculator and tables of squares should be used<br />
and compared. On the calculator the process of repeated<br />
squaring leads to the calculator limit very quickly or to zero<br />
if the number you start with is less than 1. Facility in<br />
squaring is needed in applications of Pythagoras' theorem<br />
and also in physics and in a general understanding<br />
of nature.<br />
The sequence 1, 4, 9, 16 ... should be as familiar as 10, 20,<br />
30, 40 and is much more interesting.<br />
The P units introduce notion of square roots and inverse<br />
operations. It is not necessarily true that [ill 6ZI[D will give<br />
you back n exactly. This will depend on the circuitry.<br />
Nevertheless [ill Ix 2 1 6ZIwill give n back again if n is not too<br />
large. Results should be compared with those obtained from<br />
tables. Some discussion of finding square roots without a<br />
calculator or without 6ZIwill lead to approximating methods<br />
of which 'divide and average' is simplest.<br />
Algebraic squares link spatial instinct with algebraic notation<br />
and the distributive law of numbers. There are many<br />
interesting patterns and puzzles relating to square numbers.<br />
The graph of y = x 2 will come later.<br />
Answers<br />
M31 A.<br />
1. 324<br />
6. 24·01<br />
2. 1849<br />
7. 14·44<br />
3. 3025 4. 12100<br />
8. 1789·29 9. 26·3169<br />
5. 27889<br />
10. 806·56<br />
26
B. 1. 484 2. 1296 3. 1936 4. 3249 5. 3721<br />
6. 77·44 7. 56·25 8. 4·84 9. 75·69 10. 98·01<br />
c. It is true because the squares end in 0, 1, 4, 5, 6 or 9.<br />
M32 A. 1. 0·09 2. 0·49 3. 0·81 4. 0·36 5. 0·64<br />
6. 0·7225 7. 0·3969 8. 0·1764 9. 0·0225 10. 0·0064<br />
B. 1. 0·04 2. 0·16 3. 0·25 4. 0·0484 5. 0·5625<br />
6. 0·6724 7. 0·8836 8. 0·0324 9. 0·0064 10. 0·0036<br />
c. 0·2~0·04 0·71 ~ 0·50 0·4~0·16<br />
0·5~ 0·25 0·63~0·40<br />
0·3~0·09 0·25~0·06<br />
M33 A. 1. 49 2. 144 3. 144 4. 324 5. 900<br />
6. 36 7. 22·09 8. 88·36 9. 156·25 10. 174·24<br />
B. If you have a calculator without I x 2 1 or brackets you will have to use<br />
something like ~ [K] la El ~ [±] ~ 5J ~ ~ for the decimal<br />
squares.*<br />
1. 25 (appears as 24·999999.)<br />
2. 74 3. 72 4. 170 5. 452 6. 18·72<br />
7. 11·45 8. 44·26 9. 79·93 10. 101·7<br />
(a + b)2 = a 2 + b 2 is only true if a = 0 or b = O.<br />
c. 1. II III IV V VI VII VIII<br />
(col lv-col<br />
VII)<br />
a b a+b (a + b)2 a 2 b 2 a 2 + b 2 (a + b)2 - (a 2 + b 2 )<br />
2 3 5 25 4 9 13 12<br />
4 7 11 121 16 49 65 56<br />
5 3 8 64 25 9 34 30<br />
2·5 4 6·5 42·25 6·25 16 22·25 20<br />
1·7 2·8 4·5 20·25 2·89 7·84 10·73 9·52<br />
3·6 6·6 10·2 104·04 12·96 43·56 56·52 47·52<br />
4·2 0·8 5·0 25 17·64 0·64 18·28 6·72<br />
2. The numbers in col VIII are twice the product of a and b, e.g.<br />
(2 x 3) x 2 = 12, (1·7 x 2·8) x 2 = 9·52. This is explained in the next<br />
section.<br />
*If you press ~ [K] [;] [±] [ft] [K] [;] you get the square of (a 2 + b).<br />
27
M34 A. All drawings.<br />
B. These should be demonstrated by the teacher and related to drawings.<br />
The actual steps are quite simple, but important.<br />
C. 1. They are all examples of (a + b)2 where a + b = 10.<br />
2. (c) should be d + 14c +49.<br />
3. This exercise shows that one can get by sometimes without using<br />
a calculator.<br />
P22 A. 1. 14~ 196 7·6~57·76 33~ 49~2401<br />
3·7~ 13·69 2-3~ 5·29 8·8~ 77·44<br />
60~3600 0·83~ 0·6889 0·35 ~ 0·1225<br />
2. (a) 84100 (b) 8-41 (c) 0-0841 (d) 0·000841<br />
3. (a) 532900 (b) 53-29 (c) 0·5329 (d) 0·005329<br />
4. (a) 1444 (b) 144400 (c) 0·1444 (d) 0·001444<br />
B. 1. 16 2. 5041 3. 4624 4. 1369<br />
5. There isn't one unless you count 7744.<br />
C. 1. 44 2. 65 3. 9·3 4. 9·8 5. 22<br />
6. 20 7. 1·5 8. 0·43 9. 0-66 10. 0-75<br />
P23 This exercise introduces the new concept of a square root. If a calculator<br />
with a ) button is not available tables will be needed. The best type are<br />
Basic Mathematical Tables (Bell & Hyman) which give) together with<br />
other functions in a form suited to calculator use. Some care will be<br />
needed where, for example, square roots of 3 figure numbers are to be<br />
found. The tables will give the square root of 8,3, 83 and 830 but if )835 is<br />
28<br />
required we use )8·3 = 2,881, )8·4 = 2·898 so )8·35 = 2·890 (about halfway).<br />
Thus )835= 28·9. A more complicated method of finding square<br />
roots using the multiplication tables or 4 function calculator is divide and<br />
average, taking a rough estimate from the ) tables. E.g. wanted ) n.<br />
Estimate x. Now follow the sequence [Q] El [K] [±] [K] El [2J ~. This gives<br />
at least two more decimal places of the square root. The process is<br />
repeated with the latest answer used for x and so on until a constant result<br />
is achieved. (This may not be as easy as it sounds.)<br />
A.<br />
B.<br />
1. 7 2. 11 3. 22 4. 40 5. 44<br />
6. 85 7. 58 8. 92 9. 4·1 10. 7·5<br />
11. 6-1 12. 5·8 13. 1·3 14. 1·9 15. 1-2<br />
16. 2·1 17. 0-46 18. 0·85 19. 0·9 20. 0-49<br />
1. 7 2. 17 3. 21 4. 23<br />
5. 25 6. 32 (31-62) 7. 57 8. 75 (74-53)
C. 4, 9, 25, 64, 100, 144, 256, 361, 400, 484, 625<br />
P24 A. Diagrams<br />
B. (x + 4)2~X2 + 8x + 16<br />
(x + 5)2~X2 + 10x + 25<br />
(x + 6)2~X2 + 12x + 36<br />
(x + 7)2~X2 + 14x + 49<br />
(x + 8)2~X2 + 16x + 64<br />
C. 1. (x + 2)2 2. (x + 5)2 3. (x+ 9)2 4. (x + 10)2<br />
5. (3 + X)2 6. (4 + X)2 7. (7 + X)2 8. (6 + X)2<br />
E23 A. See also the discussion of the product of negative numbers in <strong>IMS</strong><br />
Book C, Problem Number 17.<br />
B. See P22 A.<br />
C. See P22 B.<br />
D. See P22 C.<br />
E24 See P23 and ... C 2<br />
(a) True (b) Not true (c) True (d) True (e) True<br />
(f) True (g) Not true (h) True (i) Not true<br />
E25 A. See P24 A.<br />
B. 1. See P24 B. 2. See P24 C.<br />
C. 1. (a) x 2 -4x +4 (b) x 2 -8x + 16<br />
(c) x 2 -10x+25 (d) x 2 -14x+49<br />
(e) x 2 -16x + 64 (f) x 2 - 20x + 100<br />
2. Use the diagram for (x + 1)2, but make x the full side of the square.<br />
We then have x 2 = (x - 1)2 + 2x - 1. The small unit square is<br />
counted twice in the 2x strips and therefore 1 has to be subtracted.<br />
This leads to (x - 1)2 = x 2 - 2x + 1.<br />
D. 1. (a) 56 (b) 117 (c) 180 (d) 66 (e) 1512 (f) 3233<br />
2. (a + b)2 - (a - b)2 = 4ab<br />
Note: The table of squares can be constructed from differences<br />
I 3 5 7 etc.<br />
1 4 9 16<br />
So the whole process of multiplication can be converted to<br />
addition, subtraction and halving.<br />
29
Unit 10, page 45<br />
This unit blends together practical work, measurement,<br />
geometry and calculations around the topics of right-angled<br />
triangles and Pythagoras' theorem. The various puzzle-type<br />
demonstrations of Pythagoras' theorem make very decorative<br />
work for the classroom and also enjoyable puzzles. The<br />
Pythagorean triples 3, 4, 5 ... etc., will later provide interesting<br />
investigations which again link geometry, algebra<br />
and number theory.<br />
The generalizations of Pythagoras' theorem are very<br />
interesting for the faster pupils. Geometrical proofs should<br />
not be presented at this stage but those children who ask for<br />
proof can be shown the traditional one with the areas<br />
moving by stages. There are quite a number of films on<br />
Pythagoras' theorem.<br />
Familiarity with right-angled triangles is a basic skill of<br />
secondary <strong>mathematics</strong> as many applications depend on it.<br />
Answers<br />
M35 A. 2. (a) DC = 31 mm (b) PO = 3·2 em (e) WX =40mm<br />
BC = 15mm PS = 2·5em WZ =33mm<br />
BAC = 26°<br />
PSO = 52°<br />
XWY= 39·5°<br />
BCA = 64°<br />
DAC = 64°<br />
SQR = 52°<br />
OSR = 38°<br />
WYX = 50·5°<br />
YWZ = 50·5°<br />
30<br />
3. There are 5 different sizes of right-angled triangles in this figure.<br />
Their hypotenuses are 15 mm, 21 mm, 36 mm, 51 mm, 72 mm<br />
B. 1. (a) 2em 2 (b) 2·1 em 2 (e) 212·5 mm 2 (d) 261 mm 2<br />
(e) 275 mm 2 (f) 3·4em 2 (g) 162mm 2 (h) 4·2 em 2<br />
(i) 198mm 2 (j) 536·5 mm 2 (k) 666 mm 2<br />
2. (a) 4·32em 2 (b) 546 mm 2 (e) 15.62 m 2 (d) 0.3645 m 2<br />
(e) 1400 em 2 (f) 53·2 em 2 (g) 4860 m 2 (h) 5·7575 km 2<br />
c. 1. (a) 50° (b) 58° (e) 62° (d) 46° (e) 49°<br />
(f) 30° (g) 39° (h) 58° 0) 60° ( j) 47°<br />
2. (a) Yes (b) No (e) Yes (d) No<br />
(e) Yes (f) No (g) Yes (h) Yes<br />
3. (a) BAD = 53° DAC = 37°<br />
(b) BAD = 51° BOA = 90° BCD = 39°<br />
(e) BAD = 18° BOA = 90° DAC = 18°<br />
(d) BCD = 45° BOC = 90° DCA = 40°
M36 A. 1. Drawing to confirm Pythagoras' theorem.<br />
2. Checking drawings<br />
3. Note that (a) If A < 90°, BC 2 < AC 2 + AB 2<br />
(b) If A> 90°, BC 2 > AC 2 + AB 2<br />
B. 1. (a) 4·47 em (b) 5·83 em (c) 7·21 em (d) 4·95 em<br />
2. Use b 2 = c 2 - a 2 or a 2 = c 2 - b 2<br />
(a) a = 9·75 em (b) b = 8·66 em (c) a = 18·3 em<br />
(d) b=93·67cm (e) b=27·2mm (f) 16km<br />
M37 A. 1. 5·83 m<br />
2. Yes, the base will be 4·5 m from the wall<br />
3. 5·6m<br />
B. 1. 4·7 m, 4·3 m 2. 21·6 m<br />
3. Roughly 88 m<br />
C. 1. 5·83 units 2. 4·24 units 3. 3·6 units<br />
5. 3·6 units 6. 6·4 units 7. 5 units<br />
4. 1·41 units<br />
8. 6·3(25) units<br />
P25 A. 1. c = 3·97 em so area = ~ac = 8·93 cm 2<br />
2. C = 48° 3. No, the third angle is 80°<br />
B. 1. DAX = 55°<br />
ADX = 35° ABY = 55°<br />
XDC = 55° YBC = 35°<br />
XCD = 35° BCY = 55°<br />
2. The angles are either 90°, 65° or 25°<br />
3. BAD = 30° DAC = 20° ADC = 90°<br />
4. ABD = 45° DCA = 43°<br />
4. 9·22 em<br />
C. 1. True<br />
2. Not true except in special cases where the angles are 45°, 45° and<br />
90°<br />
3. True 4. True<br />
P26 A., B. and C. practical work<br />
D. 1. (a) 8·06cm<br />
2. 4·39 m<br />
3. (a) 5·385 units<br />
(b)<br />
(b)<br />
38·01cm<br />
2·83 units<br />
(c) 6·74 em<br />
(c) 3·61 units (d) 5·83 units<br />
E26 A. See P25 A.<br />
31
B. See P25 B. and 5. CDA = 90°<br />
CAD = 50°<br />
DAB = 40°<br />
ABD = 50°<br />
6. DCA = 54°<br />
DAC = 36°<br />
DBC = 54°<br />
C. 1. (a) They are congruent (b) Rectangle (c) All equal<br />
(d) Equal (diagonals of a rectangle)<br />
2. Following question 1, the middle point of AC is the same distance<br />
from A, Band C. So the centre of the circle through A, Band C is<br />
the mid-point of AC.<br />
3. ~ABX and ~BXC are 45°, 45°, 90° triangles, so ~ABC also has two<br />
angles of 45° and B = 90°<br />
E27 A., B., C. all practical work<br />
D. 1. Yes. Heights of the triangles are<br />
J3 J3 J3<br />
-xa -xb -Xc<br />
2 ' 2 ' 2<br />
. J3 J3 J3<br />
SO their areas are 4a2, 4b2, and 4C2<br />
Since the triangle is right-angled a 2 + b 2 = c 2<br />
Thus J3 a2 + J3 b2 = J3 c2<br />
444<br />
2. Not so simple, depending on the size of right-angled triangle you<br />
start with (3, 4, 5 is easy)<br />
3. Pythagoras' theorem extends to both hexagons and semi-circles<br />
for the same reasons as given in question 1<br />
32<br />
Unit 11, page 51<br />
This unit approaches the trial and error method of solving<br />
problems. The salient feature of this method is that the error<br />
is used to arrive at a better guess. This will be developed<br />
later into numerical methods of finding solutions that use<br />
iteration. Further exploration of this is to be found in <strong>IMS</strong><br />
Book C, Problem Number 28 (Iteration). Children should<br />
not be discouraged from 'guessing' but instead, encouraged<br />
to scrutinize their guesses and use them to approach an<br />
answer. They should also be encouraged to be interested in<br />
the processes by which they themselves attempt to solve<br />
problems.
