Cambridge Primary Mathematics Teacher's Resource, Emma Low, Cambridge University Press by Cambridge International Education

20 Graphs, charts and tables

Quick reference

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Handling data Graphs and tables

Core activity 20.1: Tables and line graphs (Learner’s Book p•••••) Learners extract information from and represent information in tables and lines graphs.

Water (millilitres)

700 600 500 400 300 200 100

Core activity 20.2: Pie charts (Learner’s Book p•••••) Learners learn to interpret pie charts.

line graph: a graph that uses one or more lines to join points that represent data.

Time (minutes)

Key Key

130

Euro (€) 1 2 5 10 20 50 100

Volleyball was more Volleyball popular than horse riding

Less than 10% of the friends voted for trampolining.

(a) What is the most common type of goldfish? (b) If a total of 50 goldfish were were sold in that week, estimate the number of each type of goldfish sold.

US Dollars ($) 130

pie chart: a graph using a divided circle where each section represents part of the total.

Swimming had over Swimming 40% of the votes.

1 A pet shop sells a range of fish. This pie chart were sold in one week. shows the fish that were

1 Copy and complete the currency conversion table. Japanese Yen (¥)

Vocabulary Vocabulary

Clues

Holiday activities

ready reckoner: a table of numbers used to help calculate or convert between units.

800

sft2 Pie ft2 charts Activity A ti itt title titl

Let’s investigate friendsthe voted onon what wanted to do sft4 George’s Complete labels the they Carroll diagram so during that the numbers are inComplete the correct their holiday. thespaces. key from the clues.

Vocabulary

Let’s investigate This graph shows how much water had dripped from a leaky tap in five minutes. Use the graph to work out how much water would have dripped from the tap in an hour.

Goldfish

Oranda goldfish

Fantail goldfish

Common Goldfish ____________

Fantail F antail Goldfish

____________

Oranda Goldfish

____________

Common goldfish

2 (a) How many Japanese Yen are equivalent to €20? €20?

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(b) How many US Dollars are equivalent to ¥1300? (c) How many Euros are equivalent to $6.50?

(d) How many Japanese Yen are equivalent to €6? €6?

(e) How many US Dollars are equivalent to ¥3900? (f) How many Euros are equivalent to $71.50?

Prior learning

Objectives

• Answer a set of related questions by collecting, selecting and organising relevant data; draw conclusions from their own and others’ data and identify further questions to ask. • Draw and interpret frequency tables, pictograms and bar charts, with the vertical axis labelled for example in twos, fives, tens, twenties or hundreds. Consider the effect of changing the scale on the vertical axis. • Construct simple line graphs, e.g. to show changes in temperature over time. • Understand where intermediate points have and do not have meaning, e.g. comparing a line graph of temperature against time with a graph of class attendance for each day of the week. • Relate finding fractions to division and use to find simple fractions of quantities. • Understand percentage as the number of parts in every 100 and find simple percentages of quantities. • Express halves, tenths and hundredths as percentages. • Know that US ‘cup’ measurements can be used for mass and capacity.

Vocabulary

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Unit 2C: Core activity 20.1 Tables and line graphs

Unit 2C: Core activity 20.2 Pie Cambridge charts Primary Mathematics Stage 6: Number: Unit 2C: Core activity 20.1 Tables and line graphs

Unit 2C: Organising, categorising and representing data 6Dh1 – Solve a problem by representing, extracting and interpreting data in tables, graphs, charts and diagrams for example: line graphs for distance/time, a price ‘ready reckoner’, currency conversion; frequency tables and bar charts with grouped discrete data. Unit 2C: Using understanding and strategies in solving problems 6Ps6 – Make sense of and solve word problems, single and multi-step (all four operations), and represent them, e.g. with diagrams or on a number line; use brackets to show the series of calculations necessary.

line graph • ready reckoner • pie chart

Cambridge Primary Mathematics 6 © Cambridge University Press 2014

Unit 2C

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Core activity 20.1: Tables and line graphs

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Resources: Conversion tables and graphs photocopy master (pXXX). (Optional: recipe books with amounts in US cup measures (or access to the internet). Equipment for experiment of choice.) Give learners the Conversion tables and graphs sheet. Talk about the ready reckoner for converting between US cups and millilitres and the line graph for cups and grams. You may need to remind learners that the US cup can be used as a measure of both capacity and 600g). mass. Then tell the learners to complete question 1 on the sheet (Answer: 960ml and 600g). Demonstrate how to answer question 1(b), using a straight edge to draw a line up from 4 cups on the graph to find the amount in grams. Point out the key features of a line graph, i.e. the title, the axis labels, the axis scales going up in regular intervals. Learners should check that they found the same answers. Ask learners for any advice they can share with the class for accurately reading the table or the graph. Remind learners that the line on the graph shows continuous data. Demonstrate that any point on the line can be read as cups and as grams.

