15 Distance, area and volume 350 6 The base of a triangular pyramid is an equilateral triangle with base length 6cm and perpendicular height 5.2 cm. The sides of the triangular pyramid are isosceles triangles with base length 6cm and perpendicular height 8.7 cm. Work out the surface area of the pyramid. 7 This triangular prism has a volume of 180 cm3 . The area of the triangular cross-section of the prism is A. Use the information given to work out the surface area of the triangular prism. l=5 cm h= A b=2h x=2 1 5 ×l y=1.4×l b y l x h Summary checklist I can calculate the surface area of a triangular prism. I can calculate the surface area of a pyramid. Tip Draw a diagram to help you. 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
351 15 Distance, area and volume Check your progress 1 Write true (T) or false (F) for each statement. a 22 miles is further than 22km. b 50km is exactly the same distance as 50 miles. c 200km is not as far as 200 miles. 2 a Convert 112km into miles. b Convert 205 miles into km. 3 Work out the area of each shape. a 4cm 7cm b 24m 8m 16m 4 Work out the volume of this triangular prism. 3cm 9cm 4cm 5 a Sketch the net of this shape. b Use your net to work out the surface area of the shape. 12cm 5cm 10 cm 13cm 6 a Sketch a net of this square-based pyramid. All the triangular faces of the pyramid are the same size. b Use your net to work out the surface area of the pyramid. 10 cm 7 cm 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
Project 6 Biggest cuboid Start with a 12cm by 12cm square of paper. Draw six rectangles that can be cut out and fitted together to make a cuboid. For example, these six rectangles could be joined to make this 2cm by 3cm by 5cm cuboid: There are lots of gaps between the rectangles, so perhaps we could have made a cuboid with a bigger surface area and a bigger volume. Can you find a cuboid that uses more of the paper? What is the volume of your cuboid? What different volumes of cuboid can you make from a 12cm by 12cm square? Can you find any cuboids that use the whole square of paper? What is the biggest volume of cuboid you can make? 352 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
353 Getting started 1 The frequency diagram shows the masses of the members of one family. a How many members of this family have a mass in the 80–120kg group? b How many members are in this family altogether? c What fraction of the family members have a mass in the 0–40kg group? 2 Look at the following sets of data. Which type of diagram, graph or chart do you think is best to use to display each set of data? Justify your choice. a The proportion of different flavour potato chips sold in a shop one day. b The sales of coats each month for a year. c The number of girls and boys going to an after-school club each day for one week. d The heights of 200 students in a college. 3 120 people were asked how they travel to work. The pie chart shows the results. a What percentage of the people travel to work by car? b What fraction of the people travel to work by train? Write your answer in its simplest form. c How many of the people travel to work by bus? 4 These are the weekly wages, in dollars, of the workers in an office. 500 525 650 510 500 495 740 630 450 500 a Work out the mode, median and mean weekly wages. b Which average weekly wage best represents this data? Give a reason for your choice of average. c Work out the range in the weekly wages. 16 Interpreting and discussing results 0 2 4 6 8 40 80 Mass (kg) Mass of family members Frequency 120 160 Transport to work 90 ° 132 ° 30 ° 108 ° Car Bus Train Bicycle 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
354 16 Interpreting and discussing results When you study statistics, you need to be able to draw and understand charts, graphs, tables and diagrams. A ‘picture’ of the data can make it easier to understand the information. For example, look at the table on the right. It shows the number of boxes of breakfast cereal sold at a grocery store each month from January to June. It also shows which shelf the boxes were on in the store. There is a lot of information in the table and it is difficult to understand all this information just by looking at the table. Now look at this bar chart, which shows the total monthly sales. You can easily see that May had the largest number of sales, by quite a long way, while the total sales in the other months were all very similar. When the data is put into a line graph, showing the monthly sales and the positions on the shelf, you can see that sales from the middle shelf were always greater than sales from the other shelves. The sales from the top and bottom shelves were quite close to each other on some occasions. All this information could be important to a grocery store when it is planning where to place items to maximise sales. It could also help the store to identify the months in which it needs to order extra stock. Number of boxes of breakfast cereal sold Jan Feb Mar Apr May Jun Top shelf 30 33 28 23 44 22 Middle shelf 32 52 46 40 65 51 Bottom shelf 26 10 20 35 24 14 Jan 0 20 40 60 Monthly breakfast cereal sales at a grocery store Number of boxes sold Month 80 100 120 140 Feb Mar Apr May Jun 10 20 30 40 50 60 70 0 Jan Feb Mar Month Apr May Jun Number of boxes sold Top shelf Middle shelf Bottom shelf Monthly breakfast cereals sales at a grocery store from May to June 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
355 16.1 Interpreting and drawing frequency diagrams In this section you will … • draw and interpret frequency diagrams for discrete and continuous data. 16.1 Interpreting and drawing frequency diagrams A frequency diagram shows how often particular values occur in a set of data. One example of a frequency diagram is a bar chart. In a bar chart, the bars are used to represent the frequency. When you draw a bar chart for grouped data, you must use suitable classes and have equal class intervals. When you draw a bar chart for discrete data, you should make sure: • the bars are all the same width • the gaps between the bars are equal • you label each bar with the relevant data group • you give the frequency diagram a title and label the axes • you use a sensible scale on the vertical axis. When you draw a bar chart for continuous data, you should make sure: • the class intervals are all the same width • there are no gaps between the bars • you use a sensible scale on the horizontal axis • you give the frequency diagram a title and label the axes • you use a sensible scale on the vertical axis. Key words class interval classes continuous data discrete data frequency diagram grouped 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
356 16 Interpreting and discussing results Worked example 16.1 a The frequency diagram shows how many pieces of fruit the students in class 8T ate in one week. i How many students ate 4–7 pieces of fruit? ii How many more students ate 8–11 pieces of fruit than ate 12–15 pieces? iii How many students are there in class 8T? b The frequency table shows the masses of 20 teachers. Draw a frequency diagram to show the data. Mass, m (kg) Frequency 60 < m ≤ 70 3 70 < m ≤ 80 8 80 < m ≤ 90 6 90 < m ≤ 100 4 Answer a i 6 students The bar for 4–7 has a height of 6 on the frequency axis. ii 9−4=5 students The frequency for 8–11 is 9 and the frequency for 12–15 is 4. Subtract one from the other to find the difference. iii 7+6+9+4=26 students Add together the frequencies for all the groups. b 50 2 4 6 8 60 70 80 Mass (kg) Mass of 20 teachers Frequency 90 100 110 All the bars are the same width and, as the data is continuous, there are no gaps between them. Both the horizontal and vertical axes have sensible scales. The frequency diagram has a title and the axes are labelled. 0 2 4 6 8 10 0–3 4–7 Number of pieces of fruit Number of pieces of fruit eaten by 8T in one week Frequency 8–11 12–15 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