Answers<br />
M38 The answers are less important than the process. The first number is<br />
gradually built up to the second, noting what has been added.<br />
A. 1. 128 + 300 + 40 + 7 Check 128 + 347 = 475<br />
write 300 40 7<br />
Total 428 468 475<br />
2. 23 + 100 + 70 + 4<br />
3. 245 + 800 = 1045 (too much by 4)<br />
245 + 796 = 1041<br />
4. 390 + 10 + 328 (tidy up the hundreds first)<br />
390 + 338 = 728<br />
5. 3·4 + 3 + 0·7 --)<br />
6. 5·9 + 0·1 + 6,--)<br />
7. 8·3 + 3 + 0·3 --)<br />
8. 14·4 + 0·6 + 85--)<br />
9.5·85+0·15+11--)<br />
10. 60·47 + 0·53 + 9 + 50--)<br />
11. 3852 + ,8+ 40 + 100 + 1283J<br />
3·4 + 3,7= 7·1<br />
5·9 + 6·1 = 12<br />
8·3 + 3·3 = 11·6<br />
14·4 + 85,6 = 100<br />
5·85 + 11·15 = 17<br />
60·47 + 59·53 = 120<br />
3852 + 1431 (adding by calculator) = 5283<br />
12. 10·45 + 18·9 = 29·35<br />
(add 19 and subtract 0·1)<br />
B. Similar methods to A (or direct subtraction)<br />
1.40-10-3}<br />
40 -13 = 27<br />
or40-27=13<br />
2. 47 3. 3·05 4. 5·7 5. 62·5<br />
7. 371 8. 1·15 9. 26·9 10. 2·132<br />
12. 2·228<br />
C. 1. m is somewhat less than 20, try 18<br />
2. 12 x 10 will be too large; try 12 x 9<br />
6. 58<br />
11. 2229<br />
3. 8 x 100 will be too large, try 8 x 90, then 8 x 89<br />
4. 22 x 40 = 880, this leaves 44 so try 22 x 42<br />
5. 42 x 100 is too big, 42 x 90 is too big (subtract 84) 42 x 88<br />
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<br />
7. We know 42 -;-7 = 6 so try 0·77 x 60. Too big, subtract 42·35, rem.<br />
3·85. Try 0·77 x 54 = 41·58, add 0·77. Try 0·77 x 55<br />
8. 9 x 6 = 54 so try 9·1 x 5·9. Well done!<br />
9. 741. We know 25 x 3 = 75 so try 241 x 2·9. This gives 698·9<br />
(1·1 too small). Try 241 x 2·905 ... near enough<br />
10. 10 is too small so try 3·65 x 11. Too small by 1·85, so try<br />
3·65 x 11·5,error is 0·025 so try 3·65 x 11·508.Too large by 0·0042<br />
so try 3·65 x 11·507. Error 0·00055 so try 3·65 x 11·5068, etc.<br />
33
11. Try 7·0 error 0·021, about 0·05 x 0·4. Try 0·403x6·95=<br />
12. 0·1 will be too large: try 0·09 etc.<br />
D. Can be solved by straight division (see subtraction)<br />
1. 9 2. 8 3. 18 4. 9<br />
5. 55 6. 71 7. 28 8. 28<br />
9. 8·5 10. 0·47 11. 9·6 12. 96<br />
M39 A. 1. x = 1 2. x = 4·5 3. x = 3 4. x = 0·75<br />
5. x = 2 6. x = 5 7. x = 6 8. x = 1·5<br />
9. x=2 10. x=0·5 11. x=3 12. x=1·9<br />
Of course ax + b = c may be solved by the calculator sequence [Q] B<br />
[hJ El [ill EJ but the object of the exercise is to improve guessing.<br />
B Solved by [Q] [±] [hJ El [ill EJ<br />
1. x = 6 2. 7·5 3. 9 4. 8<br />
5.<br />
5<br />
3 = 1·6666 6.<br />
16<br />
'3 = 5·3333 7.<br />
5·5<br />
3 = 1·8333 8.<br />
4<br />
3 = 1·3333<br />
9. 2 10. 2·6 11. 3·4 12. 1·3<br />
C. 1. 1·2857 2. 2 3. 4·25 4. 1·5 5. 2·125<br />
6. 2·057 7. 0·1333 8. 0·3 9. 5 10. 12<br />
11. 22·857 12. 27·5<br />
M40 A. 1. 5 cm x 15 cm 2. 8 cm x 12 cm 3. 3 cm x 17 cm<br />
4. 3 cm x 17 cm gives an area of 51 cm 2 while 4, 16 gives an area of<br />
64 cm 2 • It is a good idea at this point to plot a graph of shortest<br />
side against area. You can then read off the side length which gives<br />
an area of 60 cm 2 • Otherwise use a trial and error method.<br />
3·6cm x 16·4cm has area 59·04cm 2 while 3·7 x 16·3 has area<br />
60·31 cm 2 • Try 16·32 x 3·68<br />
5. 5·53cmx14·47cm 6. 10cmx10cm<br />
7. 3·3cm x 16·7cm 8. 2·2cm x 17·8cm<br />
34<br />
B. 1. 25cmx25cm<br />
2. (a) They lie on a curve shaped like a circle inside out. This curve is<br />
ca IIed a Hyperbola.<br />
(b) The square is the largest rectangle.<br />
C. 1. B = 60° 2. 23·65 cm x 6·35 cm<br />
3. This can be done by folding as follows.<br />
(i) Estimate one-third of the strip and make a fold F,. Fold the<br />
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<br />
nearer and nearer to Exactly One-Third of the strip. (The error<br />
is halved each time.) This method can be used to fold a strip<br />
into exactly 7 parts or 13 parts or any other prime number.<br />
4. 17·32<br />
5. 7 is a bit too big. Try 6·5, too small (274·625), try 6·7 ... just too<br />
large (300·763), try 6·69 ... too small (299·418).6·7 is a satisfactory<br />
answer.<br />
6. Examples can be found in a dictionary, e.g. abacus, abdicate<br />
... etc.<br />
P27 A. 1. 42 mm<br />
2. There will be a number of lines which reduce AX + BY + CZ but<br />
the best one will be the line passing through A and C.<br />
3. For this line AX + BY + CZ is about 30 mm<br />
B. Drawing exercise<br />
C. Exact to 3 decimal places 3·302<br />
1. (a) 2·714 (b) 3·684 (c) 10<br />
2. Yes<br />
D. 62500m 2<br />
E. The shortest route is found by using reflections of X and Y in the river.<br />
x<br />
X'<br />
The line from X' to Y is the shortest possible one. Therefore X~ P~ Y<br />
is the shortest route.<br />
F. Compare results to see who can swap the counters in the smallest<br />
number of moves.<br />
E28 A. See P27 A.<br />
B. 1. The line should pass through the two points which are farthest<br />
apart. (Can you see why this will give the shortest distance from<br />
the third point to the line?)<br />
35
2. Not usually true<br />
3. Not true<br />
C. 1. The diagonals<br />
2. Can only be found by trial and error. Mark the 4 points in ink and<br />
explore with pencil lines, rule out the line each time you find a<br />
better fitting one.<br />
E2Sa A. 1, 2, 3 see P27 C.<br />
4. The suggestion is true for all numbers<br />
B. 1. 250 m x 250 m = 62 500 m 2 •<br />
2. Your drawings should approximate to (a) 72 168 m 2<br />
(b) 75 445 m 2 (c) 79 577 m 2<br />
C. 1. See P27 E.<br />
2. (i) and (iii) can be drawn, (ii) and (iv) cannot<br />
3. The shortest route varies according to his starting point; starting<br />
from A or E he can visit all 5 towns by travelling only 31 km,<br />
although he will not get back to his starting point.<br />
36<br />
Unit 12, page 54<br />
Powers are a simple concept when a calculator is available.<br />
They lead to important generalizations through experience.<br />
It has been shown that the laws of indices can be used and<br />
understood by 8-year-olds, hence the topic should not be too<br />
difficult at this stage. The first problem is to clarify the<br />
language and symbolism. It is not hard to understand why<br />
children get confused between x 2 and x2 or 3 2 an
Answers<br />
M41 A. 1.<br />
5.<br />
9.<br />
13.<br />
Some of the facts which should appear from exploration<br />
are:<br />
(i) xn~ 00 as n~ 00 if x> 1<br />
(ii) x n ~ 0 as n ~ 00 if x < 1<br />
(iii) XO = 1<br />
(iv) x m • x n = x m + n<br />
(v) (xmt = x mxn = (xn)m<br />
4 2 2. 7 2 3. 52 4. 6 2<br />
(2·5)2 6. 9x9 7. 10x10 8. 12 x 12<br />
8 3 10. 11x11x11 11. 53 12. 6 3<br />
(4'5)3 14. (2·8) cubed 15. 31 x 31 x 31<br />
B. 1.<br />
6.<br />
11.<br />
C. 1.<br />
2.<br />
3.<br />
8 2. 9 3. 125 4. 64 5. 1000<br />
2·25 7. 484 8. 4913 9. 18·49 10. 1936<br />
175·616 12. 0·64 13. 0·343 14. 0·04 15. 0·027<br />
6 2 7 2 8 2 9 2 10 2<br />
36 49 64 81 100<br />
11 13 15 17 19 ~ differences<br />
20 2 21 2 22 2 23 2 . .. etc.<br />
400 441 484 529<br />
41 43 45<br />
1.1 2 1.2 2 1.3 2 1.4 2 . .. etc.<br />
1·21 1·44 1·69 1·96<br />
0·23 0·25 0·27 ... etc.<br />
4. (a) Cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000<br />
(b) 7 19 37 61 91 ~ differences<br />
12 18 24 30. . . etc.<br />
666<br />
(c) The 6 times table appears. Differences of this are all 6s.<br />
M42 A. 1. (a) 7 7 = 823543 (b) Also 7 7 in fact 7 8 -T 7 (c) Also gives 7 7<br />
2. (a) 3 8 (b) 3 16 (c) 3 8 = 6561<br />
3. (a) (1·6)10 (b) (1·6)18 (c) (1·6)10<br />
4. (a) (0·9)12= 0·2824295 (b) (0·9)12 (c) (0·9)16<br />
B. Some calculators will set in the constant multiplier automatically.<br />
Others will require a double press on the [K] button. There are also<br />
other systems. This should be checked before starting the exercise.<br />
1. See above<br />
2. (a) 20·4, 27,2, 37·4, 54·4, 27·88<br />
(b) 2401 (c) 32 768 (d) 15·625 (e) 0·03125<br />
37
(f) 0·43046721 (9) 131621 7 (h) 2-0121964<br />
3. Same questions as 2<br />
C. 1. True 2. True 3. Not true 4. True<br />
5. Not true 6. Not true 7. Not true 8. True<br />
M43 A. 1. 7 5 2. 2 8 3. 4 10 4. 59<br />
5. 11 4 6. 16 11 7. 23 9 8. 99 18<br />
9. (0·4)5 10. (0·8)10 11. (0.36) 10 12. (0-47)21<br />
B. 1. Yes 2. Yes 3. Yes 4. Yes<br />
5. Yes 6. Yes 7. Yes 8. Yes<br />
C. 1. True 2. Not true, (a 2 )3 = a 6 3. True<br />
4. Not true (a 3 )3 = a 9 5. True 6. True<br />
P28 A. 1. 27 2. 256 3. 3125 4. 0·25 5. 0·512<br />
6. 2·197 7. 2·8561 8. 5·832 9. 2·0736 10. 1-61051<br />
B. 1. 10 % of 50000 is 5000 so population after 1 year is<br />
55 000 = 50 000 x 1-1<br />
2. 10 % of 55000 is 5500, so population after 2 years is<br />
60 500 = 50 000 x (1.1 )2<br />
3. 10% of 60500 is 6050, so population after 3 years is<br />
66 550 = 50 000 x (1.1 )3<br />
C. 1. Use constant multiplier<br />
(1·1)3 = 1·331 (1·1) 7 = 1·949<br />
(1.1 )4 = 1.464 (1·1)8 = 2.143<br />
(1-1)5=1'610 (1·1)9 = 2.358<br />
(1·1)6=1·771 (1·1 )10= 2.593<br />
2. (1·1F 3_ Between 7 and 8 years.<br />
D. Use constant multiplier<br />
1. 4 2. 5 3. 6 4. 6 5. 4<br />
6. 6<br />
P29 A. All true<br />
B. 1. Not True 2. True 3. Not True 4. True<br />
5. True 6. True 7. Not true 8. Not true<br />
C. 1. The sequence is [m [KJ ~ ~ [KJ~. Your sequence may be<br />
a a 2 a 3 a 6<br />
different if your calculator has a different mechanism for constant<br />
multiply.<br />
(a) 4096 (b) 1838·2656 (c) 3·635215<br />
(d) 0·007256 (e) 0-001838 (f) 0·94148<br />
38
2. Use [ill [K] ~ ~ [K] ~ [K] ~ but see question 1.<br />
a a 2 a 3 a 6 a 12<br />
(a) 5·350 (b)<br />
3. (a) 2·1828746<br />
(d) Correct<br />
4. (a) True<br />
(e) Not true<br />
2·252 (c) 1·601<br />
(b) 23·2 (not 232)<br />
(e) Correct<br />
(b) True (c) True<br />
(f) True<br />
(d) 8·064<br />
(c) 0·0387595<br />
(f) Correct<br />
(d) .Not true<br />
E29 A. 1.<br />
2.<br />
3.<br />
B. 1.<br />
3.<br />
C. 1.<br />
3.<br />
4.<br />
5.<br />
6.<br />
7.<br />
(b) They form the x 6 table (c) They are all 6<br />
The patterns continue<br />
No, the 4th differences are all 24, the 3rd differences are the x 24<br />
table<br />
See P28 C. 1<br />
See P28 B. 1,2 and 3<br />
30 (= 1 2 + 2 2 + 3 2 + 4 2 )<br />
36 (= 1 + 8 + 27)<br />
2. (1.1 )7<br />
4. Between 7 and 8 years<br />
2. 55,91,140,204<br />
100 (=1 3 +2 3 +3 3 +4 3 ) (Note: This also equals (1 +2+3+4)2,<br />
see front cover of the Book A2 or B2.)<br />
The numbers are (2 3 )2 = 64, (3 3 )2 = 729, (4 3 )2 = 4096<br />
x Y is a power function. [ill IxyllliJ ~ gives a b on the display<br />
(a) 1024 (b) 17·576 (c) 0·4096 (d) 3·583<br />
E30 A. See P28 D.<br />
B. See P29 A.<br />
C. See P29 B.<br />
D. See P29 C. 1,2,3<br />
E. The result is am -;- an = a m - n (a) 32 (b) 52 (c) (1·7)4 (d) (2·3)4<br />
Unit 13, page 58<br />
This is a straight unit on careful use of calculator. There are<br />
many possibilities of error and these can usually be avoided<br />
by improving technique. The work is not unrelated to the<br />
beginnings of computer programming which will interest<br />
many children. Once again various calculators will have<br />
different methods of working and this should be discussed<br />
with the class. If someone has a programmable calculator in<br />
class this will have a more subtle way of doing the calcula-<br />
39
tions. A calculator with a statistics circuit will add a collection<br />
of numbers in the memory without the (±] key. Just press<br />
in each number and M + or 'x'. The total will be recovered<br />
on pressing LX. It will also count how many numbers have<br />
been added. This can be obtained by pressing ill],<br />
There are no P or E exercises with this Unil.<br />
Answers<br />
M44 A. 1. 331·93 2. 1444·86 3. 4107 4. 3·290<br />
B. 1. £55·70 2. 33·0875 cm 3. 550·4444 g 4. 672·7 ml<br />
C. 1. Correct 2. Correct 3. Correct 4. Should be 21·48<br />
D. 1. 330 2. 4·3 3. 0·3 4. 53·4<br />
M45 A. 1. 6·8660865 2. 344·13984<br />
3. 1·0267781. This may come out to 1·0267776 in reverse because of<br />
rounding off errors<br />
4. 0·2835144. This may come out to 0·28275 in reverse<br />
B. 1. 4, (14 -7- 7) x 2<br />
2. 2·0681818, (2·6 x 3·5) -7- 4·4<br />
3. 616, (44 -7- 0.8) x 11·2<br />
4. 4·6882978, (5,65 x 3·9) -7- 4·7<br />
C. The calculation can be treated as 'straight through' if you can ignore<br />
the brackets and get the same answer as with the brackets. For<br />
example (2 x 3) x 6 is the same as [2] ~ ~ ~ [ill.<br />
1. 14 -7- (2 -7- 3) = 14 -7- (0·666666) = 21 [I] @] B [2] B ~ ~ gives<br />
2,333333, so this is not a straight through calculation<br />
2. 5·733333, not straight through.<br />
3. 0,0009, straight through<br />
4. 1067·7333, not straight through<br />
M46 A. 1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
6.<br />
2464 @] [I] [±] I]] [ill ~ ~ [lJ. The change of order is possible<br />
because a x b = b x a<br />
76, straight through<br />
18·395, straight through<br />
46·64, change order<br />
71, change order<br />
0·0105, change order<br />
40
B.<br />
c.<br />
7. 26·75, straight through<br />
8. 1·0231481, straight through<br />
If you have a calculator with brackets you can use those instead of<br />
memory.<br />
1. 176 2. 28·53 3. 93·6 4. 6·9384<br />
5. 8·1374092 6. 1·475 7. 3·0425531 8. 0·7430703<br />
1. 5·92 2. 0·0175 3. 23355 4. 3·6644736<br />
5. 482<br />
Answers<br />
M47 A.<br />
Unit 14, page 62<br />
The topic of area is made very easy by the use of the<br />
calculator. The child must be sure that moving boundaries<br />
will not change the area they enclose and must have the<br />
concept of conservation of area before he can tackle<br />
trapezia, etc. There are many applications in physics,<br />
geography and <strong>mathematics</strong> itself. In the non-school world,<br />
area is, of course, of vital importance in the measurement of<br />
land, floor coverings, etc. though these matters may not<br />
light too much flame of interest in the pupils. They may be<br />
more interested in air resistance (getting your head down<br />
when cycling or even running), sailing and windmills, space<br />
per child in the classroom and the four-colour map problem.<br />
There are ancient mathematical puzzles which involve<br />
areas ... the proof of Pythagoras' theorem, the fact that the<br />
side of a square whose area is 2 cm 2 cannot be measured with<br />
any ruler, the fact that a circle and a square cannot have the<br />
same area. Finally, areas will be used to explain other things<br />
in <strong>mathematics</strong> so that it is most important that the fundamental<br />
ideas are grasped at this stage.<br />
1. 253 mm 2 (2·53 cm 2 ) 2. 810 mm 2 3. 320 mm 2<br />
4. 400 mm 2 5. 640 mm 2<br />
6. 12 cm 2 (can be calculated as a 5 x 5 square with a 2 x 2 square and<br />
a 3 x 3 square cut away).<br />
B. 1. 10cm 2. 12·5cm 3. 6~ cm 4. 16~ cm<br />
6. 26·3 cm 7. 45·45 cm 8. 2·86 cm 9. 21·7 cm<br />
11. 2·3cm 12. 250cm<br />
5. 22·2cm<br />
10. 9·3 cm<br />
41
C. 1. 112 cm 2 2. 6 cm 2 3. 1·62cm 2 4. 6 cm 5. 6·66<br />
6. 4·88 m 7. 7·5 m 8. 2·5 cm 9. 0·11 cm 10. 72 cm<br />
(Note: These answers could be written in terms of different units, e.g.<br />
1. 1120 mm or 1·12 m.)<br />
3. 2cm 2 4. 1·875cm 2<br />
M48 A. 1. 3cm 2 2. 2cm 2<br />
5. 2 cm 2 and 6 cm 2 6. 450 mm 2<br />
5. 3·75 cm 2 6. 3·75 cm 2 7. 3cm 2 8. 4 cm 2<br />
B. 1. 17·5cm 2 2. 26·25mm 2 3. 10·4m 2 4. 210 mm 2<br />
5. 0·14 m 2 or 1400 cm 2<br />
6. 8'8cm 2<br />
C. 1. 50cm 2. 4m<br />
3. 2·94cm 4. 4·59cm<br />
2. 4·5cm 2 3. 6cm 2 4. 2·5cm 2<br />
M49 A. 1. 4cm 2<br />
C. 1. 7cm 2 2. 12 cm 2 3. 20cm 2 4. 12cm 2<br />
5. 3cm 2 6. 6cm 2<br />
B. 1. 1·5cm 2 2. 3·5cm 2 3. 3cm 2 4. 3·375cm 2<br />
5. 3·125cm 2 6. 3·2cm 2 7. 6'9cm 2<br />
P30 A.<br />
B.<br />
1. 6·9cm 2<br />
1. (a) 8·46 cm 2<br />
2. (a) 3·03 cm<br />
3. (a) 12·1cm<br />
2. 12cm 2<br />
(b) 9 cm 2<br />
(b) 1·7cm<br />
(b) 2 cm<br />
3. 11·3cm 2<br />
(c) 20·46 cm 2<br />
(c) 16·66 cm<br />
(c) 0·34cm<br />
4. 7·5cm 2<br />
(d) 14·21 cm 2<br />
(d) 5·7 cm<br />
(d) 6·57 mm<br />
P31 A. 1. 533 cm 2 , 4467 cm 2 2. 5·3 cm<br />
3. Each equilateral triangle has area 10·825 cm 2 • So estimated area of<br />
circle is 6 x 10·825 = 64·95 cm 2<br />
Error 13·59 cm 2<br />
4. They should all be correct. The rule is very accurate.<br />
42<br />
B. 1. (15 x 4) x 3 plus 2 x 4 x 4 x 0·433 (see question 4 of Exercise<br />
A) = 193·8 cm 2<br />
2. 15·5 cm 2 • This is quite a difficult trapezium to draw. Start off with<br />
the isosceles triangle 4 cm, 4 cm and 2 cm. The rest is easy<br />
3. Areas: rectangle 588 mm 2 , bows 63 mm 2 , stern 45 mm 2 .<br />
Total 696 mm 2 • Since 1mm ~ 1 metre, 1mm 2 ~ 1m 2 •<br />
Total deck area is 696 m 2<br />
C. 1. Areas 6ABC = 999 mm 2 , .6ADE = 426 mm 2 , .6AFG = 952 mm 2<br />
~BCED = 327 mm 2 , ~DEGF = 526 mm 2
2. Find the area by dividing into rectangles and right angled<br />
triangles. The double dimension enlargements will have four<br />
times the area of the figure you started with in each case.<br />
(a) Areas (i) 1430 mm 2 (ii) 680 mm 2<br />
E31 See P31 and C.2 (iii) 10·30 mm 2<br />
E32 A. 1. (a) 17'32cm 2 (b) 24cm 2 (c) 17·1 cm 2 (d) 27·7cm 2<br />
2. All the results should agree to 1 decimal place.<br />
B. 1. The largest will have one corner in a<br />
