KS3 Maths Now Textbook

201 15.2 1 Copy each of these capital letters and write down its order of rotational symmetry. a b c d e 2 Write down the order of rotational symmetry for each shape. a bc d e 3 Write down the order of rotational symmetry for each shape. a b c d e 4 State the order of rotational symmetry of each capital letter. a bc de f 5 Copy and complete the table for each of these regular polygons. a b c d e Shape Number of lines of symmetry Order of rotational symmetry a Equilateral triangle b Square c Regular pentagon d Regular hexagon e Regular octagon What do you notice? 6 Describe the rotational symmetry of each playing card shown below. a d b c © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 201 2867_P001_680_Book_MNPB.indb 201 24/03/2020 15:32 4/03/2020 15:32

202 Chapter 15 Reason mathematically Joe said the word OXO has rotational symmetry. Is he correct and, if so, what is the order of rotational symmetry? He is correct, the order of rotational symmetry is 2. 7 These patterns are from Islamic designs. Write down the order of rotational symmetry for each pattern. a b c 8 Copy these shapes and shade parts in each one so they have rotation symmetry of order 2. a b c 9 Copy each shape onto squared paper. Shade extra squares to give the shape rotation symmetry, order 4. Mark the centre of rotation. a b c 10 Eve said her name had rotational symmetry. Her friend Ziz said Eve’s name didn’t have rotational symmetry, but his name did. Who is correct? Explain your answer. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 202 2867_P001_680_Book_MNPB.indb 202 24/03/2020 15:32 4/03/2020 15:32

203 15.3 Solve problems Shade in some small triangles on this shape to give it: a rotational symmetry of order 1 b rotational symmetry of order 3. a Shade any small triangle to give order 1. b Shade in three small triangles to give order 3. Note that there are many other different ways to correctly answer these two questions. 11 Here are four identical right-angled triangles. 2 cm 4 cm Copy the four triangles and put them together to make a shape that has rotational symmetry of: a order 1 b order 2 c order 4. 12 Copy this grid. a Shade in one square to give the shape rotational symmetry order 1. b Shade in another square to give the shape rotational symmetry order 2. c Shade in more squares to give the shape rotational symmetry order 4. 13 Which capital letters have rotational symmetry higher than 1? 14 Write down four numbers that have rotational symmetry higher than 1. 15.3 Properties of triangles and quadrilaterals ● I can understand the properties of parallel, intersecting and perpendicular lines ● I can understand and use the properties of triangles ● I can understand and use the properties of quadrilaterals Any two lines either are parallel or intersect in a point. Lines that intersect at right angles are perpendicular. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 203 2867_P001_680_Book_MNPB.indb 203 24/03/2020 15:32 4/03/2020 15:32

204 Chapter 15 Triangles have three sides and three angles or vertices. Equilateral triangle (three equal sides) Isosceles triangle (two equal sides) Right-angled triangle Obtuse-angled triangle Scalene triangle (no equal sides) Quadrilaterals have four sides and four vertices. Square Trapezium Kite Arrowhead Rectangle Parallelogram Rhombus Develop fluency Describe the geometrical properties of the triangle ABC. You can see that AB = AC And that ∠ABC = ∠ACB. The triangle is isosceles. A B C 1 Match these triangles with the correct names. a b c d e equilateral right-angled obtuse-angled scalene isosceles 2 Describe the geometrical properties of the triangle ABC. A B C © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 204 2867_P001_680_Book_MNPB.indb 204 24/03/2020 15:32 4/03/2020 15:32

205 15.3 3 Describe the geometrical properties of an obtuse-angled triangle. 4 Describe the geometrical properties of a scalene triangle. 5 Describe the geometrical properties of a right-angled triangle. 6 Which quadrilaterals have the following properties? a four equal sides b two different pairs of equal sides c two pairs of parallel sides d only one pair of parallel sides 7 Describe the geometrical properties of the parallelogram ABCD. A B D C 8 Describe the geometrical properties of: a a square b a rhombus. 9 Describe the geometrical properties of: a a rectangle b a trapezium 10 Describe the geometrical properties of a kite. Reason mathematically Mandy said that a scalene triangle could also be isosceles. Is she correct? Explain your answer. No, Mandy is incorrect as a scalene triangle has three different sides and an isosceles triangle has two sides equal. 11 Explain the difference between: a a square and a rhombus b a rhombus and a parallelogram c a trapezium and a parallelogram. 12 A line that joins two vertices of a shape is called a diagonal. This quadrilateral ABCD has two diagonals, AC and BD. A B C D Name the quadrilaterals in the diagram below in which the diagonals: i are equal in length ii bisect each other iii are perpendicular to each other. a d g b c e f © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 205 2867_P001_680_Book_MNPB.indb 205 24/03/2020 15:32 4/03/2020 15:32

206 Chapter 15 13 Neil looked at this 3 by 3 pin-board and said: ‘I cannot use the dots to draw an equilateral triangle, only an isosceles triangle.’ Is Neil correct? Explain your answer. 14 Gena looked at this 3 by 3 pin-board and said: ‘I can use the dots to draw a parallelogram but not a kite.’ Is Gena correct? Explain your answer. Solve problems Draw: a an obtuse-angled scalene triangle a b an obtuse-angled isosceles triangle. b 15 How many distinct quadrilaterals can be constructed on this 3 by 3 pin-board? Draw each one and write down what type of quadrilateral it is. 16 Draw: a a scalene right-angled triangle b an isosceles right-angled triangle. 17 Draw a kite and then show how you can draw a line to create: a two isosceles triangles b two scalene triangles. 18 Show how you can cut an equilateral triangle to create four equilateral triangles. Now I can... recognise shapes with reflective symmetry find the order of rotational symmetry of a shape understand and use the properties of triangles recognise shapes that have rotational symmetry understand the properties of parallel, intersecting and perpendicular lines understand and use the properties of quadrilaterals draw lines of symmetry on a shape © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 206 2867_P001_680_Book_MNPB.indb 206 24/03/2020 15:32 4/03/2020 15:32

207 16 Solving equations 16.1 Finding unknown numbers ● I can find missing numbers in simple calculations You can use algebra to write expressions and equations in which letters represent unknown numbers. You can use these equations to find the value of the unknowns. Develop fluency a Work out the value of the letter a if a + 4 = 20. b Work out the value of the letter b if 2b + 4 = 20. a a + 4 = 20 What number do you add to 4 to make 20? The answer is 16. So, a = 16. b 2b + 4 = 20 In this case the answer is 8 because 2 × 8 + 4 = 20. So, b = 8. For questions 1–9, work out the number that each letter represents. 1 a 5 + x = 7 b 6 + y = 12 c 5 + d = 19 d 9 + g = 31 2 a 2v = 12 b 2u = 28 c 3t = 36 d 4r = 80 3 a a – 2 = 6 b b – 7 = 9 c c – 3 = 11 d d – 1 = 9 4 a a + 6.1 = 9 b r + 8.5 = 9 c w + 1.8 = 8 d t + 1.1 = 9 5 a 4x = –12 b 4y = –20 c 7z = –7 d 8w = –56 6 a a – 2 = –1 b b – 8 = –4 c c – 2 = –2 d d – 3 = –1 7 a 9 + a = 45 b 7 + r = 31 c 9 + w = 23 d 12 + t = 75 8 a 3 – e = –2 b 3 – f = –3 c 4 – g = –2 d 4 – h = –3 9 a x 6 = 2 b y 6 = 10 c 12 t = 3 d 18 w = 6 Reason mathematically In this number wall, the number in each brick is the sum of the numbers in the two bricks below it. Work out the values of a, b and c. 20 11 b 3 a c The top number is always the sum of the two numbers below it. 11 + b = 20, so b must be 9. In the same way, 11 = 3 + a so a = 8. Finally, a + c = b, so 8 + c = 9. So, c = 1. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 207 2867_P001_680_Book_MNPB.indb 207 24/03/2020 15:32 4/03/2020 15:32

208 Chapter 16 10 In this number wall, the number in each brick is the sum of the numbers in the two bricks below it. Work out the values of a, b and c. 11 Work out the values of d, e and f in this number wall. 12 This number wall has four rows and three missing numbers. Work out the values of a, b and c. Solve problems Molly thinks of a number. She subtracts 5 and then multiplies by 3. The answer is 24. a Call Molly’s number m. Write down an expression to show what Molly did. b Find the value of m that makes the expression equal to 18. a If you subtract 5 from m you get m – 5. If you multiply this by 3 you get 3(m – 5). The brackets show that you do the subtraction first. b Now you want to find the value of m that makes 3(m – 5) = 18. You can see that m – 5 = 6, so m must be 11. 13 Jason thinks of a number. He multiplies it by 5. Then he adds 3. a Call Jason’s number j and write down an expression to show what Jason did. Find the value of j, if the answer is: b 13 c 23 d 38. 14 Nina thinks of a number. She adds 6 then multiplies by 2. a Call Nina’s number k and write down an expression to show what Nina did. Find the value of k, if the answer is: b 18 c 24 d 40. 15 I think of a number. I subtract 4 from my number. a Call my number a and write down an expression to show what I did. Find the value of a, if the answer is: b 9 c 17 d 35. 16 Kahlen thinks of a number. She doubles the number and adds 3. a Call Kahlen’s number k and write down an expression to show what Kahlen did. Find the value of k, if the answer is: b 15 c 11 d 53. 25 12 b 5 a c 25 16 e 10 d f 24 a 11 b 7 4 c 4 3 1 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 208 2867_P001_680_Book_MNPB.indb 208 24/03/2020 15:32 4/03/2020 15:32

