CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 49 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 d Learner’s own discussions. For example: Yes, for all of them there is more than one combined transformation. Each object can be rotated about any point to get it in the same orientation as the image, and then you can use a translation to move it into the correct position. In part i, you could also use a refection in any vertical or horizontal line and then you can use a translation to move it into the correct position. 9 a They are both correct. When you start with triangle G and follow Sofa’s instructions, the fnal image is triangle H. When you start with triangle G and follow Zara’s instructions, the fnal image is triangle H. b For example: Refection in the line x=3 then translation − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 6 2 . For example: Translation − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 8 2 then refection in the line x=−4. c There are an infnite number of combined transformations. Learner’s own explanation. For example: G can be refected in any line x=‘a number’ then translated to H. 10 a i Learner’s own diagram. Shape B with vertices (6, 4), (8, 5), (8, 2) and (6, 2). Shape C with vertices (2, 5), (4, 6), (4, 8) and (2, 8). ii Refection in the line y=5. b i Learner’s own diagram. Shape D with vertices (5, 8), (8, 8), (8, 10) and (6, 10). Shape E with vertices (2, 5), (5, 5), (5, 7) and (3, 7). ii Rotation 90 ° anticlockwise, centre (2, 5). Activity 13.3 Learner’s own answers and discussions. 11 a A to B b A to C c B to D d C to E Reflection: a It is the same shape and size. b • corresponding lengths are equal • corresponding angles are equal • the object and the image are congruent Exercise 13.4 1 2 a Scale factor 2 b Scale factor 3 c Scale factor 4 3 a Learner’s own explanation. For example: She hasn’t enlarged the shape correctly from the centre of enlargement. She has incorrectly used the centre as one of the vertices of the triangle.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 50 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 b Activity 13.4 Learner’s own enlargements and discussions. 4 a Learner’s own diagram. Check that the shape has been enlarged correctly. Vertices of the image should be at (1, 7), (5, 7), (5, 3) and (1, 3). b Learner’s own diagram. Check that the shape has been enlarged correctly. Vertices of the image should be at (2, 6), (8, 6), (8, 0) and (2, 0). c Learner’s own diagram. Check that the shape has been enlarged correctly. Vertices of the image should be at (1, 9), (9, 9), (9, 1) and (1, 1). 5 a i Perimeters: A=8 cm, B=16cm, C=24 cm and D=32cm ii Areas: A=4 cm2 , B=16cm2 , C=36 cm2 and D=64cm2 b Squares Scale factor of enlargement Ratio of lengths Ratio of perimeters Ratio of areas A:B 2 1:2 1:2 1:4=1:22 A:C 3 1:3 1:3 1:9=1:32 A:D 4 1:4 1:4 1:16=1:42 c ratio of lengths=ratio of perimeters. d ratio of lengths squared=ratio of areas. e Yes. Yes. f Learner’s own discussions. 6 Perimeter of R=14cm → Perimeter of T=14×3=42 cm Area of R=10cm2 → Area of T=10×32=90cm2 7 Perimeter=60cm, Area=150 cm2 8 Shape G is an enlargement of shape F, scale factor 3 and centre of enlargement at (1, 2). 9 a Enlargement scale factor 2, centre (−5, 2). b Enlargement scale factor 4, centre (−6, −2). 10 Enlargement scale factor 3, centre (4, −5). 11 Learner’s own answers and justifcation. For example: Arun is incorrect. When one shape is an enlargement of another, and the centre of enlargement is inside the shapes, you can use ray lines to fnd the centre of enlargement. 12 Enlargement scale factor 3, centre (6, 5). 13 Enlargement scale factor 2, centre (4, 4). Check your progress 1 a N N 12 cm (120km) 9cm (90km) 140° 50° b Answer in range 148km–152km (accurate answer 150km). c Answer in the range 264 °–270 ° (accurate answers 267 ° to 3 s.f.) 2 a (5, 3) b (6, 10) 3 L (4, 10) 4 a i Learner’s own diagram. The vertices of triangle B should be at (3, 3), (5, 3) and (4, 4). ii Learner’s own diagram. The vertices of triangle C should be at (3, 3), (4, 2) and (4, 4). b i Rotation of 180 °, centre (3, 4). ii Rotation 90° anticlockwise, centre (2, 3).