corner of the square with a diagonal of<br />
the square as an axis of symmetry.<br />
Area = 46·4 cm 2<br />
2. The largest square in an equilateral triangle is hard to find. Draw<br />
some of the rectangles which sit on the base and calculate their<br />
area. This will have a maximum at the square. Another method is<br />
to draw as large a rectangle as you can. Multiply the sides and<br />
take the square root of the product. This will give the approximate<br />
length of the side of the square. A large square can be drawn with<br />
3 corners on the three sides and the fourth corner on the axis of<br />
symmetry, but it will not be as large as the largest square which<br />
sits on the base.<br />
C. This is a fascinating puzzle. The secret is that the rectangle is not<br />
exact. There is a thin space along the diagonal which accounts for the<br />
extra cm 2 •<br />
Unit 15, page 68<br />
This unit explores the area of a triangle, developing from<br />
!base x height to !(length of side) x (altitude standing on<br />
the side). People learning <strong>mathematics</strong> tend to be too<br />
43
inflexible about configurations and this work emphasizes<br />
that the notion of 'base' is only a practical arrangement for<br />
identifying one of the sides. Thus a more dynamic view of a<br />
triangle is encouraged with the children feeling they have<br />
more control.<br />
Altitudes are easy to draw accurately and the concurrence<br />
of the altitudes is an interesting and curious property of<br />
triangles that is quite unexpected. Why should they meet at<br />
a point? It is especially striking where the triangle contains<br />
an obtuse angle and the altitudes meet outside the triangle.<br />
This is much more subtle than symmetry and can arouse a lot<br />
of mathematical interest in children. It can also improve the<br />
drawing and measuring skills.<br />
Answers<br />
M50 A. Side Length Altitude Length Product<br />
from<br />
1. AB 43mm C 29mm 1247<br />
BC 40mm A 31 mm 1240<br />
CA 34mm B 38mm 1292<br />
2. AB 45mm C 59mm 2655<br />
BC 61 mm A 44mm 2684<br />
CA 68mm B 39mm 2652<br />
3. AB 62mm C 46mm 2852<br />
BC 53mm A 55mm 2915<br />
CA 60mm B 48mm 2880<br />
The products are roughly the same for<br />
the given triangle. The product is the<br />
area of a rectangle in which the triangle<br />
would fit.<br />
B. 1. 8cm 2 2. 7·875cm 2 3. 9cm 2 4. 10·5cm 2<br />
5. 4cm 2 6. 5·5cm 2<br />
M51 A. 1. When C is above the mid-point of AB<br />
2. Always the same length, simply the distance between the two<br />
parallel lines<br />
3. The area is always the same (it is always (base x height) -;- 2)<br />
44<br />
B. 1. 6s AQB, ARB, XQY, XRY and XSY are all equal to 6APB
M52 1. 2cm 2<br />
2. 6.PAO = 6.PBO = 6.PXO, 6.PXR = LPAR = 6.PBR,<br />
6.OBR = L OXR = 6.OYR = 6.OAR,<br />
6. RXS = 6.RYS,<br />
6. BOX = L BRX = 6. BPX, 6.OXS = 6. OYS<br />
6. 2cm 2 2. 3 cm 2 3. 2 cm 2 4. 5 cm 2<br />
7. 2·25cm 2 8. 3·75cm 2 9. 4cm 2 5. 5cm 2<br />
P32<br />
A. 1.<br />
B. 1.<br />
2.<br />
3.<br />
Drawing 2. Drawing<br />
Drawing<br />
Yes, the altitudes should meet at a point<br />
Yes, the altitudes will meet at a point (outside the triangle)<br />
P33<br />
A. 1.<br />
3.<br />
B.<br />
231 km 2. 2·8 m<br />
You construct each triangle with compasses. Start with 6.ABC.<br />
Areas are LABC 770 m 2 6.ACD 680 m 2 LADE 774 m 2<br />
Total area = 2224 m 2<br />
1. True<br />
2. (a) True<br />
3. Not true<br />
(b) True<br />
4.<br />
(c) True<br />
Not true<br />
5. True<br />
C. This is the formula Area = abc/4R and it works for all triangles. This is<br />
especially easy if you want to find the areas of a family of triangles all<br />
drawn with their vertices on the same circle. Area = 800 mm 2 •<br />
E33 A. 2. The figure made up of P and the three arc intersections '" '2 and x<br />
is a rhombus. It has 4 sides equal because that is how you have<br />
drawn them. Therefore PX and AB are diagonals of the same<br />
rhombus and must be perpendicular.<br />
3. Drawing<br />
B. 2. (b) is true<br />
3. The altitudes still meet at a point but outside the triangle<br />
C. Find points " and '2 on the line, an equal distance from P. Now draw<br />
two circles with " and '2 as centres and with the same radius. Choose<br />
the radius large enough for the two circles to meet above and below<br />
the line. The points '" '2 and the two points of intersection of the<br />
circles will form a rhombus.<br />
E34 A. B., C. see P33<br />
D. 1. Measurement 2. All th ree suggestions are true<br />
45
Unit 16, page 73<br />
Many problems can be simplified by some sort of listing<br />
process and this unit considers a number of such problems.<br />
The listing either solves the problem completely (as in<br />
probability questions) or helps towards a solution by identifying<br />
the nature of the operations involved in the problem.<br />
This work is an introduction to programming by means of<br />
flow charts and so the type of numbering utilized in BASIC<br />
is used here. Children enjoy this type of work very much and<br />
it can be developed to involve problems devised by the<br />
pupils themselves. There are bound to be some computer<br />
enthusiasts present who will already know the fundamentals<br />
of programming and these can be brought into the discussion.<br />
The idea of a loop (and decision box) is introduced in the<br />
second half of the unit but is kept fairly simple. You may<br />
wish to introduce analogous programme language at this<br />
point in which case you should refer to the BASIC book<br />
being used in your school. (Or use Eagle: An Introduction to<br />
Basic, published by Bell & Hyman.)<br />
Flow charting is a very helpful method of organizing<br />
thought and will have applications in many of the traditional<br />
activities of <strong>mathematics</strong> such as proving theorems and<br />
solving equations. In addition flow charting problems on<br />
numbers can give valuable insight into many properties<br />
which are part of number theory.<br />
Answers<br />
M53 A.<br />
B.<br />
1. Run, jump, fall 2. Get on, buy, get off<br />
3. Homework, breakfast, school (7)<br />
4. Study, learn, pilot 5. Choose, insert, play<br />
6. Add fat, heat, add egg, eat<br />
7. Cut bread, butter, marmalade, eat<br />
8. Think, double, subtract<br />
9. Think, double, add 7, subtract 4 (others possible)<br />
10. Draw line, choose A, compass at A, mark off length<br />
Just as A but number lines, 1~, 2~, 3~, etc.<br />
2. (a) 05 (b) 15 or 25 (c) 35<br />
(e) 25 or 35 (f) 25 (g) 25<br />
(i) 45 (j) 45<br />
(d) Could be 15 or 35<br />
(h) 25 or 35<br />
46
C. 1. Should give 5 more than the number you started with<br />
2. Should give the perpendicular bisector of AS<br />
3. Should give a rhombus<br />
4. 6 lines<br />
M54 A. 1. (a) Depends on the letters, e.g. E, N, O~ ONE, CONE<br />
M55 A. 1.<br />
x 2 -1 -'-10<br />
(b) For example 9~ 81~80 ~ 8 (Answer should be one<br />
less than the number you started with)<br />
(c) For example 1~~difference 198 -T 9 = 22 remainder O.<br />
The remainder is always zero<br />
B. Different answers are possible.<br />
C. Different answers are possible though the order is strict, for example<br />
stir with a spoon could not come first<br />
No<br />
Catch bus<br />
2.<br />
3.<br />
Join school<br />
team<br />
47
4.<br />
Borrow from<br />
friend<br />
B. 1.<br />
Put in sugar<br />
and stir<br />
2. Could be quite a long process. Don't forget to tryon the second<br />
shoe of a pair.<br />
3. You have to avoid over-watering the plant<br />
4. Remember that the 27 s have to be counted as well as subtracted<br />
5. In this chart you will have to find a way of getting from one<br />
number to the next<br />
C. 1. Pumping up a bicycle tyre<br />
2. Bathing a baby. Always put the cold water in first to avoid the risk<br />
of scalding the baby.<br />
3. This is a programme for using a calculator and checking as you go<br />
along<br />
48
P34 A. 1. (a) The years are counted from 0, the birth of Christ. Different<br />
countries and faiths number the years differently. Thus 1984 is<br />
1404 in the Muslim calendar and 5744 in the Jewish calendar.<br />
(b) The minutes are counted, grouped into 60s. Each 60 is 1 hour.<br />
This system of counting time goes back to the Babylonians<br />
(many thousands of years ago)<br />
(c) Homes are counted by numbering houses or flats<br />
(d) Clothes are numbered in sizes. They are also numbered by the<br />
manufacturer.<br />
2. (a) All the names with the same first letter are grouped together.<br />
These groups are ordered alphabetically. Then the second<br />
letter is used to order the names within a group. Then the third<br />
letter is used. People with the same name are grouped by<br />
initials ... See telephone directory or voters' list.<br />
(b) Similar to names but without initials<br />
(c) Letters of the alphabet have been used in the registration<br />
numbers of cars. Usually the first three letters are a code for<br />
the district in which the car was first registered. The last letter<br />
can tell the year of manufacture, e.g. X~ 1982.<br />
(d) Letters are used to describe how good or bad a pupil is on a<br />
report.<br />
3. (a) So that each user has his own telephone number<br />
(b) So that you can find your place<br />
(c) So that no-one else can take your savings<br />
(d) So that one aircraft does not get mixed up with others (and<br />
other reasons)<br />
(e) So that notes can be checked<br />
(f) So that you can tell where they are going<br />
B. 1. 30 2. 28 3. 19 4. 36 5. 30<br />
6. 19 7. 50 8. 15<br />
C. 1. No remainder, the number is divisible by 11<br />
2. Different remainders, the number is not divisible by 11<br />
3. No remainder because n + 1 is a factor of n 2 - 1 (The other factor<br />
is n - 1)<br />
P35 A. If the flow charts are satisfactory they will work on a 'test run'.<br />
B. 1. Different answers<br />
2. (a) and (b) are examples of the equation ax + b = c where a, band<br />
c are numbers. The steps are<br />
1fll calculate c - b = m<br />
2fll calculate m -T a = n<br />
49
3~ write down x = n<br />
(c) is an example of x 2 + b = c where band c are numbers. The<br />
steps are<br />
1~ calculate c-b=m<br />
2~ is m>O<br />
3~ Yes: x=)m<br />
4~ No: x 2 + b = c cannot be solved<br />
The following flow chart would solve the equations by trial and error<br />
(a)<br />
(b) Start with x = 1·5, d = 0 and change x to x - d/2, and find<br />
(3x - 2) - 1·8 = d<br />
(c) You need a flow chart for )3, see question C2, or use tables or<br />
calculator [2] button.<br />
C. 1. (a) The flow chart needs to try all the primes in turn up to 11.<br />
These are 2, 3, 5, 7 and 11. When 11 has been tried, and shown not<br />
to be a factor of 149 the flow chart should come to an end.<br />
(b) 36 is square and also a triangle number<br />
(c) Different answers<br />
50
E35 A. See P34 A.<br />
2. (a) (b) The flow charts are stupid ways of coming to decisions.<br />
There is no guide to the choice based on the results of n -;-X, or<br />
reading first line, so it is not worth following these procedures.<br />
B. 1. (a) 30 (b) 28 (c) 19 (d) 36 (e) 30<br />
(f) 19 (g) 50 (h) 15 (i) 32 (j) - 7<br />
2. (a) 50, 100 (b) 48, 98 (c) 31,61 (d) 60,120<br />
(e) 46, 86 (f) 49, 194 (g) 56, 1<br />
(h) -1 5, -160 (i) 512, 524288 (j) -37, -182<br />
C. See P34 C and<br />
4. n 2 - 1 has two factors (n + 1) and (n - 1)<br />
E36 A. There will be various answers to these. The process for 5 is to multiply<br />
both sides by 100 ...<br />
x = 3·747474 ...<br />
1OOx = 374·747474 and then subtracting to get<br />
99x = 371<br />
371<br />
so x=- 99<br />
6. Use tables or [2J to find In, otherwise use a divide and average<br />
method.*<br />
7. Use a gradual approximation process. Start somewhere between<br />
2 and 3, since 2 3 = 8 and 3 3 = 27. Perhaps 2·2 is a good starting<br />
point<br />
(2·2)3 = 10·648 (too small) (2·35)3 = 12·977 (too large)<br />
(2·3)3 = 12·167 (too small) (2·34)3 = 12·813 (too large)<br />
(2·4)3 = 13·824 (too small) (2·33)3 = 12·649 (too small)<br />
2·338 ... etc.<br />
(Answer: 2·3386049 to 7 decimal places)<br />
B. See P35 B, also (d) use a flow chart that considers x(x + 1) each time<br />
C. Check flow charts by giving them a Itest run' (see P34 C. 2)<br />
* See Basic Mathematical Tables.<br />
51
Unit 17, page 78<br />
Answers<br />
M56 A.<br />
B.<br />
Recurring decimals should be a familiar phenomenon to<br />
most of the children by now. In this unit we make use of<br />
them to strengthen concepts of fractions as well as to<br />
investigate the recurring patterns themselves. Mathematically<br />
they are a very useful tool and one can demonstrate<br />
the existence of irrational numbers by simply constructing<br />
decimals that will never recur. Some examples are<br />
0·121122111222. .. or 0·1234567891011121314.... Later<br />
this technique can be used with binary numbers. Most<br />
important is that this investigatory unit removes fear of<br />
division; once the patterns are understood there is nothing<br />
to fear. There are also beneficial side-effects such as the<br />
discussion of the difference between 2 and 1·9999 ... or the<br />
problems that arise from approximating 0·16666 to 0·16 or to<br />
0·17. There is some encouragement for the use of fractions<br />
in their ratio forms, the advantages can be seen; division by<br />
9 is seen as an interesting rather than difficult process, while<br />
division by 99 can be found both amusing and interesting.<br />
This unit also highlights some of the limitations of the<br />
calculator, for instance the repeated pattern of 1/13 comes<br />
at the end of the display. Multiplication tables show up the<br />
remainder better than the calculator and use of these should<br />
be encouraged.<br />
This unit is more about numbers than their applications<br />
but this does not mean the work is less important. Numbers<br />
are the foundation of <strong>mathematics</strong> and any understanding<br />
gleaned in this unit will be of great value later on. There are<br />
also simpler concepts which are reinforced here such as the<br />
process of division, remainder and place value. Remember it<br />
is not always helpful to view concepts only from below. A<br />
feeling for place value comes more from this sort of investigation<br />
than from endless concentration on the 'basics'.<br />
1. 0·6 2. 2·6 3. 3·3 4. 5·6 5. 0·8<br />
6. 1·3 7. 1·6 ~. 2·4 9. 0·4 10. 0·7<br />
11. 0·16 12. 0·83 13. 1·6 14. 1·16 15. 3·6<br />
16. 5·83<br />
1.<br />
2 7<br />
9<br />
1 3<br />
- 2. - 3. -=1 4. -=-<br />
9 9 9 3 9<br />
52
1 1 1 5 2 1<br />
5. -+10=- 6. - 7. - 8. -=-<br />
3 30 6 6 30 15<br />
5 1 5<br />
5 1<br />
9. -=- 10. - 11. 1~ 12. -=-<br />
60 12 9 90 18<br />
C. These work out by converting the fractions to decimals, then adding<br />
or subtracting.<br />
M57 A. All proofs<br />
B. All proofs. Note: Some of these fractions can be simplified.<br />
45 5 63 7 6 2 36 4<br />
99 = 11; 99 = 11; 99 = 33 ; 99 = 11<br />
C.<br />
1. (a) 0·090909 = 0'9~ (b) 0·181818 = 0·18<br />
(c) 0·454545 = O·~~ (d) 0·3181818 = 0·318<br />
(e) 0·242424 = 0·24 (f) 0·155555 = 0·15<br />
(g) 0-36<br />
(h) 0·2<br />
2. (a) 31 + 99 (b) 40 + 99<br />
(c) 53 + 90 (0·088888 ... = 8/90<br />
0·5 = 45/90 Total 53/90)<br />
(d) 68 (e) 5 + 6 (f) 8 + 99 (g) 33 + 90<br />
90<br />
(h) 37 + 90<br />
3. (a) This can be checked by using decimal forms on the calculator<br />
1<br />
- = 0·111111<br />
9<br />
3<br />
- = 0·03030303<br />
99<br />
14 1 11 11 3 14<br />
and this is 99' Also "9 = 99 so 99 + 99 = 99<br />
Total = 0·14141414<br />
(b), (c), (d) all similar<br />
P36 A. 1. (a) 0·5 (b) 0·3 (c) 0·25 (d) 0·2 (e) 0·16<br />
.....-----<br />
(f) 0·142857 (g) 0·125 (h) 0·;<br />
2. (a) 0·6 . (b) 0·75 (c) 0·4 (d) 0·6 (e) 0·6<br />
.....-----....<br />
(f) 0·714285 (g) 0·75<br />
53
3. Proofs. (b), (c) and (d) these can be proved by using equivalent<br />
fractions<br />
B.<br />
c.<br />
1 1 2 1 3<br />
-+-~-+-~- etc.<br />
2 4 4 4 4<br />
1. (a) 4·9 (b) 2·9 (c) 6·9 (d) 11·9<br />
2. (a) 0·0000001 (b) 0·5<br />
3. (a) 2·499999... (b) 0.39 (c) 0·249<br />
4. (a) 10·39 (b) 0·249 (c) 5·49<br />
1. (a) 0·5:7 (b) o·§{) (c) O·fa (d) 0·3f5<br />
4 5 20<br />
2. (a) 11 (b) 11 (c) 33<br />
(e) 9·9<br />
(d) 0·679<br />
(d) 0·339<br />
r---..<br />
(e) 0·527<br />
P37<br />
A. 2. (a) 285714 (b) 428571 (c) 571428<br />
(e) 857142<br />
all are the number 142857 rotated<br />
3. 999999 (though 1x 7 should be 1)<br />
B.<br />
(d) 714285<br />
1. 0·111111, O·22222, etc.<br />
43 35 17<br />
2. (a) 90 (b) 90 (c) 30<br />
5<br />
31 23<br />
(d) 55<br />
(e) 90 (f) 90<br />
90<br />
c. 1. General investigation following the work already done<br />
2. ~ = 0·422222 23 = 0·696969. .. !...- = 0·07070707<br />
45 33 99<br />
t<br />
7<br />
/'<br />
'8<br />
4<br />
2<br />
59 = 0.6555555 ~ = 0.777777 47 = 0.522222<br />
90 9 90<br />
E37 A. See P36 and 4. (a) Use calculator<br />
54<br />
B. See P36 B.<br />
1<br />
(b) Yes 13= 0·0769230769230 ...<br />
1<br />
17 = 0·05882352941176470588 ... etc.<br />
The pattern repeats after a particular remainder comes up a<br />
second time in the division process.
