100 4 Decimals 3 Complete the workings to make these calculations easier. a 52×0.9=52×(1−0.1) b 8.3×0.9=8.3×(1−0.1) =52×1−52×0.1 =8.3×1−8.3×0.1 = − = − = = 4 Use the same method as in Question 3 to work out a 28×0.9 b 17×0.9 c 4.9×0.9 Think like a mathematician 5 Work in pairs or small groups to discuss this question. Look again at the working and method for Question 3a. a Describe a similar method you could use to work out 52×9.9 b Describe a similar method you could use to work out 52×0.99 c Use your methods to work out i 26×9.9 ii 26×0.99 Check your methods and answers with other learners in the class. Discuss any mistakes that have been made. 6 The diagram shows a rectangle. The width is 6m and the length is 9.9m. Work out the area of the rectangle. 7 A new medicine was given to 3200 patients. It made 99% of them feel better. For the other 1% the medicine made no difference. Work out how many of the patients the new medicine made better. 8 Work out the answers to these calculations. Look for different ways to make the calculations easier. There are some tips in the cloud. a 2.5×32.7×4 b 02 23 0 1 . . . × c (420+360)×0.7 d 8×1.32×2.5 e (720−120)×0.07 9.9m 6m Tip 99%=0.99, so 3200×0.99= 4×2.5= 0 2 0 1 0 2 10 0 1 10 . . . . = × × = 780×0.7=78×7= 8×2.5= We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
101 4.4 Making decimal calculations easier 9 Complete the workings. Use factors to make these calculations easier. a 24×0.35=24×0.5×0.7 b 32×0.45=32×0.5×0.9 = ×0.7 = ×0.9 = ×7÷10 = ×9÷10 = ÷10 = ÷10 = = 10 Use the same method as in Question 9 to work out a 80×0.15 b 116×0.25 11 This is part of Pedro’s classwork. Question Work out 280 × 0.12 Answer Use 0.12=0.2 × 0.6 280 × 0.12=280 × 0.2 × 0.6 =280 × 2÷10 × 6÷10 =560÷10 × 6÷10 =56 × 6÷10 =336÷10 =33.6 Use Pedro’s method to work out a 180×0.14 (use 0.14=0.2×0.7) b 120×0.16 c 250×0.18 d 450×0.24 Tip For part b, you could use 0.2× 0.8 or 0.4×0.4 Think like a mathematician 12 In questions 9 and 10, all the decimals had a factor of 0.5 Did you always multiply by 0.5 first? If you did, explain why you did this. Look at Question 11. Pedro has used a factor of 0.2 and multiplied by this first. Why do you think he has not used 0.5 in these questions? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
4 Decimals 13 A grandmother gives $12 600 to be shared between her grandchildren. Abdul gets 0.35 of the money. Zhi gets 0.25 of the money. Paula gets 0.22 of the money. Yola gets 0.18 of the money. a Work out how much they each get. b Show how you can check that your answers are correct. 14 Zara and Sofia are trying to solve this puzzle. What is the missing number in this calculation? 32 0 8 0 02 800 × − ( ) = . . I think the missing number is 0.4 I think the missing number is 0.3 Who is correct? Explain how you worked out your answer. Summary checklist I can use different methods to make decimal calculations easier. 102 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
103 4 Decimals Check your progress 1 Write the correct sign, < or >, between these. a 6.75 6.7 b −4.87 −4.81 c 0.65kg 67g 2 Write these decimals in order of size, starting with the smallest. −3.482 −3.449 −3.6 −3.44 −3.06 3 Use a mental method to work out a 0.2×0.4 b 0.7×0.3 4 Work out 0.57×4.62 5 Sam works out that 365×24=8760 Use this information to write the answers to these calculations. a 365×2.4 b 36.5×2.4 c 3.65×0.24 d 8760 24 ÷ e 8760 ÷ 2 4. f 876 ÷ 2 4. 6 Work out 5. . 4 0 ÷ 9 7 a Complete the table below showing the 15 times table. 1 2 3 4 5 6 7 8 9 15 30 45 b Use the table to help you work out 38 205 . . ÷1 5 c Show how to check your answer to part b is correct. Use a reverse calculation. 8 Work out the answers to these calculations. Use the methods you have learned to make the calculations easier. a (0.7−0.4)×0.12 b 0.9×27 c 14×0.35 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
Diamond decimals The Decimal Diamond on the right was generated from the diamond on the left. Work out how the values in the squares were calculated. 4.3 6.7 3.3 10.1 4.3 6.78.4 5.57.2 3.3 6.7 5 10.1 w y z x Now look at this Decimal Diamond. Choose a number to go in the square marked a, and choose a decimal g, between 0 and 1, which will fill the gaps between the values, so that: a+g=b b+g=c c+g=d d+g=e Can you work backwards to find out what numbers need to go in the circles to complete the Decimal Diamond? Is it possible to complete a Decimal Diamond if you are given only two of the entries? Look at this Decimal Diamond. Choose a value for the top circle w, and choose a decimal g, between 0 and 1, which will fill the gaps between the values, so that: w+g=x x+g=y y+g=z Work out the values that go in the five squares. Try this a few times with different values for w and g. What do you notice? If someone told you the values they chose for w and g, can you find a quick way to work out the values in their squares? d e c ba Project 2 104 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
105 5 Angles and constructions Getting started 1 Two angles of a triangle are 55° and 70°. a Work out the third angle. b What is the name of this type of triangle? Choose the correct word. equilateral isosceles right-angled perpendicular 2 a Work out the size of angle D. A B 127° 50° 42° C D 3 There are two parallel lines in this diagram. One of the angles is 76°. Copy the diagram and write in the values of the other angles. You do not have to draw the angles accurately. 4 a Use the measurements shown to make an accurate drawing of this shape. b Measure the length of RS. b Work out the value of x. x° 154° x° 76° 100° 5cm P Q R S 2.5cm 4cm 140° We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
106 5 Angles and constructions In this section you will … • use geometric vocabulary for equal angles formed when lines intersect. The sum of the angles in a triangle is 180°. Can you remember the first time you were shown this? You may have measured the angles and added them. You may have cut a triangle out of paper and folded it. This does not prove that the sum of the angles of any triangle is 180°. It only shows that it is true for the triangles you have drawn and that it is a reasonable conclusion. A proof is a logical argument in which a reason is given for each step. Over 2000 years ago the Greek mathematician Euclid wrote a book called The Elements. He used logical arguments to prove many facts in geometry and arithmetic. His book was the most successful textbook ever written. It is still in print today. Euclid started by defining basic things such as a point and a straight line. He also made a set of statements which he thought everyone could agree with. These were called axioms. An example of one of his axioms is: Things that are equal to the same thing are equal to one another. From this simple starting point, he proved many complicated results. In this unit you will look at several proofs. 5.1 Parallel lines Key words alternate angles corresponding angles geometric transversal vertically opposite angles a b c d This diagram shows two straight lines. Angles a and c are equal. They are called vertically opposite angles. Angles b and d are equal. They are also vertically opposite angles. Vertically opposite angles are equal. Angles a and b are not equal (unless they are both 90°). They add up to 180° because they are angles on a straight line. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
