ST(P) Mathematics 1A – Teacher’s Notes and Answers 1 ST(P) MATHEMATICS 1A NOTES AND ANSWERS The book starts with a large section on arithmetic. This has been kept together because we feel that all children starting a new school with a new teacher benefit from a thorough revision of basic arithmetic. Many children arrive at secondary school not sure of what they do or do not know, and what they do know is often obscured by the use of unfamiliar words. However, many teachers will want to break up the arithmetic with other work. Tables and Networks (Chapter 13) is particularly suitable for this purpose. It is self-contained and can easily be divided into two sections that can be taught at different times. Symmetry (Chapter 10) is another self-contained unit that can be taught at an earlier stage. CHAPTER 1 Addition and Subtraction of Whole Numbers This chapter is intended to give practice in addition and subtraction of whole numbers. We have not introduced the calculator until near the end of this chapter but an earlier introduction may be felt to be appropriate; it can be used to check answers. EXERCISE 1a (p. 1) Can be used for discussion, e.g. other methods of adding several numbers such as looking for pairs of numbers that add up to ten; can also be used for mental arithmetic. 1. 10 8. 19 15. 33 22. 17 29. 26 2. 11 9. 20 16. 18 23. 20 30. 32 3. 14 10. 27 17. 25 24. 33 31. 26 4. 15 11. 15 18. 32 25. 30 32. 26 5. 17 12. 17 19. 39 26. 21 33. 40 6. 24 13. 27 20. 32 27. 21 34. 37 7. 24 14. 27 21. 24 28. 19 35. 39 EXERCISE 1b (p. 2) 1. 79 10. 2292 19. 797 28. 2764 37. 509 2. 97 11. 549 20. 1966 29. 5936 38. 857 3. 65 12. 1835 21. 183 30. 7525 39. 1087 4. 308 13. 9072 22. 177 31. 1693 40. 1832 5. 259 14. 21 829 23. 202 32. 1382 41. 2892 6. 399 15. 16 244 24. 1252 33. 1896 42. 6779 7. 882 16. 112 25. 2783 34. 5230 43. 2226 8. 2039 17. 158 26. 2062 35. 4095 44. 3569 9. 991 18. 242 27. 1267 36. 581 45. 11 932 EXERCISE 1c (p. 3) Confidence in problem solving comes from getting the answer right. More able children can be asked for some form of explanation, at least writing the answer in sentence form. Some worked examples will be necessary to indicate what they are expected to write down.
Complete Answer Book for ST(P) Maths 1A by Nelson Thornes.
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ST(P) Mathematics 1A – Teacher’s Notes and Answers 1
ST(P) MATHEMATICS 1ANOTES AND ANSWERS
The book starts with a large section on arithmetic. This has been kept together because wefeel that all children starting a new school with a new teacher benefit from a thoroughrevision of basic arithmetic. Many children arrive at secondary school not sure of what theydo or do not know, and what they do know is often obscured by the use of unfamiliar words.
However, many teachers will want to break up the arithmetic with other work. Tables andNetworks (Chapter 13) is particularly suitable for this purpose. It is self-contained and caneasily be divided into two sections that can be taught at different times. Symmetry (Chapter10) is another self-contained unit that can be taught at an earlier stage.
CHAPTER 1 Addition and Subtraction of Whole Numbers
This chapter is intended to give practice in addition and subtraction of whole numbers. Wehave not introduced the calculator until near the end of this chapter but an earlier introductionmay be felt to be appropriate; it can be used to check answers.
EXERCISE 1a (p. 1)Can be used for discussion, e.g. other methods of adding several numbers such as looking forpairs of numbers that add up to ten; can also be used for mental arithmetic.
EXERCISE 1c (p. 3)Confidence in problem solving comes from getting the answer right. More able children canbe asked for some form of explanation, at least writing the answer in sentence form. Someworked examples will be necessary to indicate what they are expected to write down.
ST(P) Mathematics 1A – Teacher’s Notes and Answers 2
1. 89p 7. 7872. 69p 8. 77cm3. 88 9. £164. £757 10. 50min5. a) 261 b) 302 c) 3056 d) 1300 11. 49576. a) three hundred and twenty-four 12. £10.23 or 1023p
b) five thousand two hundred and eightc) one hundred and fiftyd) one thousand five hundred
CHAPTER 2 Multiplication and Division of Whole Numbers
The word “product” is used at the beginning of this chapter and will need explanation.
