CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 50 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021bActivity 13.4Learner’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 = 16 cm, C = 24 cm and D = 32 cmii Areas: A = 4 cm2, B = 16 cm2, C = 36 cm2 and D = 64 cm2bSquaresScale factor of enlargementRatio of lengthsRatio of perimetersRatio of areasA : B 2 1 : 2 1 : 2 1 : 4 = 1 : 22A : C 3 1 : 3 1 : 3 1 : 9 = 1 : 32A : D 4 1 : 4 1 : 4 1 : 16 = 1 : 42c 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 = 14 cm → Perimeter of T = 14 × 3 = 42 cm Area of R = 10 cm2 → Area of T = 10 × 32 = 90 cm27 Perimeter = 60 cm, Area = 150 cm28 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 justification. 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 find the centre of enlargement.12 Enlargement scale factor 3, centre (6, 5).13 Enlargement scale factor 2, centre (4, 4).Check your progress1 aNN12 cm(120km)9cm(90km)140°50°b Answer in range 148 km–152 km (accurate answer 150 km).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).Downloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 51 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 202156 Scale factor 3, centre of enlargement at (10, 4).7 Perimeter = 54 cm and area = 180 cm2.Unit 14 Getting started1 25.13 cm2 a 27 mm2 b 21 cm2 c 78.5 m23 a 120 cm3 b 158 cm24 a 480 cm3b Learner’s own diagram. Any correct net.c 528 cm25 a 1 b 2 c 6 d 0Exercise 14.11 a 120 cm3b 130 cm3c 134.4 cm32 Area of cross-sectionLength of prismVolume of prisma 12 cm2 10 cm 120 cm3b 24 cm2 8.5 cm 204 cm3c 18.5 m2 6.2 m 114.7 m33 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 = 12( ) 8 1 + 4 4 × = 44 cm2 Volume of prism = 44 × 20 = 880 cm34 Learner’s own answers. For example:a Yes. The cross-section is a circle.b Area of circle × heightc V = πr2hd Learner’s own discussions.5 Learner’s own explanation. For example: The radius and height are in different units. She needs to change the 5 mm to cm or change the 2 cm to mm before she works out the volume. Volume = 1570 mm3 (3 s.f.) or 1.57 cm3 (3 s.f.)6 a 942.5 cm3b 353.4 cm3c 17 592.9 mm3Activity 14.1 Learner’s own cylinders, answers and discussions.7Radius of circleArea of circleHeight of cylinderVolume of cylindera 2.5 m 19.63 m2 4.2 m 82.47 m3b 6 cm 113.10 cm2 4.48 cm 507 cm3c 2.52 m 20 m2 2.5 m 50 m3d 4.56 mm 65.25 mm2 16 mm 1044 mm38 a 5.5 cm b 4.2 cm c 2.1 cm9 Learner’s own methods and answers. For example: Volume of cylinder: V = πr2h = π × 62 ×18 = 2035.75 cm3 (2 d.p.) Volume of cube: V = 83 = 512 cm3 Volume of water: 1.5 litres = 1500 mL = 1500 cm3 Volume of cube + 1.5 litres = 512 + 1500 = 2012 cm3 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. 2012 cm3 < 2035.75 cm3Reflection: Learner’s own explanations.Downloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 52 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021Exercise 14.21 Answer using rounded intermediate values:Area of circle = cm (2 d.p.)ππr222578 54= ×= .Circumference of circle cm (2 d.p.)===ππd×1031 42.Area of rectangle = 31.42 12= 377.04 cm (2 d.p.)×2Total area = 2 78.54 + 377.04= 534 cm (3 s.f.)×2Answer using accurate intermediate values:Area of circle = cmππr222578 5398= ×= . ...Circumference of circle cm == ×=ππd1031 4159. ...Area of rectangle = 31.4159... 12= 376.9911... cm×2Total area = 2 78.5398... + 376.9911...= 534 cm (3 s.f.×2)2 a SA = 477.5 cm2b SA = 322.0 cm2c SA = 4272.6 mm23 The pyramid has a greater surface area than the cylinder. 132 cm2 > 125.66 cm2. Pyramid: SA = × × × cm4 6 8 6 + × 6 132 =122 Cylinder: SA = π × 22 × 2 + π × 4 × 8 = 125.66 cm24 Learner’s own methods and answers. For example:a SA = πr2 + πr2 + 2πrhb SA = πr2 + πr2 + 2πrh = 2πr2 + 2πrh = 2πr(r + h)c SA = 2πr(r + h) = 2πr(r + 2r) = 2πr × 3r = 6πr2d i SA = 8πr2ii SA = 10πr2iii SA = 12πr2e 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.5 226 cm2 (3 s.f.)6 Learner’s own methods and answers. For example:a i The hypotenuse of the triangular cross-section.ii Pythagoras’ theoremb Learner’s own discussions.c 408 cm27 a SA = 660 cm2b SA = 1188 mm2c SA = 23.3 m2Activity 14.2 a, b Learner’s own shapes. For example: A cuboid with length 10 cm, width 10 cm and height 8 cm (V = 800 cm3, SA = 520 cm2); A triangular prism of length 33 cm with a right-angled cross-section with base length 6 cm, height 8 cm and hypotenuse 10 cm (V = 792 cm3, SA = 840 cm2); A cylinder with height 16 cm and cross-section radius 4 cm (V = 804 cm3, SA = 503 cm2).c Learner’s own answers and explanations.d Learner’s own discussions.8 754 cm29 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 = 12Downloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 53 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021Exercise 14.31 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, b Learner’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, b Learner’s own lines of symmetry. Any of these:c A cube has a total of nine planes of symmetry.d Learner’s own justification. 