107 5.1 Parallel lines Tip We usually label pairs of parallel lines with matching arrows. Tip Alternate angles are always between the parallel lines. The arrows on this diagram show that these two lines are parallel. The perpendicular distance between parallel lines is the same wherever you measure it. Here, there is a third straight line crossing two parallel lines. It is called a transversal. Where the transversal crosses the parallel lines, four angles are formed. Angles a and e are called corresponding angles. Angles d and h are also corresponding angles. So are b and f. So are c and g. Corresponding angles are equal. Angles d and f are called alternate angles. Angles c and e are also alternate angles. Alternate angles are equal. These are important properties of parallel lines. To help you remember: • for vertically opposite angles, think of the letter X • for corresponding angles, think of the letter F • for alternate angles, think of the letter Z. Exercise 5.1 1 Look at the diagram. a Write four pairs of corresponding angles. b Write two pairs of alternate angles. 2 a One angle of 62° is marked in the diagram. Copy and complete these sentences. i Because corresponding angles are equal, angle =62° ii Because alternate angles are equal, angle =62° b Write the letters of a pair of vertically opposite angles. a c b d e g f h p r s q t v w u 62° a b d c We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
108 5 Angles and constructions 3 The sizes of two angles are marked in the diagram. a Which other angles are 105°? b Which other angles are 75°? 4 Angle APY is marked on the diagram. Complete these sentences. a APY and CQY are .................. angles. b APY and XQD are .................. angles. c APX and ......... are corresponding angles. d CQX and ......... are alternate angles. e CQP and ......... are vertically opposite angles. 5 PQ and RS are parallel lines. P c d b a R Q S 136° Find the sizes of angles a, b, c and d. Give a reason in each case. 6 Look at this diagram. A S T Y D B X C 130° 140° 40° 50° Explain why AB and CD cannot be parallel lines. A C B X Y D P Q 105° 75° p q r s t u We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
109 5.1 Parallel lines 7 This diagram has three parallel lines and a transversal. a Write a set of three corresponding angles that includes angle f. b Write a pair of alternate angles that includes angle c. c Write another pair of alternate angles that includes angle c. 8 Look at this diagram. Write whether these are corresponding angles, alternate angles or neither. a a and d b b and f c c and g d d and e e a and h b c a d f g e h a c b d e g f h i k j l Think like a mathematician 9 Arun gives this explanation of why angles h and d are equal. a g h b c e f d a Arun’s explanation is not correct. Write a correct version. b Write a different explanation of why h=d that does not use corresponding angles. h=b because they are corresponding angles. b=d because they are alternate angles. Therefore h=d. 10 AB and CD are parallel. a Give a reason why a and d are equal. b Give a reason why b and e are equal. c Use your answers to a and b to show that the sum of the angles of triangle ABC must be 180°. A C D d e c a b B We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
110 5 Angles and constructions Think like a mathematician 12 ABCD is a trapezium. Two sides are extended to make the triangle AXB. a Show that the angles of triangles ABX and DCX are the same size. b Show that angles A and D of the trapezium add up to 180°. c What can you say about angles B and C of the trapezium? Give a reason for your answer. Tip Use your answer to Question 10 as a guide. 11 Show that the sum of the angles of triangle XYZ must be 180°. X Y Z A B D C A B D C X 13 ABCD is a parallelogram. a Show that opposite angles of the parallelogram are equal. b Compare your answer to part a with a partner’s answer. Can you improve his or her answer? Can you improve your own answer? A B C D Imagine you have to explain corresponding angles to someone who does not know about them. How can you convince him or her that corresponding angles are equal? Summary checklist I can recognise vertically opposite angles and I know that they are equal. I can identify corresponding angles and alternate angles between parallel lines. Tip Extend the sides of the parallelogram. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
5.2 The exterior angle of a triangle 111 In this section you will … • learn to identify the exterior angle of a triangle • use the fact that the exterior angle of a triangle is equal to the sum of the two interior opposite angles. Here is a triangle ABC. The side BC has been extended to X. Angle ACX is called the exterior angle of the triangle at C. The angles marked at A and B are the angles opposite C. We know that a+b+c=180°, the sum of the angles in a triangle. So a+b=180°−c Also d+c=180°, the sum of the angles on a straight line. So d=180°−c Compare these two results and you can see that d=a+b This shows that: The exterior angle of a triangle=the sum of the two interior opposite angles. This is true for any triangle. 5.2 The exterior angle of a triangle Key word exterior angle of a triangle A B C X a b c d A B C X We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
112 5 Angles and constructions Exercise 5.2 The diagrams in this exercise are not drawn to scale. 1 Calculate the sizes of angles a, b and c. 45° a 80° b 20° 20° c 134° 86° 2 a Work out each of the exterior angles shown in this triangle. c b a 67° 70° Worked example 5.2 Work out x and y. A x° y° D B C 105° 40° 45° Answer x is an exterior angle of triangle ABD so x=40+105=145° y is an exterior angle of triangle BCD Angle BDC=180°−105°=75° because angles on a straight line=180° So y=75+45=120° b Work out the size of the exterior angle x in this quadrilateral. x 80° 65° We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
113 5.2 The exterior angle of a triangle 3 An exterior angle of a triangle is 108°. One of the interior angles of the triangle is 40°. a Work out the other two interior angles of the triangle. b Work out the other two exterior angles of the triangle. 4 PBC is a straight line. AQ is parallel to PC. a Explain why y=c b Explain why x=a+y c Use your answers to a and b to prove that the exterior angle at B of triangle ABC is the sum of the two interior opposite angles. 5 DX is parallel to BC. ZD is parallel to AB. BDY is a straight line. a Explain why angles BAD and ADZ are equal. b Explain why angles ABD and ZDY are equal. c Use the diagram to prove that the angle sum of quadrilateral ABCD is 360°. Do not use the fact that the angle sum of a triangle is 180°. 6 AB and CD are straight lines. A C B D 150° 160° 30° 20° Explain why the angles cannot all be correct. 7 Look at the diagram. a Explain why d=a+c b Write similar expressions for e and f. c Show that the sum of the exterior angles of a triangle is 360°. 8 ABC is an isosceles triangle. AB=AC. AB is parallel to DE. Angle ABC=68° Work out the size of angle EDC. Give a reason for your answer. 108° 40° B b c a y x A Q P C A B C D Z X Y f b c a d e A D C E B 68° We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