Vocabulary

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Learners should complete the remainder of the sheet and then compare their methods and solutions with other learners in a small group.

Ask learners to reflect on and discuss which is easier to use, the ready reckoner or the line graph? As necessary, add to the discussion that for cup amounts that are on the readyreckoner it is easier to use that because they can be read straight off the table, for amounts between values it might be easier to read the conversion off the line graph.

Learners should use the information in the resource sheet line graph to make a ready reckoner for cups to grams, by selecting points on the line and recording them in both grams and cups. Ask learners to discuss, in small groups, how they will go about drawing a similar line graph to show the conversion, and the relationship, between cups and millilitres on squared paper. Each learner should then draw the graph and compare the graph with those of their group.

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20 Graphs, charts and tables

line graph: a graph that uses one or more lines to join points that represent data. ready reckoner: a table of numbers used to help calculate or convert between units.

Look out for!

• Learners who do not find the correct solutions. Encourage them to annotate the graph by drawing lines from the x-axis and the y-axis to the point on the line that they are trying to convert. Ensure that the lines they draw are at right-angles to the axis. For questions where they need to round to the nearest 12 cup, encourage learners to mark the 1 cups on the horizontal axis by measuring the 2 interval between the cup divisions and halving it.

Opportunity for display! Display the learners’ conversion graphs.

Summary • Learners will have extracted information from, and represent information in, tables and line graphs. Notes on the Learner’s Book Graphs, charts and tables (pXXX): learners extract and interpret data in tables and line graphs. They construct their own line graphs and bar charts to represent data.

More activities

Check up!

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• Tell learners that in some shops 1 kg potatoes can be bought for $1.50. Ask them to make a ready-reckoner and a line graph to show prices of potatoes from 100 g to 10 kg.

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Recipe converter (individual or pairs)

You will need recipe books with amounts in US cup measures (or access to the internet).

Learners can find recipes (online or in books) that are given in US cup amounts and convert the measurement using their ready reckoner or line graph.

Experiments (small groups or whole class) You will need equipment for experiment of choice.

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Learners could draw a line graph linked to a scientific investigation, which is not necessarily a straight line. For example: • the temperature of water as it is heated • the growth of a plant over time • the amount of liquid in a container as it evaporates.

Core activity 20.1: Tables and line graphs

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Core activity 20.2: Pie charts

LB: pxx

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Resources: Favourite foods pie charts photocopy master (pXXX). Comparing bar graphs and pie charts photocopy master (pXXX). Rulers. Squared paper. Metre sticks. (Optional: newspapers or magazines with pie charts (or access to the internet).) Display the pie charts on the Favourite foods pie charts sheet. Ask learners to use what they know about fractions of a shape to say what fraction of children chose each flavour of crisps in ‘Red Group’ and in ‘Blue Group’. Ask learners to write down the number of children that would have chosen each flavour if the groups had eight children in them. Demonstrate how the pie chart represents the data by arranging eight learners into a circle. Place three metre sticks in the circle to divide it up as the ‘Red Group’ pie chart, i.e.

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Key Learner Metre stick stick

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Ask learners to label the three sections of the human pie chart with the flavour, the fraction and the number of learners. Tell learners that the size of the sections of the pie chart are often given as percentages. Remind them that the whole set is 100% and ask three learners to label the sections with 50%, 25% and 25%. Repeat the activity with the ‘Blue Group’ pie chart. After the sections have been labelled with percentages add eight more learners to the circle, as if the size of the ‘Blue Group’ had been 16. Ask learners to rearrange the metre sticks and replace any labels that need replacing. They should find that only the number of learners has changed, the fraction and percentage of the whole is the same. Ask learners to solve the word problems on the resource sheet. They should write down the calculations they use, using brackets as necessary, and compare their methods and solutions with a partner (Answer: (1) 9, (2) 18). 18).