209 16.2 16.2 Solving equations ● I can understand what an equation is ● I can solve equations involving one operation Develop fluency When you find the value of the unknown in an equation, you are solving the equation. Solve these equations. a x + 17 = 53 b 5y = 45 a x + 17 = 53 This means ‘a number + 17 = 53’. You could try to guess the answer, but it is easier do it by subtraction. x = 53 – 17 The number is 53 – 17. = 36 b 5y = 45 This means ‘5 × a number = 45’. y = 45 ÷ 5 The number is 45 ÷ 5. = 9 Solve these equations. 1 a x + 3 = 11 b x + 5 = 13 c x – 7 = 12 d x – 9 = 0 2 a 3x = 12 b 2m = 18 c 4x = 16 d 5t = 25 3 a a 2 = 6 b b 5 = 8 c x 12 = 2 d y 4 = 20 4 a a +16 = 41 b r + 18 = 32 c w + 13 = 54 d t + 11 = 60 5 a 4v = 92 b 8u = 96 c 5t = 95 d 3r = 84 6 a a – 12 = 8 b b – 18 = 4 c c – 12 = 12 d d – 13 = 12 7 a 9 + a = 5 b 7 + r = 3 c 9 + w = 2 d 12 + t = 7 8 a 9 – a = –3 b 1 – b = –2 c 1 – c = –3 d 1 – d = –6 9 a x 6 = 42 b y 3 = 17 c 42 t = 3 d 28 w = 4 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 209 2867_P001_680_Book_MNPB.indb 209 24/03/2020 15:32 4/03/2020 15:32

210 Chapter 16 Reason mathematically a Write down an expression for the perimeter of this triangle, in centimetres. b The perimeter of the triangle is 100 cm. Write an equation to express this fact. c Solve the equation. a The perimeter is s + 45 + 32 = (s + 77) cm. b The perimeter is 100 cm so the equation is s + 77 = 100. c Solving the equation means finding the value of s. s + 77 = 100 Then s = 100 – 77 = 23 s cm 45 cm 32 cm 10 a Write down an expression for the perimeter of this triangle. b Suppose the perimeter of the triangle is 37 cm. i Write down an equation involving f. ii Solve the equation to find the value of f. c Suppose the perimeter of the triangle is 45 cm. i Write down an equation involving f. ii Solve the equation. 11 a Write down an expression for the area of this rectangle. b Suppose the area of the rectangle is 88 cm². i Write down an equation involving r. ii Solve the equation. c Suppose the area of the rectangle is 120 cm². i Write down an equation involving r. ii Solve the equation. 12 a Show that an expression for the number in the top brick of this number wall is x + 48. b Suppose the number in the top brick is 54. i Write down an equation involving x. ii Solve the equation to find the value of x. c Suppose the number in the top brick is 65. i Write down an equation involving x. ii Solve the equation to find the value of x. d Suppose the number in the top brick is 90. i Write down an equation involving x. ii Solve the equation to find the value of x. 16 cm f cm 9 cm 8 cm r cm x 18 12 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 210 2867_P001_680_Book_MNPB.indb 210 24/03/2020 15:32 4/03/2020 15:32

211 16.2 13 a Work out an expression for the number in the top brick of this number wall. Write it as simply as possible. b Suppose the number in the top brick is 50. i Write down an equation involving y. ii Solve the equation. c Suppose the number in the top brick is 100. i Write down an equation involving y. ii Solve the equation. d Suppose the number in the top brick is 86. Work out the value of y. Solve problems The area of this rectangle is 91 cm2 . What is the length y? The area is 7 × y = 7y cm². The area is 91 cm² so we have the equation 7y = 91. Then y = 91 ÷ 7 = 13 7 cm y cm 14 a Write down an expression for the sum of the angles in this quadrilateral. 120° x 65° 105° b The angles in a quadrilateral add up to 360°. Find the value of x. 15 Work out the value of x in this number wall. 100 x 15 10 8 Try to find an equation. 16 A rectangle has sides of 4 cm and h cm. The area of the rectangle is 28 cm2 . Write an equation involving h and solve it. 17 A triangle has angles of 70°, 55° and z°. Write down an equation involving z and solve it. 32 5 y © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 211 2867_P001_680_Book_MNPB.indb 211 24/03/2020 15:32 4/03/2020 15:32

212 Chapter 16 16.3 Solving more complex equations ● I can solve equations involving two operations Some equations, such as 2n + 9 = 17, have two operations. This example includes the two operations multiplication and addition. To solve equations like this, you need to use inverse operations. Develop fluency Solve these equations. a 4a – 9 = 35 b 3a + 8 = 41 c 3(k + 2) = 27 d d 4 – 6 = 14 a 4a – 9 = 35 4a = 35 + 9 First add 9 to both sides. 4a = 44 a = 44 ÷ 4 Now divide both sides by 4. a = 11 You can see that to ‘undo’ a subtraction you do an addition. To ‘undo’ a multiplication you do a division. These are inverse operations. b 3a + 8 = 41 3a = 33 First subtract 8 from both sides. a = 33 ÷ 3 Now divide both sides by 3. a = 11 Here, to ‘undo’ an addition, you do a subtraction. c 3(k + 2) = 27 k + 2 = 9 First divide both sides by 3: 27 ÷ 3 = 9 k = 7 Then subtract 2 from both sides: 9 – 2 = 7 In this case, because of the brackets, you do the division first. The inverse of ‘multiply by 3’ is ‘divide by 3’. The inverse of ‘add 2’ is ‘subtract 2’. d d 4 – 6 = 14 d 4 = 20 First add 6 to both sides: 14 + 6 = 20 d = 80 Then multiply both sides by 4: 20 × 4 = 80 The inverse of ‘divide by 4’ is ‘multiply by 4’. Solve each equation. 1 a 2x + 5 = 13 b 4m + 3 = 19 c 2m + 17 = 33 d 7x + 5 = 19 2 a 2x – 5 = 13 b 4m – 5 = 15 c 2m – 17 = 13 d 7x – 2 = 19 3 a 2(x + 1) = 10 b 3(y + 4) = 21 c 4(z + 4) = 40 d 2(w + 3) = 30 4 a 3(x – 1) = 12 b 4(y – 4) = 20 c 3(z – 4) = 27 d 5(w – 3) = 55 5 a 4(8 – x) = 16 b 5(3 + y) = 30 c 4(15 – z) = 28 d 6(4 + w) = 54 6 a x 3 + 4 = 10 b x 5 – 1 = 2 c x 4 – 3 = 2 d m 3 – 2 = 1 7 a 3x + 4 = 19 b 3(x + 4) = 30 c m 4 + 3 = 12 d 5n – 7 = 23 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 212 2867_P001_680_Book_MNPB.indb 212 24/03/2020 15:32 4/03/2020 15:32

213 16.3 8 a 3(x + 1) = –6 b 4(y + 4) = –12 c 5(z + 4) = –40 d 3(w + 3) = –9 9 a 5(x – 1) = –10 b 6(y – 4) = –18 c 5(z – 4) = –25 d 7(w – 3) = –56 10 a 6(8 – x) = 60 b 7(3 + y) = –35 c 2(15 – z) = 36 d 8(4 + w) = –48 Reason mathematically In this pentagon, four of the sides are the same length. a Write an expression for the perimeter of the pentagon, in terms of a. b The perimeter of the rectangle is 98 cm. Write down an equation involving a. c Solve the equation to find the value of a. a The perimeter is a + a + a + a + 30 = 4a + 30 cm. b If the perimeter is 98 cm, then 4a + 30 = 98. c 4a + 30 = 98 4a = 68 Subtract 30 from both sides: 98 – 30 = 68 a = 17 Divide both sides by 4: 68 ÷ 4 = 17 30 cm a cm a cm a cm a cm 11 a Write down an expression for the perimeter of this hexagon. The perimeter of the hexagon is 152 cm. b Aaran said: ‘The value of x is 18.’ Is Aaran correct? Explain your answer 12 In this octagon, six sides are the same length. a Write down an expression for the perimeter of the octagon. b The perimeter of the octagon is 214 cm. Write down an equation for this. c Show that the longest side is 38 cm. 13 Joy thinks of a number, she doubles it and adds 15 to get 99. Theo said the number was 84. Is Theo correct? Explain your answer. 14 Ben was asked his age. He said: ‘If I add 12 to my age and multiply by 5, then I get 100.’ How old is Ben? 40 cm x cm x cm x cm x cm 40 cm y cm y cm y cm y cm y cm y cm y + 15 cm y + 15 cm © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 213 2867_P001_680_Book_MNPB.indb 213 24/03/2020 15:32 4/03/2020 15:32

214 Chapter 16 Solve problems a Work out an expression involving x for the number in the top brick of this number wall. b The top number is 21. Write an equation and solve it to find the value of x. a The expressions on the second row are x + 8 and x + 3. The expression in the top brick is x + 8 + x + 3. This simplifies to 2x + 11. b The equation is 2x + 11 = 21. 2x = 21 – 11 First subtract 11 from both sides. ‘Subtract 11’ is the inverse of ‘add 11’. 2x = 10 x = 10 ÷ 2 Then divide by 2. ‘Divide by 2’ is the inverse of ‘multiply by 2’. x = 5 8 x 3 15 For each of these number walls: i use the number in the top brick to write an equation involving x ii solve the equation to find the value of x. Part a has been done for you. a 21 x + 4 x + 7 4 x 7 b 26 3 x 11 c 26 8 x 4 d 19 x 5 2x e 21 2x x 3x f 40 x x 7 16 For each of these number walls: i use the number in the top brick to write an equation involving x ii solve the equation to find the value of x. a b 31 x + 4 x 3 33 x + 5 x x + 4 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 214 2867_P001_680_Book_MNPB.indb 214 24/03/2020 15:32 4/03/2020 15:32

215 16.4 17 This is a page from Emma’s exercise book. She has made a mistake when solving each equation. Write down a correct solution to each problem. 18 The length of a rectangle is x cm longer than its width of 5 cm. a Write an expression for the length of the rectangle in terms of x. The area of the rectangle is 35 cm2 . b Write an equation involving w. c Solve this equation. d What is the perimeter of the rectangle? 16.4 Setting up and solving equations ● I can use algebra to set up and solve equations Many real-life problems can be solved by first setting up an equation. Develop fluency Mark thinks of a number, doubles it and subtracts 8. a Use m to represent Mark’s number. Write down an expression for the answer. b Mark’s answer is 42. Write down an equation and solve it to find Mark’s initial number. a Mark’s number is 2m – 8. b The equation is 2m – 8 = 42. This solves to 2m = 50. Giving Mark’s number as 25. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 215 2867_P001_680_Book_MNPB.indb 215 24/03/2020 15:32 4/03/2020 15:32

