CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE Learner’s Book answers 1 Numbers and the number system Getting started 1 There is no unique answer but possible answers are: −5, −2, 1, 4, . . . is the only one that includes negative numbers. 9, 12, 15, 18, . . . is the only one that includes multiples of 3. 5 a subtract 5 c subtract 100 a six hundred and one b two hundred and ninety-nine c one hundred and eleven 3 a 364 b 909 aExample: 30, 33, 36, 39, . . . (make sure they know they can use numbers outside the 3 × table). 4 a 562 = 500 + 60 + 2 b 305 = 300 + 5 b Example: 1 , 4 , 7 , 10 , . . . 5 a 160 b 10 c Not possible as the sequence must be odd, even, odd, even, etc. d Example: 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, . . . 2 b 4 add 10 Exercise 1.1 1 a 1046 b 948 d 8999 e −1 +2 1 2989 1 3 5 7 9 3 5 7 9 11 5 7 9 11 13 7 9 11 13 15 9 11 13 15 17 Possible answers are: 2, 4, 6, 8, . . . and 2, 5, 8, 11, . . . both have a first term of 2 but 3, 5, 7, 9, . . . has a first term of 3. 2, 4, 6, 8, . . . and 3, 5, 7, 9, . . . both have a term-to-term rule of ‘add 2’ but 2, 5, 8, 11, . . . has a term-to-term rule of ‘add 3’. 2, 5, 8, 11, . . . and 3, 5, 7, 9, . . . both have a second term of 5 but 2, 4, 6, 8, . . . has a second term of 4. 1 2 1 2 1 2 6 No, together with an explanation: They could keep subtracting 3, but it would take a very long time and they are quite likely to make errors. You might encourage them to think about multiples of 3 (3, 6, 9, 12, . . .). If the sequence ended at 0 it would have to include multiples of 3. 397 ÷ 3 leaves a remainder, therefore Abdul is not correct. 7 Linear sequences: +2 2 3 c 1 2 a Add five – the next term is 19. b Subtract four – the next term is 8. All of these sequences have a term-to-term rule that generates successive terms with the same difference between them. Non-linear sequences: The other sequences have different differences between successive terms and are therefore non-linear sequences: c Add one more each time: 2, 3, 5, . . . ­Differences are 1, then 2 so the next ­difference will be 3 giving 8 as the fourth term. Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE Multiply by three: 2, 6, 18, so the next term is 18 × 3 which is 54. e Subtract one less each time: 50, 41, 33, . . . Differences are 9, then 8, so the next ­difference will be 7 giving 26 as the fourth term. 1 2 25 025 Divide by 2: 32, 16, 8, so the next term is 8 ÷ 2 which is 4. 3 a 805 469 = 800 000 + 5000 + 400 + 60 + 9 b 689 567 = 600 000 + 80 000 + 9000 + 500 + 60 + 7 c 508 208 = 500 000 + 8000 + 200 + 8 f 8 Exercise 1.3 d The next term in the pattern is: 4 a nine hundred thousands b fifty thousands The largest 5-digit number is 99 999. One hundred thousand is 100 000. 100 000 – 99 999 = 1 so Bruno is correct. 5 This represents 16; the sequence shows square numbers. 6 a 8, 9, 10(27 ÷ 3 = 9 which is the middle number) b 3, 4, 5, 6, 7(25 ÷ 5 = 5 which is the middle number) 7 a 670 b 4 c 36 d 4150 e 35 f 3500 606 × 10 = 6060 Answers should include: Same: digits 6 and 6 remain. Exercise 1.2 Different: all digits change their place value. 1 a −4 2 a A = −5 B = −2 C = 3 D = 5 b B −6 c d 0 −3 Think like a mathematician a The numbers are: 15, 24, 33, 42, 51 3 a = −8, b = −2, c = 11 114, 123, 132, 141, 213, 222, 231, 312, 321, 411 4 −6 ° C 5 −4 ° C 1113, 1122, 1131, 1212, 1221, 1311, 2112, 2121, 2211, 3111 6 ANTARCTICA 7 −5 ° C is colder than −4 ° C. Marcus has not taken any notice of the negative signs. He should place his numbers on a number line to help him correct the mistake. b The largest number is 111 111 c The smallest number is 15. a 2° C b −4 ° C 1 430 d –3 ° C e 4° C 2 520 3 Any justified answer, for example: 8 11 112, 11 121, 11 211, 12 111, 21 111 111 111 c Think like a mathematician 1° C Check your progress 6, 8, 10,12, . . . and 1, 3, 5, 7, . . . both have a term-to-term rule of ‘add 2’ but 8, 11, 14, 17, . . . has a term-to-term rule of ‘add 3’. Learners’ posters based on their own investigations. 2 = 30 All the other missing numbers are 300. Think like a mathematician b × 100 = 3000 so 4 −32 °C 5 a 335 271 b 105 050 c 120 202 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 6 aThree hundred and seven thousand two hundred and one. b c Five hundred and seventy seven thousand and six. Seven hundred and ninety thousand three hundred and twenty. Exercise 2.2 1 45 minutes 2 10 hours 3 09:00 to 09:25 is 25 minutes but 09:25 to 10:00 is 35 minutes. 