Tell learners that pie charts provide similar information to bar graphs. They can both be used to easily compare different parts of a set of data, but with a pie chart it is easier to also

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Vocabulary

pie chart: a graph using a divided circle where each section represents part of the total.

Look out for! • Learners who are unsure about fractions and percentages. As necessary, remind them of their learning in Stage 5 about finding simple fractions and percentages of quantities.

compare parts to the whole set of data. Show them the Comparing bar graphs and pie charts sheet as an example. Ask learners to conduct their own survey of ten children where they will choose between three options of food. They should represent the data collected by completing the third pie chart on the Favourite foods pie charts sheet (each section is one tenth or 10% on the chart). Learners should write statements about the data represented. They could use the sentence templates such as: • __(fraction/percentage)__ of the group chose ______. • More than __(fraction/percentage)__ of the group chose ______ .

Look out for!

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• Learners who quickly complete the ten section pie chart. As appropriate, challenge learners to use a different size set of data that can easily be marked on the pie chart, e.g. 5, 20, 30 or 40. Ask 1 interval on them how many people each 10% or 10 the pie chart represents. represents.

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Opportunity for display!

Display learners’ pie charts with the statements they have made about the data represented in them.

Summary • Learners will have learnt to interpret pie charts.

Notes on the Learner’s Book Pie charts (pXXX): learners estimate the number that sectors of a pie chart represent. They extract and interpret data from a pie chart and present it in the form of a bar chart.

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More activities

In the news (individual or pairs)

Check up!

• Ask learners to talk about, and answers questions on, the pie chart they have made.

You will need newspapers or magazines with pie charts (or access to the internet).

Learners can find examples of pie charts online or in newpapers or magazines. They should write a series of statements that are true about the data represented in each chart.

Different charts (individual)

Learners could represent the data collected for their pie chart (in the core activity) using different charts and graphs. They can compare the representations for clarity, precision, and efficiency.

Games book (ISBN 9781107667815)

The pie chart game (pXXX) is a game for two players. Each player starts with 50% of a pie chart. They add on percentages using a spinner. The player with the largest percentage of the pie chart at the end of the game is the winner. Core activity 20.2: Pie charts

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21 Statistics

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Average and range

mode: a type of average, the value in a set of data that occurs the most. mean: a type of average, calculated by finding the total of all the values in the set of data and dividing by the number of values. median: a type of average, the middle value in a set of values ordered from least to greatest. range: from the lowest to the highest value.

What number is on the fifth card? Kali, Summer, Benji and Kyle and learning to skip. While they were practising they recorded how many skips they did in a row. Here are their attempts: 1st try

Let’s investigate

2nd try

This is what they found out about how 9 year old and 13 year old learners get to school. Key Ke y

Key Ke y

walk

bus

car

cycle cycle cycl e

other

Describe the difference between how 9 year olds and 13 year olds go to school. What could be the explanations for these differences?

Fratania

3rd try

4th try

5th try

6th try

7th try

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Summer

Benji

Kyle

1 Copy and complete this table: Range

The council are planning transport for learners to and from school.

The four countries of Fratania, atania, Spanila, Brimland and Gretilli celebrate a dry weather festival during the months of January to May. They are each trying to encourage tourists to visit their own countries. Here are graphs of each country’s country’s rainfall last year for the five months of the festival.

The mean average of the numbers on these cards is 5.

Bus

Using statistics

Vocabulary

statistics: the collection, organization, presentation, interpretation and analysis of data.

Let’s investigate

Mode

Median

Mean

Kali

Summer Benji Kyle

Cambridge Primary Mathematics Stage 6: Number: Unit 2C: Core activity 21.1 The three averages

Spanila

120

110

Rainfall in mm

Core activity 21.2: Using statistics to persuade (Learner’s Book p•••••) Learners explore how statistics are used, and use statistics themselves to form a persuasive argument.

average: a measure used to find the middle of a set of data.