EXERCISE 2a (p. 12)Discussion of the properties of odd and even numbers is useful here, e.g. is the product of twoeven numbers even or odd and why? These properties can be used as simple checks onanswers.
EXERCISE 2e (p. 16)If it has not been done earlier, this is an appropriate place to introduce the more able pupils toa more formal setting down of answers.
EXERCISE 2i (p. 21)Not intended for use with a calculator. If calculators are used to check answers, tuition ontheir use for mixed operations will be needed and will vary with the type of calculator used.A simple four-function calculator does not usually give priority to x and but a scientificcalculator usually does and if pupils have a calculator with this facility it should be used.
1 7 21 35 35 21 7 121. 3524. a) 1, 4, 9, 16 b) 25 c) 36, 49 d) 7, 9, these differences go up by 2 each time25. a) 1, 3, 6, 10, 15, 21, 28 b) 2, 3, 4, 5, 6, 7 c) 1, 1, 1, 1, 126. 3, 8, 13, 18, …, 38, …27. 1, 2, 4, 8, …, 32, …28. a) (i) 20, 24, 28 (ii) 4 (iii) 0
b) (i) 24, 29, 34 (ii) 5 (iii) 0c) (i) 32, 64, 128 (ii) 2, 4, 8, 16, 32, 64 (iii) 2, 4, 8, 16, 32
ST(P) Mathematics 1A – Teacher’s Notes and Answers 7
d) (i) 162, 486, 1458 (ii) 4, 12, 36, 108, 324, 972 (iii) 8, 24, 72, 216, 648in (ii) and (iii), multiply by 3 each time
29. 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …30. 1, 2, 2, 4, 8, 32, 256, 8192, …31. 15 and 33. …add 6 each time32. 1 and ¼. …divide by 2 each time33. 3 and 9. …multiply by 3 each time34. a) 9 b) 1535. a) 15 b) 36
Simplifying fractions: this is the first time that the word “factor” is used. It will needexplanation and much discussion to clarify its meaning, e.g. is 2 a factor of 14; what are thefactors of 6? Factors are discussed again in Chapter 12, and Exercise 12a could be done now.Children not familiar with simplifying fractions need a lot of discussion before they do anythemselves. Discussion of the other words used for simplifying is needed, i.e. reducing andcancelling. (Cancelling really means the act of removing the common factors.)
EXERCISE 3e (p. 42)
1. 31 7. 3
1 13. 51 19. 5
3 25. 54
ST(P) Mathematics 1A – Teacher’s Notes and Answers 9
2. 53 8. 3
2 14. 52 20. 5
2 26. 74
3. 31 9. 2
1 15. 72 21. 9
5 27. 31
4. 21 10. 4
1 16. 31 22. 11
7 28. 119
5. 31 11. 7
2 17. 21 23. 4
3 29. 43
6. 21 12. 10
3 18. 51 24. 11
3 30. 54
EXERCISE 3f (p. 43)
1. 43 8. 5
2 15. 21 22. 23
15 29. 17
9
2. 21 9. 21
11 16. 109 23. 9
8 30. 1912
3. 115 10. 2
1 17. 43 24. 3
2 31. 3013
4. 1310 11. 13
11 18. 1911 25. 5
4 32. 95
5. 2319 12. 5
4 19. 21 26. 5
2 33. 21
6. 73 13. 7
6 20. 52 27. 31
23 34. 9925
7. 53 14. 17
9 21. 116 28. 14
11
Addition and subtraction of fractions: many pupils try to add or subtract at the same timeas changing denominators and are then baffled by their inevitable mistakes. This is a casewhere they should be encouraged to write down each step, as shown in the worked examples,so that they separate the two operations.