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.b6 a2D regular polygonNumber of lines of symmetry3D prismNumber of planes of symmetryTriangle 3 Triangular 4Square 4 Square 5Pentagon 5 Pentagonal 6Hexagon 6 Hexagonal 7Octagon 8 Octagonal 9b 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 13d 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 infinite number of planes of symmetry. A circle has an infinite 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 infinite number of vertical ones.Reflection: Learner’s own answers.Downloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 54 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021Check your progress1 120 cm32 14 m23 452 cm34 The square-based pyramid has the greater surface area. Pyramid: SA = 340 cm2, Cylinder: SA = 320.44 cm2, 340 > 320.445 a The shape has two vertical, one horizontal and two diagonal planes of symmetry.b Learner’s own diagrams showing the five planes of symmetry correctly as described in the answer to part a.Unit 15 Getting started1 a Age, a (years) Frequency10 < a⩽ 15 315 < a⩽ 20 620 < a⩽ 25 725 < a⩽ 30 4b 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 112 a Class 9P test results012342 4 6 7 8 92 3 4 4 6 7 83 8 90 1 6 8 9 9 90 0Key: 0 3 means 03 marksb 32%c15d 143 a Mean Median Mode RangeHistory 12.9 13 16 7Chemistry 14 16 18 15b 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 15.11 a Height, h (cm) Frequency Midpoint140 ⩽h < 150 7 145150 ⩽h < 160 13 155160 ⩽h < 170 6 165170 ⩽h < 180 2 175b 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 Midpoint40 ⩽m < 50 4 4550 ⩽m < 60 12 5560 ⩽m < 70 8 65b 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 24d23e 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 60 kg ⩽m < 70 kg.Downloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 55 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 20213 a Learner’s own frequency table. For example:Age, a (years) Frequency10 ⩽a< 25 625 ⩽a < 40 940 ⩽a< 55 755 ⩽a< 70 470 ⩽a< 85 2b 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) Frequency10 ⩽a< 20 420 ⩽a < 30 830 ⩽a <40 940 ⩽a< 50 3b 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 SurgeryTime, t (minutes) Frequency Midpoint0 ⩽t < 10 25 510 ⩽t < 20 10 1520 ⩽t < 30 12 2530 ⩽t < 40 3 35Birchfields SurgeryTime, t (minutes) Frequency Midpoint0 ⩽t < 10 8 510 ⩽t < 20 14 1520 ⩽t < 30 17 2530 ⩽t < 40 11 35c 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. Birchfields 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 Birchfields surgery. More people waited over 10 minutes in Birchfields surgery compared to Oaklands surgery.6 a, b Learner’s own comments. For example: Using Sofia’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 first. 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 20 cm 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.Downloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 56 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021c 40 boys and 50 girlsd 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) Frequency190 ⩽l < 230 13230 ⩽l < 270 18270 ⩽l < 310 9ii 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 first 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 12ii No, Arun cannot fill in the correct frequencies in his table. Learner’s own explanation. For example: From the first table Arun knows that there are five turtles between 190 and 210 cm. But this does not tell him how many turtles there are between 190 and 200 cm and how many turtles there are between 200 and 210 cm, so it is impossible for him to complete his table. He would have to find the original data, before it was grouped, in order to use the groups he wants to.Exercise 15.21 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 fit. Strong negative correlation.d Correct answer from learner’s line of best fit. 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 fit.f Learner’s own comments. For example: It is not possible to predict from a line of best fit 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.Downloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 57 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021b 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 6 km in 16 minutes. Learner’s own explanation. For example: It should have taken less time, so the taxi might have been delayed in traffic.