7 C: 1000 + 606 + 4 = 1610 4 a 7 hours b 45 minutes 8 55 500 5 a 51 minutes b 1.17 p.m. or 13:17 9 a 540 ÷ 10 = 54 b 307 × 10 = 3070 6 83 years c 60 × 100 = 6000 d 3400 ÷ 100 = 34 7 a 20 minutes b 15 minutes 4 °C b −10 °C c 2 hours 10 a Think like a mathematician 2 Time and timetables a a years c minutes b 2 9.15 and quarter past 9 3 3.05 4 a b months hours 30 c 2 a 180 b 330 c 49 d 36 e 54 f 450 g 5 60 4 Missing values are: 4 weeks 5.07 p.m. 2 11:45 3 15:30 is the only time that must be an afternoon time. 4 a C c 15:30 is the only 24-hour digital time. hours 1 3 3 months 1 Exercise 2.1 b b Check your progress Getting started 1 5 days d 12 60 a 5 years c 12 weeks b 3 days 5 40 minutes 6 a Monday b 3 November c 25 November d 1 and 15 November 7 a 10 minutes b 8.55 a.m. 8 a 12 minutes b 35 minutes c The 15:13 bus 7.15 a.m. 9.45 p.m. 3.20 p.m. 5 11.45 a.m. 6 17:10 7 06:00 and 18:00 8 Correct answer is 21:00. Ava added 10 to the hours. She should have added 12 to the hours. Think like a mathematician 3 3 Addition and subtraction of whole numbers Getting started a Ten past one or one ten. 1 78 b Other possible times: 02:20 05:50 10:01 11:11 12:21 21:12 22:22 15:51 20:02 2 86 3 94 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 4 Less than 10 Think like a mathematician Greater than 10 Even 8 12 Odd 7 13 25 4 3 Exercise 3.1 1 a 44 b 16 c 22 d 24 e 13 f 14 b 63 c 9 2 55 + 45 = 100 3 a 89 5 7 6 Think like a mathematician Findings may include: 4 100 450 450 Answers across the two diagonals are always the same. Answers are always even. 300 200 250 400 Smallest possible answer (1 in top left-hand corner) is 18. Largest possible answer (31 in bottom right-hand corner) is 46. 350 Exercise 3.2 5 a 6 a b 28 + 72 = 100 55 = 70 − 15 25 55 5 b 25 216 b 595 c 278 d 336 2 Rajiv is correct. See the Teacher’s Resource for different ways of explaining the answer. 3 86 chairs 4 340 g 5 606 stamps 6 111 7 The largest 2-digit number is 99. Exercise 3.3 50 4 a 99 + 99 = 198 which has 3 digits. 100 7 1 Any three numbers that sum to 10, for example, 2, 5 and 3 1 Own examples. 2 even 3 Own examples. 4 Own examples. 5 Martha is adding even numbers. Even + even + even = even Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 6 7 Counter-example (sufficient to show that the statement is not always true), e.g. 1 + 5 = 6 and 6 is even, or 4 Probability General case: Salem is not correct. If you add 5 to an even number the answer is odd, but if you add 5 to an odd number the answer is even. Getting started Counter-example (sufficient to show that the statement is not always true), e.g. 5 − 3 = 2 or General case: Heidi is not correct because: odd − odd = even 1 a It will not happen b It will happen c It might happen 2 a B 3 Spinner A. Learners may find it helpful to use number counters on a grid. Odd comes from odd + odd + odd or odd + even + even. The various solutions must conform to this pattern: E O E O O O O O E O E O O O O 1 2 5 a Certain b No chance c Poor chance d Good chance e Even chance b No chance Learners’ own answers. 3 Total Heads Tails O 5 Check your progress red Exercise 4.1 4 O c blue There is a greater chance of getting a red spin on spinner A because it has fewer equally likely outcomes than spinner B. Or The section for red is larger on spinner A so there is a greater chance of the pointer landing on it. Think like a mathematician Even comes from even + even + even or odd + odd + even. b 11 9 a Poor chance c Certain Spinner with 4 approximately equal sections coloured red, blue, yellow and purple. No section coloured green. 1 42 2 78 Think like a mathematician 3 76 chairs a There is a poor chance of rolling a 3. 4 466 books b There is no chance of rolling a 7. 5 26 + 34 or 24 + 36 c There is an even chance of rolling an odd number. 6 69 d 7 Own examples showing the sum of three even numbers is even. The chance of rolling a number less than 10 is certain. 8 No. Counter example, e.g. 3 + 4 = 7 which is odd. Check your progress 1 a False b False c False d True e False Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 2 There is an even chance of flipping a tail. Tally Total Head IIII IIII IIII IIII IIII III 28 Tail IIII IIII 32 IIII IIII IIII IIII II 5 1 box containing 50 apples 2 boxes containing 25 apples 5 boxes containing 10 apples 50 boxes containing 1 apple 25 boxes containing 2 apples 10 boxes containing 5 apples 6 18, 27, 36, 45, 90 Other answers are possible. 