C. See P36 C and<br />
3. You must get to a point at which the remainder has occurred<br />
before. From then on the pattern of division will repeat. An exact<br />
division will produce recurring zeros. ; = 0·5000000 ...<br />
4. Since every rational number must have some sort of repeating<br />
pattern, a decimal that does not repeat cannot have come from a<br />
rational number.<br />
5. The pattern is changing all the time.<br />
6. This is a very ancient discovery (2000 years). If a length is to be<br />
measured it must be measured with a ruler and in units. The<br />
smaller divisions are formed by dividing the units into an equal<br />
number of parts, i.e. rational numbers fit the marks on the ruler,<br />
1~, 11>1etc. A length which cannot be fitted to a rational number,<br />
e.g. the diagonal of a square side 1 unit, cannot therefore be<br />
exactly measured by an ordinary ruler, no matter how finely it is<br />
graduated!<br />
E38 A, B.<br />
4.<br />
C. 1.<br />
2.<br />
3.<br />
4.<br />
See P37 and some additional investigations<br />
Can be proved by simply listing all possible values for a and b.<br />
There are no regular patterns in 7T.<br />
There are 8 ones, 12 twos, 11 threes, 11 fours, 8 fives, 8 sixes,<br />
9 sevens, 12 eights, 14 nines, 8 zeros.<br />
No! If 7T could be represented by a fraction there would be a<br />
pattern in the decimal form.<br />
Investigation<br />
Unit 18, page 80<br />
Multibase arithmetic is more than a rather unimportant<br />
mathematical game. It is a way of investigating the important<br />
rules which govern ordinary arithmetic and the system<br />
of relationships which control all arithmetical operations.<br />
For those children who have learned their arithmetic facts<br />
blindly this work can be very illuminating. Simple binary<br />
arithmetic is important in understanding electronic calculators<br />
and storage systems and is best seen as another counting<br />
system. The relationships between numbers in base 2, 4 and<br />
8 are also revealing. Children find the addition and multiplication<br />
tables in different bases easy to understand and<br />
55
amusing because everything looks wrong. Even though the<br />
numbers change, the properties of commutativity and associativity<br />
are clearly seen in all base systems. So are primes,<br />
number patterns, and fractional systems.<br />
It is interesting to realize tha.t for generations we have<br />
asked children to work in multi base systems in their work<br />
with fractions, even before the age of 8! 1+ -! is the addition<br />
of two numbers from different base systems, base 3 and base<br />
4, yet we have expressed surprise that many children become<br />
confused and frightened when asked to manipulate<br />
fractions in primary school.<br />
Answers<br />
M58 A. 1. dots: 178 10 =262 8 =454 6<br />
3. dots: 89 10 =131 8 =225 6<br />
B.<br />
c.<br />
1. 11 2. 13<br />
6. 33 7. 23<br />
11. 50 12. 44<br />
1. (a) (i) 18 (ii) 36<br />
(v) 324 (vi) 612<br />
(b) (i) 1 box + 4 packets<br />
(ii)<br />
(iii)<br />
(iv)<br />
(v)<br />
3. 29<br />
8. 42<br />
13. 88<br />
(iii) 144<br />
(vii) 744<br />
4. 36<br />
9. 63<br />
14. 157<br />
(iv) 216<br />
(viii) 1104<br />
2 boxes + 4 packets + 4 eggs<br />
5 boxes + 3 packets + 2 eggs<br />
4 boxes + 1 packet + 5 eggs<br />
2 crates + 2 boxes + 3 packets + 3 eggs<br />
2. (a) (i ) 16 (ii) 48 (ii i) 44<br />
(iv) 64 (v) 160 (vi) 204<br />
(b) (i) 1 carton + 2 boxes (ii) 2 cartons<br />
(iii) 3 cartons + 2 glasses<br />
(iv) 1 case + 2 cartons + 1 box + 3 glasses<br />
5. 28<br />
10. 69<br />
15. 229<br />
M59 A. 1. (a) 113 6 (b) 201 6 (c) 244 6 (d) 532 6 (e) 1103 6 (f) 1443 6<br />
Note: The easy calculator check. 1443 6<br />
56<br />
[I]~[ID[±]@]~[ID[±]@]~[ID[±]~E]<br />
iii<br />
i<br />
1 443<br />
2. (a) 115 8 (b) 137 8 (c) 240 8<br />
(d) 321 8 (e) 674 8 (f) 1365 8
Check 1365 8<br />
[I]~lID[±][JJ~lID[±][[]~lID[±][]J~<br />
Note: Some calculators will require ~ after each operation.<br />
3. (a) 13 5 (b) 32 5 (c) 124 5<br />
(d) 214 5 (e) 1212 5 (f) 2132 5<br />
B. 1. 133 6 is wrong 2. 2112 4 is wrong 3. 2445 6 is wrong<br />
c. 1. (a) 7 (b) 14 (c) 24<br />
(d) 44 (e) 60 (f) 237<br />
2. (a) 19 (b) 29 (c) 39<br />
(d) 77 (e) 122 (f) 407<br />
3. (a) 23 (b) 63 (c) 64<br />
(d) 168 (e) 219 (f) 1253<br />
M60 A. 1. 13 8 2. 12 8 3. 5 4. 5<br />
5. 36 8 6. 61 8 7. 43 8 8. 6 8<br />
B. The 5 times table in base 6 is 5, 14, 23, 32, 41, 50. Similar to the<br />
ordinary 9 times table. There is a similar pattern for x7 in base 8 and<br />
x9 in base 10. In general the pattern is the same for x (n - 1) in base n.<br />
c. 1. 1,4, 13,24,41, 100. Note the pattern 1 434 1<br />
2. Base 4 1 10 21 100 (121)<br />
pattern of endings 1 0 1 0 1 0 ...<br />
Base 5 1 4 14 31 100 (121)<br />
pattern of endings 1 4 4 1 0 ...<br />
Base 7 1 4 12 22 34 51 100<br />
pattern of endings 1 4 2 2 4 1<br />
Base 8 1 4 11 20 31 44 61 100<br />
pattern of endings 1 4 1 0 1 4 1 0<br />
3. The numbers 9, 16,25 will answer this question whatever the base<br />
(a) 12 2234 (b) 11 20 31 (c) 10 17 27<br />
P38 A. 1. 40 6 , 20 9 , 20 8 , 30 5 2. 200 8 , 400 6<br />
3. 120 4 , 110 3<br />
B. 1. 243 8 2. 1302 5 3. 11031 4 4. 628 9<br />
C. 1. 160, no remainder when divided by 8<br />
2. 56, remainder 2 when divided by 6<br />
3. 69, remainder 4 when divided by 5<br />
57
4. (a) 4 (b) 0 (c) 2 (d) 2<br />
The number tells you the remainder by the last digit<br />
P39 A. 1. +<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
1 2 345 678<br />
2 3 4 5 6 7 8 10<br />
3 4 5 6 7 8 10 11<br />
4 5 6 7 8 10 11 12<br />
5 6 7 8 10 11 12 13<br />
6 7 8 10 11 12 13 14<br />
7 8 10 11 12 13 14 15<br />
8 10 11 12 13 14 15 16<br />
10 11 12 13 14 15 16 17<br />
2. (a) 16<br />
(b) 10<br />
(c) 13<br />
(d) 4<br />
(e) 1<br />
(f) 7<br />
3. (a) 6<br />
(b) 6<br />
(c) 5<br />
(d) 4<br />
(e) 6<br />
(f) 5<br />
B. 1. x<br />
1 234 5 678<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
1 234 5 678<br />
2 4 6 8 11 13 15 17<br />
3 6 10 13 16 20 23 26<br />
4 8 13 17 22 26 31 35<br />
5 11 16 22 27 33 38 44<br />
6 13 20 26 33 40 46 53<br />
7 15 23 31 38 46 54 62<br />
8 17 26 35 44 53 62 71<br />
All numbers in the table are base 9<br />
2. (a) 44 (b) 40 (c) 38 (d) 7 (e) 7<br />
(f) 6 (g) 6 (h) 4 (i) 13<br />
(40g -:- 6 so 80g -:- 6 = 6 x 2 = 13g)<br />
3. (a) Is like the 9 times table. 9, 18, 27, ....<br />
going down by 1 in the units and up by 1 in the 10s.<br />
(b) The last figures of the squares are 1 4 0 717 0 4 1 in base 9. In<br />
base 10 they are 1 4 9 6 $ 6 9 4 1. Both patterns are<br />
symmetrical.<br />
(c) The 3 times table, base 9 is like the 5 times table in base 10<br />
with a repeating simple pattern.<br />
58
E39 A. 1,2, 3, see P38<br />
4. (a) 2438 (b) 3022 4 (c) 2313 5 (d) 628 9 (e) 1031 7<br />
(f) 112211 3 (g) 100100112 (h) 2034 6<br />
B. 1. 161'0; zero 2. 56 10 ; 2<br />
3. (a) 4 (b) 5 (c) 0 (d) 3 (e) 9 (f) 2 (g) 2<br />
(h) 1 (i) 4<br />
Explanations: 568 9 -7- 9 leaves a remainder of 8 so 568 9 -7- 3 will<br />
leave the same remainder as 8 -7- 3, i.e. 2<br />
C. 1. 265 is 324 9 and 100211 3<br />
2. 3 9 2 9 4 9<br />
10 3 02 3 11 3<br />
3. 363 is 11223 4 and (0)1011010112, Each digit of 11223 4 is replaced<br />
by its base 2 form<br />
4. 363 is 553 to base 8<br />
101 ........., 101 ........., 011 ........., each base 8 digit corresponds to a<br />
5 5 3 group of 3 binary digits<br />
5. This is a quick method of converting to base 2 (especially for large<br />
numbers).<br />
(a) 100~ 1448~ 11001002<br />
Note: The first group of 3 binary digits is 001 but we can<br />
leave the zeros off the front<br />
(b) 260 ~ 4048 ~ 1000001002<br />
(c) 359 ~5478 ~1011001112<br />
(d) 826 ~14728 ~11001110102<br />
(e) 6160~ 140208~ 11000000100002<br />
(f) 7665~ 167618~ 11101111100012<br />
E40 A. See P39 for base 9 tables<br />
1. (a) 16 9 (b) 57 9 (c) 124 9 (d) 6 (e) 6 (f) 31<br />
2. (a) 6 (b) 15 9 (c) 16 9 (d) 15 9 (e) 28 9 (f) 45 9<br />
In these questions it is easier to 'build up' the smaller number than<br />
to subtract by the usual rules<br />
3. (a) 44 9 (b) 40 9 (c) 80 9 (d) 6 (e) 7 (f) 13 9<br />
4. (a) They are similar to the 9 times table in base 10<br />
(b) The last digits follow the pattern 1 4 0 7 7 0 4 1<br />
(c) The 3 times table, base 9, is like the 5 times table in base 10<br />
with a simple repeating pattern<br />
B. Base 12 tables<br />
59
c.<br />
2. (a) 14'2<br />
(f) 30'2<br />
(k) =9:<br />
(b) 13'2<br />
(g) 13'2<br />
(I) 9 R 3<br />
(c) 13'2<br />
(h) 71'2<br />
(e) 42'2<br />
(j) 4 R 1<br />
2 1 1 7 1 5 13<br />
1. (a) "8 = 4 = 0.2510 (b) 2 (c) "8 (d) "8 + 64 = 64<br />
1 3 3 21 7 5 4 39 3<br />
(e) "2 (f) "6 + 36 = 36 = 12 (g) "7 + 49 = 49 (h) 25<br />
2. (a) 0.4 8 (b) 0-6 8 (c) 0-146314631 8 (d) 0-46314631 8<br />
(e) 0-525252... (f) 0·252525 ...<br />
Note: There are repeating and recurring octimals.<br />
; is 0.111111 8 so 0·22222 ... is ~ and so on.<br />
3. j is a base 3 number, ~ is a base 5 number. We can, however,<br />
convert them both to base 15 by using equivalent fractions.<br />
60
Unit 19, page 84<br />
The purpose of this unit is to convince children that the<br />
<strong>mathematics</strong> they are learning is indeed useful. Usually,<br />
everything that is learned has to be 'stored' for later, but in<br />
this section mathematical skills are applied immediately to<br />
other areas of school work.<br />
In some ways other subjects are more 'real' to the pupils<br />
than the external world of commerce, etc. The problems<br />
they are asked to solve are not rates and taxes, or HP, but<br />
problems presented to them by other teachers in the course<br />
of learning subjects such as geography and science. Geography<br />
uses a considerable amount of modern <strong>mathematics</strong><br />
for example, set theory, networks in the study of communications,<br />
transformation in theories of rock formation, etc.<br />
Science uses the more traditional <strong>mathematics</strong> of graphs,<br />
equations and metric units. All applications can make use of<br />
the calculator and it is in the <strong>mathematics</strong> department that<br />
children will learn the' effective use of this instrument.<br />
Further applications can be collected from the departments<br />
concerned (they will have their favourite mathematical skill<br />
which the children have failed to learn) and from the<br />
children who may have interesting applications of their own<br />
outside of school. (Hobbies, jobs, etc.)<br />
Answers<br />
M61 A.<br />
1. 12·75 million<br />
2. 43·26 million<br />
3. 103·53 million<br />
4. 471·5 million<br />
B. 1. 13·53 million, 47·27 million, 108·259 million, 507·754 million (use<br />
constant multiplier)<br />
2. Populations grow by 1·03 million, 5·27 million, 6·259 million,<br />
47·754 million<br />
C. 1. The populations are 12·36,12·73,13·11,13·51,13·91,14·33,14·76,<br />
15·2, 15·66, 16·13. Not double.<br />
2. After 20 years it will be 21·7 million at this rate of growth. Not<br />
double.<br />
61
3.<br />
Country<br />
1980 (est.)<br />
(millions)<br />
1990 (est.)<br />
(millions)<br />
Algeria<br />
Nigeria<br />
Turkey<br />
India<br />
Japan<br />
Denmark<br />
USA<br />
Gt. Britain<br />
19·37<br />
71·09<br />
45·68<br />
710·95<br />
111·56<br />
5·26<br />
228·70<br />
59·18<br />
26·80<br />
91·90<br />
59·63<br />
918·99<br />
124·46<br />
5·52<br />
255·14<br />
62·21<br />
Use constant multiplier or XV button if you have one.<br />
M62 There are different ways of tabulating the data in these questions. The<br />
tables should be clear and tidy with careful labels so that the information<br />
can be seen at a glance.<br />
M63 1. The angles at the centre of the pie charts are given. 10 % = 36°<br />
(a) 216° bat ber, 144° home haircut<br />
(b) 108° hairdresser, 252° home haircut<br />
2. Angles: Nitrogen 280-8°, Oxygen 75'6°, other gases 3-6°<br />
3. Ang les: Developed countries Developing countries<br />
protein 38° protein 34°<br />
fat 142° fat 34°<br />
sugar 57° sugar 17°<br />
starch 123° starch 275°<br />
4. Angles: land 105°, water 255°<br />
M64 Draw these graphs with care. They should be drawn in pencil with a<br />
smooth curve and labelled along the axes.<br />
Note: Exercises in which questions are asked about data are to be found<br />
in the P and E units<br />
P40 A. 1. (a) 77250 (b) 81 954 (c) 86945<br />
2. (a) Yes (b) No (c) No<br />
3. (a) 5250, 5512, 6381 (b) 15 years<br />
62<br />
B. 1. Heights (cm) Weights (kg)<br />
121-130 10 21-30 1<br />
131-140 7 31-40 10<br />
141-150 6 41-50 16<br />
151-160 7 51-60 3
2. Boys Girls Boys Girls<br />
Heights (em) only only Weights (kg) only only<br />
121-130 5 5 21-30 0 1<br />
131-140 4 3 31-40 5 5<br />
141-150 2 4 41-50 6 10<br />
151-160 3 4 51-60 3 0<br />
3. (a) The average weig ht of the girls is 41-5625, average weight of<br />
boys is 44-0714 so the statement does not agree with the data.<br />
(b) Agrees with the data<br />
(c) Does not agree with the data<br />
(d) Does agree with the data<br />
(e) Does not agree<br />
(f) Agrees (g) Does not agree (h) Agrees<br />
c. 1. The data can be tabulated giving diameter and number of acorns<br />
with that diameter.<br />
2. This can be treated in various ways, e_g. dry, a little rain, wet, very<br />
wet.<br />
P41 1. (a) 33-8 % (b) 54·g o /0 (c) 11·3%<br />
2.<br />