216 Chapter 16 1 Dan thinks of a number, doubles it and adds 5. a Use d to represent Dan’s number. Write down an expression for the answer. b Dan’s answer is 59. Write down an equation and solve it to find Dan’s initial number. 2 Mike has m coloured pencils. Jon has 12 fewer pencils than Mike. a Write down an expression, in terms of m, for the total number of pencils. b They have 48 pencils altogether. Write an equation and solve it to find the value of m. c How many pencils does each boy have? 3 Josie is 24 years older than her son Tom. a If her son is t years old, write down an expression for Josie’s age. b Their total age is 122 years. Write an equation, in terms of t, to show this. c Solve the equation to find the value of t. d How old is Josie? 4 Marie thinks of a number. She adds 14 then multiplies by 6. a Call Marie’s number m. Write down an expression for her answer. b Marie’s answer is 120. Write down an equation in terms of m. c Work out Marie’s original number. 5 There are y girls in a school. The number of girls is 28 fewer than the number of boys. There are 1176 students altogether. a Write down an equation, in terms of y, based on this information. b Work out the number of girls and the number of boys in the school. 6 Ollie thinks of a number. He multiplies the number by 6 and subtracts 11. a Use x to represent Ollie’s number. Write down an expression for the result. b Ollie’s result is 37. Write down an equation and solve it to find his initial number. 7 Andrew has nine more marbles than Amber. Between them they have 39 marbles. a Using m to represent the number of marbles that Andrew has, write down how many marbles Amber has. b Write down an expression for how many marbles they have between them. c Work out how many marbles Andrew has. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 216 2867_P001_680_Book_MNPB.indb 216 24/03/2020 15:32 4/03/2020 15:32

217 16.4 8 Joy i s fifteen years older than James. a Using a to represent Joy’s age, write down an expression, in terms of a, for James’s age. b Their combined age is 73 years. Write down an equation and use it to work out how old each is. 9 The teaching staff of a school is made up of 71 men and women. There are 19 more women than men. a Where w is the number of female teachers, write down an expression, in terms of w, to represent the number of male teachers. b Set up and solve an equation to work out the number of male teachers in the school. 10 Mae thinks of a number. She multiplies it by 8 then adds 27. a Where m is Mae’s number, write down an expression for her result. b Mae’s result is 83. Work out Mae’s original number. Reason mathematically The cost of renting a hostel is £25 per night plus £30 per person staying. a Write an expression for the cost, in pounds, for x people. b The bill for a group of people for one night is £415. Show how you can use this information to find out there were 13 people in the group. a The cost in pounds is 30x + 25. b Set up the equation 30x + 25 = 415. 30x = 415 – 25 = 390 x = 390 ÷ 30 = 13 There were 13 people in the group. 11 This is the plan of the floor of a room. The lengths are in metres. a Work out an expression for the area of this shape. b The area is 29 m². Write an equation to show this. c Solve the equation to find the value of a. Find the area of each rectangle. 12 On Monday k cm of snow fell. The snowfall on Tuesday was 5 cm less than on Monday. On Wednesday there was twice as much snow as on Tuesday. a Write an expression, in terms of k, for the number of centimetres of snow on Wednesday. b In fact there were 30 cm of snow on Wednesday. Write an equation for this. c Solve the equation to find the value of k. d How many centimetres of snow fell altogether on the three days? a 6 4 3.5 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 217 2867_P001_680_Book_MNPB.indb 217 24/03/2020 15:32 4/03/2020 15:32

218 Chapter 16 Solve problems Kate thinks of a number. First she subtracts 13, then she multiplies by 4. a Call Kate’s number k. Write an expression for her answer. b Kate’s answer is 72. Write an equation to show this. c Solve the equation to find Kate’s initial number. a After subtracting 13 from her number, Kate has k – 13. She then multiplies this by 4 to get 4(k – 13). The expression for Kate’s answer is 4(k – 13). b The equation for Kate’s answer is 4(k – 13) = 72. c 4(k – 13) = 72 k – 13 = 18 First divide both sides by 4: 72 ÷ 4 = 18 k = 31 Then add 13 to both sides: 18 + 13 = 31 Kate’s initial number is 31. 13 Mrs George said, ‘Think of a number, subtract 3 and then multiply the answer by 4. Tell me your answer.’ Alfie’s answer was 52 and Harry’s answer was 96. a Set up an equation and work out the number Alfie thought of. b Set up an equation and work out the number Harry thought of. Choose a letter for Alfie’s number. 14 Buns cost £b each. Coffees cost £1.79 each. a Ravi buys four buns and six coffees. Write down an expression for the total cost, in pounds. b The cost for Ravi is £19.34. Write down an equation to show this. c Work out the cost of one bun. 15 Gary said, ‘Robert Percy scored twelve more goals than Dwayne van Mooney last season!’ Mike answered, ‘Yes but between them they scored 56 goals.’ Set up and solve an equation to work out how many goals Robert Percy scored last season. 16 Louis has £8. He buys five ice-creams that cost c pounds each. He gets 75p change. Set up and solve an equation to work out the cost of one ice-cream. Now I can... find missing numbers in simple calculations solve equations involving one operation use algebra to set up and solve equations understand what an equation is solve equations involving two operations © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 218 2867_P001_680_Book_MNPB.indb 218 24/03/2020 15:32 4/03/2020 15:32

219 17 Using data 17.1 Interpreting pie charts ● I can work out the size of sectors in pie charts by their angles at the centre Pie charts show proportions in a set of data. They are a very good way to clearly show the distribution of the data. Develop fluency The pie chart shows the types of housing on a new estate. All together there were 540 new houses built. How many were: a detached b semi-detached c bungalows d terraced? You need to work out the fraction of 540 that each sector represents. a 90 360 × 540 = 135 detached b 120 360 × 540 = 180 semi-detached c 40 360 × 540 = 60 bungalows d 110 360 × 540 = 165 terraced Detached Semi-detached Terraced Types of housing Bungalows 90° 40° 120° 110° 1 All 900 pupils in a school were asked to vote for their favourite subject. The pie chart illustrates their responses. How many voted for: a PE b Science c Maths? 2 One weekend in Edale, a café sold 300 drinks. The pie chart illustrates the proportions of different drinks that were sold. Of the drinks sold that weekend, how many were: a soft drinks b tea c hot chocolate d coffee? 3 Priya did a survey about fruit and nut chocolate. She asked 30 of her friends. The pie chart illustrates her results. How many of Priya’s friends: a never ate fruit and nut chocolate b sometimes ate fruit and nut chocolate c said that fruit and nut was their favourite chocolate? Favourite subjects PE Science Maths 60° Café drinks Soft drinks Hot chocolate Tea 60° 120° Coffee Sometimes eat it 60° 120° Never eat it Favourite Fruit & Nut © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 219 2867_P001_680_Book_MNPB.indb 219 24/03/2020 15:32 4/03/2020 15:32

220 Chapter 17 4 In one week, a supermarket sold 540 kilograms of butter. The pie chart shows the proportions that were sold on different days. How many kilograms of butter were sold on: a Monday b Tuesday c Wednesday d Thursday e Friday f Saturday g Sunday? 5 The pie chart shows the daily activities of Joe one Wednesday. It covers a 24-hour time period. How long did Joe spend: a at school b at leisure c travelling d asleep? 6 On a train one morning, there were 240 passengers. The guard inspected all the tickets and reported how many of each sort there were. The pie chart illustrates his results. How many of the tickets that he saw that morning were: a open returns b season tickets c day returns d travel passes e super savers? 7 The pie chart shows the results of a survey of 216 children, when they were asked about their favourite foods. How many chose: a chips b pizza c pasta d curry? 8 150 children went on one of four school summer holidays. How many children chose: a camping b France c pony trek d Disney World? Butter sales Mon Tue Fri 40° 60° Sun 30° 30° 60° 50° Wed Thu Sat School Travelling Leisure 60° 120° Sleep What Joe did Open return Season ticket Day return 30° 45° Travel pass 45° Super saver Train tickets Pizza Curry Pasta 120° 75° Chips 75° Favourite food Disney World Camping France Pony trek 96° 36° 120° 108° © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 220 2867_P001_680_Book_MNPB.indb 220 24/03/2020 15:32 4/03/2020 15:32

221 17.1 Reason mathematically The pie chart shows the results of a survey of 180 pupils, when they were asked about their summer holiday. a What type of holiday was chosen by 45 people? b Tara said: ‘A third of them stayed in the UK for their holiday.’ Is Tara correct? Explain your answer a 45 is a quarter of 180, and 900 is a quarter of the pie chart, so a cruise was chosen by 45 pupils. b The UK selection is shown by a sector of 1200, this represents 120 360 = 1 3, so Tara is correct. 120á 30á 120á Abroad Cruise None UK Holidays 9 The pie chart shows the results of a survey of 144 children, when they were asked about their favourite TV programmes. Which statements are correct? Explain your answers. a Jess said one third of them chose comedy. b Just as many voted for Drama as soaps. c 90 children voted for Sci Fi. 10 The pie chart shows the results of a survey of 120 children, when they were asked about their favourite sports. a Which sport was chosen by 20 people? b Dara said: ‘Tennis is more popular than athletics.’ Is Dara correct? Explain your answer 11 A group of children were asked what their favourite colour was. The pie chart shows the results. a What colour was chosen by one quarter of the children? b Jenny said that three times as many people chose yellow than orange. Is Jenny correct? Explain your answer. Comedy Sci Fi Soaps 120° 75° Drama 75° Favourite TV Tennis Cricket Athletics 60° 120° Football Favourite games Favourite colour Blue Orange Yellow 60° © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 221 2867_P001_680_Book_MNPB.indb 221 24/03/2020 15:32 4/03/2020 15:32