5 Multiplication, multiples and factors 7 57 + 7 = 64 57 + 15 = 72 57 + 23 = 80 etc. 8 Getting started 1 × 1 5 factors of 30 10 10 50 100 20 5 5 25 50 10 2 2 10 20 4 1 1 5 10 2 5, 10, 15, 20 3 9 4 39 × 3 + 39 × 7 = 39 × (3 + 7) = 39 × 10 = 390 57 with any method shown. 5 2 10 2 5 6 8 7 Think like a mathematician 3 × 4 = 4 × 3 = 12 3 × 5 = 5 × 3 = 15 3 × 6 = 6 × 3 = 18 4 × 5 = 5 × 4 = 20 4 × 6 = 6 × 4 = 24 5 × 6 = 6 × 5 = 30 Exercise 5.2 Exercise 5.1 6 factors of 40 1 17 2 21 and 42 3 1, 2, 4, 8, 16, 32 4 The dates for Saturdays are 6, 13, 20 and 27. Bruno is right because: 6 + 1 = 7 which is 1 × 7 13 + 1 = 14 which is 2 × 7 20 + 1 = 21 which is 3 × 7 27 + 1 = 28 which is 4 × 7 1 2 2 Any reason as the methods are essentially the same. They both use factors 4 = 2 × 2 and 15 = 5 × 3. Other valid methods can be based on the worked example. 3 a 235 b 116 c 267 d 444 a 47 × 3 = 4 × 40 7 3 120 21 120 + 21 = 141 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE b 93 × 4 = × 4 c d 90 9 3 360 + 12 = 372 360 12 51 × 5 = × 50 1 5 250 5 4 80 7 123 615 345 1725 567 2835 6 2D shapes 320 + 28 = 348 Getting started 320 28 1 5 280 a rectangle hexagon 6 380 cents or $3.80 2 pentagon 7 32 × 5 = 160 3 8 a 696 b 903 Shape C is not a hexagon because it has 7 sides and 7 vertices and a hexagon has 6 sides and 6 vertices. c 567 d 952 4 This pentagon is regular because it has 5 equal length sides and 5 equal angles. 5 a yes b no d yes e yes Think like a mathematician Answer: 897 × 3 = 2691 Check your progress 7 OUT 250 + 5 = 255 87 × 4 = × IN b c triangle c yes Exercise 6.1 1 7 × 8 = 56 or 8 × 7 = 56 2 Fatima is not correct. Multiples of 5 end in 5 or 0. 3 3200 4 3 and 4, 5 and 6, 8 and 9 5 6 16 × 2 × 5 16 × 2 × 5 = 32 × 5 = 16 × 10 = 160 = 160 Igor chose the better method. 6 × 2 × 15 6 × 2 × 15 = 12 × 15 = 6 × 30 = 180 = 180 Ingrid chose the better method. 75 and 30 7 The factors of 16 are 1, 2, 4, 8 and 16. 8 The factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 20 are 1, 2, 4, 5, 10 and 20. 16 has an odd number of factors because it is a square number. a 632 b 3852 c 1169 1 2 a The four triangles make a square. b The two pentagons make a hexagon. c The four triangles make an irregular quadrilateral (parallelogram). aAny shape with at least one right angle, for example, a square. b Any shape with at least one curved side, for example, a semicircle. c Any shape with at least one pair of parallel sides, for example, a rectangle. d Any shape with at least seven vertices, for example, an octagon. e Any shape that is not a polygon, for example, a circle. 3 a no 4 a hexagons b squares and triangles c octagons and squares d squares, hexagons and octagons 5 b yes c yes d no Yes, all the triangles tessellate. Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE Think like a mathematician Check your progress a 1 2 triangles 2 quadrilaterals (trapeziums) 1 square and 1 rectangle 1 triangle and a pentagon 1 triangle and 1 quadrilateral (trapezium) b–e Learners’ own investigations. Exercise 6.2 1 2 3 One possible answer: a 2 b 4 c 1 d 4 e 0 f 2 g 4 h 0 a 2 b 0 c 4 d 1 e 2 f 1 g 4 h 1 a 1 b 1 c 1 d 0 3 Drawing of hexagon tessellating. e 4 f 5 g 0 h 4 4 a yes b yes c no 5 a 4 b 3 c 2 6 a 6 b 2 c 1 d 0 e 1 4 B, D, F 5 The parallelograms that have diagonal lines of symmetry all have all four sides the same length. a No b C and E c 7 d yes 8 B 7 Fractions Think like a mathematician 8 2 Shape Name Sides Vertices Lines of symmetry Getting started A Regular (equilateral) triangle 3 3 3 Short version of answers (see teacher guide for fuller information). B Regular 4 quadrilateral (square) 4 4 C Regular pentagon 5 5 5 D Regular hexagon 6 6 6 E Regular heptagon 7 7 7 F Regular octagon 8 8 8 G Regular nonagon 9 9 9 H Regular decagon 10 10 10 1 Parts must be equal in size 2 1 4 3 1 6 < 1 3 Draw on the same diagram to compare different fractions. Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE Exercise 7.1 1 1 3 2 1 8 3 B1 4 5 and 1 but accept 1 , 1 , 1 , 1 , 1 5 7 8 9 10 6 or 1 11 instead of 3 1 3 Parts are equal in area so each part is a quarter of a whole. 6 Same: 4 parts each part is a quarter of the whole 3 3 3 3 12 8 6 4 Individual answers. Exercise 7.2 4 1 2 a 3 a 3 4 b 1 8 b 9 1 = $12 2 1 = $3 8 1 4 of 80 = 20 1 5 of 50 = 10 Think like a mathematician 27 (from 1 of 27 = 9) Check your progress 1 A 2 5 6 3 10 c 1 = $8 3 $32 4 1 1 1 1 , , , 6 5 4 3 For unit fractions, the larger the denominator 4 quantity, divide the quantity by four. 6 1 5 7 a 8 12 16 6 3 = $24 4 The larger the denominator the more the smaller the fraction. To find 1 of a 1 = $6 4 1 = $8 4 1 parts the fraction is divided into, making each part smaller. 