357 16.1 Interpreting and drawing frequency diagrams Exercise 16.1 1 The frequency diagram shows the number of phone calls made by all the employees of a company on one day. a How many employees made 10–19 phone calls? b How many more employees made 30–39 phone calls than made 0–9 phone calls? c How many employees are there in the company? Explain how you worked out your answer. 2 The frequency table shows the number of cups of coffee sold each day in a coffee shop during one month. a Draw a frequency diagram to show the data. b Which month do you think your frequency diagram represents? Explain your answer. c Read what Marcus says. The frequency diagram shows that the most cups of coffee sold was 99. Is he correct? Explain your answer. 0 0–9 10–19 20–29 30–39 2 4 6 Frequency Number of phone calls made by the employees of a company on one day Number of phone calls 8 10 Number of cups of coffee sold Frequency 0–19 2 20–39 3 40–59 6 60–79 12 80–99 5 Think like a mathematician 3 Work with a partner or in a small group to answer this question. Ryan recorded the number of text messages he sent each day for one month. Here are his results. 23 17 19 0 16 18 7 17 15 18 12 10 18 14 14 4 12 20 9 13 20 11 19 1 20 20 24 2 a Record this information in a frequency table. Choose suitable classes. Make sure you have equal class intervals. b Draw a frequency diagram to show the data. c Compare your frequency table and diagram with those of other groups. Discuss the classes used. Which classes do you think are best to show this data? Explain why. 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
358 16 Interpreting and discussing results 4 Erin recorded the number of emails she sent each day for one month. Here are her results. 31 17 37 11 35 34 36 15 33 22 31 18 34 12 28 14 30 21 39 16 13 38 34 29 10 19 39 32 38 15 Marcus, Arun and Zara discuss what classes to use. Read what they say. a Who do you think has chosen the most suitable classes, Marcus, Arun or Zara? Explain why. b Explain why you think the classes chosen by the other two are not suitable. c Record the information in a frequency table. d Draw a frequency diagram to show the data. I would use the classes 0–4, 5–9, 10–14, etc. I would use the classes 10–19, 20–29, 30–39, etc. I would use the classes 10–14, 15–19, 20–24, etc. Think like a mathematician 5 Work with a partner or in a small group to answer this question. The frequency table shows the ages of the members of a choir. a Explain what you think the class 10 ≤ a < 20 means. b Explain why you cannot use the classes 10–19, 20–29, etc. c In which class would you include someone aged exactly 30 years? d Draw a frequency diagram to show the data. e Discuss and compare your answers to parts a–d with other groups in your class. Age, a years Frequency 10 ≤ a < 20 12 20 ≤ a < 30 8 30 ≤ a < 40 15 40 ≤ a < 50 6 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
359 16.1 Interpreting and drawing frequency diagrams 6 The frequency table shows the speeds of cars passing a speed camera on one day. The speeds are recorded in kilometres per hour (km/h). a Draw a frequency diagram to show the data. b The speed limit is 80 km/h. How many cars are travelling over the speed limit? c Read what Sofia says. The frequency diagram shows that the slowest car was travelling at 50 km/h. Is she correct? Explain your answer. 7 Here are the heights, in centimetres, of some plants. 25 32 30 26 34 22 33 34 31 28 39 20 27 33 37 32 25 24 30 29 a Record this information in a frequency table. Use the classes 20 ≤h<25, 25≤h<30, 30≤h<35 and 35 ≤h<40. b Draw a frequency diagram to show the data. c How many of the plants are at least 25 cm high? Explain how you worked out your answer. Speed of car, s (km/h) Frequency 50 < s ≤ 60 2 60 < s ≤ 70 3 70 < s ≤ 80 6 80 < s ≤ 90 12 90 < s ≤ 100 5 Tip At least 25cm means 25cm or more. 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
360 16 Interpreting and discussing results 8 10 0 0 20 40 Age (years) Age (years) 60 80 100 20 30 40 50 60 70 80 90 The frequency diagrams show the population of a village by age group in 1960 and 2010. Population of a village by age group, 1960 Population of a village by age group, 2010 Frequency 10 0 0 20 40 60 80 100 20 30 40 50 60 70 80 90 Frequency a Look at the graphs. Write two sentences to compare the age groups in the population of the village in 1960 and 2010. b Read what Marcus says. Approximately 25% of the population were over the age of 40 in 1960, compared with approximately 60% in 2010. Is Marcus correct? Show your working to support your answer. In this section, you have recorded information in a frequency table. You have had to choose your own classes. a Explain to a partner the method you use to decide what classes to use. b Does your partner use the same method as you or a different one? c Discuss your methods and decide which is better/easier to use. Summary checklist I can draw and interpret frequency diagrams for discrete and continuous 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
361 16.2 Time series graphs In this section you will … • draw and interpret time series graphs. 16.2 Time series graphs A time series graph is a series of points, plotted at regular time intervals and joined by straight lines. Time series graphs are used to show trends, which tell you how the data changes over a period of time. When you draw a time series graph, make sure you: • put time on the horizontal axis • use an appropriate scale on the vertical axis • plot each point accurately • join the points with straight lines • give the time series graph a title and label the axes. Worked example 16.2 The table shows the value of a car over a period of five years. a Draw a time series graph to show the data. b During which year did the car lose the most value? c Describe the trend in the value of the car. d Use the graph to estimate the value of the car after 2 1 2 years. Answer a 0 1 5 000 0 10 000 15 000 20 000 25 000 30 000 2 Age of car (years) Value of car over five years Value of car ($) 3 4 Age of car (years) goes on the horizontal axis. Value of car ($) goes on the vertical axis. The vertical axis has a sensible scale that is easy to read. All the points are plotted accurately and joined with straight lines. The graph has a title and the axes are labelled. Age of car (years) Value of car ($) 0 25000 1 20000 2 17000 3 14900 4 13400 Key words time series graph trend We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
362 16 Interpreting and discussing results Exercise 16.2 1 The time series graph shows the profit made by a company each year for a six-year period. a How much profit did the company make in i 2006 ii 2007? b In which year did the company make the largest profit? c Between which two years was the greatest increase in profit? d Between which two years was the greatest decrease in profit? e Describe the trend in the company profits over the six-year period. 2 The time series graph shows the value of a house over a ten-year period. a What was the value of the house in i 2000 ii 2010? b In which year did the house reach its greatest value? c Between which two years was the greatest increase in the value of the house? d Describe the trend in the value of the house over the ten-year period. e Use the graph to estimate the value of the house in i 2003 ii 2009. 20072006 2008 2009 1 2 3 0 2010 2011 Year Company profit Profit ($ million) 2000 2002 2004 120 100 160 140 200 180 0 2006 2008 2010 Year House value 2000–2010 Value of house ($ thousands) Continued b During the first year The greatest loss is $5000, in the first year. c The value of the car decreases every year, but the loss each year is less than in the year before. The losses are $5000, $3000, $2000 and $1500 so each year, the loss is less than in the year before. d $16 000 (see red line on graph) Read up from 2 1 2 on the horizontal axis to the line, then across to the vertical axis to read off the value. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