100

January January

February

March March

April

100

May

January January

Brimland

120

110

100

January January

February

March Mar ch

February

March March

April

May

Gretilli

120

Rainfall in mm

Vocabulary

Core activity 21.1: The three averages (Learner’s Book p•••••) Learners find the mode, median, mean and range of sets of data. They use brackets to show the series of calculations necessary to work out the mean of a set of data.

Rainfall in mm

Quick reference

Rainfall in mm

April

May

100

January January

February February

March Mar ch

Cambridge Primary Mathematics Stage 6: Number: Unit 2C: Core activity 21.2 Using statistics to persuade

April April

May

Prior learning

Objectives

• Answer a set of related questions by collecting, selecting and organising relevant data. • Draw conclusions from their own and others’ data and identify further questions to ask. • Find and interpret the mode of a set of data.

Unit 2C: Organising, categorising and representing data 6Dh2 – Find the mode and range of a set of data from relevant situations e.g. scientific experiments. 6Dh3 – Begin to find the median and mean of a set of data. 6Dh4 – Explore how statistics are used in everyday life. Unit 2C: Using understanding and strategies in solving problems 6Ps2 – Deduce new information from existing information and realise the effect that one piece of information has on another. 6Ps6 – Make sense of and solve word problems, single and multi-step (all four operations), and represent them, e.g. with diagrams or on a number line; use brackets to show the series of calculations necessary.

Vocabulary

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average • mode • mean • median • range • statistics

Cambridge Primary Mathematics 6 © Cambridge University Press 2014

Unit 2C

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Core activity 21.1: The three averages

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Resources: Sticky notes. Finding mode and median averages photocopy master (pXXX). Rolling cars photocopy master (pXXX). Calculators. Resources for learners’ experiments.

Pose learners this question to discuss in pairs (display the question written on the board): “On average, how many days are there in a month?”

Vocabulary

Ask three or four pairs to recount their discussion to the class. Ask learners to describe what they think ‘average’ is in a sentence on a sticky note of paper and display the definitions by sticking the notes to the board. If possible, choose some notes that accurately describe ‘average’ and read them to the class. Briefly explain that ‘average’ is a term used for one piece of information that provides useful information about a whole set of data.

mode: a type of average, the value in a set of data that occurs the most.

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Remind learners that in Stage 5 they worked out the mode of a set of data. This is a type of average that provides information about the most popular or frequent value in a set of data.

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Ask learners to write down the number of days in each month (to simplify, do not consider leap years), starting with January. (Answer: 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31)

Which is the most frequent number of days in a month? month?”” Explain that 31 is the Ask learners: “Which mode, there are on average 31 days in a month. Discuss with learners whether ‘31’ provides a useful piece of information about the set of numbers of days in months. This discussion should include: • yes, it tells us the most common number of days in a month • no, five month have less than 31 days, and no months have more than 31 days.

Write this set of data on the board: 10, 12, 13, 15, 15, 18, 19, 19, 20, 31, 31, 31 Ask learners to imagine that instead of the months we use, another civilization divided their calendar up into these months. Again the mode of this set would be 31. Although 31 is the most common number of days, it does not helpfully represent the whole set of data. Remind learners that in Stage 5 they learnt about ‘range’. Ensure learners understand that range is not an average, it gives us the difference between the lowest and highest value in a set of data.

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average: a measure used to find the middle of a set of data.

mean: a type of average, calculated by finding the total of all the values in the set of data and dividing by the number of values. median: a type of average, the middle value in a set of values ordered from least to greatest. range: from the lowest to the highest value.

Teacher note: Use learners’ description of ‘average’ to informally assess prior knowledge and understanding.

Ask learners to work in pairs to look at the two sets of data (the real and imaginary days in the months) and write down the range of each set of data. Ask them to write a statement for each set of data using the mode and range, e.g. ‘The average number of days in the months is 31, the number of days is from 28 to 31 so the range is 3.’ and ‘The average number of days in the imaginary months is 31, the number of days is from 10 to 31 so the range is 21.’

Look out for!

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• Learners who are unsure how to write their statements. As necessary, provide sentences templates/stems to help learners to express that sentence they have found out as a sentence.

Tell learners that by presenting the average and the range they are making a useful statement that provides information about the whole set of data. Tell learners that they are going to learn about other types of average. Explain that the median average is the middle value when the data is written in order from least to greatest. Arrange the learners in height order, from shortest to tallest. Count the number of learners, find the middle point in the line and declare the learner (or learners) at this point the median average height for the class. The range is the shortest to the tallest learner.