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 51 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 5 6 Scale factor 3, centre of enlargement at (10, 4). 7 Perimeter=54cm and area=180 cm2 . Unit 14 Getting started 1 25.13cm 2 a 27mm2 b 21cm2 c 78.5m2 3 a 120cm3 b 158cm2 4 a 480cm3 b Learner’s own diagram. Any correct net. c 528cm2 5 a 1 b 2 c 6 d 0 Exercise 14.1 1 a 120cm3 b 130cm3 c 134.4cm3 2 Area of cross-section Length of prism Volume of prism a 12cm2 10cm 120cm3 b 24cm2 8.5cm 204cm3 c 18.5m2 6.2m 114.7m3 3 a Learner’s own explanation. For example: Yusaf hasn’t used the correct crosssection. Instead of using the trapezium as the cross-section, he has used the side rectangle (which is not the cross-section of the prism). b Area of trapezium= 1 2 ( ) 8 1 + 4 4 × = 44 cm2 Volume of prism=44×20=880cm3 4 Learner’s own answers. For example: a Yes. The cross-section is a circle. b Area of circle×height c V=πr2 h d Learner’s own discussions. 5 Learner’s own explanation. For example: The radius and height are in different units. She needs to change the 5mm tocm or change the 2cm tomm before she works out the volume. Volume=1570mm3 (3 s.f.) or 1.57cm3 (3 s.f.) 6 a 942.5 cm3 b 353.4 cm3 c 17592.9mm3 Activity 14.1 Learner’s own cylinders, answers and discussions. 7 Radius of circle Area of circle Height of cylinder Volume of cylinder a 2.5m 19.63m2 4.2m 82.47m3 b 6cm 113.10cm2 4.48cm 507cm3 c 2.52m 20m2 2.5m 50m3 d 4.56mm 65.25mm2 16mm 1044mm3 8 a 5.5 cm b 4.2 cm c 2.1cm 9 Learner’s own methods and answers. For example: Volume of cylinder: V=πr2 h=π×62 ×18 =2035.75cm3 (2 d.p.) Volume of cube: V=83=512cm3 Volume of water: 1.5 litres=1500mL=1500cm3 Volume of cube+1.5 litres=512+1500 =2012cm3 The total volume of the cube and water is less than the volume of the cylinder, so the water will not come over the top of the cylinder. 2012cm3<2035.75 cm3 Reflection: Learner’s own explanations.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 52 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 Exercise 14.2 1 Answer using rounded intermediate values: Area of circle = cm (2 d.p.) π π r 2 2 2 5 78 54 = × = . Circumference of circle cm (2 d.p.) = = = π π d ×10 31.42 Area of rectangle = 31.42 12 = 377.04 cm (2 d.p.) × 2 Total area = 2 78.54 + 377.04 = 534 cm (3 s.f.) × 2 Answer using accurate intermediate values: Area of circle = cm π π r 2 2 2 5 78 5398 = × = . ... Circumference of circle cm = = × = π π d 10 31.4159... Area of rectangle = 31.4159... 12 = 376.9911... cm × 2 Total area = 2 78.5398... + 376.9911... = 534 cm (3 s.f. × 2 ) 2 a SA=477.5 cm2 b SA=322.0 cm2 c SA=4272.6mm2 3 The pyramid has a greater surface area than the cylinder. 132 cm2>125.66 cm2 . Pyramid: SA = × × × cm ⎛ ⎝ ⎜ ⎞ ⎠ 4 6 8 6 ⎟ + × 6 = 132 1 2 2 Cylinder: SA=π×22×2+π×4×8=125.66 cm2 4 Learner’s own methods and answers. For example: a SA=πr2+πr2+2πrh b SA=πr2+πr2+2πrh=2πr2+2πrh= 2πr(r+h) c SA=2πr(r+h)=2πr(r+2r)=2πr×3r= 6πr2 d i SA=8πr2 ii SA=10πr2 iii SA=12πr2 e Add 1 to the number in front of the r, then double it. This gives you the number in front of the πr2 . So, 19+1=20, 20×2=40, so SA=40πr2 . f Learner’s own discussions. 