363 16.2 Time series graphs For questions 4 and 5, work in groups of three or four. i On your own, draw a time series graph to show the data. Then describe the trend in the data and answer the question. ii Compare your time series graph, description and answers with those of the other members of your group. Discuss the different scales you have used in your graphs and compare the descriptions you have given. Decide which are best. Check your answers to the questions are the same. Think like a mathematician 3 The time series graph shows the average price of crude oil, per barrel, every ten years since 1965. 0 5 10 15 20 25 30 35 40 45 50 55 1965 1975 1985 1995 2005 2015 Price per barrel ($) Year Average price of crude oil (to the nearest dollar) Marcus and Sofia are trying to work out in which year the average price of crude oil was at its highest. Read what they say. I think the average price of crude oil was at its highest in 2005. I don’t think you can tell from this graph in which year the average price of crude oil was at its highest. What do you think? Explain why. Discuss your answers and explanations with other learners in your class. 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
364 16 Interpreting and discussing results 4 The table shows the number of people staying in a guest house each month for one year. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Number of people 8 6 11 15 17 20 24 26 18 14 8 7 Between which two months did the number of people at the guest house change the most? 5 The table shows the average price of silver and copper, per ounce, every four years since 1990. The prices are rounded to the nearest $0.10. Year 1990 1994 1998 2002 2006 2010 2014 2018 Average price of silver $ 4.80 5.30 5.50 4.60 11.60 20.20 19.10 15.70 Average price of copper $ 1.20 1.10 0.80 0.70 3.10 3.10 3.20 3.10 a Draw a time-series graph to show both sets of data. b Write true (T) or false (F) for each statement i Between 1990 and 2002 the prices did not change very much. ii Silver increased by the greatest amount between 2002 and 2006. iii We can use the graph to predict an accurate price of copper and silver in 2022. c Use your graphs to estimate the price in 2008 of: i silver ii copper. d Marcus says ‘The price of both silver and copper went up in 2002.’ Explain why Marcus may not be correct. Think like a mathematician 6 Zara goes to a one-hour cycling class every Monday, Wednesday and Friday evening. She records the distance she cycles during each class. Zara wants to draw a time series graph and look at the trend in her data over one year. Read what Zara and Sofia say. I think I will plot every distance I have recorded over the year. If you do that, you’ll have more than 150 points to plot! I think I would plot fewer points. What do you think? Explain why. Discuss your answers and explanations with other learners in your class. 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
365 16.2 Time series graphs 7 A sports shop sells the rugby shirts of two teams, Scarlets and Dragons. The time series graph shows the number of rugby shirts the shop has in stock each week over an 8-week period. a Describe the trend in the sales of i Scarlets rugby shirts ii Dragons rugby shirts. b Do you think the shop has enough Scarlets rugby shirts in stock for week 9? Explain your answer. c Do you think the shop has enough Dragons rugby shirts in stock for week 9? Explain your answer. 8 The time series graph shows the number of hotel rooms booked in a seaside town. It shows the number booked in spring, summer, autumn and winter from 2018 to 2020. 0 10 20 30 40 50 60 70 80 90 Spring Summer Autumn Winter Spring Summer Autumn Winter Spring Summer Autumn Winter 2018 2019 2020 Number of hotel rooms booked (100s) Year and season Number of hotel rooms booked in seaside town from 2018 to 2020 a Describe how the number of hotel rooms booked changes over the seasons during 2018. b Do similar changes over the seasons that you have noticed in 2018, also happen in 2019 and 2020? Explain your answer. c Describe the yearly trend in the number of hotel rooms booked. d Use your graph to predict the number of hotel rooms that will be booked in Autumn 2021. e Explain why your answer to part d may be incorrect. 0 2 4 6 Week number Number of rugby shirts in stock over an 8-week period 8 109 10 20 30 40 50 Number of rugby shirts in stock 1 3 5 7 60 70 80 Scarlets Dragons Summary checklist I can draw and interpret time series graphs. 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
366 16 Interpreting and discussing results ‘Ordered’ means you write the numbers in order of size from smallest to largest. In this section you will … • draw and interpret stem-and-leaf diagrams. 16.3 Stem-and-leaf diagrams A stem-and-leaf diagram is a way of showing data in order of size. When you draw a stem-and-leaf diagram, make sure: • you write the numbers in order of size from smallest to largest • you write a key to explain the numbers • you keep all the numbers in line vertically and horizontally. Key words mean median mode range stem-and-leaf diagram Key: 2 3 means 23 cm 2 3 4 7 8 3 0 6 7 7 8 4 1 9 9 stem leaves Worked example 16.3 Here are the temperatures, in°C, recorded in 20 cities on one day. 9 19 26 35 6 17 32 21 30 16 14 16 18 29 27 8 25 32 21 32 a Draw an ordered stem-and-leaf diagram to show this data. b How many cities had a temperature over 28 °C? c Use the stem-and-leaf diagram to work out i the mode ii the median iii the range of the data. Answer a Key: 1 0 9 6 8 9 7 6 4 6 8 6 1 9 7 5 1 1 2 3 5 2 0 2 2 9 means 19°C First, draw an unordered stem-and-leaf diagram. Start by writing a key. You can use any of the numbers to explain the key. Write in the stem numbers. In this case, they are the tens digits, 0, 1, 2 and 3. Now write in the leaves, taking the numbers from the table above. For example, 9 has 0 as the stem and 9 as the leaf; 19 has 1 as the stem and 9 as the leaf; 26 has 2 as the stem and 6 as the leaf; and so on. Complete for all 20 temperatures. 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
367 16.3 Stem-and-leaf diagrams Think like a mathematician 2 Work with a partner to answer this question. a A cafe keeps a record of the number of cups of coffee sold each day. The stem-and-leaf diagram shows the number of cups of coffee sold each day for one month. i Which month does the stem-and-leaf diagram represent? ii What is the largest number of cups of coffee sold on one day? iii What is the modal number of cups of coffee sold on one day? 11 2 4 6 8 0 0 1 2 4 5 6 9 1 4 4 5 8 9 12 13 14 2 7 7 7 15 0 1 1 3 6 8 Continued Key: 1 0 6 8 9 4 6 6 7 8 9 1 1 5 6 7 9 1 2 3 0 2 2 2 5 9 means 19°C Now draw an ordered stem-and-leaf diagram. Rewrite the diagram with all the leaves in order, from the smallest to the biggest. Make sure the stem numbers are in line vertically and the leaves are in line vertically and horizontally. b 6 cities You can see from the stem-and-leaf diagram that 29 °C, 30 °C, 32 °C, 32 °C, 32 °C and 35°C are all over 28 °C. c i mode=32°C You can see that 32 °C is the temperature that appears most often. ii median=21°C There are 20 temperatures, so the median temperature is the average of the 10th and 11th values. These are both 21 °C, so the median is 21 °C. iii range=35−6 = 29 °C You can easily see from the diagram that the highest temperature is 35 °C and the lowest is 6 °C. Subtract to find the range. Exercise 16.3 1 Shen listed the playing times, to the nearest minute, of some CDs. He recorded the results in a stem-and-leaf diagram. a How many CDs did Shen list? b What is the shortest playing time? c How many of the CDs had a playing time longer than 60 minutes? d Work out i the mode ii the median iii the range of the data. Key: 4 4 5 5 7 9 0 2 5 6 8 9 1 2 4 6 7 5 6 5 means 45 minutes 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