114 5 Angles and constructions 9 This pentagon is divided into a triangle and a quadrilateral. a Show that the angle sum of the pentagon is 540°. b Compare your explanation with a partner’s. Do you both have a similar explanation? 10 PQRS is a parallelogram. a Explain why x must be 22°. b Work out angle y. 11 ABCD is a parallelogram. Show that p+q=r p q r B C D A 12 a Show that w+y=a+b+c+d w d c a b y 13 Work out angles a, b and c. a b c 40° 80° 95° 39° R P Q x y S 22° b Show that w+x+y+z=360° w y z x Tip Use the exterior angle property of a triangle for each angle. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
115 5.3 Constructions Think like a mathematician 14 a Explain why x=b+d b Explain why y=c+e c Show that the sum of the angles in the points of the star, a+b+c+d+e=180° Summary checklist I can identify the exterior angles of a triangle. I can use the fact that the exterior angle of a triangle is equal to the sum of the two interior opposite angles. a c b d e x y You need to be able to draw a triangle when you know some of the sides and angles. You can do this using computer software. You can also do it using a ruler and compasses. Here are four different examples of how to construct triangles. 5.3 Constructions In this section you will … • construct triangles • learn how to draw the perpendicular bisector of a line • learn how to draw the bisector of an angle. Key words arc bisector construct (geometry) hypotenuse We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
116 5 Angles and constructions 1 When you know two angles and the side between them, this is known as ASA. Step 1: Draw the side. Draw an angle at one end. 60° A B 8 cm Step 2: Draw the angle at the other end. Where the two lines cross is the third vertex of the triangle. 60° A B C 50° 8 cm 2 When you know two sides and the angle between them, this is known as SAS. Step 1: Draw the angle first. 42° A 60° A B C 50° 8 cm 42° 10cm A 12 cm B C We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
117 5.3 Constructions Step 2: Open your compasses to 12 cm. Put the point of the compasses on A and draw an arc to mark B. Mark C in a similar way. Draw the side BC. 42° 10cm A 12cm B C 3 When you know the three sides but no angles, this is known as SSS. Step 1: Draw one side. Open your compasses to the length of a second side. Put the point of the compasses on one end of the side and draw an arc. A 5cm B Step 2: Open your compasses to the length of a second side. Put the point of the compasses on the other end of the side and draw another arc. Where the arcs cross is the third vertex. Draw the other two sides. C B 4.5cm 4 cm A 5cm C B 4.5cm 4 cm A 5cm Tip An arc is part of a circle. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
118 5 Angles and constructions 4 When one angle is a right angle, and you know the length of the hypotenuse and one other side, this is known as RHS. Step 1: Draw the side. Draw a right angle at one end. A 7cm B Step 2: At the other end, draw an arc equal to the hypotenuse. Draw the third side. C 7cm 9cm A B There are two other constructions that you need to be able to do using a ruler and compasses: 1 Construct the bisector of a line segment. This is a line through the mid-point of the line segment and perpendicular to it. Step 1: Draw the line segment. Open the compasses to about the same length as the line. (You do not need to measure this exactly.) Draw arcs from one end of the line on both sides of the line. A B Tip The hypotenuse is the side opposite the right angle. C 7cm 9cm A B We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
119 5.3 Constructions Step 2: Do the same thing at the other end of the line segment. Do not change the angle between the arms of the compasses. A B 2 Construct the bisector of an angle. This is a line that divides an angle into two equal parts. Step 1: Open the compasses to a few centimetres. You do not need to measure this. Put the point of the compasses on the angle and draw arcs that cross each of the lines. Step 2: Put the compass point on each of the crosses and draw an arc between the two lines. Do not change the angle between the arms of the compasses. Draw a line through the angle and the last cross. This is the perpendicular bisector of the angle. The two angles marked are equal. Tip When you do any construction, do not rub out your construction lines. Draw them faintly and leave them on your drawing. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
120 5 Angles and constructions Exercise 5.3 1 a Draw an accurate copy of this triangle. A C B 42° 6cm 65° b Measure the length of AC and BC. 2 a Draw an accurate copy of this triangle. Z Y X 36° 100° 5cm b Measure the length of XY and XZ. 3 a Draw an accurate copy of this triangle. P 50° Q R 7 cm 10cm b Measure angle Q 4 a Draw an accurate copy of this triangle. 117° 6 cm D 10cm E F b Measure angle F. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
121 5.3 Constructions 5 The hypotenuse of a right-angled triangle is 12.5 cm. One of the other sides is 10 cm. a Make an accurate drawing of the triangle. b Measure the third side. c Measure the other two angles. 6 The sides of a triangle are 7 cm, 8.5 cm and 9.7cm. a Make an accurate drawing of the triangle. b Measure the largest angle of the triangle. 7 The sides of a triangle are 5.8 cm, 7.8cm and 7.1 cm. a Make an accurate drawing of the triangle. b Give your triangle to a partner to check the accuracy of your drawing. If necessary, correct your drawing. Activity 5.3 All the angles and sides of this triangle are shown. a Choose either 2 sides and the angle between them (SAS) or 2 angles and the side between them (ASA). Use your chosen measurements to draw an accurate copy of the triangle. b Measure the three values you did not choose. Was your drawing accurate? If not, where did you go wrong? 63° 44° 73° 6.3cm 4.6cm 5.9 cm Think like a mathematician 8 Two sides of a right-angled triangle are 10.5cm and 8.3cm. Zara and Arun give different answers. The third side is 6.4 cm The third side is 13.4 cm Use accurate drawings to show that both of them could be correct. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