5 226cm2 (3 s.f.) 6 Learner’s own methods and answers. For example: a i The hypotenuse of the triangular cross-section. ii Pythagoras’ theorem b Learner’s own discussions. c 408cm2 7 a SA=660cm2 b SA=1188mm2 c SA=23.3m2 Activity 14.2 a,b Learner’s own shapes. For example: A cuboid with length 10cm, width 10cm and height 8 cm (V=800 cm3 , SA=520cm2 ); A triangular prism of length 33cm with a right-angled cross-section with base length 6cm, height 8cm and hypotenuse 10 cm (V=792cm3 , SA=840cm2 ); A cylinder with height 16cm and cross-section radius 4 cm (V=804cm3 , SA=503cm2 ). c Learner’s own answers and explanations. d Learner’s own discussions. 8 754cm2 9 15 labels is the maximum using Method 1 below. Method 1: 120÷23.6=5 whole lengths 35÷10=3 whole lengths Number of labels=5×3=15 Method 2: 120÷10=12 whole lengths 35÷23.6=1 whole length Number of labels=12×1=12
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 53 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 Exercise 14.3 1 a, b and c Learner’s own drawings. Check that the planes of symmetry are drawn correctly. Shapes a and b have vertical planes of symmetry. Shape c has a horizontal plane of symmetry. 2 a, b Learner’s own drawings. Check that the planes of symmetry are drawn correctly. Shape a has one vertical and one horizontal plane of symmetry. Shape b has two vertical and one horizontal plane of symmetry. 3 a, bLearner’s own drawings. Check that the plane of symmetry is drawn correctly. The plane of symmetry should be vertical. c The plane of symmetry is a vertical plane of symmetry. 4 a, bLearner’s own lines of symmetry. Any of these: c A cube has a total of nine planes of symmetry. d Learner’s own justifcation. All nine diagrams shown in the answer to part b. e Learner’s own discussions. 5 a There are two vertical and one horizontal planes of symmetry. b 6 a 2D regular polygon Number of lines of symmetry 3D prism Number of planes of symmetry Triangle 3 Triangular 4 Square 4 Square 5 Pentagon 5 Pentagonal 6 Hexagon 6 Hexagonal 7 Octagon 8 Octagonal 9 b Learner’s own answers and explanations. For example: Number of planes of symmetry=number of lines of symmetry+1. This happens because the planes of symmetry can be drawn, the length of the prism, in the same place as the lines of symmetry on the cross-section of the prism. There is then the extra plane of symmetry that divides the prism halfway along its length. c i 11 ii 13 d Learner’s own discussions. 7 a,b Learner’s own diagram. Check that the plane of symmetry passes through the circular ends of the cylinder, dividing the circular cross-section into two identical semi-circles. c Learner’s own diagram. Check that the plane of symmetry passes halfway along the height, splitting the cylinder into two identical cylinders. d Learner’s own answers and explanations. For example: It has an infnite number of planes of symmetry. A circle has an infnite number of lines of symmetry, so this is the same in 3D for the cylinder. When the cylinder is placed upright there is always one horizontal plane of symmetry, but an infnite number of vertical ones. Reflection: Learner’s own answers.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 54 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 Check your progress 1 120cm3 2 14m2 3 452cm3 4 The square-based pyramid has the greater surface area. Pyramid: SA=340 cm2 , Cylinder: SA=320.44cm2 , 340>320.44 5 a The shape has two vertical, one horizontal and two diagonal planes of symmetry. b Learner’s own diagrams showing the fve planes of symmetry correctly as described in the answer to part a. Unit 15 Getting started 1 a Age, a (years) Frequency 10<a⩽15 3 15<a⩽20 6 20<a⩽25 7 25<a⩽30 4 b Learner’s own diagram. Frequency diagram showing the data in part a. Make sure the axes are labelled correctly and that a sensible scale is used. Make sure the bars are the correct width and height. c 11 2 a Class 9P test results 0 1 2 3 4 2 4 6 7 8 9 2 3 4 4 6 7 8 3 8 9 0 1 6 8 9 9 9 0 0 Key: 0 3 means 03 marks b 32% c 1 5 d 14 3 a Mean Median Mode Range History 12.9 13 16 7 Chemistry 