368 16 Interpreting and discussing results 3 These are the file sizes, in kilobytes (kB), of 30 files on Greg’s computer. 101 128 117 109 154 139 166 155 117 145 135 162 117 168 125 131 140 160 151 125 152 108 139 130 165 158 103 130 110 148 a Draw an ordered stem-and-leaf diagram to show this data. b How many of the files are larger than 150kB? c Which average, the mode or the median, better represents this data? Explain why. d Greg works out that the range in his file sizes is 65kB. Is Greg correct? Explain your answer. Continued b Discuss these questions with other learners in your class. i Were you able to answer all of the questions in part a? ii What is missing from the stem-and-leaf diagram? iii Can you work out the answers to part a, even though something is missing from the stem-and-leaf diagram? Think like a mathematician 4 Work with a partner to answer this question. Opaline counted the number of birds in her garden each day for 10 days. The stem-and-leaf diagram shows her results. This is how Opaline works out the mean number of birds in her garden each day. From my stem-and-leaf diagram: 1st line mean= 4+8+9 3 = 21 3 =7, mean=7 2nd line mean=2+4+9 3 = 15 3 =5, mean=15 3rd line mean=0+2+3+7 4 = 12 4 =3, mean=23 Overall mean= 7+15+23 3 = 45 3 =15 a Is Opaline’s method a correct method? Explain your answer. b What do you think is the best method to use to work out the mean from a stem-and-leaf diagram? Key: 0 0 4 8 9 2 4 9 0 2 3 7 1 2 4 means 4 birds 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
369 16.3 Stem-and-leaf diagrams 5 Ashish counted the number of cars passing his school between 8.30 a.m. and 9 a.m. each day for 12 days. The stem-and-leaf diagram shows his results. a Use the stem-and-leaf diagram to work out i the mode ii the median iii the mean of the data. b Which average, the mode, median or mean, best represents this data? Explain why. 6 The students in class 8B took a test. The stem-and-leaf diagram shows their scores out of 40. Key: 1 0 6 8 8 9 9 6 5 6 8 8 9 1 2 3 0 1 2 3 3 5 6 7 8 8 4 0 0 0 0 8 means 18 a What percentage of the students had a score greater than 32? b What fraction of the students had a score less than 25%? c Any student scoring less than 40% must re-sit the test. How many students do not have to re-sit the test? Key: 0 0 1 2 3 6 7 7 9 7 0 4 4 4 1 2 1 means 1 car Summary checklist I can draw and interpret stem-and-leaf diagrams. Discuss the answers to these questions with a partner. a Do you think stem-and-leaf diagrams are a good way to represent data? Explain why. b Do you think stem-and-leaf diagrams are easy to understand? Explain why. c What are the advantages and disadvantages of drawing and interpreting stem-and-leaf diagrams? 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
370 16 Interpreting and discussing results You already know how to draw and interpret a pie chart. You can use pie charts to compare different sets of categorical data data, but remember that a pie chart shows proportions, not actual amounts. In this section you will … • compare pie charts. 16.4 Pie charts Key words categorical data pie chart proportions sector Worked example 16.4 The pie charts show the proportion of male and female teachers in two schools. Gender of teachers in Oak School 108 ° 252 ° Male Female Gender of teachers in Elm School 216 ° 144 ° There are 20 teachers in Oak School. There are 45 teachers in Elm School. a Which school has the greater proportion of female teachers? b Which school has the greater number of female teachers? Show your working. Answer a Oak School has the greater proportion of female teachers. The blue sector (female teachers) in the pie chart for Oak School is larger than the blue sector in the pie chart for Elm School. b Oak: 252 360 7 10 × = 20 × 20 = 14 female teachers Elm: 144 360 2 5 × = 45 × 45 = 18 female teachers Elm School has the greater number of female teachers. Work out the number of female teachers in each school. Start by writing the fraction of female teachers, then simplify this fraction and multiply by the number of teachers. Although the proportion of female teachers is greater in Oak School, there are more female teachers in Elm School. 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
371 16.4 Pie charts Exercise 16.4 1 The pie charts show the proportion of boys and girls in two swimming clubs. Gender of children in the Dolphins swimming club Girls Boys 150 ° 210 ° Gender of children in the Seals swimming club 240 ° 120 ° There are 120 children in the Dolphins swimming club. There are 72 children in the Seals swimming club. a Which swimming club has the greater proportion of girls? b Which swimming club has the greater number of girls? Copy and complete the working. Dolphins: 150 360 × 120= girls Seals: 240 360 × 72= girls The ................... swimming club has the greater number of girls. 2 The pie charts show the proportion of Ivan’s income that he made from gardening, washing windows and painting houses in 2009 and 2019. Ivan’s business in 2019 90 ° 135 ° 135 ° Ivan’s business in 2009 Gardening Washing windows Painting houses 180 ° 45 ° 135 ° a What fraction of Ivan’s income came from gardening in i 2009 ii 2019? 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
372 16 Interpreting and discussing results b Copy and complete these sentences. Choose from the words in the rectangle. doubled stayed the same tripled halved more than tripled i In 2019 the proportion of Ivan’s income that came from gardening had ...................... compared to 2009. ii In 2019 the proportion of Ivan’s income that came from painting houses had ...................... compared to 2009. iii In 2019 the proportion of Ivan’s income that came from washing windows had ...................... compared to 2009. In 2009, Ivan’s total income was $12 000. In 2019, Ivan’s total income was $24 000. c Show that Ivan earned the same amount of money from gardening in 2009 and 2019. d Show that Ivan earned six times as much money from washing windows in 2019 as in 2009. e How much more money did Ivan earn from painting houses in 2019 than in 2009? 3 The pie charts show the results of a survey about the types of chocolate preferred by men and by women. 480 men took part in the survey. 600 women took part in the survey. a How many men chose plain chocolate? b How many women chose plain chocolate? c Hassan thinks that more men than women like milk chocolate. Is Hassan correct? Show how you worked out your answer. d The ‘Caramel’ sector is the same size for men and for women. Without doing any calculations, explain how you know that more women than men chose ‘Caramel’. Tip Use the fractions you found in part a. Men’s favourite chocolate Plain 60 °135 ° Milk White Caramel Women’s favourite chocolate 120 ° 84 ° 81 ° Milk Plain White Caramel 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
373 16.4 Pie charts 4 The pie charts show the favourite sports of the students in two schools. Castlehill School Riverside School 12% 14% 38% 15% 7% 14% 2% Rugby Football Tennis Hockey Cricket Other 25% 18% 13% 10% 32% There are 1600 students in Castlehill School. There are 1100 students in Riverside School. Which school had the larger number of students who chose tennis as their favourite sport? Show your working. Think like a mathematician 5 Work with a partner or in a small group to answer these questions. The pie charts show the proportions of different makes of car sold by two garages in 2019. Make of car sold in Kabir’s garage Kia Ford Seat Nissan 90 ° 120 ° 120 ° 30 ° Make of car sold in Ekta’s garage 45 ° 40 ° 155 ° 120 ° In 2019, Kabir sold 600 cars in total. a How many cars does Ekta sell in total if she sells i the same number of Kia cars as Kabir ii the same number of Ford cars as Kabir iii the same number of Nissan cars as Kabir? 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