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Ask learners to write out the real number of days in each month again, this time in order of size from least to greatest, i.e. 28, 30, 30, 30, 30, 31, 31 , 31, 31, 31, 31, 31 As there is an even number of pieces of data in this set, the median is half way between the middle values. In this case the median average is 31, the same as the mode average. Tell learners to find the median average of the imaginary number of days in the month listed on the board (Answer: 18.5).

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Give learners the Finding mode and median averages sheet. Ask learners to complete the sheet, with a partner where asked.

Tell learners that the third average they are learning about in this session is the ‘mean’. It is the type of average that most people are referring to when they say ‘average’. The mean average is found by finding the total of all the data, then dividing that total by the number of pieces of data. Learners might find it useful to think about it as ‘if all the values were shared equally, what would every number be?’ Demonstrate calculating the average number of days in a month on a calculator. Give learners calculators so that they can calculate the mean at the same time, following the same steps on the calculator: (28 + 30 + 30 + 30 + 30 + 31 + 31 + 31 + 31 + 31 + 31 + 31) ÷ 12 = 30.416667. Write the calculation on the board and explain what each part means.

Look out for! • Learners who may have difficulty keeping track of where they are in a sequence of numbers when inputting them into the calculator, or who might make a mistake halfway through the list and feel they need to start at the beginning. Some calculators, including computer on-screen calculators, list all of the entries as they are made. Show learners how to use the ‘Clear Entry’ function on their calculator (sometimes shown as ‘CE’ or ‘C’) to only clear the last entry and carry on with the calculation.

Tell learners that to work out the median and mean averages the data must be numerical. Core activity 21.1: The three averages

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Show learners the Rolling cars sheet. Calculate the mean of Ramp A results together, using written, mental or calculator methods (Answer: 6.1). They should record the number sentence needed to calculate the mean time for each ramp. Learners should complete the

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resource sheet and then discuss the different solutions learners have found for question 3. Small groups of learners should carry out their own experiment (you may wish to do this in a separate session), either the same experiment as on the Rolling cars sheet or choose another one, for example: • How far do different designs of paper aeroplane fly? • How long does it take for an object to drop to the ground with different designs of parachute from an upstairs window? The experiment must be repeatable and the data must be numerical for this activity. Groups should devise their own table to record the results and then use average and range to describe their conclusions.

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• Learners have found the mode, median, mean and range of sets of data. • They have used brackets to show the series of calculations necessary to work out the mean of a set of data. Notes on the Learner’s Book Average and range (pXXX): learners calculate the mode, median, mean and range of sets of data. They make sets of numbers that have a particular, mode, median and mean.

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Groups can make a display of their experiment, including photographs of the experiment being carried out, the table of results and written conclusions using average and range.

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Ask learners to revisit the sticky notes. They should reflect on and, as necessary, improve their definition of average.

Summary

Opportunities to display!

Check up!

• Generate a set of data by throwing a dice five times. Ask learners to find the mode, median, mean and range for the set of data. • Tell learners that earlier you threw the die five times and the mode was 2, the median was 3, the mean was 3.2 and the range was 3. Ask them to work out what five numbers you threw.

More activities About my class (groups)

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Small groups of learners can investigate, and find the averages and range, for information about their classmates, e.g. • number of pets • number of siblings • age of all siblings of learners in the class • shoe size • distance from home to school.

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Games book (ISBN 9781107667815)

The mean game (pXXX) is a game for two to four players. Players select numbers from a grid to try to make a set of data with a mean closest to 15.

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Core activity 21.1: The three averages

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Core activity 21.2: Using statistics to persuade

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Resources: The Daily Gossip photocopy master (pXXX). 1 to 20 cards photocopy master (pXXX). Hotel reviews photocopy master (pXXX). (Optional: newspapers (or access to online news sites). Access to online hotel review sites. Changing the World with charts (CD-ROM).) Tell learners that they are going to investigate what is meant in statements that use averages to provide information. Give the class the following statements to discuss, one at a time: • Cars in our car park hold 4.3 people on average.