14 16 18 15 b The Chemistry group has better marks on average, because the mean, median and mode are all greater than for the History group. c The History group has more consistent marks because the range is lower. Exercise 15.1 1 a Height, h (cm) Frequency Midpoint 140⩽h<150 7 145 150⩽h<160 13 155 160⩽h<170 6 165 170⩽h<180 2 175 b Learner’s own diagram. Frequency polygon with points (145, 7), (155, 13), (165, 6) and (175, 2) joined with straight lines. Make sure that the axes are labelled correctly and that a sensible scale is used. 2 a Mass, m (kg) Frequency Midpoint 40⩽m<50 4 45 50⩽m<60 12 55 60⩽m<70 8 65 b Learner’s own diagram. Frequency polygon with points (45, 4), (55, 12) and (65, 8) joined with straight lines. Make sure that the axes are labelled correctly and that a sensible scale is used. c 24 d 2 3 e Arun is incorrect. Learner’s own explanation. For example: You do not know how heavy the heaviest student is. You only know that their mass is in the interval 60kg⩽m<70kg.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 55 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 3 a Learner’s own frequency table. For example: Age, a (years) Frequency 10⩽a<25 6 25⩽a <40 9 40⩽a<55 7 55⩽a<70 4 70⩽a<85 2 b Learner’s own diagram. Frequency polygon with points (17.5, 6), (32.5, 9), (47.5, 7), (62.5, 4) and (77.5, 2) joined with straight lines. Make sure that the axes are labelled correctly and that a sensible scale is used. c Learner’s own discussions. 4 Learner’s own frequency tables and polygons. For example: a Time, t (minutes) Frequency 10⩽a<20 4 20⩽a <30 8 30⩽a <40 9 40⩽a<50 3 b Learner’s own diagram. Frequency polygon with points (15, 4), (25, 8), (35, 9) and (45, 3) joined with straight lines. Make sure that the axes are labelled correctly and that a sensible scale is used. 5 a 50 at each surgery. Oaklands Surgery Time, t (minutes) Frequency Midpoint 0⩽t<10 25 5 10⩽t<20 10 15 20⩽t<30 12 25 30⩽t<40 3 35 Birchfields Surgery Time, t (minutes) Frequency Midpoint 0⩽t<10 8 5 10⩽t<20 14 15 20⩽t<30 17 25 30⩽t<40 11 35 c Learner’s own diagram. Two frequency polygons drawn on one grid. Oaklands points (5, 25), (15, 10), (25, 12) and (35, 3) joined with straight lines. Birchfelds points (5, 8), (15, 14), (25, 17) and (35, 11) joined with straight lines. Make sure that the axes are labelled correctly and that a sensible scale is used. d Learner’s own comments. For example: Over three times as many people waited less than 10 minutes in Oaklands surgery compared to Birchfelds surgery. More people waited over 10 minutes in Birchfelds surgery compared to Oaklands surgery. 6 a,b Learner’s own comments. For example: Using Sofa’s method you don’t need to work out the midpoints. When you have drawn the bars it is easy to join the midpoint of each bar with straight lines. Her method will take longer though, as you have to draw all the bars frst. Using Zara’s method is quicker as you don’t have to draw all the bars, but you do need to work out the midpoints, and if you make a mistake with one of the midpoints you might not notice when you plot the point. c Learner’s own discussions. 7 a Learner’s own diagram. Frequency polygon with points (5, 2), (15, 4), (25, 8) and (35, 6) joined with straight lines. Make sure that the axes are labelled correctly and that a sensible scale is used. b Learner’s own comments. For example: The plants that were grown in the greenhouse grew higher than the plants that were grown outdoors. 14 of the plants grown in the greenhouse were over 20cm tall, whereas only six of the plants grown outdoors were over 20 cm tall. 8 a Learner’s own diagram. Two frequency polygons drawn on one grid. Boys’ points (2, 5), (6, 10), (10, 15), (14, 7) and (18, 3) joined with straight lines. Girls’ points (2, 7), (6, 8), (10, 12), (14, 18) and (18, 5) joined with straight lines. Make sure that the axes are labelled correctly and that a sensible scale is used. b Learner’s own comments. For example: More girls spend between 0 and 4 and between 12 and 20 hours doing homework each week. More boys spend between 4 and 12 hours doing homework each week.