374 16 Interpreting and discussing results 6 The pie charts show the proportions of different sizes of T-shirt sold in a shop on two days. Sizes of T-shirts sold on Monday 80 ° 220 ° 60 ° Small Medium Large Sizes of T-shirts sold on Tuesday 40 ° 140 ° 180 ° On Monday, the shop sold 144 T-shirts. On Tuesday, the shop sold the same number of small T-shirts as on Monday. a How many small T-shirts did the shop sell on Tuesday? b How many T-shirts did the shop sell altogether on Tuesday? 7 The pie charts show the proportions of types of rice sold by two shops in May. Type of rice sold in Shop A Black Brown Red White 30 ° 120 ° 60 ° 150 ° Type of rice sold in Shop B 20 ° 180 ° 30 ° 130 ° In May, Shop A sold 6kg of black rice. Continued b Explain how you worked out your answers to part a. Can you describe a rule to follow to answer this type of question? c Discuss your answers and methods with other groups in your class. 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
375 16.4 Pie charts Summary checklist I can compare pie charts. a Copy and complete this table to show the amounts of rice sold by Shop A. Amounts of rice sold by Shop A Type of rice Degrees in pie chart Kilograms sold black 30º 6kg brown 120º red 60º white 150º Total: 360º In May, Shop B sold the same number of kilograms of red rice as shop A. b Copy and complete this table showing the amounts of rice sold by Shop B. Amounts of rice sold by Shop B Type of rice Degrees in pie chart Kilograms sold black 20º brown 180º red 30º white 130º Total: 360º c Read what Sofia says. Explain why Sofia is incorrect. Tip Use the fact that 6kg of rice is represented by 30º in the pie chart. Tip Use the fact that the number of kilograms of red rice sold is the same in both tables. In Shop A, 60° of the pie chart represents red rice. In Shop B, 30° of the pie chart represents red rice. This means that, without doing any calculations, I can say that the total amount of rice sold in Shop A is double the total amount of rice sold in Shop 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
376 16 Interpreting and discussing results When you represent data using a diagram, graph or chart, you need to decide which type is best to use. This table will help you to decide. 16.5 Representing data In this section you will … • choose how to represent data. Key word justify What type of diagram / graph / chart? When do I use it? What does it look like? Venn or Carroll diagram When you want to sort data or objects into groups with some common features Bar chart When you want to compare discrete data Dual bar chart When you want to compare two sets of discrete data Compound bar chart When you want to combine two or more quantities into one bar, to look at individual amounts and total amounts Frequency diagram When you want to compare continuous data Time series graph When you want to see how data changes over time 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
377 16.5 Representing data What type of diagram / graph / chart? When do I use it? What does it look like? Scatter graph When you want to compare two sets of data points Pie chart When you want to compare the proportions of each sector with the whole amount Infographic When you want to show some information in a quick way that is easy to understand 40% delivered on time 60% delivered late Stem-and-leaf diagram When you want to compare data that is grouped, but you still want to see the actual values Key: 2 2 3 4 7 8 0 6 7 7 8 1 9 9 3 4 3 means 23 cm Worked example 16.5 Look at the following sets of data. Which type of diagram, graph or chart do you think is best to use to display the data? Justify your choice. a The value of gold every Monday morning b The ingredients in 100ml of two different salad dressings c The ages of the people on a bus Answer a Time series graph – so you can see how the value of gold changes over time. b Compound bar chart – so you can easily compare the amounts of each ingredient. c Stem-and-leaf diagram – so you can see the data as grouped data but also see the exact values. 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
378 16 Interpreting and discussing results Exercise 16.5 1 Look at the following sets of data. Which type of diagram, graph or chart do you think is best to use to display the data? Justify your choice. a The percentage of the members of two running clubs that are men, women, girls and boys b The ages and heights of the horses at a riding school c The scores, out of 50, of 30 students in a spelling test d The mass of a baby chimpanzee each week 2 A group of 30 students study science at advanced level. Four students study physics, biology and chemistry. Five students study only chemistry and biology, three study only chemistry and physics, and two study only physics and biology. Six students study only physics, seven study only biology and three study only chemistry. a Draw a diagram, graph or chart to represent this data. b Justify your choice of diagram, graph or chart. c Make one comment about what your diagram, graph or chart shows you. 3 The table shows the monthly average mass of a baby girl from newborn to one year old. Month 0 1 2 3 4 5 6 7 8 9 10 11 12 Mass (kg) 3.2 4.2 5.1 5.8 6.4 6.9 7.3 7.6 7.9 8.2 8.5 8.7 8.9 a Draw a diagram, graph or chart to represent this data. b Justify your choice of diagram, graph or chart. c Make one comment about what your diagram, graph or chart shows you. 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
379 16.5 Representing data Think like a mathematician 4 The table shows the ingredients of two different cans of beans. Ingredient beans water tomato paste sugar salt Can A 48g 32g 17g 2g 1g Can B 67g 30g 18g 8g 2g Read what Arun and Zara say. I think it is best to use a compound bar chart to represent this data. I think it is best to use a pie chart to represent this data. Discuss the answers to these questions in a small group, and then with other groups in the class. a If you represented this data in a compound bar chart i what parts of the data would it be easier to compare? ii what parts of the data would it be more difficult to compare? b If you represented this data in a pie chart i what parts of the data would it be easier to compare? ii what parts of the data would it be more difficult to compare? c Complete each statement with either ‘compound bar chart’ or ‘pie chart’. i When you are comparing individual and total amounts, it is better to use a ............... ii When you are comparing proportions, it is better to use a ............... 5 These are the numbers of pages of a book that Daylen reads each day for four weeks. 25 5 18 34 16 35 12 12 20 14 8 27 39 9 30 11 22 19 7 27 10 32 27 33 11 24 17 22 a Draw a diagram, graph or chart to represent this data. b Justify your choice of diagram, graph or chart. c Make one comment about what your diagram, graph or chart shows you. d Work out i the mode ii the median iii the range of the 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
16 Interpreting and discussing results 380 6 Zara recorded the number of minutes she spent doing homework each evening for one month. The frequency table shows her results. Time, t (minutes) Frequency 0 ≤ d < 20 1 20 ≤ d < 40 6 40 ≤ d < 60 2 60 ≤ d < 80 8 80 ≤ d < 100 14 a Draw a diagram, graph or chart to represent this data. b Justify your choice of diagram, graph or chart. c Make one comment about what your diagram, graph or chart shows you. 7 A scientist measured the length and mass of 12 sea turtles. The table shows her results. Length (cm) 87 99 92 84 108 105 109 94 85 95 100 90 Mass (kg) 125 150 135 112 175 163 188 132 115 144 158 128 a Draw a diagram, graph or chart to represent this data. b Justify your choice of diagram, graph or chart. c Make one comment about what your diagram, graph or chart shows you. Summary checklist I can choose how to represent 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