5 1 = $16 2 7 10 1 2 0 10 $24 1 = $4 6 1 = $4 8 of 60 = 30 3 $4 2 4 3 Think like a mathematician 1 3 of $15 = $5 and 1 of $24 is $6 so I would choose 1 of $24. 7 Different: parts are a different shape 9 Finding one-third is equivalent to dividing by 3 and so on. 6 B 1 Finding one-half is equivalent to dividing by 2. 1 6 In each case, the fractions are acting as operators. 5 7 Yes. 9 25 b 30 c 15 24 No, 1 of $30 = $15 and 1 of $60 = $20 2 3 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE Think like a mathematician 8 Angles Maryam could draw a line through an obtuse angle that gives a right angle and an acute angle or an obtuse angle and an acute angle, so she won’t always end up with two acute angles. Getting started 1 Right angle 2 B and E 3 a 4 Exercise 8.3 b c 0 d 1 3 Exercise 8.1 1 a D 2 a c b c G True b True True d False E 3 L, J, K 4 r, p, t, q, s 5 Angles A and B are the same size. The lines for angle A are longer, but that does not mean that the angle is greater. You could convince Sam by tracing one of the angles and placing it on top of the other angle to check they are the same. 1 a 2 After four right angles you are facing in the same direction as when you started. You have turned a full circle. 3 a Estimate between 20 and 40 degrees. b Estimate between 70 and 89 degrees. a Estimate between 100 and 120 degrees. b Estimate between 150 and 170 degrees. 4 6 No answer (talking activity). 270 c 360 Either 10 degrees or 20 degrees because using the decision tree the angle is between 0 degrees and 45 degrees, and using the angle diagram we can tell it is much smaller than 45 degrees. 6 Carly could use the decision tree to work out that the angle is between 135 and 180 degrees. By using the angle diagram she could see that the angle is much closer to 135 degrees than 180 degrees so a better estimate would be closer to 135 degrees. Think like a mathematician Estimate between 110 and 125 degrees for the first angle. Estimate between 65 and 80 degrees for the second angle. Exercise 8.2 1 Independent learner activity. Check your progress 2 a acute b obtuse 1 B c obtuse d right angle 2 D, G, C, F, E e acute 3 4 a An angle drawn less than 90 degrees. 4 b An angle drawn between 90 and 180 degrees. 5 An obtuse angle is between 90 degrees and 180 degrees. or An obtuse angle is larger than a right angle, but smaller than a straight line. Estimate between 60 and 80 degrees. 6 Estimate between 160 and 179 degrees. 3 4 A right angle is an angle of 90 degrees. An acute angle is smaller than 90 degrees. An obtuse angle is greater than 90 degrees and smaller than 180 degrees. 10 b 5 Think like a mathematician Many possible answers. For example, 1 minute past 12 o’clock would create one of the smallest acute angles; 1 minute past 6 o’clock would create one of the largest obtuse angles. (Note: Learners are not expected to use the words ‘acute’ or ‘obtuse’.) 180 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 9 Comparing, rounding and dividing 7 Think like a mathematician Getting started The ones digit forms a repeating pattern in every case, for example: 1 17 ÷ 4 and 21 ÷ 5 • 2 18 3 145 4 a 216 > 126 c 216 < 226 149 150 153 • b 226 > 216 • 1 a 50 000 b 20 000 c 50 000 2 a 100 000 b 900 000 c 200 000 3 a 5000 b 5200 c 5210 4 335, 336, 337, 338, 339, 340, 341, 342, 343, 344 5 a 5500 b 5500 c 5000 d 5500, 6000 e The answers are different, 5000 and 6000. 6 645 123 < 645 213 7 a 2228 5895 6194 6962 8848 b 2200 5900 6200 7000 8800 Think like a mathematician a A = 5500 E = 5505 B = 5050 C = 5045 Remainder 1 when divided by 4: 5, 9, 13, 17, 21, 25, 29, 33, 37, . . . Ones digit: 5 → 9 → 3 → 7 → 1 → 5 (repeat) Remainder of 1 when divided by 5: 6, 11, 16, 21, 26, . . . Ones digit 6 → 1 → 6 (repeat) Exercise 9.1 Remainder of 1 when dividing by 6: 7, 13, 19, 25, 31, 37, 43, . . . Ones digit: 7 → 3 → 9 → 5 → 1 → 7 (repeat) Learners may go on to investigate other numbers and patterns. Check your progress 1 a 16 787, 16 976, 32 622, 48 150, 150 966 b 17 000, 17 000, 33 000, 48 000, 151 000 2 43 3 6162, 6164, 6166, 6168 4 42 ÷ 6 = 7 because all the other answers are 8. 