122 5 Angles and constructions 9 a Draw this diagram accurately. b Construct the perpendicular bisector of AB. c The perpendicular bisector of AB intersects AC at D. Label D on your diagram and measure AD. 39° A B C 7.5 cm 10 cm Think like a mathematician 10 RST is a triangle. RS=5cm, RT=6cm. a If ST=9cm, use a diagram to show that angle R is obtuse. Write the size of angle R. b If angle R is obtuse what can you say about the length of ST? Give reasons for your answer. 5 cm 6 cm S T R 11 a Draw this triangle accurately. b Construct the bisector of angle A. c The bisector of angle A intersects BC at X. Mark X on your triangle and measure BX. 12 Read what Marcus and Sofia say. If you draw the perpendicular bisectors of each side of a triangle, they intersect at a single point. The bisectors of each angle of a triangle meet at a single point. a Draw a triangle. Construct the perpendicular bisector of each side to test Marcus’s theory. b Look at the triangles that other learners have drawn. Do you think Marcus is correct? c Draw another triangle. Construct the bisector of each angle to test Sofia’s theory. d Look at the triangles that other learners have drawn. Do you think Sofia is correct? A B C 10cm 8cm 7cm We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
5.3 Constructions 123 Summary checklist I can draw a triangle when I know two angles and the side between them (ASA). I can draw a triangle when I know two sides and the angle between them (SAS). I can draw a triangle when I know the three sides (SSS). I can draw a right-angled triangle when I know the hypotenuse and one other side (RHS). I can draw the perpendicular bisector of a line segment. I can draw the bisector of an angle. You can draw a triangle if you know three side (SSS) or 2 sides and the angle between them (SAS) or two angles and the side between them (ASA). Can you draw a triangle if you only know the 3 angles? Why is this different from the other examples? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
124 5 Angles and constructions Check your progress 1 a Write the correct words to complete these sentences. i c and h are ................... angles. ii f and l are ................... angles. iii g and k are ................... angles. b Explain why e=j+k c Explain why c=i+j 2 ABCD is a trapezium. 64° 105° B A C D Work out the angles of the trapezium. Give a reason for each answer. 3 Work out x and y. 60° 118° 50° y x a b e i j k l n m f c g h d We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
125 5 Angles and constructions 4 Show that triangle ABC is isosceles. B C E 40° 70° A D 5 The sides of a triangle are 5.1 cm, 6.8 cm and 8.5cm. a Draw the triangle. b Construct the bisector of the smallest angle. c The angle bisector divides one of the other sides into two parts. How long is each part? 6 a Draw triangle ABC accurately. A B C 75° 8cm 6 cm b Construct the perpendicular bisector of AC. The perpendicular bisector meets BC at P. c Measure PC. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
126 Getting started 1 Give an example of a discrete data b continuous data c categorical data. 2 You want to choose a sample of 3 boys and 3 girls from your class. Describe three different ways to do this. 3 a Write one advantage of a large sample size. b Write one disadvantage of a large sample size. 4 You want to find the number of brothers and sisters of a group of children. Describe two different ways you could collect this data. In the United States of America, elections are held every two years to choose members for the House of Representatives. This is one of the groups of people who run the country. Each American state chooses representatives. The number they can choose depends on how many people live in that state. The more people live in a state, the more representatives they can choose. It is therefore very important to keep accurate records of the number of people living in each state. To maintain accurate records, a census is held every 10 years. A census is a way of collecting data. It is a questionnaire that must be filled in by every household in the USA. 6 Collecting data We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
127 6 Collecting data There was a census in 2010. The questionnaire had only 10 questions, which people were able to answer in about 10 minutes. The results of the 2010 census showed that there were 308 745 538 people living in the USA. There are 435 representatives in the House of Representatives. This means there is one representative for roughly every 710 000 people in the USA. The map shows the number of representatives in each state of the USA. You can see that California has the largest number of representatives, even though it is not the largest state by area. This is because more people live in California than in any other state in the USA. WA (12) OR (7) ID (4) NV (6) CA (55) UT (6) WY (3) MT (3) AZ (11) NM (5) AK (4) HI (3) CO (9) TX (38) OK (7) KS (6) NE (5) SD (3) ND (3) MN (10) IA (6) MO (10) AR (6) LA (8) MS (6) AL (9) GA (16) FL (29) SC (9) KY (8) TN (11) IN (11) IL (20) WI (10) MI (16) OH (18) PA (20) NY (29) VT (3) NH (4) ME (4) MA (11) RI (4) CT (7) NJ (14) DE (3) MD (10) DC (3) VA (13) WV (5) NC (15) In this unit, you will learn more about collecting data. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
128 6 Collecting data In this section you will … • select a method to collect data to answer a number of linked questions • consider the different types of data • consider different sampling methods. 6.1 Data collection To answer questions in statistics, you must collect data. First, you decide which data you need to collect. You need to know if it is discrete, continuous or categorical. Next, you must decide how to collect the data. If you need to question people, you could use a questionnaire that they fill in by themselves. Alternatively, you could interview them and write down the answers. Sometimes, you need to make observations. For example, you might record times or count vehicles. In this case, you need a sheet to record your observations. You may not be able to interview or give a questionnaire to everyone. In this case, you need to take a sample. You must think carefully about the best way to choose your sample. Whenever you collect data, you need to choose a method and explain why you think that method is the best one to use. Worked example 6.1 The head teacher wants answers to the following questions: • Are learners happy with the length of lessons? Should they be longer or shorter? • Are learners happy with the length of their lunch break? Your task is to investigate these questions using a sample of the learners in the school. a What data will you collect? b How will you choose your sample? c How will you collect the data? Give reasons for your answers. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