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 fit 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 70809010011012013014015018 20 22 24 26 28 30Number of fishTemperature (ºC)Number of fish at different points in the Red Seab Weak negative correlation.c Learner’s own line of best fit, and correct answer from their line, for number of fish 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 fit to predict the number of fish 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 fish 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.Downloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 58 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021Exercise 15.31 aShop A Shop B012389 96 3 2 5 6 0 1 29 5 4 7 78 91 4 3 99 0 7 7 4 6 2 81 0Key: 9 0 means 9 years old Key: 3 6 means 36 years oldb Shop A Shop B012389 90 1 2 2 3 4 6 64 5 7 7 99 89 4 3 19 8 7 7 6 4 2 01 0Key: 9 0 means 9 years old Key: 3 6 means 36 years oldc Learner’s own checks.d i Shop Aii Shop Be Learner’s own answers. For example: Shop A sells clothes for younger people and shop B sells clothes for older people.2 aKey: For the Beach car park, 5 4 means 45 ice-creams For the City car park, 3 0 means 30 ice-creamsBeach car park City car park34560 4 92 5 5 5 78 8 94 6 99 7 7 6 7 6 6 6 52 2 1 0 0bc 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 RangeBeach car park 46 57 17City car park 45 46 39Downloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 59 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 20213 a i Mode ii Median iii Range iv MeanBoys’ times 17.4 s 16.3 s 2.9 s 16.56 sGirls’ times 16.8 s 17.5 s 4 s 17.72 sb 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 14, B 23b 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 = 409 g B: mean = 395.6. g, median = 395 ge Learner’s own answers. For example:Location A because on average the mass of the hedgehogs is greater.5 aWebsite A Website B12131415168 94 6 84 5 6 7 7 86 7 8 95 5 5 6 6 84 3 0 08 7 6 5 5 5 2 11 09 8 5 3 3 2 2Key: For Website A, 0 13 means 130 hits For Website B, 12 8 means 128 hitsb 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 15.41 a i 150 cm ⩽h < 160 cmii 150 cm ⩽h < 160 cmb 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 40 cmdMidpoint Frequency Midpoint × frequency145 7 145 × 7 = 1015155 13 155 × 13 = 2015165 6 165 × 6 = 990175 2 175 × 2 = 350Totals: 28 4370Estimate of mean = cm437028= 1562 a i 50 kg ⩽m < 60 kgii 50 kg ⩽m < 60 kgb i 56.6. kg or 57 kgii 30 kgc 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 MeanWebsite A145 147 31 147.1Website B145 148 41 149.9Downloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 60 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 20213 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.bHospitalModal class intervalClass interval where the median liesEstimate of meanThe Heath10 ⩽ t < 20 minutes10 ⩽ t < 20 minutes17.25 minutesMoorlands 0 ⩽ t < 10 minutes20 ⩽ t < 30 minutes19.4 minutesc 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 2b 13c Mean = 7.15, Median = 8, Mode = 3d Table AScore Tally Frequency2–4 llll l 65–7 lll 38–10 llll lll 811–13 lll 3Table BScore Tally Frequency2–5 llll l 66–9 llll lll 810–13 llll l 6eModal class intervalClass interval where the median liesEstimate of meanTable A 8–10 8–10 7.2Table B 6–9 6–9 7.5f 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 reflected at all in either of the modal class intervals, which are totally different.g Learner’s own discussions.Activity 15.4 a 1b 36c–i Learner’s own data, tables, answers and discussions.6 a i 750 g ⩽m < 800 gii 750 g ⩽m < 800 gb i 798 g ii 400 gc Mass, m (g) Frequency600 ⩽m < 700 7700 ⩽m < 800 19800 ⩽m < 900 18900 ⩽m < 1000 6d i 700 g ⩽m < 800 gii 700 g ⩽m < 800 ge i 796 g ii 400 gDownloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922
CAMBRIDGE LOWER SECONDARY MATHEMATICS 9: TEACHER’S RESOURCE 61 Cambridge Lower Secondary Mathematics 9 – Byrd, Byrd & Pearce © Cambridge University Press 2021f 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 progress1 a 60bKabayan SupermarketTime, t (minutes) Frequency Midpoint0 ⩽t < 15 5 7.515 ⩽t < 30 8 22.530 ⩽t < 45 38 37.545 ⩽t < 60 9 52.5Shoprite SupermarketTime, t (minutes) Frequency Midpoint0 ⩽t < 15 32 7.515 ⩽t < 30 13 22.530 ⩽t < 45 10 37.545 ⩽t < 60 5 52.5c 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 five 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, 13 500), (3, 12 500), (15, 3500), (1, 15 000), (12, 4000), (5, 10 000), (9, 6500) and (4, 12 000) plotted.b Negative correlation.c Learner’s own line of best fit and correct estimate of the value of a car that is six years old. Answer should be within range 9600–10 400.3 a i–ivi Mode ii Median iii Range iv MeanBoys’ times 67 s 69 s 32 s 69.1 sGirls’ times 56 s 63 s 32 s 64.5 sb 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 hoursii 6 ⩽t < 8 hoursb i 7.26. hours or 7 hours 16 minutes or 7.3 hoursii 6 hoursDownloaded by Tanishka Doshi ([email protected])lOMoARcPSD|16792922