4. Yes, the large (41-50) classes have almost disappeared on the pie<br />
chart.<br />
E41 A. 1. 3-9735 x 10 13 km 2. 45360 years<br />
3. 226800000 km per hour<br />
63
4. There would be enormous problems of control as the rocket<br />
would be so far away. There would be enormous energy problems<br />
in escaping from the Sun's gravitational field and many<br />
more besides.<br />
B. 1. 14080 m. Lighter because of no human, no breathing apparatus,<br />
and no need to keep a human warm; 0·008 of earth's radius.<br />
2. 1·6 x 10 8 gallons per day, 5·84 x 10'0 gallons per year<br />
= 2·774 x 10" litres per year<br />
= 2·774 x 10 8 m 3 per year<br />
Thus a tank 28 m deep, 100 m wide and 100 km long would be<br />
needed to store all this water.<br />
3. 3 x 10 9 litres, enough for about 4 days.<br />
4. 1140. Not realistic as it would produce an oyster explosion!<br />
C. See P40 A. for 1 and 2, and 3 is just for interest. Of course such<br />
growth could never happen because the flies would overcrowd their<br />
habitat within the first 3 generations and start to die off.<br />
E42 A. 1.<br />
2.<br />
64
3.<br />
Med.3°<br />
4. Starts with (a) above and divides each section in proportion for<br />
the continents and oceans.<br />
B. Drawings<br />
Unit 20, page 88<br />
Although the use of logarithm tables for multiplication and<br />
division will give way to the use of calculators, simple<br />
multiplication tables will always be important. * They demonstrate<br />
important principles which the hidden mechanism<br />
* This unit makes use of Basic Mathematical Tables (Bell & Hyman).<br />
65
of the calculator never reveals. They also provide a simple<br />
and quick calculating aid for use when two-figure accuracy<br />
only is required. A further valuable feature is that any child<br />
could construct these tables for himself. He will also use his<br />
mental arithmetic.<br />
In learning to use multiplication tables to tackle more<br />
extended calculations, the pupil comes face to face with the<br />
basic laws of arithmetic which make calculation techniques<br />
feasible. Thus the distributive and associative laws of arithmetic<br />
will be met, fractions can be converted to decimal<br />
form, algebraic patterns such as (a + b? can be confirmed<br />
and other patterns such as (n + 1)(n - 1) = n 2 - 1 will be<br />
noticed. Searching for primes is a revealing activity. First, if<br />
n is being tested, an area up to J n is cordoned off and the<br />
search restricted to that area. If n does not appear then it is<br />
definitely a prime number.<br />
The tables give a useful check on the calculator (whereas<br />
pencil and paper checking is beyond the powers of many of<br />
the pupils) and the manipulations needed to deal with<br />
decimals bring the pupil closer to understanding the numbers<br />
less than 1, something very difficult to glean from<br />
calculator usage alone.<br />
Answers<br />
M65 A. 1. 350 2. 1504 3. 5005 4. 948 5. 3969<br />
6. 3168 7. 2499 8. 6596 9. 9 10. 11·48<br />
11. 30-03 12. 25·92 13. 1-665 14. 1·2 15. 8·64<br />
16. 0·2844<br />
B. 1. 2415 2. 930 3. 430-2 4. 16650<br />
5. 28820 6. 2553-6 7. 16-8078 8. 148-608<br />
C. 1. 11 978710 2. 11425674 3. 30080424 4. 40344854<br />
5. 449·2345 6. 499·6986 7. 279244·8 8. 48230·7<br />
M66 A. 1. 9 2. 6 3. 8 4. 35 5. 35<br />
6. 88 7. 67 8. 157 9. 56 10. 45<br />
11. 35 12. 42 13. 56 14. 63 15. 54<br />
16. 88<br />
66<br />
B. 1. 54 R6 2. 48 R3 3. 64 R4 4. 55 R7 5. 49 R9<br />
6. 38 7. 82 R9 8. 90 R15 9. 87 R44 10. 60<br />
11. 74 R63 12. 75 R60 13. 85 R55 14. 83 R39 15. 85 R25<br />
16. 60 R17
C. 1. 45·88 2. 47·22 3. 72·08<br />
5. 20·83 6. 34·76 7. 7·16<br />
9. 168·47 (subtract 100 x 17 first)<br />
11. 82·07 (not 82·7) 12. 75·41<br />
14. 129·86 15. 93·73 16. 90·91<br />
4. 27<br />
8. 52·33<br />
10. 99<br />
13. 180·91<br />
M67 A. 1. The first two numbers go up 1 at a time while the last number<br />
goes down 1 at a time, e.g. 279288. The sum of the digits is 9 or 18<br />
for each number.<br />
2. The numbers are 300 + (15 x 1), (15 x 2) ... etc. so the 15 times<br />
table repeats. The sum of the digits is a multiple of 3 and each<br />
number ends in 0 or 5.<br />
3. Contains the numbers 222, 444, 666, etc. because 74 is 2 x 37.<br />
Endings have pattern 4 8 2 6 0 4 8 2 6 0 ... (like 4 itself).<br />
4. With the exception of 99, 209, 308 and 319, the outer two digits add<br />
to the middle one.<br />
B. Digital sums Digital roots<br />
1. 6, 12, 18, 6, 12, 18... 1. 6, 3, 9, 6, 3, 9 ...<br />
2. 18, 18, 18, 18... 2. 9, 9, 9, 9, 9 ...<br />
3. 8, 7, 6, 14, 13, 3, 11, 10, 9, 8. . . 3. 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, ...<br />
4. 9, 9, 9, 9, 9, 9, 9, 9, 9, 18, 9, 4. 9, 9, 9, 9, 9, ...<br />
18, ...<br />
5. 9, 9, 9, 9, 9, 18, 18, 18, 18, 9, 5. 9, 9, 9, 9, 9, ...<br />
18, 18, 9<br />
6. 10, 11, 12, 13, 5, 15, 16, 8, 18, 6. 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, ...<br />
10, 11, 21 ...<br />
Digital roots are often used as spot checks for errors in computing.<br />
(Ask a bank!)<br />
M68 A. 1. 81<br />
6. 2025<br />
B. 1. 49284<br />
5. 533~61<br />
2. 196 3. 484 4. 1089 5. 900<br />
7. 5184 8. 8649 9. 6·76 10. 18·49<br />
2. 119025 3. 271 441 4. 407 044<br />
6. 1840·41 7. 3203·56 8. 3956·41<br />
Note: The calculator is the easiest tool to use to answer these<br />
questions but it is important to know that the answers can be found<br />
with just a little more trouble using ordinary multiplication tables.<br />
M69 A. 1. (a) 13<br />
(f) 36<br />
(k) 64<br />
(b) 17<br />
(g) 42<br />
(I) 72<br />
(c) 22<br />
(h) 48<br />
(m) 75<br />
(d) 29<br />
(i) 50<br />
(n) 81<br />
(e) 33<br />
(j) 56<br />
(0) 88<br />
67
2. (a) 1·4<br />
(f) 7·2<br />
3. (a) 1·183<br />
(f) 7·720<br />
(b) 3·4<br />
(b) 1·924<br />
(g) 8·173<br />
(c) 4·3 (d) 5·2<br />
(c) 3·578 (d) 5·477<br />
(h) 8·660<br />
(e) 6·3<br />
(e) 6·870<br />
P42 A. 1. 374 2. 3060 3. 15200 4. 25470 5. 304·2<br />
6. 7·84 7. 2·96 8. 9·12<br />
B. 1. 5 RO 2. 5 R10 3. 44 4. 25 R19 5. 26 R26<br />
6. 13 R29 7. 14 R19 8 7 R57 9. 40 10. 70 R41<br />
11. 118 R1 12. 156 R9<br />
C. 1. 1·9 2. 140 3. 69 4. 1·2 5. 21<br />
6. 1600<br />
D. These can be checked by noting that the remainder stays the same:<br />
e.g. 7079 = 7 R7, i.e. the 70 continues.<br />
70079 = 77 R7 this leads to 0·70079 which again gives the same<br />
remainder, so the division keeps on repeating. This is what happens<br />
with all the questions.<br />
E. 1. 1764 2. 3481 3. 18496 4. 67·24 5. 44·89<br />
6. 15·21 7. 0·2304 8. 0·8649 9. 0·7225 10. 0·0016<br />
11. 0·0081 12. 0·0049<br />
F. 1. 14 2. 22 3. 27 4. 35 5. 47<br />
6. 54 7. 65 8. 73 9. 6·3 10. 7·1<br />
P43 A. 1. £12 2. £41·16 3. 6·29 kg 4. 10·11 kg 5. 2774m 2<br />
6. 0·4125m 2 7. (a) 696 km (b) 725 km (c) 5220 km<br />
8. 498cm 2<br />
68<br />
B. 1. £2·20 per kg 2. 11 P 3. 91·75 kg 4. 44<br />
5. 47em 6. 36km 7. 33 metres 8. 3'6kg<br />
C. 1. (a) 11·56 cm 2 (b) 7.29 m 2 (c) 0·4225 m 2<br />
(d) 729 mm 2 (e) 4356 mm 2 (f) 0·5625 mm 2<br />
2. (a) 18cm (b) 28 em (c) 3·3 em<br />
(d) 4·2 m (e) 68·67 m (f) 18·4m<br />
3. The pattern continues 7, 7, 9, 4,1,9,1,4,9,7,7, 9oo.etc.<br />
4. 7, 5, 3, 1, 8, 6, 4, 2, 7, 9, 5, 3, 1 ... etc.<br />
5. Different answers
E43 A. 1. 374 2. 15200 3. 304-2 4. 2·96 5. 5<br />
6. 25·6786 7. 14·5278 8. 70·5325 9. 1·8947 10. 20·506<br />
11. 68·387755 (Interesting? What are the next decimal places?)<br />
12. 1620·91 13. 1764 14. 67·24 15. 0·2304 16. 0·0016<br />
17. 14 18. 27 19. 47 20. 6·3<br />
B. 1. (a) 11·56cm 2 (b) 7·29m 2 (c) 0·4225m 2<br />
(d) 729 mm 2 (e) 4356 mm 2 (f) 0·5625 mm 2<br />
2. (a) 18 cm (b) 28 cm (c) 3·3 cm<br />
(d) 4·2 m (e) 68·67 m (f) 18·44 m<br />
3. (a) The pattern continues. This can tell you when not to expect an<br />
exact square root.<br />
(b) 7, 5, 3, 1, 8, 6, 4, 2, 7, 9, 5, 3, 1 ... etc.<br />
(c) Your own choice.<br />
4. a, b, c are all equal to )10, thus a 2 , b 2 and c 2 are all equal to 10.<br />
E44 A. 1. This is not necessarily true. For example a = 2, b = -4 so a> b.<br />
But b 2 = 16, a 2 = 4, so b 2 > a 2 .<br />
2. Again not necessarily true. Same example. Both of the statements<br />
1 and 2 are true if we restrict a and b to be positive numbers.<br />
3. True.<br />
4. Not necessarily true. Example. Suppose a = 9, n = 27. a 2 = 81 so n<br />
is a factor of 81 but not of 9.<br />
5. All are true. This is easily proved by using algebra as follows. If n<br />
is a factor of a then a = xn where x is a whole number and<br />
similarly b = yn. Thus a + b = (x + y)n.<br />
So a + b is a multiple of n. Also a 2 + b 2 = x 2 n 2 + y2n2 = (x 2 + y2)n 2<br />
and (a + b)2 = (x + y)2n2 so both a 2 + b 2 and (a + b)2 are multiples<br />
of n.<br />
6, 7, 8 are all true and can be verified by using any numbers you like<br />
for a and b.<br />
B. 1. Not true<br />
6. True<br />
2. True 3. True 4. True 5. True<br />
69
Unit 21, page 92<br />
This unit reinforces work on Pythagoras' theorem. It considers<br />
applications, solution of triangles and (in the E Unit)<br />
proof. The work brings together algebra, arithmetic,<br />
geometry and real world applications. It is therefore ideal<br />
for fitting the various parts of <strong>mathematics</strong> together. The<br />
application on using Pythagoras' theorem with coordinates<br />
links with geography and navigation. Other aspects such as<br />
Pythagorean triples will be considered later in books M 1 and<br />
N1. All work on Greek <strong>mathematics</strong> gives opportunity to<br />
indicate the thread of mathematical discovery which has run<br />
through human history. There are many sources of information<br />
and legends in books such as Men of Mathematics by<br />
E. T. Bell or The World of Mathematics by James R.<br />
Newman though these will have to be edited for children's<br />
use.<br />
Answers<br />
M70 A.<br />
B.<br />
Because of the limitations of measurement it is only possible to<br />
confirm Pythagoras' theorem roughly by these examples.<br />
1. 20 mm, 2-0mm, 28 mm" ... squaring ... 400 + 400 = 800 = 784<br />
(=28 2 )<br />
2. 30 mm, 20 mm, 36 mm 900 + 400 = 1300 = 1296 (= 36 2 )<br />
3. 40 mm, 15 mm, 43 mm 1600 + 225 = 1825 = 1849 (=43 2 )<br />
4. 20 mm, 22 mm, 30 mm 400 + 484 = 884 ~ 900 (= 30 2 )<br />
5. 25 mm, 15 mm, 29 mm 625 + 225 = 850 = 841 (=29 2 )<br />
6. 35 mm, 23 mm, 42 mm 1225 + 529 = 1754 = 1764 (=42 2 )<br />
7. 28 mm, 28 mm, 40 mm 784 + 784 = 1568 = 1600 (=40 2 )<br />
1. Rectangle 30 mm x 20 mm diagonal 36 mm<br />
900 + 400 = 1300 = 1296 (=36 2 )<br />
2. Rectangle 21 mm x 35 mm diagonal 41 mm<br />
441 + 1225 = 1666 = 1681 (=41 2 )<br />
3. Rectangle 15 mm x 50 mm diagonal 52 mm<br />
225 + 2500 = 2725 = 2704 (= 52 2 )<br />
4. Rectangle 25 mm x 30 mm diagonal 39 mm<br />
625 + 900 = 1525= 1521 (=39 2 )<br />
70
5. Rectangle 26 mm x 18 mm diagonal 31·5 mm<br />
676 + 324 = 1000 = 992·25 (=31'5 2 )<br />
6. Rectangle 45 mm x 22 mm diagonal 50 mm<br />
2025 + 484 = 2509 = 2500 (= 50 2 )<br />
M71 A.<br />
B.<br />
c.<br />
1. 38·9mm 2. 31·8mm 3. 48·8mm 4. 51 mm 5. 51·2mm<br />
6. 44·8mm<br />
1. 43·7 mm 2. 47·7 mm 3. 48·7 mm 4. 40 mm, 25 mm, 47 mm<br />
5. 56·6mm 6. 34·4mm<br />
1. 213·6mm 2. 18km 3. 71·2m 4. 73·3m<br />
P44 A. There are many different right angled triangles in the figure.<br />
Pythagoras' theorem works for all of them, of course.<br />
B. AD=8·5cm, CO=~AD=4·24cm, DX=6·7cm, DY=6·2cm<br />
C. 1. Draw a triangle with sides 6cm, 8cm and 10cm with careful<br />
construction. Any other lengths can be used as long as a 2 + b 2 = c 2<br />
2. (a) Not, AC should be 108 mm (b) Yes (c) Yes<br />
(d) Not, AC should be 10·8 (e) Yes<br />
P45 A. 1. 8'6cm, BC, EG, FH<br />
2. 6·4 cm, BG, AF and CE<br />
3. CG = 8·06, GH = 5<br />
CH = )90 = 9·49 cm<br />
90 = 7 2 + 4 2 + 52<br />
H<br />
c<br />
4. The diagonals are AG, CH, ED and SF. All four have the same<br />
length.<br />
B. Different answers<br />
71
E45 A. 1. (a) Use a triangle whose sides are 5 em and 3 em<br />
(b) Use a triangle whose sides are 8 em and 5 em<br />
(c) Use a triangle whose sides are 5 em and 6 em<br />
2. Drawing. For this drawing accurate right angles should be drawn.<br />
B. 1. (a) ABD = 40°, DBC = 50°<br />
2. (a) BAD = 25°, (b) DAC = 65°, (c) A~D = 25°<br />
3. (a) POR = 59°, (b) QPS = 31°, (c) RPS = 59°<br />
4. If B = x o , BAD = (90 - x)O, DAC = X O and DCA = (90 - x)O<br />
Thus all three triangles have angles 90°, X O and (90 - x)O<br />
AB<br />
AD<br />
C. AC = AB => AC x AD = AB 2 by cross-multiplication<br />
CD = BC => AC x CD = BC 2<br />
BC AC<br />
Now add (AC x AD) + (AC x CD) = AB 2 + BC 2<br />
But AC(AD + CD) = AC(AC) = AC 2<br />
So AB 2 + BC 2 = AC 2 ... Pythagoras' theorem<br />
E46 A. 1. AD = 8·6 em also the length of BC, FH and EG<br />
DH = 6·4 em also the length of BG, AF and CE<br />
2. CH = )90 = 9·49 em. AG, ED and BF are also )90 em<br />
B. Different answers<br />
72<br />
Unit 22, page 96<br />
Percentages are widely used in everyday communication<br />
from wage deals to unemployment figures. They are also<br />
frequently quoted in science, geography, economics, home<br />
economics and so on. In spite of this many people would say<br />
that they did not understand 'percentages'. It is hard to<br />
understand why not as the topic is extremely simple. It must<br />
be surmised that some sort of confusion lingers on from<br />
failure to understand fractions and that this is the barrier.