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 56 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 c 40 boys and 50 girls d Learner’s own comments. For example: No, as there were 10 more girls than boys surveyed. There should have been the same number of boys and girls in order to make a fair comparison. 9 a Learner’s own diagram. Frequency polygon with points (200, 5), (220, 8), (240, 11), (260, 7), (280, 5) and (300, 4) joined with straight lines. Make sure that the axes are labelled correctly and that a sensible scale is used. b i Length, l (cm) Frequency 190⩽l<230 13 230⩽l<270 18 270⩽l<310 9 ii Learner’s own diagram. Frequency polygon with points (210, 13), (250, 18) and (290, 9) joined with straight lines. Make sure that the axes are labelled correctly and that a sensible scale is used. c Learner’s own answers and explanations. For example: The frst frequency polygon gives you better information because there are more groups so it shows you more information on the lengths of the turtles. The second frequency polygon only has three groups so less information can be taken from the graph. d i 12 ii No, Arun cannot fll in the correct frequencies in his table. Learner’s own explanation. For example: From the frst table Arun knows that there are fve turtles between 190 and 210cm. But this does not tell him how many turtles there are between 190 and 200cm and how many turtles there are between 200 and 210cm, so it is impossible for him to complete his table. He would have to fnd the original data, before it was grouped, in order to use the groups he wants to. Exercise 15.2 1 a Learner’s own scatter graph. Horizontal axis showing ‘Hours doing homework’ and vertical axis showing ‘Hours watching TV’. Points (14, 7), (11, 12), (19, 4), (6, 15), (10, 11), (3, 18), (9, 15), (4, 17), (12, 8), (8, 14), (6, 16), (15, 7), (18, 5), (7, 16) and (12, 10) plotted. Make sure that the axes are labelled correctly and that a sensible scale is used. b Negative correlation. The more time the student spends doing homework, the less time they spend watching TV. c Student’s line of best ft. Strong negative correlation. d Correct answer from learner’s line of best ft. Answer should be within range 16–17. 2 a Learner’s own answer and explanation. b Learner’s own scatter graph. Horizontal axis labelled ‘Maximum daytime temperature’ and shown from 25 to 35. Vertical axis labelled ‘Number of cold drinks sold’ and shown from 20 to 40. Points (28, 25), (26, 22), (30, 26), (31, 28), (34, 29), (32, 27), (27, 24), (25, 23), (26, 24), (28, 27), (29, 26), (30, 29), (33, 31) and (27, 23) plotted. c Positive correlation. The higher the temperature, the more cold drinks were sold. d Learner’s own answer. e Learner’s own line of best ft. f Learner’s own comments. For example: It is not possible to predict from a line of best ft a value higher or lower than the data given, as there are no data to show that the correlation is the same after or before these points. With a temp of 44°C the store might not sell many drinks as people might not go outside in that temperature. g Learner’s own discussions. 3 a Learner’s own answer and explanation.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 57 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 b Learner’s own scatter graph. Horizontal axis labelled ‘History result’ and shown from 0 to 100. Vertical axis labelled ‘Music result’ and shown from 0 to 100. Points (12, 25), (15, 64), (22, 18), (25, 42), (32, 65), (36, 23), (45, 48), (52, 24), (58, 60), (68, 45), (75, 68), (77, 55), (80, 42), (82, 32) and (85,76) plotted. c No correlation. Getting a good result in one subject does not mean a student will get a good, or bad, result in the other subject. d Learner’s own answer and explanation. 