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Ask learners to discuss with their partners: ‘How many people can travel in an average car in our car park?’

• On average, children in Class 6 watch 1.8 films at the cinema per year.

Ask learners to discuss with their partners: ‘Does this mean that most children in Class 6 leave the cinema before the end the second time that they go in a year?’

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• Average number of legs for human beings is 1.99.

Ask learners to discuss with their partners: ‘How many legs does the average person have?’

For each of these examples, discuss what the ‘average’ means. Tell learners that these averages are all mean averages. The mean average is a very useful statistic for giving an understanding of a set of data, but it does not necessarily relate to a ‘real-life’ typical example of a value in a set of data.

Display The Daily Gossip sheet with the headline ‘Half of all students get a lower than average mark in National Tests!’ Tell learners that the intention of the newspaper story is that we should be shocked by how poor the students’ marks were in the test. Ask the learners, “Is Is this shocking?” shocking?”

In small groups, learners investigate different marks, by generating a number each using cards numbered 1 to 20 (card replaced each time), they might have got in a test with 20 marks. They should work out the mean, mode and median averages and decide whether, with those marks, half of them would have got lower than the average score. Ask groups to write a statement explaining whether or not they should be shocked by the headline.

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21 Statistics

Vocabulary

statistics: the collection, organization, presentation, interpretation and analysis of data.

Look out for!

• Learners who need support organising their investigation. Suggest they follow this sequence: • shuffle the 1 to 20 cards • take one card, write down the number, then replace the card • continue taking and replacing cards until six numbers are written • work out the mode, median and mode of the numbers • 3 is half of 6, so check if three of the numbers written down are lower than each of the averages • repeat with further sets of six cards. • discuss the outcomes of the investigation and try to generalise the results.

Assign one of the hotels on the Hotel reviews sheet to each pair of learners. They should argue a case for visiting that hotel, rather than the others, by choosing data to represent in a graph or chart and presenting an argument using the statistics, including at least one measure of average. Remind learners to think carefully about the scale on their graph, so that it will be most persuasive (but it must not be misleading). Pairs can make a poster showing why their hotel is the best.

Look out for!

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• Learners who have difficulty starting on this challenging activity. Give them one piece of information from the resource sheet about their hotel e.g. • The Seaview Hotel has the highest average score for location. • The Riverside Lodge has the highest average score for service. • The Plaza Hotel has the highest average score for rooms.

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Opportunity to display!

Display the persuasive posters. Make arrow labels that point to where and how learners have used graphs, charts or statistics in their arguments.

Summary

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• Learners have explored how statistics are used, and used statistics themselves to form a persuasive argument.

Notes on the Learner’s Book Using statistics (pXXX): learners read bar charts of rainfall and then calculate the mode, median and mean averages, choosing the average that best supports an argument for each graph. Learners work together in a group to produce a project. They use data collection and representation and written persuasive statements to argue for an improvement to the school or local area.

Check up!

• Ask learners to talk about the statistics they used in their poster and how they chose to represent the data. Ask them to reflect aloud about how they could improve how they communicate the data on the poster.

Core activity 21.2: Using statistics to persuade

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More activities Statistics in the news (individuals or pairs)

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You will need newspapers (or access to online news sites).

Ask learners to look through print or online newspaper for stories for statistics that are used to argue a point of view. Make a collection of these and display them in the classroom to discuss.

More hotels (individuals or pairs)

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You will need access to online hotel review sites.

Look at a hotel review website and use real data (reviews) to compare hotels/holiday/tourist attractions.

Change the world (pairs or small groups) You will need Changing the World with charts (CD-ROM).

Tell learners that they are going to explore how to use statistics to make a persuasive argument. In pairs, or small groups, ask learners to look closely at the chart on Changing the World with charts, which has information about the 19th century statistician and founder of modern nursing, Florence Nightingale. They should work together to write some statements about the data presented in the chart and explain why this chart might persuade people to improve hospital conditions.

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The Daily Gossip

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Half of all students get a lower than average mark in National tests!

Everyone was shocked today as these frightening statistics were revealed about our failing children.

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Original Material ÂŠ Cambridge University Press, 2014

ÂŠ Cambridge University Press 2014

1 to 20 cards

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Original Material ÂŠ Cambridge University Press, 2014