Answers<br />
Revision Exercises<br />
M72 A. 1. 70p<br />
5. 10·08 km<br />
9. 42·75 mg<br />
B. 1. 50% 2.<br />
6. 5% 7.<br />
C. 1. 1<br />
(a) -<br />
2<br />
In this unit a number of different applications are explored<br />
and the percentage is presented in fraction and decimal<br />
form. The decimal form is easier to use in problems especially<br />
as the calculator is available. The percentage button<br />
on the calculator is best ignored, or perhaps explored after<br />
the concept has been grasped. It is not necessary at this<br />
stage that the applications should be learned, the purpose<br />
is to experience percentages being used and for the pupils<br />
to feel they can cope.<br />
The work can be made more interesting if it is extended to<br />
include problems produced by members of the class. These<br />
may concern such varying topics as how to invest their<br />
pocket money to their chance of surviving in a war with<br />
inhabitants of outer space. Probabilities are traditionally<br />
expressed as percentages and make sense if they are related<br />
to a lottery with 100 tickets. The chance of winning is 50% if<br />
you buy 50 tickets. This emphasizes the 'out of' aspect of a<br />
statement in percentage terms (see Unit 23).<br />
2. £6<br />
6. 18·75 litres<br />
3. 33·33%<br />
8. 27%<br />
3. £16·50<br />
7. 5·28 kg<br />
1<br />
4. 10 %<br />
9. 58·666%<br />
(d) ~ 5<br />
4. 1·7kg<br />
8. £7·04<br />
2<br />
(e) - 5<br />
(f)<br />
7<br />
- 10<br />
(b) ~ 4<br />
4<br />
(g) 5<br />
(c) -<br />
4<br />
66<br />
(h) 100<br />
2. (a) 50%<br />
(f) 12·5%<br />
M73 A. 1. £30 2.<br />
6. £147·50 7.<br />
B. Interest<br />
1. £150<br />
2. £490<br />
3. £1760<br />
4. £395·20<br />
(b) 25%<br />
(g) 80%<br />
£88<br />
£135<br />
(c) 75%<br />
3. £104·50 4. £135<br />
8. £243<br />
Amount repaid altogether<br />
£650<br />
£1190<br />
£2560<br />
£1155·20<br />
(e) 70%<br />
5. £72·45<br />
73
M74 A. Profit (cash)<br />
1. £ 4<br />
2. £100<br />
3. £ 12<br />
4. £ 34<br />
B. Profit<br />
1. £16<br />
2. £50<br />
3. £8750<br />
4. £15·40<br />
C. 1. £1384·61<br />
Profit %<br />
Selling price<br />
(4 -;-75) x 100 = 5·33%<br />
23·8<br />
36·36<br />
20·36<br />
£80 + £16 = £96<br />
£175<br />
£43750<br />
£43·40<br />
2. £113·33 3. £55 4. £140<br />
M75 A. 1. North 7·9, South 8·10/0 2. North 55·7%, South 49·1<br />
3. North 3.6 % , South 4·6%<br />
4. Yes<br />
B. 1. 8% 2. 7·9% 3. No, very close<br />
M76 3. 19·7% 4. Yes<br />
P46 A. 1. (a) 0·25 (b) 0·36 (c) 0·08 (d) 0·12 (e) 0·18 (f) 0·165<br />
2. (a) 33% (b) 15% (c) 20% (d) 4% (e) 26% (f) 37;%<br />
3. (a) 100 % per month = 1200% per year<br />
(b) 25% (c) 20% per week = 1040% per year<br />
(d) 20% per year<br />
B. 1. Profit 50p, percentage profit 33·3%<br />
2. Loss £15, percentage loss 20% (15/75)<br />
3. £162 4. £62 (simple interest)<br />
P47 A. 1. 77·77% of rivers were unpolluted in 1975.<br />
2. Of people hurt in road accidents 1·9% were killed, 23·6% were<br />
seriously injured and 74·59% were slightly injured.<br />
3. In 1978 the quantity of oil was 58% of the 1973 quantity but it cost<br />
269% of the 1973 cost: the price had increased by 360%.<br />
4. 1981 children aged 5-14 form 14·3% of all people<br />
2001 children aged 5-14 estimated 14·8% of all people<br />
B. 1. 3·375 kg<br />
2. Gold was worth £3·2 per g. Value of gold in the ring £24.<br />
3. 144g 4. 24g 5. 48ml pure alcohol<br />
74
E47 A. 1. £13 2. £22·50 3. £58·65 4. £945 5. £92·80<br />
6. £198·55 7. £924 8. £649·08<br />
B. 1. £103·70 (= 85x 1·22) 2. £136·80 3. £492·80<br />
4. £4602 5. £5368 6. £7735<br />
C. Interest Principal + interest (0.5) .<br />
1. £ 54 £ 129<br />
2. £ 288 £ 608<br />
3. £ 968 £1408<br />
4. £5292 £8892<br />
E48 A. 1,2. See P47 A, 1,2<br />
3. The figures for 1968 and 1977 are very similar in all respects.<br />
4,5. See P47 A, 3,4<br />
B. 1. 337·5 g 2. £24 3. 144 g 4. 12 g<br />
C. 1. The answers depend on the choice of rough estimate.<br />
2. (a) 0·079% (tanker), 0·6% (car). The second is the greater percentage<br />
error.<br />
(b) 2·78% (London-New York), 2·5% (London-Paris). The first<br />
percentage error is the greater.<br />
3. Error (a) is 1/1600 Error (b) is 1/3600 so (b) has smaller percentage<br />
error and is therefore the better estimate.<br />
4. 31 x 11 x 2·5 = 852·5m 3<br />
30 x 10 x 3 = 900 m 3 (error 5.6 % )<br />
30 x 12 x 2 = 720 m 3 (error 15.6 % )<br />
So 30 x 10 x 3 is the better estimate<br />
5. (a) Percentage error 0·0666% (b) Percentage error 0·8333%<br />
So it looks as if the first scientist is more accurate even though he<br />
is working to the nearest 100000 kilometres!!<br />
Unit 23, page 100<br />
Probability is one of the most important applications of<br />
arithmetic in that it is so often the basis for decisions. The<br />
use of probability is much wider than in mere gambling<br />
situations and ranges from 'shall I call the doctor?', 'when is<br />
the best time to take a holiday?', 'what are my chances of<br />
75
promotion?', 'will my bike be "nicked"?, to vital questions<br />
of health and family planning.<br />
The concepts have been proved to be difficult to understand<br />
and it is not certain that understanding in one situation<br />
such as tossing coins can be transferred to other situations<br />
such as the chance of inheriting a particular genetic weakness.<br />
This unit is essentially one of discussion and experiment<br />
rather than obtaining answers to calculations and reinforcement<br />
of percentages. The probability as a percentage is to be<br />
found in common language in such terms as 'fifty-fifty' or 'I<br />
am 100% sure that ... ' and these expressions should be<br />
related to the work of the unit. A great deal of the<br />
conceptual difficulty of this topic is due to careless language,<br />
but in tightening up the language it is very easy to lose the<br />
thread of meaning. Later on <strong>IMS</strong> will use the model of a<br />
raffle with 100 tickets to relate probability and percentage.<br />
Answers<br />
M77 The question refers to the probability of future events.<br />
A. 1. Likely<br />
4. Likely (7)<br />
7. 7<br />
10. Definitely not<br />
2. Unlikely<br />
5. ?<br />
8. ?<br />
3. Unlikely<br />
6. Unlikely<br />
9. Definitely not<br />
B. Different answers. Not really right or wrong here as the figures are a<br />
matter of opinion (except 8 which will not happen)<br />
c. Different answers<br />
M78 A. 1. 50%<br />
6. 23·1%<br />
2. 7·7%<br />
7. 30·80/0<br />
3. 7·7%<br />
8. 3·8%<br />
4. 7·7% 5. Under 1·9%<br />
B. All experimental resu Its.<br />
P48 A. 1. They are all certain or have happened already<br />
2. You could argue some of these. For example you could (a) forget<br />
to clear the calculator; (b) if you went by boat you would walk on<br />
the boat, etc.<br />
(c) Impossible because we do not allow it to happen.<br />
(f) Impossible because you must score at least 2 with two dice.<br />
76<br />
B. 1. (a) This is to be expected, say 70% chance, if your class is not all<br />
boys
(b) This is unlikely but not impossible (especially in Wales where<br />
so many people are Jones or Davies ... )<br />
(c) Almost 100% certain. (They are only half-brothers if, for<br />
example, they have different fathers)<br />
(d) This is more likely than you would expect. At least 50%<br />
(e) There is no probability to this event. It either was a fine day or<br />
was not.<br />
(f) This is about the chance of having two fine days, one after the<br />
other. The chance is about 40% in Britain, though this will be<br />
different for different parts of the country.<br />
2. (a) More likely to be an 8 because this can happen as 1 + 7,6 + 2,<br />
3 + 5 or 4 + 4<br />
(b) More likely to be a spade (13 of them, only 4 aces)<br />
(c) 1 head, 1 tail is more likely<br />
P49 A. Experimental results<br />
(d) A mixture is more likely<br />
(e) More likely to just win the 100 m race<br />
B. Experimental results<br />
C. Experimental results<br />
E49 A., B., C., D. Experimental results<br />
ESO A. 1, 2, 3 Drawing. Simple tree diagrams<br />
4. (a) Around 14% chance<br />
(b) More likely to pick up a left and right (16-12) chance, i.e. 4n)<br />
(c) 3·6% chance (1/28). This problem can be simulated using a<br />
pack of cards (use two cards from each suit)<br />
B. 1. (a) No (b) The probability is ,1. Suppose her friend chooses a<br />
shop, then Jenny has 12 shops to choose from but only one will<br />
be right.<br />
(c) iJ<br />
2. 9 different ways. There are 3 different choices of the second part<br />
of the journey for each choice of New York to Chicago<br />
3. 80<br />
4. 125, probability 0·8% (small chance!)<br />
C. Different answers<br />
77
Unit 24, page 103<br />
Mathematics is all about proof and all children following a<br />
<strong>mathematics</strong> course should have some experience of this<br />
central idea. Perhaps in the past, the formalities of proof<br />
were over-emphasized in geometry but in discarding formal<br />
geometry from today's syllabuses we are in danger of leaving<br />
out the main concept forming experience. Proof itself is<br />
about argument. There is an argument, to win it something<br />
stronger than a point of view is needed. All children<br />
appreciate this (though they may resort to physical means of<br />
winning an argument) and they are instinctively appreciative<br />
of an irrefutable argument. At this stage the language of<br />
proof is introduced, especially the symbol =} which is<br />
interpreted as a shorthand for 'if ... then'. The word 'implies'<br />
follows rather than leads as it is a new word whose<br />
meaning has to be felt before it can be understood.<br />
Discussion is of particular importance in this unit. Not<br />
only is implication being discussed but also the problems will<br />
usually contain profound mathematical concepts. Do not<br />
rush this work, the ideas will be presented again during the<br />
main courses. Do not over-emphasize the formal structure.<br />
Answers<br />
M79 A.<br />
1. n > 100 =;> n > 99 2. n < 6 =;> n < 10<br />
3. n = 10 + 5 =;> n = 20 - 5<br />
4. n = k x 6 =;> n = I x 3 where k and I are whole numbers.<br />
5. (n is a multiple of 2) and (n is a multiple of 5)<br />
=;> n is a multiple of 10.<br />
6. ;~~ }=;>X
5. If a number is more than 6 then its negative is less than -6.<br />
-6<br />
I I<br />
-n (less<br />
than -6)<br />
-5 -4 -3 -2 -1 0 2 3 4 5 6<br />
I I I I I I I I I I I I<br />
n (more<br />
than 6)<br />
6. If £x is more than £y then £(x - y) is positive.<br />
If I have £x, I can buy something for £y and still have something<br />
left over.<br />
C. Different answers.<br />
M80 A. 1. 10 0 5. True<br />
6. True if SAC is a straight line<br />
C. 1. It depends how you cook it.<br />
2. It depends on the line.<br />
3. This is only true if you start with a cylinder and cut perpendicular<br />
to the axis.<br />
4. What about 12·5, i.e. only true if you are thinking of whole<br />
numbers.<br />
5. Depends on how worn the coin is.<br />
M81 Remembering that anything already proved can be used in a new proof ...<br />
A. 1. 2n + 1 is odd (proved), adding 2 to an odd number produces<br />
another odd number, 2n + 3 is odd.<br />
2. 2n + 1 is odd so 2n - 1 is odd as well, being two less.<br />
3. Since 2n + 1 is odd and 2n - 1 is odd, (2n - 1) + (2n + 1) will be<br />
even (since odd + odd = even).<br />
4. 2n - 1 + 2n + 1 = 4n which is a multiple of 4 (verify by example, if<br />
n = 5, 2n - 1 = 9, 2n + 1 = 11,9 + 11 = 20 which is a multiple of 4).<br />
5. If n is even, n + 1 is odd, (n + 1)2 is odd (since odd x odd = odd). If<br />
n is odd, n + 1 is even, (n + 1)2 is even (since even x even = even).<br />
B. 1. If two angles are 60° the third must be 180°-(60° +<br />
60°) = 180° - 120° = 60°. Thus the triangle is equilateral.<br />
2. The third angle is 180°-(45° + 45°) = 180° - 90° = 90°<br />
3. The other angles add up to 180° - 20° = 160°. Since the angles are<br />
both equal, each one is 80°,<br />
79
4. Since a rhombus has opposite angles equal, the angle opposite<br />
90° is also 90°. This leaves the other pair of angles with 360°-180°<br />
to share. So both of these are 90° as well.<br />
P50 A. 1. Logical 2. Should be sister 3. Logical<br />
4. Logical 5. Logical<br />
6. Not true. For example, I have £10, you have £4 and Jan has £6.<br />
One example which breaks the implication is enough to make it<br />
false.<br />
B. 1. True<br />
2. The statement should be !n is even? n is even.<br />
3. Not true, something else may be wrong. Should be the bulb is<br />
broken? the light does not come on.<br />
4. Not true. Even the reverse is not always true as the bus may stop<br />
to let people off!<br />
5. (a) True (b) True (c) Not true<br />
(d) Not true. The score could go 0-1, 0-2, 0-3, 0-4, 0-5, 1- 5 ,<br />
1-6, ... etc.<br />
(e) Must be true.<br />
P51 A. 1. Since odd + odd = even and even + even = even, two numbers<br />
adding to 21 must be one odd and one even.<br />
2. Since two consecutive numbers must be odd and even or even<br />
and odd, their sum will be odd.<br />
3. Four consecutive numbers are made up of two odd pairs<br />
e.g. 3 + 4 + 5 + 6 and odd + odd = even<br />
odd<br />
odd<br />
4. The numbers are n, n + 1 and n + 2. Their sum is 3n + 3 but<br />
3n + 3 = 3(n + 1) so their sum is a multiple of 3.<br />
B. 1. Put 8 = X O BAD = 90° - X O DCA = 90° - X O DAc = X O<br />
So 6ABD, ACD and ABC have angles x O , 90° - X O and 90°<br />
2. Put BDC = X O and ADB = yO then DBC = yO and DBA = X O<br />
A = C = 180° - (XO + yO), so both triangles have angles x o , yO and<br />
1800 - (X O + yO)<br />
3. DAB + DCB = X O + yO + ZO + W O = ADC + ABC<br />
All four angles add up to 360° = 2x + 2y + 2z + 2w<br />
Therefore DAB + DCB = 180°<br />
E51 A. 1. Logical 2. Should be sister 3. Logical<br />
4. Logical 5. Logical 6. Wrong<br />
7. Not true, e.g. I have £10, you have £4 and John has £6. Only one<br />
counter-example is needed to break an implication.<br />
8. Logical 9. Logical 10. Logical<br />
80
B. 1., 2., 3., 4. see P50 B.<br />
5. (a) Not true (b) True<br />
(c) Somewhere during the set.<br />
(d) Not true, e.g. the score could be 0-1, 0-2, 0-3, 0-4, 1-5, 1-6<br />
C. Different answers from different people.<br />
E52 A. See P5i A.<br />
B. See P51 Band 4. Simple proof<br />
Unit 25, page 106<br />
The concept of set is fundamental to <strong>mathematics</strong> but that<br />
does not mean it can be taught in the abstract at an<br />
elementary stage. Classification, belonging to, not belonging<br />
to and their consequences have a much wider application<br />
than <strong>mathematics</strong>. The basic idea is that if a set is defined to<br />
have certain properties then an element will have those<br />
properties if it can be shown to be a member of the set. A<br />
good example is the classification of verbs in French. If a<br />
verb is an 're' verb then we know how it will form its past<br />
and future tenses. Belonging to the family carries a lot of<br />
information. Social sciences make use of the set concept in<br />
classifying society, chemistry classifies elements by the<br />