5 11 melons 10 Collecting and recording data D = 5455 b–c Round to 5000: 5046, 5047, 5048, 5049 Round to 5100: 5051, 5052, 5053, 5054 Getting started Exercise 9.2 1 a Shoe size Tally Total 5 weeks 30 I 1 2 5 packs 31 II 2 3 a 32 II 2 4 33 II 2 Remainder 1: 25 ÷ 3, 7 ÷ 3, 1 ÷ 3 another example: 31 ÷ 3 (any correct answer) 34 IIII I 6 Remainder 2: 20 ÷ 3, 23 ÷ 3, 14 ÷ 3, 2 ÷ 3 7 boats another example 32 ÷ 3 (any correct answer) 35 III 3 36 III 3 37 I 1 1 5 6 11 Any valid reason that matches their chosen method. 14 b 16 c 12 d 14 4 bags (the last bag will only have 3 apricots in it) b 6 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 2 Check your progress a–b Answers depend on the class. Exercise 10.1 b 1 1 a 2 a–c Individual investigations. 3 a–c Individual investigations. 32 c Cerys 113 a Number of biscuits b 3 c 3 2 4 7 8 9 10 11 12 Number of biscuits 3 1 2 3 4 5 6 a Completed sentences. b Table drawn with labelled heading columns, e.g. Number of names, Number of people. Number of books 5 11 Fractions and percentages Getting started 3 4 5 6 7 8 9 10 Scores in the spelling test 6 a 4 b d Completed sentence, beginning ‘The dot plot shows that . . .’ 0 c 4 Think like a mathematician 1 1 3 5 7 8 8 8 8 2 3 5 3 6 10 4 A and C 8 12 Answers depend on learners’ choices in collecting, representing and interpreting the data. For example, they might find that Book 1 is easier to read than Book 2 because Book 1 has only 1 word with more than 8 letters and Book 2 has 5 words with more than 8 letters on the pages shown. Individual investigations. 5 Exercise 11.1 1 1 3 = = 6 3 = 9 = 18 4 12 24 4 12 24 2 1 5 = 2 10 3 3 4 Learners’ own tables/preferences. 7 >1 and 3 = 30 10 100 9 12 out OR are equivalent so 3 4 and odd one out 4 3 6 or 7 35 = 10 50 4 6 4 8 = 4 10 5 10 4 6 is left is the odd one both contain 4 so 9 12 is the 6 12 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 5 1 3 1 5 3 7 4 8 2 8 4 8 5 3 = 30 = 6 10 100 20 6 1 4 6 a 15% b 30 7 The larger the denominator the smaller the parts, so eighths are smaller than quarters. Three quarters are equivalent to six eighths 7 a 42% b 70% 8 2% 3 8 < < 5 16 <3 8 Think like a mathematician There are 9 different fractions: Getting started 1 3 1 3 = 1 = 4 1 = 5 1 = 2 = 3 9 2 6 2 8 2 10 3 6 1 2 1 2 2 4 2 6 3 6 = 5 = 10 3 = 6 = = 4 8 3 9 4 8 Exercise 11.2 1 a 2 a 75% 3 50% 4 a 25% b 75% 5 a 30% b 70% 6 a 50% b 1 2 7 75% b 25% Bigger than 1 4 c c 1 4 3 4 than 80% 14% 50% 85% 2 triangles 4 25% 6 Exercise 12.1 3 4 aA triangular prism has 3 rectangular faces and 2 triangular faces. b A cuboid has 6 rectangular faces and 0 triangular faces. c A square-based pyramid has 1 rectangular faces and 4 triangular faces. d A cone has 0 rectangular faces and 0 triangular faces. a 7 b 2 pentagons and 5 rectangles 3 1 hexagon and 6 triangles 4 a 5 aTwo of the possible common properties are: A and C have the same number of vertices, or B and E are both pyramids. 6 b 3 4 Sets are: and not 4 13 b 4 C – sphere 50% <3<5 3 4 A and B have 3 of the shape shaded. 4 A – cuboid 2 Check your progress 3 3 Bigger 35% 9 1 3 a E – triangular prism 8% 45% 2 2 1 but smaller than 8 2 3 6 B – cylinder 74% 1 1 D – square-based pyramid Smaller than b 9 12 Investigating 3D shapes and nets 3 4 1 4 so 3 is the odd one out. 1 2 = =3=4 5 10 15 20 8 b B and E c 5 c d 12 d A, C and D 8 B and E 6 Descriptions of hexagonal prism. May include that it has 8 faces, 18 edges and 12 vertices. 7 a 6 d For example: Cubes and cuboids both have 6 rectangular faces. Cubes need to have 6 square faces, but cuboids do not. b 0 c 2 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE Think like a mathematician 5 Hexagon-based pyramid No 3D shapes with 1 to 5 straws. 6 a Triangular prism b Rectangle 6 straws – tetrahedron 7 straws – no shape 13 Addition and subtraction 8 straws – square-based pyramid 9 straws – triangular prism 10 straws – pentagon-based pyramid 11 straws – no shapes 12 straws – cuboid (including cube), hexagonbased pyramid, octahedron Exercise 12.2 1 a Cuboid b 2 a Hexagonal prism b 8 c 2 hexagons and 6 rectangles 6 3 C 4 A 5 a Cone b Tetrahedron c Cylinder d Square-based pyramid c 6 rectangles 1 a 743 b 107 2 a 207 b 225 3 a 395 b 684 4 a 3 5 b 2 8 5 a 4 5 b 5 8 a 1245 b 1632 c 1134 2 a 333 b 245 c 48 3 889 4 256 5 a882 − 435 = 447. The student has always subtracted the smallest digit from the largest digit. A hexagon-based pyramid has 7 faces. An octagon-based pyramid has 9 faces. The number of faces is one more than the number of sides of the base shape of the pyramid. c–d For prisms: The number of faces is two more than the number of sides of the base shape of the prism. 