129 6.1 Data collection Continued Answer a You will need to collect data for: opinion about length of lessons; opinion about whether lessons should be longer, shorter or the same; opinions about length of lunch break. There could be differences of opinion between boys and girls and between younger and older learners so you also need to collect data about the age and gender of the learners sampled. b If there are equal numbers of boys and girls in the school, you could choose 2 girls and 2 boys from each tutor group. Use the register and choose two names at random (for example, in one tutor group, you might speak to the third and sixth girl and the third and sixth boy in the register). A random sample is more likely to be representative. c Individual interviews would be best because this will allow you to get an answer from everyone. You could also talk to a group of students together so they can discuss their ideas. However, this might take too long or be inconvenient, in which case you could use a questionnaire. Exercise 6.1 1 How would you collect data to answer these questions? State the type of data each time. a When a drawing pin is dropped, is it more likely to land point up or point down? b How many people visit a particular shop before 09:00? c How many brothers and sisters do the members of your class have? d How long do learners spend doing homework each night? e What is the average number of words in a sentence in a book? f How long do learners take to get to school each day? 2 Ahmad uses a gym. He asks these questions. • Do people visit this gym every week? • Do people come at a particular time of day? • Is there a difference between the habits of men and women? He interviews a sample of people at the gym. a What data does he need to collect? b What type of data is it? c Here is his data collection sheet. Name How often do you visit the gym? Do you prefer to visit in the morning or the evening? What is wrong with this data collection sheet? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
130 6 Collecting data d Design a better data collection sheet. e He asks the first people who come into the gym each morning for a week. Why is this not a good way to choose his sample? Describe a better way. 3 A cinema manager asks these questions. • How often do people visit the cinema? • Do younger people visit more often than older people? • What type of film do people like? a What data is required? b Describe two ways to collect the data. c The manager decides to give a questionnaire to a sample of customers. She gives it to all the customers on one night. Why is this not a good way to choose the sample? d Describe how the manager can get a representative sample. e Compare your answer to part d with another group’s answer. Can you improve your answer? Can you improve theirs? 4 Xavier has a simple puzzle for children. He asks these questions: • How long does it take to solve the puzzle? • Can girls solve it more quickly than boys? • Can older children solve it more quickly than younger ones? a What data must Xavier collect? What type of data is it? b Xavier gives the puzzle to a sample of children. Design a data collection sheet for Xavier. Activity 6.1 You are going to plan and trial the collection of data from learners in your school. a Think of three questions you can ask about aspects of school life. b What data do you need to answer these questions? c Decide how you will collect the data you need. You could use a questionnaire or a data collection sheet. d Test your data collection method on a few learners. Does it work well? Can you improve it? e You need to choose a sample of learners. How can you do this? What is the best way and why? f Compare your answers with those of another group. Can you see a way to improve your work or their work? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
131 6.1 Data collection 5 Sofia surveys cars using a busy road. She wants to answer these questions: • What percentage of drivers are male? • What percentage of cars carry only one person? • What is the average number of people in a car? a What data does Sofia need to collect? b What type of data is this? c Design a data collection sheet for Sofia. 6 Anders is comparing two books, X and Y. He thinks book X is harder to read than book Y. a What things make a book hard or easy to read? b What statistical questions can you ask to compare how easy it is to read each book? c What data can you collect to answer your questions? d How would you collect data to answer your questions? e Choose a book and use it to test your data collection method. Does it give you the data you need? Can you improve your method? f Compare your answers with those of another group. If you have chosen different approaches, which do you prefer? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
132 6 Collecting data In this exercise, you have thought about collecting data to answer questions. a What is the most important thing to remember? b Explain in your own words what a representative sample is. Summary checklist I can choose a method to collect data to answer a number of questions and justify my choice. I can decide on the best way to select a sample and justify my choice. In this section you will … • understand the advantages of different sampling methods. The word ‘population’ usually refers to the people living in a town or country. In a statistical investigation, however, it means the people you are interested in. If you are investigating your class, the population means the people in your class. If you are investigating your school, the population is all the learners in your school. Often you cannot question the whole population. In this case, you need to choose a sample. There are different ways to choose a sample. In any investigation you need to decide on the best way to choose your sample. Sometimes, your investigation is not about people. For example, you might be investigating the traffic going past your school. If you collect data about some of the vehicles you are still taking a sample. In this case, the population is all the vehicles passing your school. 6.2 Sampling Key word population We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
133 6.2 Sampling Worked example 6.2 A palace has visitors every day. You are doing a survey to find out what visitors think of the visit. You want to talk to a sample of 100 people. The survey must be done on one day. a What factors might affect a person’s opinion of the visit? b Describe how you can choose a sample. Take account of the factors you identified in part a. Give any advantages and disadvantages of your method. Answer a Age and gender are two factors. b Choose about five different age bands. Choose people arriving and ask them what age band they are in as one of the questions. Make sure you speak to an equal number of men and women. Select people at several different times during the day. An advantage is that this includes a range of people. If you spoke to mostly older people or mostly men, for example, the answers would not represent all the visitors. A disadvantage is that this method will take longer than simply asking the first 100 people you see. You might need to speak to more than 100 people to make sure you cover all the different age bands. Exercise 6.2 1 A manager wants to find customers’ opinions about his shop. The manager wants to choose a sample of 50 customers. a The sample could be the first 50 customers in the shop after it opens. i Write one advantage of this method. ii Write one disadvantage of this method. b The sample could be 10 customers chosen at random every 2 hours until 50 have been chosen. i Write one advantage of this method. ii Write one disadvantage of this method. c The manager thinks that the opinions of men and women could be different. Explain how he should choose the sample to take account of this. d Can you think of another factor that might affect customers’ opinions? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