periodic tables and then discusses properties of the whole<br />
group. Mathematics itself does the same type of classification,<br />
for example, rational and irrational numbers; prime<br />
and composite numbers; transformations of various types,<br />
equations of various types, shapes of various types. The key<br />
question in a problem is often does ... belong to this set.<br />
In this unit and the next the two ideas of union and<br />
intersection are explored. The only symbols needed are n,<br />
U, E, $. and::} and these should be examined carefully and<br />
committed to memory after discussion. During the exploration<br />
of the uses of these symbols the work will touch upon<br />
numerous important mathematical ideas and in this way the<br />
work on sets does unify the <strong>mathematics</strong> syllabus though not<br />
in a formal, theoretical way.<br />
81
Answers<br />
M82 A. 1. The sets are the different states of Russia and Siberia. The<br />
elements are people who live in the states.<br />
2. The sets are different engineering trades unions which have been<br />
joined to form one big union. The sets are the small unions, the<br />
elements are workers.<br />
4.<br />
5.<br />
B. 1.<br />
2.<br />
3.<br />
C. 1.<br />
2.<br />
3.<br />
3. This is a trade union of people who work in local government. The<br />
elements are workers. Here union does not mean the union of<br />
4.<br />
sets. It is another name for a set of people.<br />
As above, but the workers are teachers.<br />
As above, but the workers are school-children.<br />
This is not a union because only parts of the sets were joined.<br />
The shoes they threw out formed a set which vvas the union of<br />
Mary's unwanted shoes with Liz's unwanted shoes.<br />
Not a true union since the new set is not made up of all the parts of<br />
the old sets.<br />
The set of all squares and circles.<br />
The set of all rectangles (including squares)<br />
The set of numbers 2 n where n is a whole number, positive or<br />
negative.<br />
All words beginning with A or B. The first two sections of the<br />
dictionary.<br />
M83 A.<br />
B.<br />
1. U 2. E<br />
6. x, x 7. y, Y<br />
1. U 2. E, E<br />
5. Cubes, primes<br />
3. P<br />
8. Y<br />
4. E 5. =?<br />
3. W, V 4. x<br />
6. Odds U evens<br />
M84 A. 1.<br />
_~_A~<br />
3. Same as question 2.<br />
5.<br />
A<br />
4. Same as question 1.<br />
x•<br />
B<br />
82
B.<br />
• Ahmed<br />
• Fred<br />
B<br />
• Jenny<br />
C. 1. (a), (b), (c), (e) belong to AU B<br />
2. (a) (b) (c, possibly), (e, possibly), (f, possibly)<br />
M85 A. 1. {Dave, Pete} 2. {Britain, Netherlands}<br />
3. {Red, Blue} 4. {The point D}<br />
B. 1. (a) A teacher with a child of his/her own in the school<br />
(b) Nobody<br />
(c) A parent that works in the school<br />
(d) Nobody<br />
2. (a) People with brown eyes and fair hair<br />
(b) Sea animals such as whales which suckle their young<br />
(c) Letters that have a vertical and a horizontal axis of symmetry,<br />
e.g. X, 0<br />
(d) Flats with 3 bedrooms and gardens<br />
(e) Team sports in which a ball is used<br />
3. (a) (D·1, 0·2, 0·5, 1·0) (b) All prime numbers except 2<br />
(c) All even square numbers<br />
(d) (1, 2, 3, 4 .... ) since all these numbers appear in both sets.<br />
)1 = 1, )4=2, )9=3, ... and so on<br />
P52 A. 1.<br />
2.<br />
3.<br />
4.<br />
5.<br />
B. 1.<br />
2.<br />
{ABCDE 12345}.<br />
Teams "in 1st divisi.on together with teams in 2nd division<br />
(a), (b) only<br />
Different choice of answers, 6 even numbers would do or 6<br />
multiples of 1D or a mixture<br />
Circle, parallelogram, rhombus ... etc.<br />
Different answers.<br />
(a) Mary belongs to the Girl Guides<br />
(b) Roger does not belong to the Naval Cadets<br />
(c) Patrick is a Catholic<br />
(d) Elsa is not human (she was a famous lion),<br />
(e) Blanco is not a drink<br />
(f) Deadly nightshade is a poisonous plant<br />
83
3. (a) Is true (b) Is not true. Examples will be different.<br />
4. (a) Jan is not at nursery school<br />
(b) Peter is under five<br />
(c) Francis is not under 5<br />
(d) Gary is under 5 or at nursery school<br />
(e) Joanne is not under five or at nursery school<br />
(f) If a child is at nursery school then he or she is under five years<br />
old.<br />
C. 1. (a), (b), (e), (f), (g)<br />
2. (a) Countries which are in Africa or contain desert.<br />
(b) No, e.g. Arabia. (c) No.<br />
(d) China, USA, Nigeria ... China and USA have desert land.<br />
Nigeria is in Africa.<br />
3. (a) Is true (b) Is true (c) Not always true, he uses cigarettes<br />
or alcoholic drinks but maybe both.<br />
(d) Not true (e) True (f) Not true<br />
P53 A. 1. (a) A (b) A (c) x, y (d) z<br />
(e) No points in A and in B<br />
2. (a) Agrees (b) Does not agree (c) Agrees (d) Agrees<br />
3. (a) Yes (b) Yes (c) Yes (d) Yes (e) No<br />
(f) Yes<br />
B. 1. (a) (iii) and (iv) (b) (i), (ji), (iv)<br />
(c) 5 triangles (d) Triangle (iv) is counted twice<br />
2.<br />
c.<br />
1. A set made up of the members which belong to both sets<br />
2. (a) A n B is squares<br />
(b) No<br />
(c) Yes<br />
(d) Rectangles which are not squares<br />
3. A<br />
A<br />
8 8<br />
(a)<br />
(b)<br />
A<br />
(e)<br />
(d)<br />
4. (a) True. If x E A n a then x E A and therefore x E A U a<br />
(b) A n B = (10, 20, 30, 40 ... )<br />
Au B (2, 4, 5,6,8, 10, 12, 14, 15 ... ) every member of An B<br />
belongs to A U B. We can describe this situation as<br />
xEAnB=?XEAUa<br />
(c) A U is all the people of Canada who speak French or English<br />
A n a is all the people of Canada who speak both French and<br />
English<br />
x E A n B =? x E A, people who speak French and English<br />
speak<br />
English<br />
x E A n B =? x E a, people who speak French and English<br />
speak French<br />
8<br />
E53<br />
A., B., see P52<br />
C. 1. (a) All even numbers are whole numbers<br />
(b) Isosceles triangles are 'triangles'<br />
(c) Potatoes are vegetables<br />
(d) Pupils in the class belcung to the school<br />
Not that the reverse of each statement is not true<br />
2. (a) Not true (b) Not true (because of 2)<br />
(c) True<br />
(d) True<br />
3. (a) A is a subset of A u a because if you belong to A you must<br />
belong to the union of A with B<br />
85
(b) If you belong to B you must belong to the union of B with A<br />
(c) If you belong to A n B you must belong to A (and also to B)<br />
(d) If you belong to A you must belong to A, i.e. A c A (this seems<br />
odd!)<br />
E54<br />
1. The next perfect number is 28, the next is 496. After that they take<br />
quite a long time to find as the next two are 8128 and 33550336.<br />
2. (a) It happens because 26 is 25 + 1 so 48 x (25 + 1) gives<br />
1200 + 48 and the 48 reappears<br />
(b) 76 has the same property, so has 51<br />
3. (a) 37 is a factor of 111 so it is a factor of all numbers such as 222,<br />
333, etc.<br />
(b) Multiples of 37 are either of the form nnn or the sum of their<br />
digits is the same as n + n + n. Multiples of 37 such as 2664<br />
have first and last digits adding to the same as the central<br />
digits of the number<br />
4. (a) The number is just rearranged but the numbers remain in the<br />
same order<br />
(b) Another number is 0·076923 which comes from ,1 or better<br />
still 0·0588235294117647 which comes from ,j<br />
5. Investigation<br />
Unit 26, page 110<br />
The ideas of set algebra are taken a little further in this unit<br />
to consider the universal set and the empty set. Most of the<br />
exercises concern the language. In many ways set algebra is<br />
easier than traditional algebra because the symbols are quite<br />
distinctive and it is not so heavily concerned with manipulation.<br />
Some writers say that this type of exercise is not relevant<br />
to non-academic children. Careful thinking 'develops' the<br />
mind and if work on sets is omitted at this stage children's<br />
potential for analytical thinking could well remain undeveloped.<br />
Remember this work does not have to be 'learned'.<br />
It is experience of a simple symbolic language.<br />
Answers<br />
M86 A.<br />
1. n<br />
6. F<br />
2. n<br />
7. xEQ<br />
3. and 4. E<br />
8. n, A, Q<br />
5. M<br />
86
B. 1. If x does not belong to A then x does not belong to A n B. True<br />
2. If x belongs to A then x belongs to A n B. Not true<br />
3. If x does not belong to B then x does not belong to A n B. True<br />
4. If x belongs to A n B then x belongs to A. True<br />
5. If x does not belong to A n B then x does not belong to A. Not true<br />
6. If x does not belong to A n B then x does not belong to B. Not true<br />
7. If x does not belong to A n B then x does not belong to A and also<br />
x does not belong to B. Not true<br />
8. If x does not belong to A and x does not belong to B then x does<br />
not belong to A n B. True<br />
C. 1. Yes,0 2. Not 0 3. 0 4. Not 0 5. 0<br />
6. 0 7. Not 0 (squares) 8. Not 0<br />
M87 A. 1.<br />
4. y could be in Q instead<br />
5. g does not belong to the football<br />
team or the hockey team<br />
6. xn Y squares<br />
B. 1. (a) y<br />
(f) x, k<br />
2. (a) 6131<br />
(b) x (c) z (d) x, y, z (e) k<br />
(g) z, k<br />
(b) 8270 (c) 1875 (d) 6395 (e) 4256 (f) 10651<br />
87
3. people with both<br />
A<br />
7<br />
13<br />
4 P<br />
C. 1. 13 children<br />
2. (a) Numbers which are multiples of 3 and 4, i.e. the multiples of<br />
12.<br />
(b) (i) Yes (ii) Yes (iii) Yes (iv) No (v) Yes<br />
Maa A. 1. Fruits<br />
3. Whole numbers<br />
6. Animals<br />
B. 1. True<br />
6. True<br />
11. True<br />
2. True<br />
7. False<br />
12. True<br />
4. Words<br />
7. Fish<br />
2.<br />
3. True<br />
8. False<br />
Numbers and fractions<br />
5. Names<br />
a. Motor cars<br />
4. True<br />
9. True<br />
5. True<br />
10. True<br />
C. ·1. All the numbers except 1, 2, 3, 4<br />
2. The odd numbers<br />
3. All numbers which do not end in 2<br />
4. All numbers which do not begin with 2<br />
P54 A. 1. A U B = things that are red or edible, or both. A n B = red food.<br />
88<br />
2. (a) Yes (b) Yes (c) Yes<br />
(d) Could be (depends whose blood).<br />
3. (a) No (b) No (c) No<br />
4. More in A U B<br />
5. (a) True (b) Not true (c) Not true<br />
(d) True (e) Not true (f) True<br />
B. All different answers.<br />
C. 1. (a) 6 E X n Y (b) 10 E X n Y<br />
(c) 7 $. X n Y (d) 8 E X n Y<br />
2. True (a), (d); not true (b), (c)
3. (a) 10 belongs to the set X implies 10 belongs to the union of X<br />
with any other set<br />
(b) If 19 does not belong to X it does not belong to the intersection<br />
of X with any other set<br />
(c) If 9 does not belong to Y then 9 does not belong to the<br />
intersection of X and Y<br />
(d) 25 belongs to the union of X with Y implies that 25 belongs to<br />
X or to Y (or perhaps to both)<br />
4. (a) 5 ~ Y::} 5 ~ X n Y<br />
(b) {1a E X and 1a E Y}::} 1a E X n Y<br />
(c) xn Y= YnX<br />
P55 A. 1.<br />
x<br />
y<br />
2.<br />
3. You cannot like tea best and<br />
like beer best at the same<br />
time.<br />
00<br />
4°CltersQ )<br />
5.<br />
B. 2. P n Q is shaded horizontally and vertically,<br />
P U Q is anywhere that is shaded at all<br />
(h)<br />
E55 A., B. See P54.<br />
3. (b) Horizontally and vertically<br />
(c) Yes (d) A nB=A (e) A nB=0<br />
(f) A n B = B (g) A n B = A ... if you think squares are a subset<br />
of the set of rectangles<br />
An B = B<br />
C. 1. (a) 0' = {trees other than oaks}<br />
(b) B' = {trees other than beeches}<br />
(c) 0' u B = 0' (d) S' U 0 = S'<br />
89
2. (a), (b) 0 u 0' = e by definition: every tree either is an oak or<br />
is not an oak<br />
(c), (d) 0 n 0' = 0 since no tree is both an oak (E 0) and not an<br />
oak (E 0')<br />
3. Different universal sets are possible, giving different complementary<br />
sets.<br />
4. The lopposite' picks out a special set which has a special opposite<br />
quality. The complement consists of everything else besides the<br />
set in question. For example the complement of the set Itall<br />
people' is the set Ipeople who are not tall'. This is not the same as<br />
the set of short people.<br />
5. (a) (b) (c)
2. (a) 7 (b) 9 (c) 21<br />
3. Drawings<br />
B. 1. (a)<br />
0<br />
0 2 3 4<br />
(b)<br />
0<br />
-1 0 2 3<br />
(e)<br />
0<br />
-4 -3 -2 -1 0 2<br />
!<br />
•<br />
(d)<br />
0<br />
-3 -2 -1 0<br />
(e)<br />
~ 0<br />
2 3 4 5 6<br />
(f)<br />
0 0<br />
-4 -3 -2 -1 0 2 3<br />
2. (a) The part of the number line between 3 and 5, not including 3<br />
or 5<br />
(b) the whole number line<br />
(c) x : x < 3, and 3 itself<br />
(d) x: x > 5, and 5 itself.<br />
C. 1. (a) (b) (e)<br />
~ ~ ~<br />
91
(d) (e) (f)<br />
2. When the diagrams are shaded it is easy to see that the sets are<br />
equal<br />
3. (a) Diagram (b) (i) 3 (ii) 7 (iii) 5 (iv) 0 (v) 42<br />
Unit 27, page 114<br />
The first step in statistics is to learn to look at collections of<br />
data without feeling overwhelmed. The terms data and<br />
frequency are important and the idea of a variable should be<br />
discussed at this stage. It is not necessary to define these<br />
terms but children will get used to using them in context.<br />
Data which is of particular topical interest (e.g. Olympic<br />
Games) or which is proposed by the class is of great value in<br />
giving the feeling of importance to this work. There are<br />
several useful collections of data some of which will be<br />
known to the children. A small library of data might include<br />
the Guinness Book of Records, Whitaker's Almanack, the<br />
Penguin Book of Facts, World Statistics in Brief (United<br />
Nations) and Annual Statistics of the EEC. Interesting<br />
alternative approaches may be found in the problems of the<br />
Schools Council Statistics in Education Project. All data<br />
should be presented in a very clear form. (Remember the<br />
school micro will have ways of presenting data which will be<br />
of interest to your pupils.)<br />
Answers<br />
M89 A.<br />
1. 4 2. 7 3. 4 4. 11 5. 19<br />
6. (a) 204 (b) 235 (c) 235 (d) 228 (e) 194 (f) 172<br />
7. 13 8. 1974 9. 1980<br />
10. 1970-74; 1154 students 1976-80; 1010 students<br />
92
B.<br />
c.<br />
1. 17<br />
4. 13<br />
7. 1st quarter<br />
10. 320/0<br />
1.<br />
5.<br />
6.<br />
34 2. 76<br />
3 to 4 hours<br />
More than 5 hours<br />
2. 22<br />
5. April<br />
8. 4th quarter<br />
3. 13<br />
6. November<br />
9. The total number of children<br />
3. 40 4. 27<br />
M90. A. 1. Physics 2. Biology 3. 253 4. 74<br />
5. 185 6. 119<br />
7. Chemistry 104, physics 117, biology 174, Rural science 30,<br />
Domestic science 68, Engineering science 44,<br />
General Science 51<br />
8. Biology 86 difference 9. Rural science<br />
10. Boys and girls expect to have different jobs<br />
B. 1. 77 2. 108 3. 175<br />
4. A higher percentage of boys can swim than girls<br />
5. A higher percentage of girls can swim over 5 lengths<br />
6. Once a swimmer can swim more than 3 lengths he or she is likely<br />
to swim longer distances<br />
M91 1. (a) 73p (b) 47p<br />
(c) 22p (d) 16p<br />
2. 73p, January<br />
3. (a) March and November<br />
(b) January, February<br />
(c) August, September<br />
(d) April, October<br />
4. October<br />