6 Learners’ spoken explanations. 6 b 531 + 278 = 809. The student has forgotten to add the carrying digit. a 1173 b 381 Think like a mathematician The answer is 1110 for all lines that pass through 5 in the middle of the array. Two of the other lines give palindromic numbers (they read the same forwards or backwards): 741 + 147 = 888 123 + 321 = 444 Check your progress 14 4 1 A heptagon-based pyramid has 8 faces. b or 1 Exercise 13.1 Think like a mathematician a Getting started Exercise 13.2 1 8 2 1 pentagon and 5 triangles 3 A tetrahedron has 6 edges, 4 vertices and 4 faces. 4 A 1 5 6 2 a 1 4 b 7 8 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 3 a 1 1 + =1 2 2 2 1 + =1 3 3 1 3 + =1 4 4 6 2 + =1 8 8 5 4 + =1 9 9 5 2 + =1 7 7 3 3 + =1 6 6 4 1 + =1 5 5 3 712 − 486 = 226. The student has always ­subtracted the smallest digit from the ­largest digit. Then write each addition as two ­subtractions. b 1 3 + =1 4 4 1 7 + =1 8 8 1 1 + =1 2 2 6 3 + =1 9 9 2 1 + =1 3 3 2 4 + =1 6 6 2 5 + =1 7 7 3 2 + =1 5 5 456 + 352 = 808. The student has forgotten to add the carrying digit. 4 506 is 500 to the nearest hundred. 789 is 800 to the nearest hundred. 800 + 500 is 1300. Leroy’s answer of 1295 is close to 1300 so his answer is reasonable. 5 a 6 A and C are correct 3 = Then write each addition as two subtractions. 4 10 9 a b 5 3 7 c d 4 12 10 9 4 9 5 9 1 2 + 5 5 Think like a mathematician 1 1 2 +2=1 3 4 1 2 3 1 4 +4=1 Check your progress 1 345 2 455 biscuits 7 1 +1=1 2 2 1 2 does not equal 3 . 10 It is not correct to add the denominators. b is incorrect as the denominators have been added. Fatima is correct. Parveen has incorrectly added denominators. 1 4 6 10 1 2 6 + = 5 5 10 9 1 2 5 2 8 So Yuri should add the numerators but not the 1 2 b 3 6 = 5 10 denominators to give 5 . 7 12 8 1 2 3 + = 5 5 5 6 9 1 9 3 9 6 3 5 The student did not work out an approximation for either calculation. 2 7 9 + = 3 + 6 = 9 4 + 5 = 9 8 8 8 8 8 8 8 8 8 14 Area and perimeter Getting started 7 8 1 8 7 1 8 +8=1 1 The perimeter is 16 cm. 2 Rectangle drawn 4 cm by 6 cm. 3 The area of the blue shape is 5 square units. 4 The area of the red shape is 8 square units. Exercise 14.1 1 a A Between 14 cm and 22 cm B Between 12 cm and 24 cm C Between 6 cm and 12 cm b A 18 cm B 18 cm C 9 cm 15 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 2 3 a 5000 m b c i 12 m ii 1200 cm a m2 b cm2 c km2 d mm2 60 mm c Answer dependent on learners’ investigations. The smallest possible area with whole square centimetres is 9 cm2. 9 a c 16 cm2 29 cm2 4 15 km2 Check your progress 5 14 cm2 1 20 m 6 Jo will get the best estimate. Lee’s estimate will be too low. Zaid’s estimate will be too high. 2 90 mm 3 18 km2 or 18 1 km2 4 5 Rectangle with area of 10 cm2 drawn, e.g. 5 cm by 2 cm. Perimeter matches the rectangle drawn, e.g. 14 cm. a 18 cm2 b 16 cm2 c 18 cm2 6 66 m2 Exercise 14.2 1 aRectangle accurately drawn with sides 2 cm and 5 cm. b 2 c 14 cm 10 cm2 aRectangle accurately drawn with sides 6 cm and 3 cm. b 18 cm c 18 cm2 3 a 15 cm2 b 24 cm2 4 a 16 cm2 b 21 cm2 5 a 6 7 8 2 15 Special numbers Getting started length width area 3a 5 cm 3 cm 15 cm2 1 6 and 12 b 6 cm 4 cm 24 cm2 2 16 4a 4 cm 4 cm 16 cm2 b 7 cm 3 cm 21 cm2 3 6, 12, 18 and 24 4 a −5 ° C, −2 ° C, −1 ° C, 0 ° C, 1 ° C, 3 ° C b −5 ° C, −2 ° C, −1 ° C, 0 ° C b We have found out that the area is equal to the length multiplied by the width. a 14 cm2 b 24 cm2 c 36 cm2 d 35 cm2 There are three other rectangles that can be drawn using whole numbers of centimetres: 1 cm by 24 cm; 2 cm by 12 cm; and 4 cm by 6 cm Learners should discover that Lila’s method works, as long as the two sides measured meet at a vertex. 5 −3 6 A, C, D Exercise 15.1 1 −7, −6, −5, −2, −1 2 a −4 ° C b −4 ° C, −2 ° C, −1 ° C, 3 ° C a −18, −12, −6, 0, 6, 12 b The numbers count on in 6s; they are ­multiples of 6. c 121 will not be in the pattern because it is odd, and 6, which is even, is repeatedly added onto the even numbers in the pattern so the terms will always be even. 4 a < 5 −4 or −3 3 Think like a mathematician 16 b 16 cm2 a Possible rectangles include: 4 cm by 4 cm, 3 cm by 5 cm, 2 cm by 6 cm, 1 cm by 7 cm. b Answer dependent on learners’ investigations. The largest possible area with whole square centimetres is 25 cm2. b > c < d > Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 6 −11 or −10 or −9 or −8 or −7 or −6 or −5; −3 or −2; any number greater than −1. 