222 Chapter 17 12 These pie charts show how two companies first contacted new customers. 120 ° Tech Net 15° 135 ° 60° Infoflow 480 customers 264 customers 45° 45° Letters Emails Mobile telephone calls Office telephone calls a Work out how many letters each company sent. b Which company sent the most emails? Explain your answer. c Make two more comparisons between the companies. Solve problems The pie chart shows how one country dealt with 3000 kg of dangerous waste in 2013. How much waste did the country dispose of by: a putting it into landfill b burning it c dumping it at sea d chemical treatment? a 1 2of the waste was put into landfill. 1 2of 3000 kg = 1500 kg So, 1500 kg of the waste was put into landfill. c The angle of this sector is 30°. Then the fraction of the circle is: 30 360 = 1 12 Therefore 1 12 of the waste was dumped at sea. 3000 kg ÷ 12 = 250 kg So, 250 kg of the waste was dumped at sea. b 1 4 of the waste was burned. 1 4 of 3000 kg = 3000 kg ÷ 4 = 750 So, 750 kg of the waste was burned. d The angle of this sector is 60°. Then the fraction of the circle is: 60 360 = 1 6 Therefore 1 6of the waste was treated by chemicals. 3000 kg ÷ 6 = 500 kg So, 500 kg of the waste was treated by chemicals. Disposing of dangerous waste Dumped in sea Landfilled Chemical treatment Burnt © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 222 2867_P001_680_Book_MNPB.indb 222 24/03/2020 15:33 4/03/2020 15:33

223 17.2 13 To motivate the pupils, a headteacher placed this pie chart on the school noticeboard. Emma decided to use the chart to find out how much money each year group had raised. Help Emma by estimating how much each year group raised. 14 Brendan saw this pie chart in a magazine. Northern Ireland W ales Scotland England UK reservoirs total 634 megalitres of w ater Northern Ireland 11° Wales 53° Scotland 58° England 238° He measured and recorded the angles of the pie chart in the table shown. Calculate how many megalitres of water there were in the reservoirs of each country. Give your answers correct to two significant figures. 15 A dog rescue centre provides refuge for four types of dog as shown in the pie chart. They currently have six collies. Estimate the angles for each sector and work out how many of each breed there are and how many dogs they have in total. 16 An international event was attended by people from five European countries as shown in the diagram. There were 240 Germans. Estimate how many people attended all together. 17.2 Drawing pie charts ● I can use a scaling method to draw pie charts Sometimes you will have to draw a pie chart to display data that is given in a frequency table. The size of the sector will depend on the frequency for that sector. You can use the scaling method to find the angle for each sector. Y7 Y8 Y9 Y10 Y11 Christmas charity collection total so far … £1690 Greyhoun d Lurcher Collie Whippet British Spanish French German Italian © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 223 2867_P001_680_Book_MNPB.indb 223 24/03/2020 15:33 4/03/2020 15:33

224 Chapter 17 Develop fluency Draw a pie chart to represent the data showing how a group of people travel to work. Set the data out in a frequency table and write the calculations in it. Now draw the pie chart. When drawing a pie chart, draw the smallest angle first and try to make the largest angle the last one you draw, then any cumulative error in drawing will not be so noticeable. Walk Bus 126° 72° Cycle 36° 48° 78° Car Train Travelling to work Sector (type of travel) Frequency Calculation Angle Walk 24 24 240 × 360° = 36° 36° Car 84 84 240 × 360° = 126° 126° Bus 52 52 240 × 360° = 78° 78° Train 48 48 240 × 360° = 72° 72° Cycle 32 32 240 × 360° = 48° 48° Total 240 360° 1 The results of a transport survey show the various ways students go school. copy the table and fill in the gaps. Then draw a pie chart to show the results. Transport Frequency As 120 pupils are represented by 360°, each pupil will be represented by 360 ÷ 120 = ° Car 23 23 × 3 = 69° Bus 17 17 × 3 = Train 25 25 × = Bicycle × 3 = Walk 40 × = Total 120 Check the total = 360° 2 Draw a pie chart to represent the numbers of birds spotted on a field trip. Bird Crow Thrush Starling Magpie Other Frequency 19 12 8 2 19 3 Draw a pie chart to represent the favourite subjects of 36 pupils. Subject Maths English Science Languages Other Frequency 12 7 8 4 5 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 224 2867_P001_680_Book_MNPB.indb 224 24/03/2020 15:33 4/03/2020 15:33

225 17.2 4 Draw a pie chart to represent the types of food that 40 people usually eat for breakfast. Food Cereal Toast Fruit Cooked Other None Frequency 11 8 6 9 2 4 5 Draw a pie chart to represent the numbers of goals scored by an ice hockey team in 24 matches. Goals 0 1 2 3 4 5 or more Frequency 34754 1 6 Draw a pie chart to represent the favourite colours of 60 Year 8 pupils. Colour Red Green Blue Yellow Other Frequency 17 8 21 3 11 7 a Draw a pie chart to represent the sizes of dresses sold in a shop during one week. Size 8 10 12 14 16 18 Frequency 3 7 10 12 6 2 b What size are a quarter of the dresses? c What fraction of dresses are size 16 or above? d What percentage are size 14? Reason mathematically Andrew was given data about the numbers of bottles of water sold at a test match. Day Thursday Friday Saturday Sunday Monday Number of bottles 190 350 440 410 30 He said: ‘I can’t draw a clear pie chart as the Monday angle will be too small.’ Comment on Andrew’s statement. He is wrong as the angle needed for Monday on the pie chart will be just under 8°, which although small will still show the comparison. 8 Trains arriving at Blackpool station were monitored to see how well they ran on time. The results are shown in the table. a Draw a pie chart to display the results. b How many trains were late? c What proportion were early? The railway promises that no more than 10% of trains will be more than 5 minutes late, and no more than 5% will be over 10 minutes late. d Write a brief report to explain whether or not they achieved this. Early 4 On time 18 Up to 5 minutes late 14 5 to 10 minutes late 3 Over 10 minutes late 1 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 225 2867_P001_680_Book_MNPB.indb 225 24/03/2020 15:33 4/03/2020 15:33

226 Chapter 17 9 Paul has to draw a pie chart to represent the eye colours in his school from the following data. Colour Blue Brown Green Hazel Grey Amber Frequency 192 130 20 8 9 1 Paul says: ‘The angle for Amber will be too small to draw, shall I combine some colours to make a category of “other colours” to make a clear pie chart?’ comment on Paul’s statement. 10 Helen is asked to draw a pie chart to represent the time she spent on activities in a day. Activity Sleeping Lessons Travelling Eating Playing Reading Time (hours) 9 5 1 25 2 Helen said: ‘The angle for travelling will be 15°, so the angle for sleeping will be 130°.’ Is Helen correct? Explain your answer. 11 There are 120 pupils in a year group. 12 of these pupils wear glasses. a The pie chart to show this is not drawn accurately. What should the angles be? Show your working. Exactly half of the 120 pupils in the school are boys. b From this information, is the percentage of boys in this school that wear glasses 5%, 6%, 10%, 20%, 50% or is it not possible to tell? Solve problems This is data about the number of siblings each pupil had in a school. a Why would it be extremely difficult to draw a pie chart to illustrate this information? b Combine some groups so that you could draw a pie chart to represent this information. Number of siblings Number of applicants None 323 One 672 Two 40 Three 3 Four 1 a The data for three and four siblings would require a very small angle to be drawn. b Number of siblings Number of applicants None 323 One 672 More than one 44 W ear glasses Do not wear glasses Who wears glasses © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 226 2867_P001_680_Book_MNPB.indb 226 24/03/2020 15:33 4/03/2020 15:33

227 17.2 12 A player entered an 18-hole golf tournament. The table shows how well she did on the first day (par is the target, birdie is one less, eagle is two less, bogey is one over par and double bogey is two over par). Eagle 1 Birdie 4 Par 11 Bogey 2 Double bogey 0 a Draw a pie chart to illustrate her results. b Was her overall score better than par or worse? By how much? c Par for one round on this course is 72. Major tournaments play four rounds of 18 holes. If she repeats these results each day, what would her total score be? 13 This is an estate agent’s waiting list for rental flats on one day in a large city. a Why would it be extremely difficult to draw a pie chart to illustrate this information? b Combine the groups so that you could draw a pie chart to represent this information. c Draw the pie chart. Type of flat Number of applicants 1 bedroom 646 2 bedrooms 1344 3 bedrooms 80 4 bedrooms 6 5 bedrooms 2 14 This diagram was shown in a business magazine. It illustrates the funds received by a charity one year. Redraw it as an accurate pie chart. 15 Match the pie charts with their equivalent bar charts. 1 2 3 4 20 0 40 60 80 100 120 A 10 0 20 30 40 50 60 B 5 0 10 15 20 25 30 C 5 0 10 15 20 25 30 D Donations £815000 37% Funds received Legacies £24 7000 11% Profit on sale of assets £10 8000 5% Investment income £87000 4% Trading income £956000 43% © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 227 2867_P001_680_Book_MNPB.indb 227 24/03/2020 15:33 4/03/2020 15:33

228 Chapter 17 17.3 Grouped frequencies ● I can understand and use grouped frequencies Sometimes you are given too many different values to make a sensible bar chart, so you need to organise them into a grouped frequency table. You sort the data into groups called classes. Where possible, you should always keep the classes the same size. You can then draw a bar chart. You cannot find a mode for grouped data so, instead, you must use the modal class. This is the class with the highest frequency. Develop fluency A group of pupils are asked the number of times they have walked to school this term. Here are their replies. 6 3 5 20 15 11 13 28 30 5 2 6 8 18 23 22 17 13 4 2 30 17 19 25 8 3 9 12 15 8 a Organise the data into a grouped frequency table. b Draw a bar chart. c What is the modal class? a Times walked to school 1–5 6–10 11–15 16–20 21–25 26–30 Frequency 76 6 5 3 3 b 1 Frequency 1–5 6–10 11–1 5 Number of times walked to school 0 2 3 4 5 6 7 16–2 0 21–2 5 26–3 0 c The modal class is 1–5 times. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 228 2867_P001_680_Book_MNPB.indb 228 24/03/2020 15:33 4/03/2020 15:33