381 16.6 Using statistics In this section you will … • use mode, median, mean and range to compare sets of data. 16.6 Using statistics You can use an average to summarise a set of data. This could be the mode, median or mean. You can use the range to measure the spread of the data. The larger the range, the more varied the data. You already know how to work out the mode, median, mean and range. Here is a reminder: • The mode is the most common value or number. • The median is the middle value, when they are listed in order. • The mean is the sum of all the values divided by the number of values. • The range is the largest value minus the smallest value. You can use these statistics to compare two or more sets of data. Key words mean median mode range Worked example 16.6 A health club recorded the masses (in kilograms) of eight men and six women. Men: 65, 79, 68, 72, 77, 77, 81, 67 Women: 68, 52, 47, 49, 50, 58 Calculate the mean and the range of each set of data and use these values to compare the two sets. Answer The mean for the men is 73.25kg. 65 79 68 72 77 77 81 67 8 + + ++++ + =586 8 =73.25kg The mean for the women is 54kg. 68 52 47 49 50 58 6 + + + + + =324 6 =54kg On average, the men are 19.25kg heavier than the women. 73.25−54=19.25kg The range for the men is 16kg. 81−65=16kg The range for the women is 21kg. 68−47=21kg The women’s masses are more varied than the men’s. 21kg is greater than 16kg. 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
382 16 Interpreting and discussing results Exercise 16.6 1 In the 2010 football World Cup, Spain won and Brazil was knocked out in the quarter finals. Spain: 0, 2, 2, 1, 1, 1, 1 Brazil: 2, 3, 0, 3, 1 The numbers of goals they scored in their matches are shown. a Work out the mean score for each team. b Use the means to state which team scored more goals, on average, per match. c Work out the range for each team. d Use the ranges to state which team’s scores were more varied. 2 A teacher measured the heights of two groups of children. Here are the results. Group A: 84 cm, 73cm, 89 cm, 80cm, 77cm Group B: 77 cm, 85cm, 75 cm, 69cm, 82cm, 67cm, 72cm a For each group i write the heights in order of size ii write the median height iii work out the range in heights. b Use the medians to state which group is taller, on average. c Use the ranges to state which group’s heights are less varied. 3 The maximum daytime temperature (ºC) was recorded in Madrid and Cartagena during one week in August. Here are the results. Madrid: 38, 34, 36, 32, 35, 37, 36 Cartagena: 30, 32, 29, 30, 28, 30, 33 a For each city i write the temperatures in order of size ii write the modal temperature iii work out the range in temperatures. b Use the modes to state which city is hotter, on average. c Use the ranges to state which city’s temperatures are more varied. 4 A nurse measured the total mass of 20 baby boys as 64kg. The total mass of 15 baby girls was 51kg. Which babies were heavier on average, the boys or the girls? Give a reason for your answer. 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
383 16.6 Using statistics 6 Nialls recorded the temperatures in two experiments. First experiment (ºC) 29, 28, 21, 33, 30 Second experiment (ºC) 28, 29, 28, 33, 32, 31, 32, 29 a Work out the mean, median and range for each experiment. b State whether each of these statements is True (T) or False (F). Justify your answers. i The temperatures in the first experiment are higher, on average, than the temperatures in the second experiment. ii The temperatures in the first experiment are more varied than the temperatures in the second experiment. c Is it possible to work out the modal temperature for each experiment? Explain your answer. Think like a mathematician 5 The test marks of two groups of students are shown. Maths: 77, 89, 75, 80, 80, 91, 78, 76, 76, 76 Science: 72, 79, 77, 87, 81, 62, 75, 87 a Copy and complete this table. Mean Median Mode Range Maths Science b In which group, Maths or Science, do you think the students did better on average? c In which group, Maths or Science, do you think the students had more consistent scores? d Compare your answers to parts b and c with those of other learners in the class. Discuss these questions. i Which average did you use to compare the scores? Why did you use this average? Why did you not use the other averages? ii What does ‘more consistent’ mean? What statistic did you use to decide which group had more consistent scores? e Now you have discussed the answers of other learners in your class, which average do you think is the best to use to compare these scores? Explain why. 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
16 Interpreting and discussing results 384 Think like a mathematician 7 Work with a partner or in a small group to answer this question. You are going to roll two dice and subtract the numbers on the dice to give a score. Always subtract the numbers to give you a positive, or zero, score. a What is the smallest score you can get? b What is the largest score you can get? You are going to roll the dice 40 times. c Draw a table to record the scores you get. Your table needs a ‘Tally’ column and a ‘Frequency’ column. d Now roll the dice 40 times and record all your scores. When you have finished, make sure the frequency column adds up to 40. e Work out i the mode ii the median iii the mean score for your data. f Which average best represents your data? Give a reason for your choice of average. g Compare your data and averages with other learners in the class. Do you have different averages? Do you have the same averages? Have you chosen the same average to represent your data? Discuss your answers. Tip For example, if you roll these numbers you get a score of 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
385 16.6 Using statistics Summary checklist I can use mode, median, mean and range to compare sets of data. I can decide when it is more appropriate to use mean, mode or median to compare sets of data a Show that Marcus, Zara and Arun could all be correct. b Which average do you think best represents the data in the tables. Explain why. Who do you agree with, Marcus, Zara or Arun? 8 The frequency tables show the number of goals scored in each match by two hockey teams in 20 matches. Team A Team B Number of goals 0 1 2 3 4 5 Number of goals 0 1 2 3 4 5 Frequency 4 1 4 2 4 5 Frequency 0 6 1 5 4 4 Read what Marcus, Zara and Arun say. I think that, on average, team A scored more goals per match. I think that, on average, team B scored more goals per match. I think that, on average, they scored the same number of goals per match. 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
386 16 Interpreting and discussing results Check your progress 1 Here are the ages, to the nearest year, of the members of a bowling club. 15 12 16 20 24 28 23 14 10 22 29 23 17 14 27 22 25 24 20 29 a Record this information in a frequency table. Use the classes 10 ≤ a < 15, 15 ≤ a < 20, 20 ≤ a < 25 and 25 ≤ a < 30. b Draw a frequency diagram to show the data. c How many of the members are at least 20 years old? Explain how you worked out your answer. 2 The students in class 8T took a test. These are the results, marked out of 50. 28 12 50 28 24 39 46 27 18 50 49 28 36 45 34 43 8 28 36 37 18 39 29 38 9 a Draw an ordered stem-and-leaf diagram to show the scores out of 50. b What percentage of the students had a score greater than 35? c What fraction of the students had a score less than 25? d Any student scoring less than 40% must re-sit the test. How many students do not have to re-sit the test? e Read what Zara says. I had the average score, because my mark was 28 out of 50. i Show that Zara could be correct. ii Show that Zara could be wrong. 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