EXERCISE 3g (p. 45)
1. 1513 9. 42
19 16. 10033 23. 15
13 30. 12. 40
23 10. 4241 17. 20
19 24. 43 31. 40
39
3. 3011 11. 99
82 18. 85 25. 20
19 32. 1813
4. 3529 12. 90
47 19. 98 26. 24
17 33. 2017
5. 3029 13. 10
7 20. 1813 27. 20
19 34. 1817
6. 5639 14. 16
13 21. 2013 28. 12
11 35. 3019
7. 4225 15. 21
17 22. 2213 29. 7
6 36. 32
8. 2120
EXERCISE 3h (p. 47)
1. 32 6. 7
3 11. 157 16. 12
1 21. 81
2. 21 7. 13
5 12. 31 17. 100
9 22. 41
3. 175 8. 5
3 13. 5518 18. 56
19 23. 61
4. 2011 9. 21
5 14. 91 19. 16
3 24. 154
5. 52 10. 21
5 15. 263 20. 15
4
ST(P) Mathematics 1A – Teacher’s Notes and Answers 10
EXERCISE 3i (p. 49)
1. 83 6. 12
5 11. 43 16. 16
1 21. 10019
2. 75 7. 5
3 12. 21 17. 9
2 22. 41
3. 161 8. 18
17 13. 181 18. 20
7 23. 185
4. 125 9. 50
17 14. 121 19. 8
1 24. 301
5. 509 10. 2
1 15. 51 20. 3
1
EXERCISE 3j (p. 50) Intended for the above average; can be used for discussion with others.
1. 1513 , 15
2 2. 1511 , 15
4 3. 31 , 12
1 4. 83 , 8
7 5. 4011 , 20
19 , 407
EXERCISE 3k (p. 52)
1. 412 5. 9
79 9. 5225 13. 9
413 17. 3213
2. 434 6. 2
13 10. 11410 14. 6
115 18. 5213
3. 616 7. 4
36 11. 8513 15. 11
107 19. 3124
4. 1035 8. 8
15 12. 7611 16. 6
512 20. 1094
EXERCISE 3l (p. 52)
1. 313 5. 7
57 9. 311 13. 5
19 17. 1019
2. 433 6. 5
33 10. 211 14. 9
43 18. 320
3. 1017 7. 7
20 11. 537 15. 4
35 19. 859
4. 998 8. 6
25 12. 922 16. 7
73 20. 10101
EXERCISE 3m (p. 53)
1. 715 4. 2
12 7. 3213 9. 6
18 11. 527
2. 659 5. 5
216 8. 917 10. 10
710 12. 216
3. 1184 6. 4
17
EXERCISE 3n (p. 54)Again it is important to encourage the writing down of each step so that only one operation isperformed at a time.
1. 435 7. 20
94 13. 14311 19. 10
110 25. 5215
2. 653 8. 14
33 14. 1618 20. 10
111 26. 5415
3. 40235 9. 10
77 15. 16112 21. 2
111 27. 1005114
4. 949 10. 21
1713 16. 10911 22. 7
317 28. 321317
5. 36295 11. 16
1310 17. 1038 23. 16
317 29. 7222
6. 614 12. 3
16 18. 2118 24. 18
121 30. 2122
ST(P) Mathematics 1A – Teacher’s Notes and Answers 11
EXERCISE 3p (p. 56)
1. 851 9. 10
71 16. 28252 23. 4
3 30. 211
2. 15131 10. 35
113 17. 431 24. 35
271 31. 652
3. 611 11. 15
22 18. 2073 25. 8
31 32. 872
4. 43 12. 4
13 19. 3593 26. 10
72 33. 1093
5. 1255 13. 10
33 20. 3326 27. 2
13 34. 32
6. 211 14. 63
42 21. 2833 28. 2
12 35. 611
7. 1451 15. 24
73 22. 851 29. 9
7 36. 21162
8. 1032
EXERCISE 3q (p. 56)
1. a) 2151 b) 24
11 c) 7235 d) 6
12 e) 1211
2. a) 412 b) 5
133. a) 7
3 b) 3017
4. a) 21 , 5
3 , 2013 , 10
7 b) 127 , 3
2 , 43 , 6
5 c) 53 , 10
7 , 10071 , 20
17
5. a) < b) > c) >6. a) 11
3 b) 227 c) 11
9
EXERCISE 3r (p. 57)
1. a) 152 b) 10
71 c) 223 d) 12
76 e) 21 f) 20
1322. a) 8
7 b) 651 c) 13
12
3. a) 10013 b) 366
233
4. a) > b) < c) <5. a) 10
3 , 207 , 8
3 , 52 b) 10
3 , 52 , 15
7 , 21 c) 32
17 , 169 , 8
5 , 43
6. a) 2815 b) 7
2
EXERCISE 3s (p. 57)
1. a) 14043 b) 45
17 c) 81 d) 12
13 e) 0 f) 52. a) 8
31 b) 522 c) 16
5
3. a) < b) <4. a) 2
1 , 53 , 4
3 , 65 b) 2
1 , 95 , 3
2 , 65
5. a) 607 b) 3
1 c) 7938
6. a) 1917 b) 19
13
EXERCISE 3t (p. 58)
1. a) 611 b) 8
5 c) 121 d) 20
92 e) 1211 f) 3
232. a) 8
34 b) 81 c) 7
42
ST(P) Mathematics 1A – Teacher’s Notes and Answers 12
3. a) 245 b) 10
1 c) 125
4. a) > b) <5. a) 11