229 17.3 1 Two classes carried out a survey to find out how many text messages each pupil had sent the day before. These are the results of this survey. 4 7 2 18 1 16 19 15 13 0 9 17 4 6 10 12 15 8 3 14 2 14 15 18 5 16 3 6 5 18 12 5 9 19 5 17 17 16 5 10 19 7 10 17 16 10 7 19 3 16 16 18 6 5 8 9 3 a Copy and complete this grouped frequency table with a class size of 5. Number of texts Tally Frequency 0–4 5–9 10–14 15–19 Total: b Draw a bar chart of the data. c What is the modal class? 2 A doctor kept a record of the waiting times, in minutes, of patients. These are the results of the survey. 5 8 3 19 2 17 15 16 14 1 10 18 5 7 11 13 16 9 4 15 3 15 16 19 18 17 6 11 19 8 11 18 17 11 8 15 a Copy and complete this grouped frequency table with a class size of 5. Number of minutes Tally Frequency 0–4 5–9 10–14 15–19 Total: b Draw a bar chart of the data. c What is the modal class? © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 229 2867_P001_680_Book_MNPB.indb 229 24/03/2020 15:33 4/03/2020 15:33

230 Chapter 17 3 A leader of a youth club asked her members: ‘How many times this week have you played video games?’ These were their responses. 3 6 9 2 23 18 6 8 29 27 2 1 0 5 19 23 13 21 7 4 23 8 7 1 0 25 24 8 13 18 15 16 3 7 11 5 27 23 6 9 18 17 6 6 0 6 21 26 25 12 4 24 11 11 5 25 a Create a grouped frequency table with a class size of 5. Number of times Tally Frequency 0–4 5–9 10–14 15–19 20–24 25–29 Total: b Draw a bar chart of the data. c What is the modal class? 4 Penny kept a record of how many minutes she had to wait for her bus every day for one month. These are the results. 2 5 8 1 12 17 5 7 28 26 1 0 0 4 18 22 12 10 6 3 22 7 6 0 1 24 4 7 3 17 14 15 2 6 10 4 a Create a grouped frequency table with a class size of 5. b Use the data above to complete your table. c Draw a bar chart of the data. d What is the modal class? 5 In a class sponsorship, the pupils raised these amounts of money, in pounds (£). 12.25 06.50 09.75 23.00 01.86 05.34 16.75 11.32 06.45 02.50 05.00 18.65 05.90 04.34 02.17 08.89 07.86 19.70 21.55 13.87 23.12 14.67 11.98 13.60 04.75 19.00 16.41 01.90 06.89 08.33 Create a grouped frequency table: a with a class size of £4, i.e. £0–£4, £4.01–£8, £8.01–£12,… b with a class size of £6, i.e. £0–£6, £6.01–£12, £12.01–£18,… . © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 230 2867_P001_680_Book_MNPB.indb 230 24/03/2020 15:33 4/03/2020 15:33

231 17.3 6 Theo recorded how much in pounds (£) he spent at a shop each day one month. 11.15 05.40 08.65 14.00 00.76 04.24 15.65 10.22 05.35 01.40 04.00 17.55 04.80 03.24 01.07 07.79 06.76 18.60 20.00 12.77 18.02 13.57 10.88 12.50 03.65 18.00 17.31 00.80 05.79 07.23 05.75 a Create a grouped frequency table: i with a class size of £4, i.e. £0–£4, £4.01–£8, £8.01–£12,… ii with a class size of £6, i.e. £0–£6, £6.01–£12, £12.01–£18,… . b What is the modal class for each group? Reason mathematically Look back at Question 6. a Select the better class size to illustrate the data. b Explain why you chose that class size. a ii b The class shows a greater difference between frequencies. 7 Look back at Question 5. a What is the modal class for each table that you created in your answer? b The teacher was asked to make a display illustrating how well the pupils had done. Select the better class size to illustrate the data. c Explain why you chose that class size. 8 The average ages of families at a small holiday resort are shown below. 31 17 31 21 32 29 27 39 25 25 40 27 36 28 46 19 38 32 23 28 35 19 41 30 24 34 55 26 20 36 43 51 a Which of the options shown is the most sensible format for this data? Explain your answer. b Copy and complete the table you chose in part a. c Draw a bar chart to illustrate the data. Average age Tally Frequency 10–14 15–19 etc. Average age Tally Frequency 0–19 20–39 etc. Average age Tally Frequency 10–19 20–29 etc. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 231 2867_P001_680_Book_MNPB.indb 231 24/03/2020 15:33 4/03/2020 15:33

232 Chapter 17 9 In a club sponsorship, the members raised these amounts of money (£). 24.50 13.00 19.50 45.00 03.72 10.68 03.50 22.64 12.90 05.00 10.00 37.10 11.80 08.68 04.34 17.78 15.72 39.40 43.10 27.74 44.24 29.34 23.96 27.20 09.50 38.00 32.82 03.80 13.78 16.66 15.85 08.15 a Create a grouped frequency table: i with a class size of £5.00, i.e. £0.00–£5.00, £5.01–£10.00, £10.01–£15.00,… ii with a class size of £10.00, i.e. £0.00–£10.00, £10.01–£20.00, £20.01–£30.00,… . b What is the modal class for each group? c Select the best class size to illustrate the data. Explain your decision. Solve problems This grouped bar chart shows the number of people attending a yoga class over the year. 1 Frequency 0–5 6–10 11–1 5 Number of pupils attending 0 2 3 4 5 6 7 8 9 10 16–2 0 21–25 a What is the modal class? b How many times did fewer than 11 people attend? c How many times did over 15 people attend? d How many classes were there over the year? a 16–20 is the modal class. b Add the two classes less than 11 giving 3 + 4 = 7. c Add the two classes over 15 giving 9 + 6 = 15. d Add the frequency in each class to give 3 + 4 + 7 + 9 + 6 = 29 classes. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 232 2867_P001_680_Book_MNPB.indb 232 24/03/2020 15:33 4/03/2020 15:33

233 17.3 10 This grouped bar chart shows the lifetimes of some batteries in a test. 2 Number of batteries 21–25 26–30 31–3 5 Lifetime (hours) 0 4 6 8 10 12 14 16 18 20 36–4 0 41–4 5 46–5 0 Lifetime of batteries a Make a frequency table for the data. b What is the modal class? c What is the total number of batteries tested? d How many batteries had a lifetime of 40 hours or less? e What is the maximum and minimum possible range? 11 A weather station records the minutes of direct sunlight each day for a month. Class interval Tally Frequency 0–39 / 1 40–79 //// 5 80–119 //// //// // 12 120–159 //// /// 8 160–199 //// 4 200–239 / 1 Sum = 31 a What could the longest amount of direct sunshine have been, in hours and minutes? b At what time of year do you think this recording took place? c What was the chance of having no direct sunlight at all: 1 in 31, 1 in 39 or 1 in 40? d Estimate how many days you think will have had 3 hours or more sunshine. 12 Mr Roberts gave his class a test and made the grouped bar chart below from the data. 2 Frequency 0–20 21–40 41–60 Maths in test 0 4 6 8 10 12 14 16 18 20 61–8 0 81–100 a What is the modal class? b How many pupils scored fewer than 41 marks? c How many pupils scored more than 60 marks? © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 233 2867_P001_680_Book_MNPB.indb 233 24/03/2020 15:33 4/03/2020 15:33

234 Chapter 17 13 Tom’s morning train was always late. He made this grouped bar chart. 2 Frequency 0–10 11–20 21–30 Time (minutes) 0 4 6 8 10 12 14 16 18 20 31–4 0 41–50 a What is the modal class? b How many times was the train more than ten minutes late? c How many times was the train over thirty minutes late? d From this data what’s the highest possible number of times the train was on time? 17.4 Continuous data ● I can understand and work with continuous data Data such as masses or heights of all the pupils in a class often has a wide range of possible values. This type of data is continuous data and you have to group it together, to find any pattern. In a grouped frequency table, you arrange information into classes or groups of data. You can create frequency diagrams from grouped frequency tables, to illustrate the data. Develop fluency These are the journey times, in minutes, for a group of 16 railway travellers. 25 47 12 32 28 17 20 43 15 34 45 22 19 36 44 17 Construct a frequency table to represent the data. Looking at the data, 10 minutes is a sensible size for the class interval. There are six times in the group 10 < T ≤ 20: 12, 17, 20, 15, 19 and 17. There are three times in the group 20 < T ≤ 30: 25, 28 and 22. There are three times in the group 30 < T ≤ 40: 32, 34 and 36. There are four times in the group 40 < T ≤ 50: 47, 43, 45 and 44. Note how you write the class interval in the form 10 < T ≤ 20. 10 < T ≤ 20 is a short way of writing the time interval of 10 minutes to 20 minutes. The possible values for T include 20 minutes but not 10 minutes. Put all the information in a table. Time, T (minutes) Frequency 10 < T ≤ 20 6 20 < T ≤ 30 3 30 < T ≤ 40 3 40 < T ≤ 50 4 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 234 2867_P001_680_Book_MNPB.indb 234 24/03/2020 15:33 4/03/2020 15:33