134 6 Collecting data 3 Zalika is investigating the number of people in each car on a busy road. She predicts that most cars will contain only one person, the driver. Zalika says, ‘I will start at 08:00 and observe 200 cars.’ a Write one advantage and one disadvantage of Zalika’s method. b Describe a better way to take a sample of 200 cars. Explain why your method is better than Zalika’s. 4 You have been asked to carry out an investigation. You want to find out if learners would like to change the school homework policy. You will choose a sample of about 50 learners. Explain how you will choose your sample. Explain why you have chosen this method. 5 Arun and Sofia carry out a survey of parents about school homework. One conjecture is that parents want more homework. To test this, each asks this question of a sample of 50 parents: Is the amount of homework your child gets too little / about right / too much ? (choose one) Think like a mathematician 2 Work with a partner to answer this question. You are going to find the lengths of the words in a novel. Choose a book to use. You want a sample of 50 words. a Describe three different ways of choosing a sample of 50 words. b Use one of your methods from part a to sample 50 words from your chosen novel. Use a tally chart to record the number of letters in each word. c Did your sampling method give you a sample that was representative of the whole book? Could you improve your method? d Try one of your other sampling methods. e Compare your first method with your second method. Can you improve the second method? Was one better than the other? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
6.2 Sampling 135 This chart shows the results: a Do Arun’s results support this conjecture? Give a reason for your answer. b Do Sofia’s results support this conjecture? Give a reason for your answer. c Give a possible reason why the results of the two surveys are different. 6 A large factory has a restaurant where employees go for lunch. Arun is investigating ways to improve the restaurant. a Give one disadvantage of Arun’s method. b Describe a better way of doing the survey. 7 Here is a conjecture about the cars using a particular road: The modal number of people in a car is 1. Marcus, Zara and Sofia each do a survey of cars on the road. They count the number of people in each car, including the driver. Each person does the survey for 15 minutes. They do their surveys at different times of day. The results are in the graph on the right. a Do the results of each survey support this conjecture? b Describe any similarities or differences between the surveys. c Why do the samples give different results? 8 An examiner has marked 250 examination papers. To check the accuracy of her marking, a sample of 10 papers will be re-marked. a Describe three different ways of choosing the sample. b Which of the three ways do you think is the best? Explain why you think so. 5 0 10 15 20 25 Arun Sofia Too little About right Too much Frequency I will give a questionnaire to the first 50 workers visiting the restaurant this lunchtime. Tip The mode is the number with the highest frequency. 10 0 1 2 3 Number of people 4 5 6 20 30 40 50 60 Marcus Zara Frequency Sofia Summary checklist I can describe different sampling methods. I understand the advantages and disadvantages of different sampling methods. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
136 6 Collecting data Check your progress You want to answer these questions: • Are girls or boys better at throwing a ball through a basketball hoop? • Does height make a difference? 1 What data could you collect to answer these questions? What type of data is it? 2 How would you collect the data you need? 3 Explain how you could select a sample from your school. Justify your choice. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
137 Getting started 1 Write the correct symbol,=or ≠, between each pair of fractions. a 5 3 11 3 b 2 1 2 10 4 c 35 6 21 4 2 Write the correct symbol, < or >, between each pair of fractions. a 2 3 5 3 b 2 1 2 9 4 c 2 5 3 7 3 Work these out. Give each answer as a mixed number in its simplest form. a 2 3 2 9 4 9 + b 1 5 2 3 3 4 + 4 Work these out. Give each answer in its simplest form. a 2 3 7 8 × b 2 3 7 8 ÷ 5 Work these out using a method to make the calculation easier. Show all your working. a 1 8 × 600 b 2 5 × 320 c 19 20 × 4000 Fractions are used in everyday life more often than you might think. One important use of fractions is in music. Here is an example of a few bars of music. one bar A bar lasts a particular length of time, measured in a number of beats. Different types of musical note last for different numbers of beats. This means that the number of notes that can fit into each bar depends on the type of notes. Imagine a bar is like a cake. The number of slices (notes) into which it can be cut depends on how large the slices are (how many beats each note lasts for). 7 Fractions We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
138 7 Fractions This table shows the names of some of the different types of note. It also shows the length of time (number of beats) for which each note lasts. You can see that two minims last for the same length of time as one semibreve. You can also see that four semiquavers last for the same length of time as two quavers or one crotchet. In the piece of music below, each bar must contain three beats. Try to think of a combination of notes that would fill the third bar. 1 + 1 + 1 = 3 = 3 + +1 + 1 1 2 1 2 In this section you will … • recognise fractions that are equivalent to recurring decimals. 7.1 Fractions and recurring decimals Key words equivalent decimal improper fraction mixed number recurring decimal terminating decimal unit fraction You already know how to use equivalent fractions to convert a fraction with a denominator that is a factor of 10 or 100 to a decimal. For example: 3 5 6 10 = = 0 6. and 3 20 15 100 = = 0 1. 5 You can also use division to convert a fraction to an equivalent decimal. The fraction 5 8 is ‘five eighths’, ‘five out of eight’ or ‘five divided by eight’. To work out the fraction as a decimal, divide 5 by 8: 5÷8=0.625 The decimal 0.625 is a terminating decimal because it comes to an end. When you convert the fraction 71 99 to a decimal you get: 71÷99=0.71717171… Note Name semibreve minim crotchet quaver semiquaver whole note (1) 4 2 1 Fraction Number of beats half note 1 2 1 2 1 4 quarter note 1 4 eighth note 1 8 sixteenth note 1 16 Tip can be written as can be written as can be written as We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