5 , 21 , 44
23 , 2213 b) 9
5 , 127 , 3
2 , 43
6. a) 51 b) 15
8 c) 31
CHAPTER 4 Fractions: Multiplication and Division
If pupils have not done multiplication of fractions before, much classroom discussion isadvisable, using cake diagrams, rectangles, etc., to get across the meaning that, for example,21 x 4
EXERCISE 6f (p. 99)Discussion about quantities that can be given exactly, quantities that cannot be given exactly(e.g. measurements), quantities that can be given exactly but often are not (e.g. governmentstatistics) is useful here.
EXERCISE 6g (p. 100)Calculators should be used except by the brightest children who should use them only forchecking answers. At this point they will need to be shown how to give an answer correct to aspecified number of decimal places, by reading the display to one more place than necessary.
Division by decimals: much class discussion is necessary before pupils are asked to work ontheir own.
EXERCISE 6i (p. 102)Nos. 1–24 do not need a calculator. Nos. 25–36: benefit will be obtained from using acalculator but pupils need to be shown how to get an estimate.
EXERCISE 7a (p. 108)A good opportunity to point out the importance of eyes being directly over each end of a linewhen using a ruler to measure its length.
1. a) metres b) centimetres c) metres d) kilometres e) centimetres f) millimetres3. a) 4 b) 2 c) 5 d) 1 e) 104. (to the nearest millimetre) a) 20 b) 10 c) 4 d) 16 e) 249. 40cm 10. 900cm
In all the geometry chapters there are no instructions as to how the solutions to problemsshould be written down. An intuitive approach is best at this age and most pupils should beasked only to fill in the sizes of angles in diagrams. The teacher will decide whether or notbrighter children should be asked to write down reasoned solutions.
EXERCISE 9e (p. 130)Worth discussing the number 360, e.g. how many whole numbers divide exactly into it?Compare it with 100; which is the better number and why? Its origins are interesting: itprobably came from the Babylonians who used 60 as a number base. It is also worth notingthat 60 is the base used for time (seconds and minutes and hours).
EXERCISE 9h (p. 138)If pupils do measure each other’s angles, it is worth pointing out that protractors are notalways as accurate as they should be; an angle measured as 51º on one protractor could bemeasured as 52º on another.
EXERCISE 9i (p. 138)In No. 3 check that the pupils’ diagrams vary.
4. 150º 6. 35º 7. 65º 8. 140º 9. 160º
ST(P) Mathematics 1A – Teacher’s Notes and Answers 25
5. 20º
EXERCISE 9j (p. 140)No. 1, or a similar one, could be demonstrated by one of the children in front of the class.
1. 180º 2. 180º
EXERCISE 9k (p. 140)
1. 120º 10. 140º 19. 50º, 130º, 130º2. 155º 11. 90º 20. 60º, 120º, 120º3. 10º 12. 50º 21. 180º, 60º4. 100º 13. e & f 22. 105º, 180º5. 20º 14. m & k, j & d 23. 45º, 135º, 135º6. 130º 15. d & f, f & e, e & g, g & d 24. 180º, 155º7. 80º 16. f & g 25. 80º, 100º, 100º8. 15º 17. f & g, g & d, d & e, e & f 26. 165º, 180º9. 135º 18. n & d, d & p, p & m, m & n
This chapter can be done earlier, but should be done before Chapter 11.