235 17.4 1 This table shows the lengths of time 25 customers spent in a shop. a One of the customers was in the shop for exactly 20 minutes. In which class was this customer recorded? b Which is the modal class? Time, T (minutes) Frequency 0 < T ≤ 10 12 10 < T ≤ 20 7 20 < T ≤ 30 8 2 These are the heights (in metres) of 20 people. 1.65 1.53 1.71 1.72 1.48 1.74 1.56 1.55 1.80 1.85 1.58 1.61 1.82 1.67 1.47 1.76 1.79 1.66 1.68 1.73 a Copy and complete the frequency table. b What is the modal class? Height, h (metres) Frequency 1.40 < h ≤ 1.50 1.50 < h ≤ 1.60 1.60 < h ≤ 1.70 1.70 < h ≤ 1.80 1.80 < h ≤ 1.90 3 The table shows the waiting times for the bus of each pupil in a class. a One pupil had to wait exactly 10 minutes. In which class was this pupil recorded? b Which is the modal class? Time, T (minutes) Frequency 0 < T ≤ 10 11 10 < T ≤ 20 8 20 < T ≤ 30 5 4 The table shows the masses of the marrows in a competition. a Tim’s marrow weighs exactly 5 kilograms. In which class was this marrow recorded? b Which is the modal class? Mass, M (kilograms) Frequency 0 < M ≤ 5 3 5 < M ≤ 10 9 10 < M ≤ 15 4 5 These are the lengths, in metres, jumped in a competition. 2.75 2.64 2.81 2.62 2.48 2.85 2.65 2.66 2.90 2.53 2.47 2.72 2.51 2.78 2.56 2.87 2.89 2.78 2.76 2.64 a Copy and complete the frequency table b What is the modal class? Length, L (metres) Frequency 2.40 < L ≤ 2.50 2.50 < L ≤ 2.60 2.60 < L ≤ 2.70 2.70 < L ≤ 2.80 2.80 < L ≤ 2.90 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 235 2867_P001_680_Book_MNPB.indb 235 24/03/2020 15:33 4/03/2020 15:33

236 Chapter 17 6 These are the noon temperatures, in degrees Celsius, °C, recorded in Bude during August. 16 18 19 19 18 17 19 20 22 24 22 20 25 24 23 21 19 18 17 17 16 16 18 19 20 21 21 20 19 18 22 a Copy and complete the frequency table. b What is the modal class? c Joy was in Bude when the noon temperature was exactly 21°C. In which class was this temperature recorded? Temperature, T (°C) Frequency 15 < T ≤ 17 17 < T ≤ 19 19 < T ≤ 21 21 < T ≤ 23 23 < T ≤ 25 7 The heights, in cm, of flowers in a collection were measured. 35.2 28.3 29.1 19.5 18.9 27.1 19.7 20.8 22.3 24.5 22.2 20.9 25 24.4 27.3 23.6 19.8 18.9 27.3 27.1 18.2 26.3 28.7 29.6 20.8 21.4 21.2 30.3 a Copy and complete the frequency table b What is the modal class? c How many flowers were taller than 30 cm? Height, H (cm) Frequency 15 < H ≤ 20 20 < H ≤ 25 … 8 The masses, in grams, of apples in a box were measured as below. 75.3 88.4 99.2 89.6 78.8 77.2 89.1 90.9 82.5 75.6 82.7 90.8 75.1 79.5 87.2 83.8 99.9 88.2 77.5 87.2 88.0 76.4 78.6 89.7 80.7 81.5 81.2 80.3 a Copy and complete the frequency table. b What is the modal class? c How many apples were heavier than 90 grams? Mass, M (grams) Frequency 75 < H ≤ 80 80 < H ≤ 85 … © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 236 2867_P001_680_Book_MNPB.indb 236 24/03/2020 15:33 4/03/2020 15:33

237 17.4 Reason mathematically The times, T seconds, of swimmers in a competition were: 14.3 17.4 18.2 18.6 17.8 16.2 18.1 15.9 12.5 13.6 11.7 12.8 14.0 13.5 16.2 12.8 18.9 17.2 16.5 16.2 17.0 15.4 17.6 18.7 12.7 11.5 11.2 12.3 Explain how you would create a suitable grouped frequency table from this data. Note the range, from 11.2 seconds to 18.9 seconds. You can divide this range into four suitable classes: 11 < T ≤ 13, 13 < T ≤ 15, 15 < T ≤ 17 and 17 < T ≤ 19. So the table becomes: Time, T (seconds) Frequency 11 < T ≤ 13 13 < T ≤ 15 15 < T ≤ 17 17 < T ≤ 19 9 The lengths, L centimetres, of babies born one week were recorded as: 45.7 58.1 49.3 49.9 58.2 47.4 59.8 50.2 52.7 54.2 52.1 50.9 55.3 54.3 49.9 48.3 49.7 58.1 57.6 57.8 46.7 56.2 53.7 52.8 50.4 51.2 51.9 50.8 a Explain how you would create a suitable grouped frequency table from this data. b Create that suitable table. 10 The times, T seconds, Clara took to swim 200 m were recorded as: 51.7 59.1 60.3 60.9 59.2 58.4 59.8 51.2 53.7 55.2 53.1 51.9 55.3 56.3 50.9 51.3 59.7 57.1 56.6 58.8 50.7 51.2 53.7 52.9 50.4 51.2 50.9 50.9 a Explain how you would create a suitable grouped frequency table from this data. b Create that suitable table. 11 The mass, M grams, of each grape in a bunch was recorded as 7.57 6.71 5.83 9.92 8.23 7.43 9.81 8.25 8.17 7.23 9.18 7.19 6.53 7.93 9.39 8.43 5.73 8.41 7.65 7.89 7.78 9.08 5.17 9.83 7.48 8.26 7.96 6.85 a Explain how you would create a suitable grouped frequency table from this data. b Create that suitable table. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 237 2867_P001_680_Book_MNPB.indb 237 24/03/2020 15:33 4/03/2020 15:33

238 Chapter 17 12 In a doctors’ surgery, the practice manager told each doctor that the length of most consultations should be more than 5 minutes but less than 10. She monitored the consultation times of the three doctors at the practice throughout one day. These are the results. Dr Speed (minutes): 6, 8, 11, 5, 8, 5, 8, 10, 12, 4, 3, 6, 8, 4, 3, 15, 9, 2, 3, 5 Dr Bell (minutes): 7, 12, 10, 9, 6, 13, 6, 7, 6, 9, 10, 12, 11, 14 Dr Khan (minutes): 5, 9, 6, 3, 8, 7, 3, 4, 5, 7, 3, 4, 5, 9, 10, 3, 4, 5, 4, 3, 4, 4, 9 a Did any of the doctors manage to follow the practice manager’s advice? b Write a short report about the consultation times of the three doctors. Solve problems A nurse recorded the reaction times, T seconds, of patients one morning. These are the results. 1.71 1.33 1.64 1.58 1.42 0.65 1.12 1.38 1.79 1.32 1.70 1.63 1.21 1.54 1.24 1.17 0.82 1.43 1.65 0.99 0.86 0.73 1.25 1.78 1.37 1.18 1.53 1.32 0.92 0.84 1.34 Using a class size of 0.3 seconds, work out the modal class of the patient reaction times. The shortest time is 0.65, the longest time is 1.79, so set up a table from 0.6 to 1.8. Dividing into groups of 0.3 will give four classes. Set up a group tally and table to find the results, the smallest being 0.6. Reaction time T (seconds) Tally Frequency 0.6 < T ≤ 0.9 //// 5 0.9 < T ≤ 1.2 //// 5 1.2 < T ≤ 1.5 //// //// / 11 1.5 < T ≤ 1.8 //// //// 10 You can now see that the modal class is 1.2 < T ≤ 1.5 seconds. 13 A petrol station owner surveyed a sample of customers to see how many litres of petrol they bought. These are the results. 27.6 31.5 48.7 35.6 44.8 56.7 51.0 39.5 28.8 43.8 47.3 36.6 42.7 45.6 32.4 51.7 55.9 44.6 36.8 49.7 37.4 41.2 38.5 45.9 34.1 54.3 41.3 49.4 38.7 33.2 Using a class size of 5 litres, work out the modal class of the amount of petrol that customers bought. 14 A farmer weighed his bags of potatoes in kilograms, W kg. These are the results. 6.81 6.23 6.74 6.57 6.41 6.15 6.12 6.38 6.19 6.32 6.30 6.63 6.81 6.55 6.34 6.18 6.12 6.45 6.45 6.29 6.16 6.63 6.25 6.48 6.37 6.08 6.53 6.22 6.02 6.54 6.34 Using a class size of 0.2 kg, work out the modal class of the potato bags. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 238 2867_P001_680_Book_MNPB.indb 238 24/03/2020 15:33 4/03/2020 15:33

239 17.4 15 The seasons results of Kath’s high jumps were recorded. 0.91 1.33 1.24 1.17 1.21 1.15 1.12 1.18 1.19 1.22 1.20 1.03 1.11 1.15 1.04 1.28 1.22 1.25 1.35 1.29 1.16 1.03 0.95 1.28 1.27 1.18 1.23 1.22 1.02 1.34 1.31 Using a suitable class size, work out the modal class of Kath’s height in the high jump. 16 A weights and measures team measured the amount of lemonade, in litres, contained within lemonade bottles in a supermarket. These are their results. 1.01 1.03 0.94 1.07 0.98 1.05 1.12 0.98 1.09 1.12 0.94 1.03 0.99 1.05 1.04 1.15 0.92 0.95 1.05 1.09 1.14 0.93 1.05 0.98 0.97 1.08 1.03 1.12 1.02 0.94 0.96 Using a class size of 0.05 litres, work out the modal class of the bottles of lemonade. Now I can... work out the size of sectors in pie charts by their angles at the centre understand grouped frequencies use grouped frequencies use a scaling method to draw pie charts understand continuous data work with continuous data © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 239 2867_P001_680_Book_MNPB.indb 239 24/03/2020 15:33 4/03/2020 15:33