139 7.1 Fractions and recurring decimals The number 0.71717171… is a recurring decimal as the digits 7 and 1 carry on repeating forever. You can write 0.71717171… with the three dots at the end to show that the number goes on forever. You can also write the number as 0 71. . . , with dots above the 7 and the 1, to show that the 7 and 1 carry on repeating forever. When you convert the fraction 1 14 to a decimal, you get 1÷14= 0.0714285714285714285… You can see that 714285 in the decimal is repeating, so you write this as 0 0714285. . . You put a dot above the 7 and the 5 to show that all the digits from 7 to 5 are repeated. Worked example 7.1 Use division to convert each fraction to an equivalent decimal. a 3 8 b 5 11 c 11 12 Answer a 3÷8=0.375 This answer is a terminating decimal, so write down all the digits. b 5÷11=0.45 This answer is a recurring decimal, so write it as 0.45 c 11÷12=0.916 This answer is a recurring decimal, but only the 6 is recurring, so write 0.916666… as 0.916 . . . . . . Tip You can use a written method or a calculator to do this. Tip A recurring decimal can always be written as a fraction. Exercise 7.1 1 Use a written method to convert these unit fractions into decimals. Write if the fraction is a terminating or recurring decimal. The first two have been done for you. a 1 2 b 1 3 c 1 4 d 1 5 e 1 6 f 1 7 g 1 8 h 1 9 i 1 10 j 1 11 k 1 12 Tip A unit fraction has a numerator of 1, e.g. 1 2 1 3 1 4 , , , ... Tip In part f, you will need to keep going with the division for quite a long time! 0 . 5 2 1 . 1 0 1 2 =0.5 Terminating decimal 0 . 3 3 3 ... 3 1 . 1 0 1 0 1 0 1 3 =0.3. Recurring decimal We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
140 7 Fractions Think like a mathematician 2 Work with a partner or in a small group to answer these questions. a Copy and complete this table. Use your answers to Question 1. Unit fraction 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 Decimal 0.5 0.3 Terminating (T) or recurring (R) T R b Read what Zara says. i Do you think Zara is correct? Test her idea on 1 16 and 1 32 Explain your decisions. ii What other patterns can you see in the table in Question 1? Test your ideas to see if they work. iii Discuss your ideas with other groups of learners in your class. . Tip 24=16, 25=32 The denominators of the fractions 1 2, 1 4 and 1 8 are all powers of 2. The powers of 2 are 21=2, 22=4, 23=8, etc. There is a pattern in the equivalent decimals for 1 2, 1 4 and 1 8. They are all terminating decimals. The decimals are 0.5, 0.25 and 0.125. The pattern is 0.5, 0.25, 0.125. I think that all unit fractions with a denominator which is a power of 2 will be a terminating decimal that ends in 25, apart from 1 21 which just ends in 5. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
141 7.1 Fractions and recurring decimals 3 Here are five fraction cards. A B C D E a Without doing any calculations, do you think these fractions are terminating or recurring decimals? Explain why. b Use a written method to convert the fractions to decimals. c Write the fractions in order of size, starting with the smallest. 4 Here are five fraction cards. A B C D E a Without doing any calculations, do you think these fractions are terminating or recurring decimals? Explain why. b Use a written method to convert the fractions to decimals. c Write the fractions in order of size, starting with the smallest. 3 4 7 10 11 20 3 5 5 8 2 3 7 12 5 9 3 11 5 6 Think like a mathematician 5 Work with a partner or in a small group to discuss these questions. Maddie converts four fractions to recurring decimals on her calculator. These are the answers she gets. 0.111111111 0.733333333 0.888888889 0.388888889 a Why has the calculator put a 9 at the end of two of the decimals? b Match each of the decimals to its equivalent fraction: 7 18, 1 9, 11 15, 8 9 When Maddie converts the fraction 5 18 on her calculator, this is what she types. 5 ÷ 1 8 = The answer she gets is: 5 18 Maddie then presses the button: S ⇔ D c What does this button do to the fraction? What happens when you press the same button again? d Use a calculator to work out the decimal equivalent of i 7 15 ii 8 11 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
142 7 Fractions 6 Use a calculator to convert these fractions to decimals. a 7 9 b 13 20 c 2 15 d 9 40 7 Marcus and Sofia are discussing the fraction 5 13. My calculator tells me that 5÷13= 0.38461538, so I think that 5 13 is a recurring decimal which I can write as 0.384615 . . I don’t think the calculator shows you enough decimal places to decide it is a recurring decimal. What do you think? Explain your answer. 8 Use a calculator to convert these fractions to recurring decimals. a 2 7 b 9 13 c 11 14 9 This is part of Kim’s homework. Question Write these fractions as decimals. i 5 12 ii 10 11 iii 6 7 iv 1 37 Answer i 5 12=0.416. ii 10 11 =0.9. 0 iii 6 7 =0.85. 7142. iv 1 37=0.02. 7 . a Use a calculator to check Kim’s homework. b Explain any mistakes she has made and write the correct answers. 10 Without using a calculator, write these fractions as decimals. a 4 3 b 13 6 c 19 9 d 45 11 Tip Remember, when several digits repeat in the decimal, you only put a dot over the first and the last digit of the sequence that repeats, e.g. 1 7=0.142857 . . Tip Change the improper fractions into mixed numbers first. Then use your answers to Question 1 to help. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
143 7.1 Fractions and recurring decimals 11 This is part of Ada’s homework. Question Write 2 hours and 10 minutes as a recurring decimal. Answer 10 minutes is the same as 10 60= 1 6 of an hour and 1 6=0.16. So, 2 hours and 10 minutes=2.16. hours. Use Ada’s method to write these lengths of time as recurring decimals. a 4 hours 20 minutes b 1 hour 40 minutes c 6 hours 10 minutes d 3 hours 50 minutes 12 Rajim has 8 weeks holiday a year. There are 52 weeks in a year. What fraction of the year does he have on holiday? Write your answer as a decimal. 13 Sasha is told that 1 15=0.06 and that 1 22 =0.045 Without using a calculator, she must match each yellow fraction card with the correct blue decimal card. 4 15 7 22 0.26. 0.318. . Sasha thinks that 4 15=0.26 and that 7 22 =0.318 Do you think she is correct? Explain your answer. . . . . . . In this lesson, you have used a lot of mathematical words and phrases. Write a short explanation, in your own words, for each of the following terms. a terminating decimal b recurring decimal c unit fraction d equivalent fractions and decimals Summary checklist I can recognise fractions and recurring decimals. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