5. Tomatoes from Spain and the Canary Islands are ripe<br />
6. 57p per Ib<br />
7. Not quite the same as crops are harvested at different times. Also<br />
some crops keep better than tomatoes so they can be sold over a<br />
longer period of time. This avoids a glut (very low prices) followed<br />
by a sharp rise in price.<br />
P56 A. 1. Dogs 2. 73 3. 275 4. 121 5. Rabbits<br />
B. 1. (a) 170 (b) 243 (c) 142<br />
2. Bad signals 3. Alcohol/drugs 4. 1000<br />
93
C. 1. Chocolate<br />
2. Sweet<br />
3. (a) About 100 (b) About 70 (c) About 22<br />
4. About 425<br />
5. Chocolate, savoury, ginger, shortbread, custard, cream, cheese,<br />
sweet<br />
P57 A. 1. (a) 500 000 (b) 1 450 000 (c) 300 000<br />
2. (a) 1 260 000 (b) 450 000 (c) 3 150 000<br />
3. (a) France, Italy, Spain, W. Germany, other countries<br />
(b) Austria, Ireland, Switzerland<br />
4. Spain and France<br />
5. Still Spain<br />
6. 1968; apart from other countries, Spain, France, Ireland and Italy,<br />
Austria, Switzerland and W. Germany.<br />
1978; Spain, France, Italy, Ireland, W. Germany, Austria and<br />
Switzerland<br />
France has moved well ahead of Italy and Ireland.<br />
W. Germany is more popular than Switzerland and Austria (shorter<br />
journey, lower cost of living).<br />
B. 1. (a) 71 000 (b) 17000 (c) 3000<br />
2. (a) 4000 (b) 48000 (c) 47 000<br />
3. The number of farm workers dropped from 75000 to 43 000<br />
4. (a) True (b) True<br />
5. Horses were replaced by tractors as tractors were more efficient.<br />
(Though there is some doubt about this because<br />
(i) petrol is expensive;<br />
(ii) tractors wear out;<br />
(iii) horses produce new horses;<br />
(iv) horses produce fertilizer)<br />
6. Yes, roughly<br />
7. 43 000 8. 28000<br />
9. Farms became larger so that one tractor (and ploughman) served<br />
a much greater area.<br />
E57 A. See P56 A and 6. 30%<br />
7. Hamsters, guinea pigs, fish, birds ... etc.<br />
8. Different answers<br />
B. See P56 Band<br />
5. Different answers<br />
6. Different answers<br />
94
C. 1. Chocolate<br />
2. Sweet and cheese<br />
3. (a) 100 (b) 70 (c) 20<br />
4. About 425<br />
5. £138·74<br />
6. Chocolate, savoury, ginger, shortbread, custard, cream, cheese<br />
and sweet<br />
7. 3 cheese, 10 savoury, 3 sweet, 7 shortbread, 7 creaJ'!l,<br />
14 chocolate, 7 custard and 10 ginger<br />
D. Reasonable<br />
E58 A. 1, 2, 3, 4 see P57 A<br />
5. The answer to this question depends on how you think the<br />
changes will happen. For example, Austria reduced from 35 to 18,<br />
Le. a reduction factor of 18/35. If you apply this reduction to 18000<br />
you get 18000 x 18/35 = 9257. So 9300 would be a reasonable<br />
prediction. If you say the numbers fell by 17000 and this would<br />
happen again the result would be only 1000 holidays in Austria in<br />
1988. This method seems less reasonable.<br />
Applying the first method of prediction gives estimates as<br />
follows for 1968: Austria 93000, France 3 170 000, Ireland<br />
405 000, Italy 1 040 000, Spain 5 030 000, Switzerland 108 000, W.<br />
Germany 432 000.<br />
In practice, it is unlikely that the factors producing the change in<br />
the period 1968-1978 would continue to operate in the same way<br />
for the next decade.<br />
6. Order of popularity:<br />
1968 Spain, France, Italy and Ireland, Austria, Switzerland and<br />
W. Germany<br />
1978 Spain, France, Italy, Ireland, W. Germany, Austria and<br />
Switzerland<br />
1988 Spain, France, Italy, W. Germany, Ireland, Switzerland,<br />
Austria<br />
B. 1. (a) 71 000 (b) 17 000 (c) 3000<br />
2. (a) 4000 (b) 48 000 (c) 47 000<br />
3. (a) The number dropped from 75 000 to 43 000<br />
(b) Farms become more capital intensive<br />
4. True<br />
5. Tractors are more efficient<br />
6. Yes, roughly<br />
7. (a) 43 000 (b) 28 000 8. Different answers<br />
C. Drawing and different answers<br />
95
Unit 28, page 118<br />
The idea of average is widely used for comparisons and to<br />
summarize data. However it is not always well understood in<br />
the context of problems. Many people would say that<br />
'nobody should be paid less than the average wage' without<br />
realizing that this means that no-one should be paid more<br />
than the average wage either. The concepts of average<br />
weight and height are related to health, yet a person can be<br />
far from average and be perfectly healthy. The idea of<br />
average contains the notion of acceptable range and deviation.<br />
We, as <strong>mathematics</strong> teachers, sometimes misapply the<br />
notion of average child, mixing up the idea of what an<br />
average child should be able to do (in which case 50% will<br />
not be able to do it) with the expectation of every child (in<br />
which case it will probably be such a low level as to be<br />
worthless) .<br />
The exercises of this unit emphasize the idea of average as<br />
a single number representing a set of data, the relationship<br />
between average and total and also the use of averages to<br />
make comparisons. Further interesting examples can arise<br />
from class discussion. Calculators with statistic circuits will<br />
have an 'average' button marked x.<br />
Answers<br />
M92 A. 1. 7·8 minutes 2. 7·2 minutes 3. 2·25<br />
4. 4·25 5. 10·49 seconds 6. 7·1<br />
B. 1. £5·30 2. 41·77 em 3. 3·2 kg<br />
4. 37·18°C 5. 26·07 km<br />
C. 1. 14 2. £66 3. £84·30<br />
4. 252 hours 5. 4320 6. Yes, by 46 kg<br />
M93 1. (a) 215·7, corrected to 216 as 0·7 of a child does not mean<br />
anything<br />
(b) 1974, 1975, 1976, 1977 (c) 1978, 1979, 1980.<br />
2. (a) 12·5 (b) 12·5 (c) 12·5 (d) Jan, Feb, May, Nov, Dec.<br />
(e) Dark 14·8; light 10·2<br />
(f) The dark months are more dangerous<br />
3. (a) 58·6p<br />
(b) Because the fruit does not ripen till October<br />
(c) Eggs, milk, bread ...<br />
96
4. (a) 10·1 (b) 1976, less rain<br />
(c) 1978 after two dry years the reservoirs would be low<br />
P58 A. 1. £659·17 2. £776·33 3. The second half<br />
4. March, June, July, August, September<br />
B. 1. 61·6<br />
2. 60, worse than last term so far (but better than last term's average<br />
on the first three marks).<br />
c. 1. Paul 101·7 minutes; Joanne, 106·8 minutes. Joanne's average was<br />
higher but things might have been different ifshe had timed her<br />
viewing on Thursday.<br />
2. Cod 266·33; haddock 142; plaice 35·2; herring 128-8; lobster 0·97.<br />
All the catches are declining. This is because of overfishing.<br />
P59 1. x f xf 2. x f xf<br />
0 3 0 0 42 0<br />
1 7 7 1 36 36<br />
2 14 28 2 33 66<br />
3 12 36 3 41 123<br />
4 8 32 4 6 24<br />
5 3 15 5 22 110<br />
6 3 18<br />
- -<br />
6 20<br />
-<br />
120<br />
-<br />
50 136 200 479<br />
Average 2-72 Average 2-395<br />
3. x f xf<br />
0 300000 0<br />
1 5600000 5600000<br />
2 9400000 18800000<br />
3 1300000 3900000<br />
4 265000 1 060000<br />
16865000 29360000<br />
The average is 1·74 radios per household. This seems reasonable<br />
because many people have radio alarm clocks, etc., but accurate data<br />
would be difficult to collect.<br />
97
E59 A. 1. See P58 A. 2. See P58 B.<br />
3. See P58 C. and (b) A reasonable guess would be a little below<br />
average, i.e. 90-100.<br />
4. See P58 C. 2 and (c) 85·4 million, about 1~ fish per<br />
person (d) Overfishing.<br />
B. See P59<br />
EGO A. 1. Different answers 2. Different answers<br />
B. 1. £106·92 2. £89·16<br />
3. £124·67 since (124·67 x 6) + (89·16 x 6) = (106·92 x 12). She has<br />
to make the same total if she wants the same monthly average.<br />
4. Green £75·05; Jones £71·14. They also need to know what the<br />
money was spent on and to compare the number of miles they<br />
had travelled.<br />
C. 3. Assuming that after 1000 throws the scores settled down you<br />
would expect the average score to be<br />
(1 + 2 + 3 + 4 + 5 + 6) 7 6 = 3·5<br />
5. The average score wou Id not be (1 + 2 + 3 + ... 12) 7 12 because<br />
the scores are not equally likely. The average score should be 7<br />
and not 6·5. Out of 36 throws the exact expectation is shown in<br />
this table.<br />
score<br />
f<br />
2 3 4 5 6 7 8 9 10 11 12<br />
1 234 5 6 543 2 1<br />
(7 for example can occur as 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2 or 6 + 1)<br />
Unit 29, page 122<br />
Drawing and using straight line graphs is a basic skill of<br />
elementary <strong>mathematics</strong> and of everyday life. Of these, the<br />
family y = kx is the simplest and most important. We only<br />
need to know one point and the whole line may be drawn.<br />
The point that a straight line is a mathematical ideal can be<br />
discussed. This gives a greater reality to children who<br />
98
understand that the world of commerce allows discounts for<br />
large purchases. The main purpose of this unit is to recognize<br />
the form y = kx in problems, draw and use the graph,<br />
and to see the effect of adding a constant to form y = kx + c.<br />
Answers<br />
M94 A.<br />
B.<br />
1. Drawing<br />
2. (3,4); (2,5); (1, 1·5); (2·2,3·2); (0,4) are above the line<br />
while the rest are below the line<br />
If the y-co-ordinate is greater than the x-eo-ordinate the point<br />
will be above the line.<br />
If the x-eo-ordinate is larger the point will be below the line<br />
1. Drawing<br />
2. (1·5,3); (2·1,4·2); Because the y-co-ordinate is exactly twice the<br />
x-eo-ordinate, i.e. y = 2x.<br />
3. (1,3); (0,4) are above the line. The rest are below the line. The<br />
points are above the line if the y-co-ordinate is more than twice<br />
the x-eo-ordinate<br />
4. x 0 1 2 3 4 5<br />
y 0 1·5 3 4·5 6 7·5<br />
(a) Different answers are possible<br />
(b) (6,9); (7,10·5); (1·5, 2·75), etc.<br />
(c) Only one<br />
(d) Only one<br />
C. Drawing<br />
M95 A. 1. x = number of kilos, y = total cost, k = 10 pence<br />
2. x = number of kilos, y = total cost, k = 80 pence<br />
3. x = number of metres, y = total cost, k = £1·80<br />
4. x = number of miles, y = total cost, k = 75p<br />
5. x = number of hours, y = total distance, k = 8 km<br />
6. x = number of hours, y = total distance, k = 3 km<br />
B. 1. 120p 2. 20miles 3. 96g 4. £15<br />
C. 1. (a) 17p (b) 25p (c) 42p (d) 84p<br />
2. (a) 2 (b) 3 (c) 5 (d) 12 (e) 30 (f) 42<br />
M96<br />
Drawing<br />
99
P60 A. 1. Drawing<br />
2. (a) Yes<br />
(b) £26·40<br />
(c) a = 5<br />
(d) b=8·4<br />
3. (a) Different points are possible<br />
(b) Yes<br />
(c) Different points are possible<br />
(d) This is not certainly true for all points. For example (2,2·2)<br />
would come below the line but the y-co-ordinate is greater<br />
than the x-eo-ordinate.<br />
B. 1. Drawing<br />
2. (a) Only (0,0) (b) Different points<br />
(c) Depends on (b) (d) Should be true<br />
3. (a) Different points<br />
(b) Because the y-co-ordinate is more than double but less than 3<br />
times the x-eo-ordinate, e.g. (2,5) will be between the lines<br />
y= 2x and y= 3x<br />
C. 1. (0,0); (2,3); (4,6) are allan the line y = 1~x (3,5) is the other point<br />
asked for in the question<br />
2. (6,12) 3. (0,0) 4. (7,6)<br />
P61 A. 1. Drawing<br />
2. (a) £6·40 (b) £9·80 (c) £16·20<br />
(d) £6·96 (e) £11·76 (f) £38·40<br />
100<br />
B. 1. (a) £97·75 (b) £103·50 (c) £7·48<br />
(d) £20·70 (e) £3·91 (f) £7·82<br />
2. (a) The graph would be y = 1·25x instead of y = 1·15x, i.e. the<br />
slope would be greater.<br />
(b) The line still passes through (0,0)<br />
C. 1. (a) y = 4·5x + 14<br />
(b) £68, £122, £176<br />
(c) 19 months<br />
2. The graph required is y = 0·12x + 78. This must be drawn on<br />
1 mm 2 graph paper if an accurate answer is to be obtained.<br />
100 miles cost £90<br />
250 miles cost £108<br />
400 miles cost £126<br />
1000 miles cost £198
E61 A. See P60 A.<br />
B. 1. It is the reflection in the x-axis (and also in the y-axis)<br />
2. (a) Drawing (b) They are perpendicular (c) (0,0)<br />
3. They have the same value but opposite signs (a and -a)<br />
C. 1. (3,3) 2. (0,0) 3. (1, -3) 4. (3,0)<br />
E62 A. 1. Different answers<br />
2. The cost of cement decreases as you buy larger quantities so the<br />
graph will not be a straight line<br />
B. 1. (a), (c) and (d)<br />
2. £6·40, £6·96, £9·80, £11·76, £16·20, £38·40<br />
3. (a) £6210 (b) £7425 (c) £6534 (d) £8694<br />
(e) £9315 (f) £11475 (g) £17280<br />
C. See P61 C.<br />
Unit 30, page 125<br />
Ratio is one of the least successfully taught topics in<br />
<strong>mathematics</strong>. This is because one is asking the children to<br />
hold a relationship in their head while they juggle with it. It<br />
is, however, a fundamental concept and failure will undermine<br />
many of the ideas that will be taught in science,<br />
<strong>mathematics</strong>, home economics and geography. At least, use<br />
of the calculator removes fear of arithmetic which is one of<br />
the contributory factors to failure in this topic. In all the<br />
examples the relationship between m: nand m/m + n is<br />
explored in the context of problems. These problems should<br />
be considered as material for discussion and children should<br />
be encouraged to persist until they fully understand what is<br />
going on.<br />
Applications in geometry provide an alternative approach<br />
to the concept of ratio and incidentally revise some of the<br />
more important geometrical relationships.<br />
The ratio sign is just another division sign and it is worth<br />
noting that both the fractional 'over' sign and the ratio sign<br />
are present in our division sign -:-, 1: 2, t, 1 -:-2<br />
101
Answers<br />
M97 A. Ratios of boys: girls (not girls: boys):<br />
1. 17: 19 2. 25: 40 (simplifies to 5 : 8 or 1 : 1·6)<br />
3. 6: 8 (=3 : 4) 4. 232: 250 (=0·928 : 1)<br />
5. 22: 18 (= 11 : 9)<br />
B. 1. 42: 65 2. 360: 240 (=3 : 2) 3. 480: 74 (=240 : 37)<br />
4. 15000: 31 000 (=15 : 31) 5. 194: 6 (=32·3 : 1)<br />
6. 75: 25 (=3 : 1)<br />
Note that a given ratio can be expressed in a number of ways.<br />
C. 1. (a) 1: 1·2857 (b) 1·6: 1 (c) 1: 1·2<br />
(d) 1:2·222 (e) 1:1·428 (f) 1:1·67<br />
(g) 1: 1·67 (h) 1: 4 (i) 1: 2<br />
2. (a) 1: 8 (weight of oxygen 0·120 g) (b) 1: 3<br />
(c) 5: 100 = 1: 20<br />
M98 1. (a) (i) 21:39 (ij) 9:34 (iii) 23:33 (iv) 27:21<br />
(b) 600 m<br />
(c) 3: 2·4 This is the only ratio in which you go up more than<br />
1 unit for 1 along<br />
2. (a) 200 (b) 45<br />
3. NO.4: 5 is a better ratio than 6: 8<br />
4. 78·75 mm<br />
5. (a) £296. Note: It will be larger than £185 by a scale factor<br />
.<br />
4: 2·5. Thus the fare IS<br />
185 x 4<br />
2.5<br />
305 x 2·5<br />
(b) £190·63... 4<br />
M99 Usually the best known constant ratio is 7T, the ratio of circumference to<br />
diameter of circles.<br />
102<br />
A. Drawing<br />
B. If the rectangles are the same proportion the ratios will be constant,<br />
otherwise not<br />
C. 1,2 Drawing.AB:AD=1·55:1.<br />
3. Yes, the ratio altitude: side is constant for equilateral triangles<br />
D. All drawing and measuring<br />
E. All drawing and measuring<br />
There are no P or E exercises with this Unit.
· .•··i •• ;;::{;i;......................... .·····..·..··.·.·...i.·?<br />
INTEGRATED<br />
M&JIiEMI\TICS<br />
SCHEME<br />
1"1 <strong>T2</strong><br />
Bell& Hyman limited<br />
ISBN 0 7135 1339 X