240 18 Pencil and paper calculations 18.1 Short and long multiplication ● I can choose a written method for multiplying two numbers together ● I can use written methods to carry out multiplications accurately There are many different methods of multiplying two numbers, including the grid or box method, column method or long multiplication and the Chinese method. Develop fluency Work out 36 × 43. Grid or box method (partitioning) Long multiplication (expanded working) Long multiplication (compacted working) Chinese method × 30 6 40 1200 240 1440 3 90 18 108 1548 36 × 43 18 (3 × 6) 90 (3 × 30) 240 (40 × 6) 1200 (40 × 30) 1548 36 × 43 108 (3 × 36) 1 1440 (40 × 36) 1 2 1548 3 6 1 5 4 8 4 3 1 2 2 4 0 9 1 8 1 Note the carried figure, from the addition of 9, 1 and 4. The answer is 1548. 1 Copy each of these grids and fill in the gaps. a 21 × 6 × 20 1 6 b 37 × 6 × 30 7 6 2 Use the grid method to work these out. a 18 × 6 b 18 × 8 c 22 × 9 d 22 × 7 3 Use long multiplication to work these out. a 27 × 5 b 32 × 5 c 32 × 8 d 19 × 8 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 240 2867_P001_680_Book_MNPB.indb 240 24/03/2020 15:33 4/03/2020 15:33

241 18.1 4 Use the Chinese method to work these out. a 42 × 4 b 42 × 9 c 19 × 9 d 19 × 3 5 Copy each of these grids and fill in the gaps. a 21 × 16 × 20 1 10 6 b 58 × 16 × 50 8 10 6 6 Use the grid method to work these out. a 17 × 23 b 32 × 23 c 15 × 23 d 56 × 23 7 Use long multiplication to work these out. a 27 × 15 b 32 × 15 c 32 × 18 d 19 × 18 8 Use the Chinese method to work these out. a 42 × 14 b 42 × 19 c 19 × 19 d 19 × 13 9 Use your favourite method to work these out. a 32 × 24 b 32 × 53 c 18 × 53 d 27 × 53 10 Use any method to work these out. a Each day 17 jets fly from London to San Francisco. Each jet can carry up to 348 passengers. How many people can travel from London to San Francisco each day? b A van travels 34 miles for every gallon of petrol. How many miles can it go on one tank, given that the petrol tank holds 18 gallons? Reason mathematically 12 × 5 × 17 = ■ × 15 × 17 What number goes in the box to make the equation true? 12 × 5 = 60 60 ÷ 15 = 4 4 goes in the box as 12 × 5 × 17 = 1020 and 4 × 15 × 17 = 1020. 11 15 × 16 × 17 = ■ × 8 × 17 What number goes in the box to make the equation true? © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 241 2867_P001_680_Book_MNPB.indb 241 24/03/2020 15:33 4/03/2020 15:33

242 Chapter 18 12 Follow these steps. ● Write down any three-digit number. ● Multiply the number by 7. ● Multiply your answer by 11. ● Multiply your answer by 13. ● Write down your final answer. a What do you notice? b Now try to explain why this happens. 13 a Work out these multiplications. i 34 × 11 ii 71 × 11 iii 26 × 11 iv 45 × 11 b What do you notice about the answers? c Write down the answer to 16 × 11. d Write down the answer to 85 × 11. 14 Timothy says that 15 × 31 gives the same answer as 51 × 13 because the numbers are the same, just swapped around. Is he correct? Explain your answer. 15 Show how you can use the grid method to work out 7.2 × 12.7 and explain how you have split up the numbers. Solve problems Bilal is having a birthday party at an ice-skating rink. The ice-skating rink charges £17 per guest for the party and requires a deposit of £50, with the rest of the payment due on the day of the party. He has 12 guests coming. How much must he pay for the party on the day? Work out 17 × 12. × 10 7 10 100 70 170 2 20 14 34 204 204 – 50 = 154 He must pay £154 on the day. 16 Mr Z’s class is going on a residential trip. Each student must pay £35 to go on the trip. There are 29 students in the class. What is the total cost of the trip? 17 Taylor is making cakes for the school cake sale. She is going to make five cakes and has everything she needs except for eggs. Each cake needs three eggs. A package of six eggs costs £2.89. How much will it cost to buy all the eggs to make the cakes? 18 a One lunchtime, a burger van sells 35 burger meals at £3.98 each and 17 hot dogs at £2.90 each. How much money did they take in that time? b Write down a way you can work out 35 × 3.98 and 17 × 2.9, using mental methods rather than short and long multiplication. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 242 2867_P001_680_Book_MNPB.indb 242 24/03/2020 15:33 4/03/2020 15:33

243 18.2 18.2 Short and long division ● I can choose a written method for dividing one number by another ● I can use written methods to carry out divisions accurately There are many different ways to work out a division calculation, including long division, short division and repeated subtraction. Divisions do not always give whole number answers. There is sometimes a number left over. This is called the remainder. You will often need to add some decimal places and continue the division. You may sometimes want your answer with a remainder or written as a fraction. Develop fluency Work out 970 ÷ 8. a Give your answer as a remainder and as a fraction. b Give your answer as a decimal. a Repeated subtraction Short division 970 − 800 (100 × 8) 170 160 (20 × 8) 10 − 8 (1 × 8) 2 1 2 1 r 2 1 1 8 9 7 0 As a remainder, 970 ÷ 8 = 121 r2 To write this as a fraction, put the remainder in the top and the number you are dividing by in the bottom: 121 2 8 = 121 1 4 b Short division Long division 1 2 1 . 2 5 1 1 2 4 8 9 7 0 . 00 1 2 1 . 2 5 8 9 7 0 . 00 8 17 16 10 8 20 16 40 To give your answer as a decimal, you need to write zeros after the decimal point in 970 in order to complete the division. 970 ÷ 8 = 121.25 1 Use repeated subtraction to work these out. a 84 ÷ 6 b 126 ÷ 6 c 168 ÷ 6 d 312 ÷ 6 2 Use long division to work these out. a 54 ÷ 3 b 84 ÷ 3 c 93 ÷ 3 d 114 ÷ 3 3 Use short division to work these out. a 84 ÷ 7 b 119 ÷ 7 c 105 ÷ 7 d 168 ÷ 7 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 243 2867_P001_680_Book_MNPB.indb 243 24/03/2020 15:33 4/03/2020 15:33

244 Chapter 18 4 Use any method to work these out. a 200 ÷ 50 b 200 ÷ 25 c 2000 ÷ 50 d 2000 ÷ 250 5 Use long division to work these out. Leave your answer as a remainder. a 185 ÷ 12 b 212 ÷ 12 c 370 ÷ 12 d 375 ÷ 12 6 Use long division to work these out. Write the remainder as a fraction in simplest form. a 248 ÷ 16 b 282 ÷ 16 c 500 ÷ 16 d 500 ÷ 8 7 Use short division to work these out. Write the remainder as a decimal. a 171 ÷ 12 b 219 ÷ 12 c 438 ÷ 12 d 657 ÷ 12 8 Use any method you prefer to work these out. Write the remainder as a decimal. a 684 ÷ 16 b 692 ÷ 16 c 684 ÷ 5 d 692 ÷ 12 Reason mathematically A company has 95 boxes to move by van. The van can carry 8 boxes at a time. How many trips must the van make to move all the boxes? Divide 95 ÷ 8. 1 1 r 7 1 8 9 5 In this case, there is no need to continue the division into a decimal because the question is asking how many trips the van must make. The van must make 12 trips. 9 a Professional cycling teams enter nine riders into the large races such as the Tour de France, Giro d’Italia and the Spanish Vuelta. Given that there are 198 riders in a race, how many teams have entered? b Football teams have 11 players. On one afternoon, 286 players start their matches. How many teams are playing? 10 Electric light bulbs can be packed into boxes of 16 or 24. How can 232 bulbs be packed into full boxes only? Find two possible ways. 11 3000 people go on a journey from Paris to Rome. They travel in 52-seater coaches. a How many coaches do they need? b Each coach costs €680. What is the total cost of the coaches? c How much is each person’s share of the cost? Solve problems A group of 8 friends won £3358 in the lottery and split the money equally amongst themselves. How much money did each friend get? Divide £3358 ÷ 8. 0 4 1 9. 7 5 3 1 7 6 4 8 3 3 5 8. 0 0 Each friend got £419.75. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 244 2867_P001_680_Book_MNPB.indb 244 24/03/2020 15:33 4/03/2020 15:33

245 18.3 12 The mathematics department has printed 500 information sheets about long division. They put them into sets of 30 sheets. a How many full sets are there? b How many more sheets should be printed to make another full set? 13 To raise money, a running club is doing a relay race of 120 kilometres. Each runner except the last one will run 9km. The last runner will just run the extra distance to the finish. a How many runners are needed to cover the distance? b How far does the last person run? 14 The diameter of a 2p coin is 26mm. A piece of A4 paper is 210mm wide and 297mm tall. How many 2p coins would it take to fill up the area of an A4 paper? (Hint: You must work out how many coins will fit across and how many will fit up the page.) 15 A contractor has 13 tonnes of waste to move by van. The van can carry 0.8 tonnes at a time. a How many trips must the van make to move all the waste? b A different contractor had a lorry that can take 1.1 tonnes at a time. How many fewer trips would this contractor need to shift the 13 tonnes? 18.3 Calculations with measurements ● I can convert between common metric units ● I can use measurements in calculations ● I can recognise and use appropriate metric units You need to know and use these metric conversions for length, capacity and mass. Length is measured in metres, m. Capacity is measured in litres, L. Mass is measured in grams, g. You can use this diagram to help you convert units. Learn the prefix for each sub-unit. For example, to convert mm (millimetres) to metres (the unit), divide by 1000. milli- cent- unit kilo- × 1000 × 10 ÷ 1000 × 1000 ÷ 10 ÷ 1000 × 100 ÷ 100 When adding or subtracting metric amounts that are given in different units, you need to convert so that they are in same units first. You also need to be able to convert times. © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 245 2867_P001_680_Book_MNPB.indb 245 24/03/2020 15:33 4/03/2020 15:33