2 6 42 48 284 7 −4 ° C < −2 ° C < 0 ° C < 5 ° C 3 The units digit is 0, 2, 4, 6 or 8. Anton is not correct. Any number ending in 5 is divisible by 5 but numbers ending in 0 are also divisible by 5. 4 a 1250 1050 6700 b 525 1250 1050 6775 a any number divisible by 10 Exercise 15.2 b any number divisible by 10 1 c any number divisible by 100 6 a 6105 7 48 − 23 = 25 Think like a mathematician −1, −3, −5, −7, −9 or −9, −7, −5, −3, −1 The pattern is that the numbers have a difference of 2. 5 564 2 multiples of 2 14 11 10 b 1065 89 − 64 = 25 multiples of 4 91 − 66 = 25 16 12 13 Think like a mathematician Hexagon maze 15 Maze 1: 4 36 and 64 5 Every number with a factor of 6 must also have factors of 1, 2 and 3 in any order. 2 → 5 → 60; 2 → 5 → 80; 14 → 15 → 20; 14 → 15 → 70; 18 → 15 → 20; 6 133 7 Ingrid is not correct. Number 14 ends in 4 but it is not a multiple of 4. 18 → 15 → 70; 18 → 15 → 20; 18 → 20 → 50; 18 → 20 → 50; 18 → 15 → 70; 18 → 20 → 90; 10 → 25 → 60; 10 → 25 → 40; 10 → 5 → 60; 10 → 5 → 80 32 c 35 d 18 → 20; a 33 b 2 → 5; 14 → 15; 18 → 15; 10 → 25; 10 → 5 3 30 Maze 2: Think like a mathematician Multiples Check your progress You may need to suggest that learners start by making lists of multiples. 1 There are various solutions, so learners could compete to see who can use the most cards. It is possible to use all ten cards to make multiples of 3: 3, 9, 12, 45, 60 and 78. a 100 and 700 b 100, 350 and 700 c 10, 60,100, 350, 530 and 700 d 5, 10, 60, 100, 125, 305, 350, 530 and 700 2 7536 Exercise 15.3 3 −8 or −9 1 a 100, 300, 700 4 a b 10, 40, 100, 300, 530, 650, 700 5 c all of them: 5, 10, 25, 40, 100, 300, 530, 650, 700 No. Square numbers have an odd number of factors, for example, the factors of 9 are 1, 3 and 9. 26 b 27 c 24 d 25 Multiples of 100 are also multiples of 5 and multiples of 10. 17 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 2 6 factors of 24 8 36 12 6 multiples of 3 1 7 23 3 21 11 13 27 9 17 19 24 6 15 29 12 26 18 25 30 8 22 14 2 16 5 10 28 4 multiples multiples 20 of 5 of 2 multiples of 4 24 40 16 Data display and interpretation 3 a even Getting started 1 Red Not red Triangle 2 red scalene 2 triangles that triangles are not red Not triangle 2 2 shapes that are neither red nor triangles aAny three even numbers that are also multiples of 3, e.g. 6, 12, 18. b 3 2 red shapes that are not triangles Any three odd numbers that are not multiples of 3, e.g. 5, 7, 11. a 3 b worm c ant and beetle d 5 Exercise 16.1 5 1 curly hair Norman Adith Yutu glasses Filip Petra earrings 10, 20 not a multiple of 10 2, 4, 6, 8, 1, 3, 5, 7, 12, 14, 9, 11, 13, 16, 18 15, 17, 19 There cannot be any number in the section for multiples of 10 that are not even because all multiples of 10 are even. a 24 c 2.00 p.m. to 2.15 p.m. d 2 f Between 9.00 a.m. and 9.15 a.m. was ­busiest. 46 vehicles passed the school between 9.00 a.m. and 9.15 a.m. 43 vehicles passed the school between 2.00 p.m. and 2.15 p.m. 46 is greater than 43. a Singer 1 b 0 c 3 d 26 e Answers include: Both the adults and children voted singer 1 and singer 4 as their top two favourites. f Answers include: The children voted singer 2 as their least favourite, but the adults voted singer 5 as their least favourite. g Answers include: I think that the data for the children’s vote and the data for the adult’s vote is different because people of different ages like different music. Tapu Antonella multiple of 10 b Sophie Sun 18 4 not even b 1 e 46 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 6 Check your progress a 8 b No. It is possible that all of the households in the group 4 to 6 had 4 people, and then there would be 17 households of 4 people, but we cannot tell from the data. 7 Graph 2 shows the data most clearly because the vertical scale best matches the data. 8 a b pictogram For example: Number of children Class 1 maths scores 1 2 aShape A is a regular, green quadrilateral (square). Shape B is a regular quadrilateral (square), but is not green. Shape C is a green shape that is not a quadrilateral and is not a regular polygon. 8 6 4 Shape D is not a quadrilateral, it is not green and it is not a regular polygon. 2 0 1 2 Score 3 4 5 3 Class 2 maths scores 12 10 8 b Shape H a 17 b 1 c 47 words d 51 words e Possible answer: Both books have no words that are 1 letter long. f Possible answer: Book 1 does not have words that are 9 or 10 letters long, Book 2 does. g Possible answer: I think that Book 2 might be written for older children who can read more words and read some longer words, and Book 1 might be written for ­younger children who are not experienced readers. 