387 16 Interpreting and discussing results 3 The pie charts show the favourite animals of the students in two schools. 15% 18% 5% 25% 35% 2% Ryefield School 30% 12% 15% 25% 10% 8% Haywood School Elephant Lion Giraffe Zebra Warthog Other There are 1200 students in Haywood School. There are 700 students in Ryefield School. Which school had the larger number of students choosing lion as their favourite animal? Show your working. 4 The test marks, out of 20, of two groups of students are shown. Art: 9, 16, 8, 10, 18, 16, 12, 8, 16, 9 Music: 12, 20, 6, 18, 6, 16, 11, 19 a Copy and complete this table. Mean Median Mode Range Art Music b In which group, Art or Music, do you think the students did better on average? Justify your answer. c In which group, Art or Music, do you think the students had more consistent scores? Justify your answer. 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
388 absolute change when a numerical value increases or decreases, the absolute change is the positive difference between the old and new values 227 accurate exact or correct to a given level of accuracy such as 1 d.p. or 3 s.f. 183 adapt change to make suitable for a new situation 262 advantages good points 146 algebraically using algebra 217 alternate angles two equal angles between two parallel lines on opposite sides of a transversal 107 anticlockwise turning in the opposite direction to the hands of a clock 317 approximate value a number rounded to a suitable degree of accuracy 181 arc part of the circumference of a circle 117 bearing an angle measured clockwise from north 292 bisector a line that divides a line segment or an angle into two equal parts 118 brackets used to enclose items that are to be seen as a single expression 17 categorical data data that can be divided into specific groups such as eye colour and favourite sport 370 centre of enlargement (COE) the fixed point of an enlargement 324 centre of rotation the point that remains still when a shape is turned 317 changing the subject rearranging a formula or equation to get a different letter on its own 37 circumference the perimeter of a circle 181 class interval the width of a group in a grouped frequency table/diagram 355 classes groups used for recording data in a grouped frequency table/ diagram 355 clockwise turning in the same direction as the hands of a clock 317 closed interval a range of numbers that does include its endpoints 57 coefficient a number in front of a variable in an algebraic expression or equation; the coefficient multiplies the variable 30 Glossary and index 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
389 column vector two numbers, placed vertically, that describe a translation of a point or shape; the top number describes the horizontal movement, while the bottom number describes the vertical movement 304 common factor a number that is a factor of two or more different numbers 258 compare look at and see what is the same or different 82 comparison to look at how things are the same or different 268 complementary event two events are complementary if one of them must happen and they cannot both happen 276 congruent identical in shape and size 304 conjecture a possible value based on what you know 19 constant a number on its own (with no variable) 30 construct (algebra) use given information to write an equation 51 construct (geometry) use given information to draw shapes, angles or lines in diagrams using compasses and a ruler 115 continuous data data that can take any value within a given range 355 corresponding angles two equal angles formed by parallel lines and a transversal 107 cross-section the 2D face formed by slicing through a solid shape 341 cube root the cube root of a number produces the given number when it is cubed; for example, the cube root of 125 is 5, because 53 is 125 20 decimal number a number in the counting system based on 10; a number with at least one digit after the decimal point; the part before the decimal point is a whole number, while the part after the decimal point is a decimal fraction 68 decimal places (d.p.) the number of digits after the decimal point 73 degree of accuracy the level of accuracy in any rounding 73 derive construct a formula 37 diameter a straight line between two points on the circumference of a circle (or surface of a sphere) that passes through the centre of the circle (or sphere) 181 disadvantages bad points 146 discrete data data that can only take exact values 355 enlargement a transformation that increases or decreases the size of a shape to produce a mathematically similar shape 324 equation of a line when a function is used to draw a line, the function is called the equation of the line 246 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
390 equivalent equal in value 33 equivalent calculation a calculation which uses a different method but gives exactly the same answer 70 equivalent decimal a decimal number that has the same value as a fraction 138 estimation approximation of an answer, based on a calculation with rounded numbers 88 expand brackets multiply all parts of an expression inside the brackets by the term directly in front of the bracket 43 experimental probability a probability resulting from an experiment involving a large number of trials 282 expression a collection of symbols representing numbers and mathematical operations, but not including an equals sign (=) 30 exterior angle of a triangle the angle formed by extending one of the sides of a triangle 111 factor a whole number that divides exactly into another whole number 98 factor tree a method of finding prime factors 11 factorisations expressions that have been factorised 48 factorise write an algebraic expression as a product of factors 47 formula an equation that shows the relationship between two or more quantities 37 formulae plural of formula 37 frequency diagram any diagram that shows frequencies 355 front view, front elevation a 2D drawing of a solid shape seen from the front 186 function a relationship between two sets of numbers 216 function machine a method of showing a function 216 generalise use a set of results to make a general rule 25 generate make or form 200 geometric an adjective meaning that something relates to geometry 106 gradient the slope of a line; a positive gradient slopes up from left to right; a negative gradient slopes down from left to right 246 grouped data where data values are recorded in groups, rather than as individual values 355 HCF an abbreviation for highest common factor 11 hierarchy a system in which things are arranged in order of their importance 180 highest common factor (HCF) the largest number, and/or letters, that is a factor of two or more other numbers or expressions 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