EXERCISE 10a (p. 148)
1, 3, 4 and 6
EXERCISE 10b (p. 150)
1. 2 3. 0 4. 1 5. 2 6. 22. 1
ST(P) Mathematics 1A – Teacher’s Notes and Answers 26
EXERCISE 10c (p. 152)
1. 6 2. 6 3. 0 4. 3
EXERCISE 10d (p. 153)It is advisable to point out that the amount of rotation must not be a complete revolution.
2, 3 and 59. In Exercise 10c, numbers 1, 2, 3, 4, 7 and 8 have rotational symmetry.
EXERCISE 10e (p. 155)
1. yes 3. yes 5. yes 7. no 8. yes2. no 4. yes 6. yes
EXERCISE 10f (p. 156)
EXERCISE 10g (p. 158)
1. yes 3. yes 5. yes 7. e.g. saucepan, milk bottle2. no 4. yes 6. no
CHAPTER 11 Triangles and Angles
Angles of a triangle: some teachers may prefer to use paper tearing before drawing andmeasurement of angles. This applies to angles of a quadrilateral later in the chapter.
ST(P) Mathematics 1A – Teacher’s Notes and Answers 27
EXERCISE 11f (p. 168) Some of the remaining measurements of each constructed triangle are given hereand in the following exercises to help check pupils’ drawings. Alternatively, pupils could be asked to find themfrom their own drawings.
EXERCISE 11k (p. 177)In No. 6, two tetrahedra can be stuck together to make a polyhedron with six faces. The netsfor other simple polyhedra are provided in Book 2 but are not included here because at thisstage constructions are rarely accurate enough to give satisfying results.
1. 1 x 18, 2 x 9, 3 x 62. 1 x 20, 2 x 10, 4 x 53. 1 x 24, 2 x 12, 3 x 8, 4 x 64. 1 x 27, 3 x 95. 1 x 30, 2 x 15, 3 x 10, 5 x 66. 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 67. 1 x 40, 2 x 20, 4 x 10, 5 x 88. 1 x 45, 3 x 15, 5 x 99. 1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8
10. 1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12, 6 x 1011. 1 x 64, 2 x 32, 4 x 16, 8 x 812. 1 x 72, 2 x 36, 3 x 24, 4 x 18, 6 x 12, 8 x 913. 1 x 80, 2 x 40, 4 x 20, 5 x 16, 8 x 1014. 1 x 96, 2 x 48, 3 x 32, 4 x 24, 6 x 16, 8 x 1215. 1 x 100, 2 x 50, 4 x 25, 5 x 20, 10 x 1016. 1 x 108, 2 x 54, 3 x 36, 4 x 27, 6 x 18, 9 x 1217. 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, 10 x 1218. 1 x 135, 3 x 45, 5 x 27, 9 x 1519. 1 x 144, 2 x 72, 3 x 48, 4 x 36, 6 x 24, 8 x 18, 9 x 16, 12 x 1220. 1 x 160, 2 x 80, 4 x 40, 5 x 32, 8 x 20, 10 x 16
EXERCISE 12b (p. 181)Some examples discussed with the class would be useful.
ST(P) Mathematics 1A – Teacher’s Notes and Answers 29
EXERCISE 12c (p. 181)
1. 2, 3, 5, 7, 11, 13 3. 31, 37, 41, 43, 47 5. 41, 101, 1272. 23, 29 4. 5, 19, 29, 61 6. a) F b) F c) T d) T e) F
EXERCISE 12d (p. 182)
1. 23 7. 133 12. 27 17. 81 22. 33
2. 34 8. 192 13. 25 18. 16 23. 72
3. 54 9. 27 14. 8 19. 22 24. 52
4. 75 10. 64 15. 9 20. 32 25. 25
5. 25 11. 32 16. 49 21. 23 26. 26
6. 36
EXERCISE 12e (p. 183)A calculator should be used for Nos. 11–16.