4 a Strong positive correlation. b 6km in 16 minutes. Learner’s own explanation. For example: It should have taken less time, so the taxi might have been delayed in traffc. 5 a Learner’s own answers. b Learner’s own answers. For example: Try to get an equal number of points on either side of the line (not always possible). The line can go through some of the points. Make lines long enough to go through all the data, don’t make the lines too short. Work out the mean of the data and make the line go through this point. c Learner’s own discussions. d It is not a good idea to use the line of best ft to make predictions outside the range of the data, because you do not know what happens beyond the data you are given. It could be that after a body length of 60 cm, a bird’s wingspan hardly changes in length. 6 a 70 80 90 100 110 120 130 140 150 18 20 22 24 26 28 30 Number of fsh Temperature (ºC) Number of fsh at different points in the Red Sea b Weak negative correlation. c Learner’s own line of best ft, and correct answer from their line, for number of fsh when the temperature is 27°C. Answers should be within range 74–78. d It is not a good idea to use the line of best ft to predict the number of fsh in the Red Sea when the temperature of the sea is 30°C, 35 °C or even higher, because you do not know what happens beyond the data you are given. There may be no fsh at 30°C and the number cannot keep dropping after that. e Learner’s own answers. Reflection: Learner’s own answers. 7 a Learner’s own explanation. For example: It is a coincidence that the graph shows a positive correlation. In a school the older learners might have longer feet, and they might be better at maths as they have been in school longer than the younger students, but they might not. Also, when your feet stop growing, it doesn’t mean that you are going to stop getting better at maths. Your ability in maths does not depend on the length of your foot. Your ability in maths depends on how hard you work. b Learner’s own discussions.
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 58 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 Exercise 15.3 1 a Shop A Shop B 0 1 2 3 8 9 9 6 3 2 5 6 0 1 2 9 5 4 7 7 8 9 1 4 3 9 9 0 7 7 4 6 2 8 1 0 Key: 9 0 means 9 years old Key: 3 6 means 36 years old b Shop A Shop B 0 1 2 3 8 9 9 0 1 2 2 3 4 6 6 4 5 7 7 9 9 8 9 4 3 1 9 8 7 7 6 4 2 0 1 0 Key: 9 0 means 9 years old Key: 3 6 means 36 years old c Learner’s own checks. d i Shop A ii Shop B e Learner’s own answers. For example: Shop A sells clothes for younger people and shop B sells clothes for older people. 2 a Key: For the Beach car park, 5 4 means 45 ice-creams For the City car park, 3 0 means 30 ice-creams Beach car park City car park 3 4 5 6 0 4 9 2 5 5 5 7 8 8 9 9 7 7 6 4 6 9 7 6 6 6 5 2 2 1 0 0 b c Learner’s own answers. For example: On average the vendor had better sales at the Beach car park. Their median was higher. This shows that 50% of their daily sales were 57 ice-creams or more, compared to only 46 for the City car park. Their mode was also higher. The range was smaller, showing that their sales were more consistent. However, it was at the City car park where they had their highest daily sale of 69 ice-creams. d Learner’s own answers. For example: No. The vendor’s sales were better at the Beach car park as they had a higher median and mode and sales were more consistent. e Learner’s own discussions. i Mode ii Median iii Range Beach car park 46 57 17 City car park 45 46 39