391 hypotenuse the longest side of a right-angled triangle, opposite the right angle 118 image a shape after a transformation 304 improper fraction a fraction in which the numerator is larger than the denominator 142 improve make better/easier 146 index the number of times a number is multiplied; 73 = 7 × 7 × 7 = 243, so the index is 3; the plural of index is indices 11 inequality a relationship between two expressions that are not equal 57 input a number that goes into a function machine 216 integer a whole number that can be positive or negative or zero, with no fractional parts 11 inverse the operation that has the opposite effect to another operation; for example, the inverse of ‘multiply by 5’ is ‘divide by 5’ 19 inverse function the equation that reverses a function 220 inverse operation the operation that reverses the effect of another operation 38 investigate explore an idea or method 15 justify give a reason to support your decision 268 kilometre measure of distance, approximately 5 8 of a mile 332 LCM an abbreviation for lowest common multiple 11 line segment a part of a straight line between two points 298 linear expression an expression with at least one variable, where the highest power of any variable is 1 31 linear function a function of the form y = mx + c where m and c are numbers 246 lines of symmetry a line of symmetry divides a shape into two parts, where each part is the mirror image of the other 174 lowest common multiple (LCM) the smallest positive integer that is a multiple of two or more positive integers 11 map the process of changing an input number to an output number using a function 217 mapping diagram a type of diagram that represents a function 217 mean an average of a set of values, found by adding all the values and dividing the total by the number of values in the set 156 median the middle number when a set of numbers is put in order 366 mentally a mental method, worked out in your head 88 midpoint the centre point of a line segment 298 mile measure of distance, approximately 8 5 of a kilometre 332 mirror line a line dividing a diagram into two parts, each part being a mirror image of the other 310 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
392 mixed number a number which is the sum of a whole number and a proper fraction 142 mode the most common number in a set of numbers 366 multiplier a number by which you multiply a given value 230 natural numbers 0, 1, 2, 3, 4, 5, ... 20 net a flat diagram that can be folded to form a 3D shape 346 nth term the general term of a sequence where n represents the position number of the term 210 object a shape before a transformation 304 one-step function a function that has only one mathematical operation 216 order arrange in a particular pattern, for example, from the smallest to the biggest 82 output a number that comes out of a function machine 216 partitioning a method of multiplying two numbers where the units, tens, hundreds, etc. in one of the numbers are multiplied separately by the other number 154 percentage decrease the decrease in the value of a number, written as a percentage of the original value 225 percentage increase the increase in the value of a number, written as a percentage of the original value 225 pi (π) the ratio of the circumference of a circle to the diameter of the circle 181 pie chart a circle divided into sectors, where each sector represents its share of the whole 370 place value the value of a digit in a number based on its position in relation to the decimal point 89 plot put points on a graph in order to draw a line 240 population all the possible people from whom you choose a sample 132 position number the position of a term in a sequence of numbers 205 position-toterm rule a rule that allows you to calculate any term in a sequence, given its position number 205 power 2 × 2 × 2 = 23 is called ‘2 cubed’ or ‘2 to the power 3’ 24 prime factor a factor that is a prime number 11 prism a solid 3D shape that has the same cross-section along its length 341 profit a gain in money; you make a profit if you sell something for more than you paid for it 267 proportion amount compared to the whole thing, usually written as a fraction, decimal or percentage 268 proportions fractions (or percentages) of the whole 370 quadrilateral a flat shape with four straight sides 174 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
393 radius a straight line from the circumference of a circle (or surface of a sphere) to the centre of a circle (or sphere) 182 range the difference between the largest and smallest numbers in a set 366 ratio an amount compared to another amount, using the symbol : 258 rational numbers numbers that can be written as fractions; this includes all integers 23 reciprocal the multiplier of a number that gives 1 as the result; for example, the reciprocal of 2 is 1 2 because 1 2 × 2 = 1 160 recurring decimal in a recurring decimal, a digit or group of digits is repeated forever 139 reflect draw the image of a shape as seen in a mirror 310 regular polygon a 2D shape with three or more straight sides, where all the sides are equal in length and all internal angles are the same size 174 reverse calculation a method of checking your answer by working backwards through the calculation using inverse operations 96 rotational symmetry a shape has rotational symmetry if, in one full turn, it fits exactly onto its original position at least twice 174 round make an approximation of a number, to a given degree of accuracy 73 scale factor the ratio by which a length is increased (or decreased) 324 sector a portion of a circle made from two radii and the part of the circumference that joins them 370 semicircle half a circle 185 sequence of patterns patterns made from shapes; the number of shapes in each pattern forms a sequence of numbers 205 shades a colour which is lighter or darker than a similar colour 268 share to split up into parts 263 short division a method of division where remainders are simply placed in front of the following digit 93 side view, side elevation a 2D drawing of a solid shape seen from the side 186 significant figures (s.f.) the first significant figure is the first non-zero digit in a number; for example, 2 is the first significant figure in both 2146 and 0.000 024 73 simplified written in its simplest form 158 simplify (a ratio) divide all parts of the ratio by a common factor 258 solve find the value (or values) of the unknown letter (or letters) in an equation 37 square root the square root of a number multiplied by itself gives that number; for example, the square root of 36 is 6 or −6 20 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
394 stem-and-leaf diagram a way of displaying data, similar to a horizontal bar graph but with sets of digits, in order of size, forming the bars 366 strategies methods 165 subject of a formula the letter on its own on one side (usually the left) of the formula 37 substitute replace part of an expression, usually a letter, with another value, usually a number 37 surface area the total area of the faces of a 3D shape 346 term a single number or variable, or numbers and variables multiplied together 30 terminating decimal a decimal number that does not go on forever 138 term-to-term rule a rule to find a term of a sequence, given the previous term 86 theoretical probability probability calculated on the basis of equally likely outcomes. It is equal to the number of favourable outcomes divided by the total number of possible outcomes. 282 time series graph a series of points, plotted at regular time intervals, joined by straight lines 361 top view, plan view a 2D drawing of a solid shape seen from above 186 translate transform a shape by moving each part of the shape the same distance in the same direction 304 transversal a line that crosses two or more parallel lines 107 trapezia plural of trapezium 336 trend a pattern in a set of data 361 two-step function a function that has two mathematical operations 217 unit fraction a fraction that has a numerator of 1 139 unknown a letter in an equation for which the value is yet to be found 30 upside down when you turn a fraction upside down you swap the numerator and the denominator, so 2 3 turned upside down is 3 2 160 variable a symbol, usually a letter, that can represent different values 30 vertically opposite angles two equal angles formed where two straight lines cross 106 written method worked out on paper 88 x-intercept the value of x where the graph of a line crosses the x-axis 250 y-intercept the value of y where the graph of a line crosses the y-axis 246 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