1. 22 x 72 5. 23 x 32 x 52 8. 52 x 133 11. 108 14. 362. 33 x 52 6. 22 x 3 x 112 9. 33 x 52 x 72 12. 225 15. 1803. 53 x 132 7. 32 x 5 x 74 10. 22 x 32 x 52 13. 112 16. 1264. 22 x 32 x 52
EXERCISE 12f (p. 184)
1. yes 4. yes 6. yes 8. no 10. yes2. no 5. no 7. yes 9. yes 11. yes3. yes
EXERCISE 12g (p. 185)
1. 23 x 3 3. 32 x 7 5. 23 x 17 7. 23 x 33 9. 34 x 72
2. 22 x 7 4. 23 x 32 6. 22 x 3 x 7 8. 24 x 3 x 11 10. 24 x 72
EXERCISE 12j (p. 186)These problems are difficult and should be approached with caution. They are useful fordiscussion but only the most able children should be allowed to work through them on theirown.
ST(P) Mathematics 1A – Teacher’s Notes and Answers 30
1. a) £19.20 b) £18.60 c) £35.30 d) London, Saturday + Alton Towers, weekday, orBirmingham, Sunday + Alton Towers, Saturday
2. a) £49 b) £61 c) £6000, in Area 3 d) £6000 in Area 1 or £7000 in Area 2e) £6000, in Area 2 f) Martins £7000, Barkers £6000
EXERCISE 13b (p. 190)Many other questions can be asked about these tables.
1. a) 4 b) 15 c) 22 d) 32 e) Otherwise there is no-one to be in the class2. a) 1 b) 15 c) 303. a) 9 b) 1 c) 14 d) 28 e) 234. a) Missing numbers are 4 and 9 b) 9 c) 3
Other tables can be made to show information collected in the class.
EXERCISE 13c (p. 192)
1. a) 14km b) 17km c) 22km d) 21km e) e.g. A to E to D to C, 24km f) via F2. a) 550m b) 440m c) 705m3. a) 790m b) yes, between church and school and between Post Office and school4. a) Post Office, shop, school, Daisy’s house, school, Post Office; 560m
b) Post Office, school, Daisy’s house, school, Post Office, Pete’s house, Post Office orthis route in reverse; 820m
5. a) 12m b) 33m c) 60m d) A to C to D, 32m e) A to B to D, 33m6. a) 10m b) 35min c) A to D to E to B, 30min d) B to E to D, 25min
EXERCISE 13d (p. 194)
1. drawing is possible starting at B but not at C.3. (a) and (b) are not possible.4. a) B, F, I, K, L M b) points other than those in (a)6. Diagrams with only even numbers can be drawn starting at any point.
Diagrams with two odd numbers can be drawn starting from one of the odd points.Other diagrams cannot be drawn.
ST(P) Mathematics 1A – Teacher’s Notes and Answers 31
3. a) Yes, from P, finishing at CYes from C, finishing at P. Not possible from any other point.
b) no4. a) yes b) yes c) no
EXERCISE 13f (p. 198)1.
2 a) b)
c)
ST(P) Mathematics 1A – Teacher’s Notes and Answers 32
EXERCISE 13g (p. 200)
1. a) David b) no, son c) sister d) grandfather
4. a) the relationship works both ways b) yes5. a) Philip and Martin are cousins b) Sarah is not a cousin of either Philip or Martin
c)
CHAPTER 14 Area
Plenty of class discussion is advisable before finding areas of specific objects: e.g. What is“area”? Why is area counted in squares and not in triangles? The number of squares may varybecause it is not always easy to say whether more than half a square is included.
EXERCISE 14a (p. 202)
1. 11 5. 26 8. a) A b) B 11. 50 14. 762. 16 6. 20 9. 45 12. 40 15. 623. 11 7. 21 10. 43 13. 37 16. 264. 20
1. a) 30 000 b) 120 000 c) 75 000 d) 820 000 e) 85 0002. a) 1400 b) 300 c) 750 d) 2600 e) 32503. a) 560 b) 56 0004. a) 4 b) 25 c) 0.5 d) 0.25 e) 7.345. a) 0.55 b) 14 c) 0.076 d) 1.86 e) 29706. a) 7.5 b) 0.43 c) 0.05 d) 0.245 e) 176
EXERCISE 14i (p. 215)Pupils will benefit from using a calculator.
Negative numbers as coordinates are introduced in this chapter. Some teachers may preferfirst to introduce negative numbers in general, in which case Chapter 17 should be takenbefore this one.
EXERCISE 16a (p. 237)Nos. 10–21 can be used for discussion.