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 59 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 3 a i Mode ii Median iii Range iv Mean Boys’ times 17.4s 16.3s 2.9s 16.56s Girls’ times 16.8s 17.5s 4s 17.72s b Learner’s own answers. For example: On average the boys ran faster than the girls, as their mean and median were lower. The girls had the fastest modal time, but they had a larger range showing that their times were more varied than the boys. c Learner’s own answers. For example: No, as the girls’ mean and median are both slower. This shows that on average the boys are faster. 4 a A 1 4 , B 2 3 b A 25%, B 0% c The variation is the same for A and B. They both have a range of 31 g. d A: mean=408.83. g, median=409g B: mean=395.6. g, median=395g e Learner’s own answers. For example: Location A because on average the mass of the hedgehogs is greater. 5 a Website A Website B 12 13 14 15 16 8 9 4 6 8 4 5 6 7 7 8 6 7 8 9 5 5 5 6 6 8 4 3 0 0 8 7 6 5 5 5 2 1 1 0 9 8 5 3 3 2 2 Key: For Website A, 0 13 means 130 hits For Website B, 12 8 means 128 hits b Learner’s own answers. For example: Website A and Website B both had the same mode and almost the same median. The median for Website B was only one more than Website A, so this average is almost the same. The mean was also very similar with only a difference of 2.8 hits per day. So, on average Website B had slightly more hits than Website A. Website B’s range is a lot higher than Website A, showing that the number of hits it had per day varied a lot more. c Learner’s own answers. For example: Neither website appears to be better. Website A was more consistent. Website B was only slightly better on average than Website A. d Learner’s own discussions. Exercise 15.4 1 a i 150cm⩽h<160cm ii 150cm⩽h<160cm b Learner’s own explanation. For example: You can only give the modal class and class where the median lies, because the data is grouped and you don’t know the individual values. c 40cm d Midpoint Frequency Midpoint×frequency 145 7 145×7=1015 155 13 155×13=2015 165 6 165×6=990 175 2 175×2=350 Totals: 28 4370 Estimate of mean = cm 4370 28 = 156 2 a i 50kg⩽m<60kg ii 50kg⩽m<60kg b i 56.6. kg or 57kg ii 30kg c Learner’s own explanation. For example: Answers are estimates because the data is grouped and you do not know the individual values. Mode Median Range Mean Website A 145 147 31 147.1 Website B 145 148 41 149.9
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 60 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 3 a Learner’s own answers and explanations. For example: Using the midpoint would be best. If you use the smallest value in each class the estimate of the mean will be too low, because not all the values will be the smallest value. If you use the highest value the estimate of the mean will be too high, because not all the values will be the highest value. b Learner’s own discussions. 4 a 40 at The Heath and 50 at Moorlands. b Hospital Modal class interval Class interval where the median lies Estimate of mean The Heath 10 ⩽ t<20 minutes 10 ⩽ t<20 minutes 17.25 minutes Moorlands 0 ⩽ t<10 minutes 20 ⩽ t<30 minutes 19.4 minutes c Learner’s own answers. For example: The modal class interval is lower for Moorlands than The Heath, but the class interval containing the median is lower for The Heath than Moorlands. The mean is just over 2 minutes less waiting time in The Heath than Moorlands. d Learner’s own answers. For example: The Heath, because the mean is lower and the median is lower. Even though the modal group is lower at Moorlands, on average I think waiting times will be less at The Heath. 