246 Chapter 18 1 year = 365 days 1 day = 24 hours 1 hour = 60 minutes 1 minute = 60 seconds (A leap year has 366 days, but usually questions assume 365 days in a year and ignore leap years. Ask your teacher if you are not sure.) Develop fluency Add 1.57m, 23cm and 0.092km. First convert all the lengths to the same units. In this case, metres are a sensible unit. 23cm = 0.23m 23 ÷ 100 = 0.23 0.092km = 92m 0.092 × 1000 = 92 1.57m + 0.23m + 92m = 93.8m 1 Convert each length to centimetres. a 60mm b 600mm c 6m d 0.6km 2 Convert each length to kilometres. a 456m b 4562m c 4562000cm d 45620cm 3 Convert each length to millimetres. a 34cm b 340cm c 3.4m d 0.34km 4 Convert each mass to kilograms. a 1259g b 125.9g c 1259mg d 125.9tonnes 5 Convert each mass to grams. a 4.32kg b 43.2kg c 4320mg d 432000mg 6 Convert each capacity to litres. a 237cl b 237ml c 2370ml d 23.7ml 7 Convert each capacity to millilitres. a 3650 litres b 356 litres c 35.6cl d 356cl 8 Convert each capacity to centilitres. a 862 litres b 8.62 litres c 862ml d 86.2ml 9 Convert each time to hours and minutes. a 85minutes b 185minutes c 108minutes d 208minutes 10 Add together the measurements in each group and give the answer in an appropriate unit. a 1.78m, 39cm, 0.006km b 0.234kg, 60g, 0.004kg c 2.3 litres, 46cl, 726ml d 6 hours, 24 minutes, 330 seconds © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 246 2867_P001_680_Book_MNPB.indb 246 24/03/2020 15:33 4/03/2020 15:33

247 18.3 Reason mathematically Choose a sensible unit to measure each of these. a The width of a football field b The length of a pencil c The mass of a car d A spoonful of medicine When you are asked to choose a sensible unit, sometimes you will find that there is more than one answer. a metres b centimetres c kilograms d millilitres 11 Fill in each missing unit. a A two-storey house is about 7… high. b Joe’s mass is about 47…. c Ruby lives about 2… from school. d Luka ran a marathon in 3…. 12 Asha has these jugs. How is it possible to use these jugs to measure 100ml of water? 13 Pierre buys 1kg of sugar, 750g of bananas and 1.2kg of apples. Is the total mass of the three items more than 3kg? Show your working. 14 Jenny knows that she needs 525 minutes of sleep a night in order not to feel tired the following day. She goes to bed at 11:15 p.m. and sets her alarm for 6:45 a.m. the following day. Will she get enough sleep? Solve problems Lily ties together three pieces of rope to make one longer piece. The pieces of rope are 62cm, 1.08m and 340mm. She loses 0.14m of length for each knot. How long is the new piece of rope? Convert to the same units. Choose a sensible unit. In this case, centimetres is sensible. 1.08m = 108cm 1.08 × 100 = 108 340mm = 34cm 340 ÷ 10=34 0.14m = 14cm 0.14 × 100 = 14 Add and subtract the units. 62 + 108 + 34 – (14 × 2) = 176cm 15 A kitten is born weighing 174g. The kitten loses 24050mg the first day, then gains 7g on the second day and gains 0.07562kg on the third day. How much does the kitten weigh on the third day? 1 litre 800 ml 500 ml © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 247 2867_P001_680_Book_MNPB.indb 247 24/03/2020 15:33 4/03/2020 15:33

248 Chapter 18 16 The United Arab Emirates produce 21.6 tonnes carbon dioxide per person, per year. Their population is around 9.4 million. The UK has a larger population, around 66 million, and produces around 5.7 tonnes of carbon dioxide per person, per year. Which country produces more carbon dioxide in total? 17 This is an extract from the Isle of Man TT (Tourist Trophy) Motorbike races. Moto GP 2002 Classic Junior Position No. Competitor Machine Time Speed 1 5 Bill Swallow 350 Linton Aermacchi 100.26 Bill Swallow holds the record for the fastest lap on a 350cc bike. One lap is 37.66 miles and the race was over four laps. Bill was the only racer to average over 100mph. How many hours and minutes did it take him to win the race, to the nearest minute? 18.4 Multiplication with large and small numbers ● I can multiply with combinations of large and small numbers mentally You need to be able to complete some multiplications without using a calculator. Develop fluency Work these out. a 4 × 500 b 0.4 × 0.5 c 400 × 0.05 a 4 × 5 = 20 First just multiply the non-zero digits. 4 × 500 = 2000 Because 500 is 5 × 100 you multiply 20 by 100. b 4 × 5 = 20 First just multiply the non-zero digits. 0.4 × 5 = 2 0.4 = 4 ÷ 10 so divide 20 by 10. 0.4 × 0.5 = 0.2 0.5 = 5 ÷ 10 so divide 2 by 10. c 4 × 5 = 20 First just multiply the non-zero digits. 400 × 5 = 2000 400 = 4 × 100 so multiply 20 by 100. 400 × 0.05 = 20 0.05 = 5 ÷ 100 so divide 2000 by 100. Work these out. 1 a 300 × 4 b 30 × 40 c 3 × 4000 d 300 × 400 2 a 0.4 × 7 b 0.4 × 70 c 4 × 0.7 d 40 × 0.7 3 a 0.2 × 60 b 0.02 × 6000 c 20 × 0.06 d 2000 × 0.006 4 a 0.6 × 0.3 b 0.6 × 0.03 c 0.6 × 0.8 d 0.06 × 0.08 5 a 60 × 9 b 600 × 9 c 600 × 0.9 d 600 × 0.09 6 a 0.3 × 7 b 30 × 70 c 300 × 0.7 d 0.03 × 0.007 7 a 0.3 × 0.04 b 30 × 0.4 c 30 × 400 d 0.03 × 4000 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 248 2867_P001_680_Book_MNPB.indb 248 24/03/2020 15:33 4/03/2020 15:33

249 18.4 Reason mathematically 17 × 52 = 884 Use this fact to work out 170 × 0.52. 170 = 17 × 10 0.52 = 52 ÷ 100 170 × 0.52 = (17 × 52) × 10 ÷ 100 170 × 0.52 = 88.4 8 18 × 35 = 630 Use this fact to work out: a 1.8 × 35 b 18 × 0.35 c 180 × 350 d 0.18 × 0.35. 9 232 = 529 Use this fact to work out: a 2302 b 2.32 c 0.232 . 10 882 = 7744 Use this fact to work out: a 88002 b 0.882 c 8.82 d 0.0882 . 11 0.32 × 0.15 = 0.048 Use this fact to work out: a 32 × 0.15 b 3.2 × 1.5 c 32 × 15 d 0.032 × 0.015. Solve problems A piece of paper is cut to be 200mm wide and 300mm high. Work out the area of the piece of paper in cm2 . There are two methods. Method 1 Convert each unit to cm and find the area. 200mm = 20cm 300mm = 30cm 20cm × 30cm = 600cm2 Method 2 Find the area in mm2 and convert to cm2. 1cm2 =102mm2 = 100mm2 200mm × 300mm = 60000mm2 60000mm2 = 600cm2 As 60000 ÷ 100 = 600 Method 2 is sometimes easier depending on the information given in the problem. You can use this method if you are only given an area or volume and not the individual side lengths. Note that to convert between cubic units, just cube the conversion factor. For example, 1cm3 =103mm3 = 1000mm3 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 249 2867_P001_680_Book_MNPB.indb 249 24/03/2020 15:33 4/03/2020 15:33

250 Chapter 18 12 The lengths of the sides of a large box are 0.4m, 0.5m and 0.8m. a Work out the volume, giving your answer in cubic metres (m3 ). b Work out the volume, giving your answer in cubic centimetres (cm3 ). 13 Sound travels about 0.3km in 1 second. How far does sound travel in the following times? Work in kilometres. a 200 seconds b 10 minutes c 0.02 seconds 14 A rectangle measures 60cm by 80cm. What is the area of the rectangle in: a square centimetres? b square millimetres? c square metres? d square kilometres? 15 A cuboid measures 30cm by 40cm by 50cm. Find the volume of the cuboid in: a cubic metres. b cubic kilometres. 18.5 Division with large and small numbers ● I can divide combinations of large and small numbers mentally The number you divide by is called the divisor. Changing the divisor to a whole number can make a division easier to do. Develop fluency Work these out. a 32 ÷ 0.08 b 8 ÷ 400 a 32 ÷ 0.08 = 3200 ÷ 8 Multiply both numbers by 100 to make the divisor 8. = 400 32 ÷ 8 = 4 so the answer is 400. b 8 ÷ 400 = 0.08 ÷ 4 Divide both numbers by 100 to make the divisor 4. = 0.02 For questions 1–5, work out the answer. 1 a 6 ÷ 0.2 b 60 ÷ 0.2 c 30 ÷ 0.2 d 3 ÷ 0.2 2 a 400 ÷ 20 b 40 ÷ 20 c 4 ÷ 20 d 0.4 ÷ 20 3 a 8 ÷ 20 b 8 ÷ 200 c 80 ÷ 200 d 8 ÷ 0.2 4 a 0.6 ÷ 10 b 0.6 ÷ 20 c 0.6 ÷ 30 d 0.6 ÷ 60 5 a 18 ÷ 300 b 1.8 ÷ 30 c 1.8 ÷ 3 d 1.8 ÷ 300 For questions 6–7, work out the answers and then put them in order of size, smallest first. 6 a 21 ÷ 0.7 b 0.21 ÷ 0.7 c 21 ÷ 700 d 0.21 ÷ 0.07 7 a 42 ÷ 0.6 b 420 ÷ 0.6 c 4.2 ÷ 0.6 d 42 ÷ 600 © HarperCollinsPublishers Ltd 2019. Restricted to use in schools that have purchased the KS3 Maths Now Learn & Practice Book. Sharing or copying by schools who have not purchased is an infringement of copyright law. 62867_P001_680_Book_MNPB.indb 250 2867_P001_680_Book_MNPB.indb 250 24/03/2020 15:33 4/03/2020 15:33