1. A (2,2), B (5,2), C (7,6), D (4,5), E (7,0), F (9,4), G (0,8), H (5,8)
EXERCISE 16b (p. 241)This and the next exercise use positive coordinates to investigate some of the properties ofthe special quadrilaterals. The questions are not difficult but this section can be omitted at afirst reading.
1. a) 8, 8, 8, 8, b) DC, yes c) 90º2. a) AB and DC, BC and AD b) AB and DC, BC and AD c) 90º3. a) all equal b) AB and DC, BC and AD c) A = C, B = D4. a) AB and DC, BC and AD b) AB and DC, BC and AD c) A = C, B = D5. a) none b) AB and DC c) none
EXERCISE 16c (p. 243)
ST(P) Mathematics 1A – Teacher’s Notes and Answers 36
EXERCISE 16g (p. 250)Omit this exercise if Exercise 16b and Exercise 16c were not covered. This exerciseinvestigates the properties of the diagonals of the special quadrilaterals and can be omitted,although the questions are not difficult.
1. a) parallelogram c) no d) both e) no2. a) square c) yes d) both e) yes3. a) trapezium c) no d) neither e) no4. a) rhombus c) no d) both e) yes5. a) rectangle c) yes d) both e) no6. rectangle, square7. rhombus, square8. parallelogram, rectangle, rhombus, square
ST(P) Mathematics 1A – Teacher’s Notes and Answers 37
The two algebra chapters should be done in their entirety only by above average abilitygroups, but all pupils can have some introduction to equations at this stage. We havesuggested some convenient stopping places. Equations are dealt with again in Book 2A.
ST(P) Mathematics 1A – Teacher’s Notes and Answers 40
5. 3 14. 5 23. 1 32 31. 2 7
3 39. 56. 0 15. 2 24. –1 5
4 32. 3 40. 76
7. 6 16. 3 25. 21 33. 2 5
1 41. –38. 5 17. 3 26. 4 34. –6 42. –19. –1 18. 0
This is a convenient stopping place for average ability groups.
EXERCISE 18g (p. 271)Good questions to discuss with above average ability groups but only the most able childrenshould be allowed to work through these on their own.
EXERCISE 18i (p. 275)A lot of discussion is necessary to get over the idea of “a term of an expression” and what ismeant by “like terms” and “unlike terms”.
1. a) no b) no2. Yes, measurements needed. Lengths on the drawing are not correct. 3. no
EXERCISE 19l (p. 289)
1. and 2. lines are the correct length3. a) lines are the correct length c) no d)one vertex is hidden behind another4. a) and b) lines are the correct length
ST(P) Mathematics 1A – Teacher’s Notes and Answers 43
EXERCISE 19m (p. 291)
2. a) (i) 2 (ii) 2 (iii) 4cm by 3cmb) e.g.
3. a) 6 b) two faces 1cm by 4cm, two 2cm by 1cm, two 4cm by 2cm4. b) IJ c) K and G5. a) IH b) B and D6.
7.There are a large number of arrangements of six squares and of these, 11 will make cubes.(Count reflections as the same.)
CHAPTER 20 Vectors
This unit is optional. It can be done later (it is repeated with different exercises in Book 3) oromitted completely. If a brief introduction is thought appropriate, Exercise 20a and Exercise20b form a good start.Some pupils may suggest the need to state a time in the initial paragraph (p. 294); this can bedealt with if it arises but need not be introduced otherwise.
EXERCISE 20a (p. 294)
1. scalar 2. vector 3. scalar 4. scalar 5. vector
EXERCISE 20b (p. 295)
ST(P) Mathematics 1A – Teacher’s Notes and Answers 44
1. a) b = 2a b) c = –a c) d = 3a d) e = a e) b = 2e f) d = –3c
2. a =
− 24
b =
−−
32
c =
−−
64
d =
32
e =
− 48
f =
−2
4 g =
96
h =
−4
8
e = 2a, f = –a, h = –2a, c = 2b, d = –b, g = –3b, h = –e, g = 3d, h = 2f, …
3.
128
,
−−
64
,
32
7.
210
,
−−
15
,
315
,
−−
420
4.
− 42
,
−8
4,
− 84
8.
−0
6,
04
,
−0
10,
08
5.
− 810
,
−4
5,
−1215
9.
−4
6,
−1218
,
− 23
,
−8
12
6.
63
,
−−
126
,
126
10.
−−
6018
,
8024
,
−−
103
,
10030
EXERCISE 20f (p. 303)
1.
−17
5.
010
9.
87
13.
107
17.
−−
42
2.
−2
86.
34
10.
− 46
14.
010
18.
−−
25
ST(P) Mathematics 1A – Teacher’s Notes and Answers 45
3.
− 47
7.
−−
66
11.
96
15.
−11
119.
−5
8
4.
62
8.
63
12.
117
16.
−10
220.
00
EXERCISE 20g (p. 306)
1. a)
57
b)
57
c)
68
d)
68
e)
64
f)
96
g)
910
h)
910
2. a)
23
b)
23
c)
− 50
d)
− 50
e)
−12
6 f)
−−
1220
3. a)
105
b)
2418
c)
2412
4. a)
−−
119
b)
−114
EXERCISE 20h (p. 307)
1.
35
5.
12
9.
−−
23
13.
105
16.
32
2.
60
6.
23
10.
−3
714.
− 54
17.
−11
3
3.
42
7.
911
11.
42
15.
−14
18.
−7
11
4.
−1
58.
85
12.
−11
19. a)
21
b)
−−
21
20. a)
−−
46
b)
−−
33
c)
33
21. a)
28
b)
199
c)
− 220
d)
1110
e)
30
22. a)
−18
3 b)
−0
3 c)
83
d)
− 230
e)
−22
4
23. a)
−105
b)
−14
17 c)
−1420
CHAPTER 21 More Algebra
This work should be done only with above average ability children and even then it can beleft until alter. The work in this chapter is repeated in Book 2A.
ST(P) Mathematics 1A – Teacher’s Notes and Answers 46
Multiplication of directed numbers: can be introduced in many ways. When this work isdone with average ability children they will probably benefit from a more practical approach.
4. y5 18. 5a²b² 32. 6z²5. s3 19. 3 x z x z 33. 24x²6. z6 20. 2 x a x b x c 34. 16x7. a x a x a 21. 4 x z x y x y 35. 4s³8. x x x x x x x 22. 6 x a x a x b 36. x6
9. b x b 23. 2 x x x x x x 37. y²z²10. a x a x a x a x a 24. 3 x a x a x a x a x b x b 38. 10xyz11. x x x x x x x x x x x 25. 6xz 39. a7
12. z x z x z x z 26. 6x³ 40. 8x4
13. 2a 27. 12a² 41. axyz14. 4x² 28. 6a³ 42. s7
EXERCISE 21i (p. 318)
1. 2 8. 24
2y 15. 79 or 7
21 21. 242r 27. b3
20
2. 522 or 5
24 9. 102c 16. 2 22. 2
5z 28. 13. 8
5 10. 6 17. 4 23. a32 29. 4
ay
4. 62z 11. 4
2x 18. 52 24. 1 30. x
y2
5. 103ab 12. 1 19. y
c23 25. 4
x 31. b4
6. 34 or 3
11 13. 52 20. z10
3 26. 47 or 4
31 32. yx
32
7. 3 14. 67 or 6
11
EXERCISE 21j (p. 320)
1. x = 5 3. 13 5. 4 x a x a 7. 2x – 1 8. x = 02. 4x – 11 4. x = –4 6. x = 3
11
EXERCISE 21k (p. 320)
ST(P) Mathematics 1A – Teacher’s Notes and Answers 48
1. x = – 21 3. 12 5. x = 12 7. 5x + 6y 8. x = 3
2. –2x + 15 4. 60abc 6. 1
EXERCISE 21l (p. 320)
1. x = 2 3. 6 + x + 12 = 4x; x = 6 5. 4 – x 7. 6x + 4 8. –2x + 102. a6 4. x = –3 6. 5
4
EXERCISE 21m (p. 321)
1. x = –3 3. x = 1 5. x x x x x x x x x 7. 5 – x2. 5
8 or 531 4. x + x + 2 + 8 = 18; £4 6. 4x –6
8. We get 3 = 0 which cannot be true (This problem can be used to discuss ∞.)
CHAPTER 22 Statistics
EXERCISE 22a (p. 322)If a copy of the table is made then each item in the table can be crossed out once it has been“counted”. The answers give the frequencies in each group.