5 a 2 b 13 c Mean=7.15, Median=8, Mode=3 d Table A Score Tally Frequency 2–4 llll l 6 5–7 lll 3 8–10 llll lll 8 11–13 lll 3 Table B Score Tally Frequency 2–5 llll l 6 6–9 llll lll 8 10–13 llll l 6 e Modal class interval Class interval where the median lies Estimate of mean Table A 8–10 8–10 7.2 Table B 6–9 6–9 7.5 f i Learner’s own answers. For example: When there are more groups, the estimate of the mean is closer to the accurate mean. ii Learner’s own answers. For example: The accurate median lies in both the class intervals containing the median. iii Learner’s own answers. For example: The accurate modal value is 3, but this isn’t refected at all in either of the modal class intervals, which are totally different. g Learner’s own discussions. Activity 15.4 a 1 b 36 c–i Learner’s own data, tables, answers and discussions. 6 a i 750g ⩽m<800 g ii 750 g⩽m<800 g b i 798g ii 400 g c Mass, m (g) Frequency 600⩽m<700 7 700⩽m<800 19 800⩽m<900 18 900⩽m<1000 6 d i 700 g⩽m<800 g ii 700g ⩽m<800 g e i 796g ii 400 g
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 61 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021 f i Learner’s own answers and explanations. For example: I think the answers in parts a and b are the more accurate answers because the groups are smaller in size so the individual values are more likely to be nearer the midpoints in the smaller groups than in the bigger groups. The range is the same for both sets of answers because the smallest and greatest possible values are the same. ii Learner’s own answers and explanations. For example: The answers in parts d and e were quicker to work out because there were fewer groups, so there were fewer calculations to do for the median and mean. Check your progress 1 a 60 b Kabayan Supermarket Time, t (minutes) Frequency Midpoint 0⩽t<15 5 7.5 15⩽t<30 8 22.5 30⩽t<45 38 37.5 45⩽t<60 9 52.5 Shoprite Supermarket Time, t (minutes) Frequency Midpoint 0⩽t<15 32 7.5 15⩽t<30 13 22.5 30⩽t<45 10 37.5 45⩽t<60 5 52.5 c Learner’s own diagram. Two frequency polygons drawn on one grid. Kabayan Supermarket points (7.5, 5), (22.5, 8), (37.5, 38) and (52.5, 9) joined with straight lines. Shoprite Supermarket points (7.5, 32), (22.5, 13), (37.5, 10) and (52.5, 5) joined with straight lines. Make sure that each line is labelled clearly. Make sure that the axes are labelled correctly and that a sensible scale is used. d Learner’s own answers. For example: More than six times as many employees took less than 15 minutes to travel to work to Shoprite than Kabayan, whereas nearly four times as many employees took between 30 and 45 minutes to travel to Kabayan than Shoprite. Only fve employees (8%) from Shoprite took longer than 45 minutes to travel to work, compared with nine employees (15%) from Kabayan. 2 a Learner’s own scatter graph. Horizontal axis labelled ‘Age (years)’ and shown from 0 to 16. Vertical axis labelled ‘Value ($)’ and shown from 0 to 16 000. Points (8, 8500), (10, 6000), (2, 13500), (3, 12 500), (15, 3500), (1, 15 000), (12, 4000), (5, 10000), (9, 6500) and (4, 12 000) plotted. b Negative correlation. c Learner’s own line of best ft and correct estimate of the value of a car that is six years old. Answer should be within range 9600–10400. 3 a i–iv i Mode ii Median iii Range iv Mean Boys’ times 67s 69s 32s 69.1s Girls’ times 56s 63s 32s 64.5s b Learner’s own answers. For example: The range is the same for the boys and the girls so they are both as varied as each other. The median and the mean for the boys and girls are all over 60 seconds. The boys’ mean and median are higher than the girls’. The girls’ mean and median are closer to 60 seconds. The girls’ mode is only 4 seconds under 60 seconds, whereas the boys’ mode is 7 seconds over 60 seconds. c Learner’s own answers. For example: No, the boys’ median is higher, but is further away from 60 seconds, as is their mean, so the boys are worse at estimating 60 seconds. 4 a i 6⩽t<8 hours ii 6⩽t<8 hours b i 7.26. hours or 7 hours 16 minutes or 7.3 hours ii 6 hours