6 4 2 c 14, 28 not a 1, 3, 5, 9, 11, 2, 4, 6, 8, 10, multiple 13, 15, 17, 19, 12, 16, 18, 20, of 7 23, 25, 27, 29 22, 24, 26, 30 10 0 not odd multiple 7, 21 of 7 12 0 Number of children odd 0 1 2 Score 3 4 5 Possible answer: The data for Class 1 and Class 2 show that 2 children in each class scored 2 marks. d Possible answer: The data for Class 1 and Class 2 show that no children scored 0 in Class 2, but 2 children scored 0 in Class 1. e Possible answer: I think that maybe the children in Class 2 are older than the ­children in Class 1 so they have learnt more maths, which means they could answer more ­questions correctly. Think like a mathematician 4 Pictogram showing the number of badges each child has Kevin Todd Tia Raquel Amanda Individual posters. Key: 19 = 4 badges Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE 17 Multiplication and division Exercise 17.2 Getting started 1 380 2 2916 3 19 4 12 r3 5 4 packs 6 128 ÷ 8 = 16 flowers 16 × 7 = 112 hexagons Exercise 17.1 1 96 months 2 A = 135 3 4 Estimate 400 × 6 = 2400 Calculated answer 2448 g 2304 5 3852 6 IN D = 297 615 345 1725 567 2835 4 8 $1672 9 No. Sometimes a 3-digit number multiplied by a 1-digit number gives a 4-digit answer, for example, 124 × 9 = 1116. But sometimes the answer has only 3 digits, for example 124 × 4 = 496. Think like a mathematician 121 4338 3 482 14 teams 3 9 cartons 4 12 friends 5 86 ÷ 3 = 28 r280 ÷ 3 = has a remainder which must be added to the ones before dividing the ones by 3. 6 57 ÷ 3 = 19Learners have worked from R to L and not L to R. 7 7 a 2 b 3 c The divisor = remainder + 1 28 cm 9 415 10 ÷ 2 = 5 12 ÷ 3 = 4 12 ÷ 4 = 3 10 ÷ 5 = 2 18 ÷ 6 = 3 14 ÷ 7 = 2 14 ÷ 2 = 7 18 ÷ 3 = 6 20 ÷ 4 = 5 20 ÷ 5 = 4 30 ÷ 6 = 5 21 ÷ 7 = 3 16 ÷ 2 = 8 21 ÷ 3 = 7 28 ÷ 4 = 7 30 ÷ 5 = 6 42 ÷ 6 = 7 28 ÷ 7 = 4 18 ÷ 2 = 9 24 ÷ 3 = 8 32 ÷ 4 = 8 40 ÷ 5 = 8 54 ÷ 6 = 9 42 ÷ 7 = 6 27 ÷ 3 = 9 56 ÷ 7 = 8 63 ÷ 7 = 9 16 ÷ 8 = 2 18 ÷ 9 = 2 36 ÷ 4 = 9 24 ÷ 8 = 3 27 ÷ 9 = 3 32 ÷ 8 = 4 36 ÷ 9 = 4 40 ÷ 8 = 5 54 ÷ 9 = 6 56 ÷ 8 = 7 63 ÷ 9 = 7 72 ÷ 8 = 9 72 ÷ 9 = 8 Check your progress 2905 7 Multiply the two numbers on the bottom row to give the number on the top row. 20 2 Think like a mathematician 7 363 6 packs 8 OUT 123 1 1 Yes, because 400 × 6 = 2400 2 105 plants (15 × 7 = 105) 3 a 4 7 (all the numbers are multiples of 7) 15 r3 b 558 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE One square South-West, one square SouthEast, one square South-East, one square South-West 5 2905 and 12 r1 6 a 7 12 bags 8 Missing numbers are: 8, 72, 84, 28 b 9 15 One square South-East, one square SouthWest, one square South-West, one square South-East One square South-East, one square SouthWest, one square South-East, one square South-West 18 Position, direction and movement Getting started 5 North 1 West East One square South-East, one square SouthEast, one square South-West, one square South-West A (0, 2), B (2, 1), C (3, 3), D (5, 6) 6 The coordinate marked on the grid is (1, 4). Safiya has used the wrong numbers for the horizontal axis and the vertical axis. 7 y-axis 6 South 2 D 3 B 5 4 3 4 2 1 0 North-East 2 SW / South-West 3 a Burd b i South-East ii North-East 1 2 3 4 5 6 x-axis Think like a mathematician Exercise 18.1 1 0 1–3, 5, 6, 8 and 9 Individual answers. 4 The second number is the same for all the coordinates on the line. 7 The first number is the same for all of the coordinates on the line. iii South-West 4 One square South-West, one square SouthWest, one square South-East, one square South-East One square South-West, one square SouthEast, one square South-West, one square South-East 21 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021 CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE Exercise 18.2 Check your progress 1 1 East, North-East, South, South-West 2 y-axis 6 5 4 3 2 a 2 1 0 3 0 1 2 3 4 5 6 x-axis Octagon 4 b 3 a A rectangle B irregular four-sided polygon (rhombus) C hexagon b Learners’ sketches to check predictions. 4 The lines are horizontal. The lines are parallel. 5 The lines are vertical. The lines are parallel. 6 (6, 2), (6, 5), (5, 2), (5, 5) 5 (1, 4), (4, 4), (5, 2), (2, 2) Think like a mathematician A reflected shape has the same area as the original shape. 22 Cambridge Primary Mathematics 4 – Wood & Low © Cambridge University Press 2021