144 7 Fractions In this section you will … • compare and order positive and negative fractions. 7.2 Ordering fractions When you write fractions in order of size, you must first compare them. You can compare fractions in two ways. 1 Write them as fractions which have the same denominator. 2 Write them as decimals. Key words advantages disadvantages improve Worked example 7.2 a Write these fractions in order of size, starting with the smallest: 7 3, 8 9 and 12 5 b Use decimals to decide which is smaller: −3 4 7 or − 43 12 Answer a 7 3 1 3 = 2 and 12 5 2 5 = 2 First, write the improper fractions as mixed numbers. 8 9 , … , … 8 9 is smaller than 2 1 3 and 2 2 5 so write that down first. 2 2 1 3 5 15 = and 2 2 2 5 6 15 = Now compare the other two fractions by writing them with a common denominator of 15. 8 9 , 2 1 3 , 2 2 5 2 5 15 < 2 6 15 so 2 1 3 is smaller than 2 2 5 b −3 4 7 and − = − 43 12 7 12 3 First of all, write any improper fractions as mixed numbers. 4 7 =4÷7 Use division to work out 4 7 as a decimal. 0 5 7 1 4 7 4 0 0 0 0 4 5 1 3 . . ... −3 4 7 as a decimal is −3.5714… 7 12=7÷12 Use division to work out 7 12 as a decimal. 0 5 8 3 12 7 0 0 0 7 10 4 . ... . −3 7 12 as a decimal is −3.583… −3.583… < −3.5714… As the numbers are negative, −3.58 is smaller than −3.57 − 43 12 < −3 4 7 Finally, write the answer using the original fractions given in the question. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
145 7.2 Ordering fractions Exercise 7.2 For questions 1 to 4, use the common denominator method. 1 This is part of Seren’s homework. She uses the symbol=to show that one fraction is equal to another. She uses the symbol ≠ to show that one fraction is not equal to another. Question Write the correct sign,=or ≠, between each pair of fractions. a 5 2 2 7 14 b – 11 3 –3 4 9 Answer a 5 2 =21 2 and 2 7 14= 21 2 so 5 2 =2 7 14 b – 11 3 =–3 2 3 =–3 6 9 so – 11 3 ≠ –3 4 9 Write the correct sign,=or ≠, between each pair of fractions. a 11 4 216 20 b 45 6 7 1 2 c −15 8 −2 1 8 d −8 4 5 −132 15 2 Write the correct symbol, < or >, between each pair of fractions. Parts a and e have been done for you. a 13 2 6 5 8 Working: 13 2 1 2 4 8 = = 6 6 and 6 4 8 < 6 5 8 Answer: 13 2 < 6 5 8 b 17 3 6 7 12 c 5 3 5 82 15 d 19 4 4 4 5 e −17 4 −4 5 12 Working: − =− =− 17 4 1 4 3 12 4 4 and −4 3 12 > −4 5 12 Answer: −17 4 > −4 5 12 f − 7 3 −2 5 9 g − 21 5 −4 2 15 h − 8 5 −15 7 Tip Change the improper fractions to mixed numbers first. Then compare the fractions by using a common denominator. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
146 7 Fractions 3 Zara and Sofia compare the methods they use to work out which is larger, − 8 3 or −2 4 7 They both use this method to start with. Step 1: Write both fractions as mixed numbers first: – 8 3 =–2 2 3 Step 2: Write –2 2 3 and –2 4 7 using a common denominator of 21: –2 2 3 =–2 14 21 and –2 4 7 =–2 12 21 Read what Zara and Sofia do next. My step 3 is: Without the negative signs 2 2 14 21 12 21 > So with the negative signs − < 2 2− 14 21 12 21, so − 8 3 < −2 4 7 a Write the advantages and disadvantages of each method. b Can you improve either method? c What is your preferred method for comparing negative fractions? Explain why. 4 Work out which is larger. a − 7 4 or −113 16 b − 21 5 or − 83 20 c −6 2 9 or − 37 6 My step 3 is to sketch a number line. −3 −2 −1 −214 21 −212 21 I can see that − < 2 2− 14 21 12 21, so − 8 3 < −2 4 7 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
147 7.2 Ordering fractions 6 Write the correct symbol, < or >, between each pair of fractions. a 3 11 5 11 b 7 18 5 18 c 12 7 10 7 d 8 17 8 19 e 9 13 9 10 f 15 4 15 7 Think like a mathematician 5 Work with a partner or in a small group to answer this question. a With each pair of fractions, decide which is larger. i 1 5 or 3 5 ii 7 9 or 5 9 iii 13 11 or 19 11 b Discuss your answers to part a. Copy and complete this sentence. Use either ‘larger’ or ‘smaller’. When the denominators are the same, the larger the numerator the ................... the fraction. c With each pair of fractions, decide which is larger. i 1 5 or 1 7 ii 2 9 or 2 3 iii 13 4 or 13 7 d Discuss your answers to part c. Copy and complete this sentence. Use either ‘larger’ or ‘smaller’. When the numerators are the same, the larger the denominator the ................... the fraction. Think like a mathematician 7 Work with a partner. Discuss different methods you could use to answer this question. Put these fraction cards in order of size, starting with the smallest. − 17 8 −3 1 4 − 13 6 − 7 13 What do you think is the best method to use? Explain why. 8 Put these fraction cards in order of size, starting with the smallest. −31 6 − 9 11 −19 5 −4 2 5 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
148 7 Fractions 9 Three sisters sat a maths test on the same day. Adele scored 16 25, Belle scored 13 20 and Catrina scored 63%. Who had the highest percentage score? 10 Two driving instructors compare the pass rates for their students in January. Steffan had 34 out of 40 students pass. Irena had 87% of students pass. Who had the higher pass rate for their students in January? Show how you worked out your answer. For questions 11 to 13, use the division method. 11 a Complete the workings to write each fraction as a decimal. Work out the first four decimal places. i −11 7 − = − 11 7 4 7 1 0 5 7 1 4 7 4 0 0 0 0 4 5 1 3 . . 4 7 = −14 7 =−1.571… ii −14 9 − = − 14 9 5 9 1 0 5 555 9 5 0000 5555 . . 5 9 = −15 9 = iii −19 12 − = − 19 12 7 12 1 0 5 8 3 3 12 7 0 000 7 1044 . . 7 12= −1 7 12 = b Write the fractions −11 7 , −14 9 and −19 12 in order of size, starting with the smallest. 12 a Match each fraction with the correct decimal. −4.18 −4.27... −4.16... −4.11... − 37 9 − 25 6 − 209 50 − 47 11 b Write the fractions − 37 9 , − 25 6 , − 209 50 and − 47 11 in order of size, starting with the smallest. 13 Write these fractions in order of size, starting with the smallest. −107 20 − 37 7 −53 8 − 82 15 Tip Change the fractions into percentages by writing equivalent fractions with a denominator of 100. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
149 7.2 Ordering fractions 14 One day, a farmer sells 92% of her eggs. The following day, she sells 56 out of 62 eggs. Use a calculator to work out on which day she sold the greater percentage of eggs. 15 Arun takes two English tests. In the first test he scores 65 72. In the second test he scores 35 38 Read what Arun and Sofia say. If I compare my scores using a common denominator, I will have to use a common denominator of 1368! I think you should use a calculator and change your scores into decimals or percentages. Then it will be easy to compare your scores. a Use Arun’s method to compare the scores. b Use Sofia’s method to compare the scores. c Which method do you prefer and why? d In which test did Arun get the better score? 16 In a science experiment, two groups of seeds are planted. In group A, 175 seeds are planted and 156 start to grow. In group B, 220 seeds are planted and 189 start to grow. Use a calculator to work out which group is better at growing. 17 Li has 5 improper fraction cards. He puts them in order, starting with the smallest. There are marks on two of his cards. − 20 7 − 25 9 −13 5 What fractions could be under the marks? Give two examples for each card. Tip Change 56 62 into a decimal, then multiply the answer by 100 to get a percentage. Tip Change 156 175 and 189 220 into decimals or percentages to compare. Summary checklist I can compare and order fractions. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE