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3.1 Frequency tables

StarterTell students that they are going to investigate their ability to estimate time.

Organise the students into groups of 6. Each group will design a data collection sheet to record their time estimates.

Explain that one student will hold the stop watch and call start. The others will close their eyes and put their hand up when they think one minute has passed. The holder of the stopwatch will then record the times.

Discuss how easy/difficult each group found recording the individual times.

Using individual seconds can make data collection difficult and makes interpretation of the data very difficult. Recording the data in class intervals on a tally chart would be an improvement.

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Key wordsTally chartFrequency tableClass intervals

ResourcesStopwatches

LinksInequalities:www.heinemann.co.uk/hotlinks (code 4554T)

ActiveTeach resourcesMS PowerPoint presentation 3.1 Frequency tablesChapter 3 game

Follow upStudent Book sections 3.1–3.2, and Exercises 3A and 3B

Extra practiceWorksheet 3.1 (answers on page 182)

Student BookSections 3.1–3.2

Main teachingHow could you record the length of mobile phone calls in minutes in a frequency table?

Display slide 1 of PowerPoint 3.1. What is wrong with the intervals shown? Stress < and ≤ notation, and the importance of avoiding overlap. Reveal the improved intervals and comments.

Discuss rounding and the effect this has on intervals. Which class intervals would be better for mobile phone call lengths that have been rounded?

Return to the starter activity. Each group should repeat the activity so that they have more than ten times recorded.

Ask students to look at the data collected in their group and, as a class, discuss suitable class intervals. Remind them that class intervals should not overlap.

Should we allow open-ended class intervals? Why are they sometimes useful?

Reveal slide 2 to show a basic frequency table without any class intervals and complete the Time column with the agreed intervals.

Invite two pupils from each group to record their group’s times in the tally column. Complete the frequency column as a class and discuss the results.

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ObjectivesCollect continuous dataChoose class intervals for continuous data

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48 members of a sports club kept a record of the time, in hours, that they spent exercising during one week. The results are shown below.

1.

Four of the members suggested different class intervals to help them sort the data.

Give a reason why the intervals suggested by Jo, Anton and Mona are unsuitable.Give two reasons why Sam’s class intervals are better than the others.i) Use the class intervals Sam suggested to draw a frequency table.

Draw a pie chart to illustrate these data.Explain why Sam’s class intervals would not have been suitable if times had been rounded to the nearest hour.

The table shows the diameters (to the nearest 0.1 mm) of some ball bearings.

a)b)c)

ii)d)

2.

Worksheet 3.1Frequency tables, pie chartsand stem and leaf diagrams

Student Book sections 3.1–3.2

1.3 1.6 5.4 7.2 4.6 3.2 1.5 0.8 0.9 0.7 12.2 4.64.5 10.2 2.3 2.4 6.2 2.1 3.6 2.5 3.7 4.5 6.8 2.42.0 2.5 3.6 2.6 2.6 1.8 5.2 4.8 4.6 4.7 3.6 2.32.9 4.6 3.1 9.0 2.4 3.2 2.9 5.2 3.5 2.5 4.6 3.2

Jo Anton Mona SamTime Time (hours) Time (hours) Time (hours)

0 to 59 min 0–1 0 < t ≤ 1 t ≤ 21 h to 1 h 59 min 1–2 1 < t ≤ 2 2 < t ≤ 32 h to 2 h 59 min 2–3 2 < t ≤ 3 3 < t ≤ 43 h to 3 h 59 min 3–4 3 < t ≤ 4 4 < t ≤ 54 h to 4 h 59 min 4–5 4 < t ≤ 5 5 < t ≤ 7

etc. etc. etc. 7 < t

8.1 7.2 6.9 5.3 6.8 6.8 7.6 5.9 7.7 7.1 6.6 5.96.2 7.3 5.8 7.8 6.5 6.1 8.2 7.4 8.4 7.3 6.7 6.3

Draw an unordered stem and leaf diagram for these data.Use your unordered stem and leaf diagram to help you draw an ordered stem and leaf diagram for these data.What is the range of diameters for these ball bearings?

Any ball bearings that are less than 5.5 mm or greater than 8 mm are rejected.Write down the number of ball bearings that can be used.

a)b)

c)

d)

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3.3 Cumulative frequency diagramsKey wordsCumulativeCumulative frequencyClass interval

ResourcesDice

LinksStudent Book section 2.2 Cumulative frequency tablesStudent Book section 2.14 Cumulative frequency step polygons

ActiveTeach resourcesMS PowerPoint presentation 3.3 Cumulative frequencyChapter 3 game

Follow upStudent Book section 3.3 and Exercise 3C

Extra practiceWorksheet 3.3 (answers on page 182)

Student BookSection 3.3

StarterExplain that you are going to demonstrate cumulative frequency.

Roll a six-sided dice to select a frequency and ask that many students to come to the front of the class. Then ask them to stand to one side. Roll the dice again and invite that many students to the front of the class. Then ask them to go and join the group stood to one side. How many students are there at the front of the class now?

Repeat the process and explain that each time the dice gives a frequency (number of students) and they are calculating the cumulative frequency by keeping a running total.

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Main teachingDisplay slide 1 of PowerPoint 3.3 showing the maximum temperatures measured in Oxford over a two-year period.

Discuss how you could estimate the number of days the temperature was less than 11°C. (1 + 7 = 8 days)

Ask students to calculate each cumulative frequency and move through slide 1 to reveal the answers.

How could you estimate the frequency for a maximum temperature of 12°C? Lead students towards drawing a graph.

Tell them that they are going to draw a graph with a continuous scale for maximum temperature.

Discuss the scale for continuous data along the x-axis. Reveal slide 2 to show suitable axes.

Emphasis that points must be plotted at the end of each class interval and invite students to plot the points on the board.

How should you join up the points? A curve or straight lines are both acceptable.

Display the accurately drawn graph on slide 3 and click to reveal the key points.

Pose the following questions and ask students to use the graph to answer them.

Estimate on how many days the temperature was higher than 19°C. (8)

Estimate on how many days it was between 12°C and 19°C. (7)

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ObjectivesDraw a cumulative frequency diagramUse cumulative frequency diagrams to estimate or predict values from grouped data

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The speeds of 100 cars travelling through some roadworks on a section of the M6 motorway were recorded by a speed camera.

1.

Complete the cumulative frequency table for these data.Use the information in your table to draw a cumulative frequency diagram.

The speed limit for traffic travelling through the road works was 50 mph.Estimate how many cars were breaking the speed limit.

The members of an angling club recorded the weights of the largest fish they caught this season.

a)b)

c)

2.

Worksheet 3.3Cumulative frequencydiagrams

Student Book section 3.3

Speed, s (mph) Frequency Cumulative frequencys ≤ 20 1

20 < s ≤ 30 430 < s ≤ 40 1740 < s ≤ 50 3550 < s ≤ 60 2160 < s ≤ 70 1370 < s ≤ 80 6

80 < s 3

Weight, w (kg) Frequency Cumulative frequencyw ≤ 1 2

1 < w ≤ 5 145 < w ≤ 10 8

10 < w ≤ 15 515 < w ≤ 20 420 < w ≤ 25 425 < w ≤ 30 3

Complete the cumulative frequency table for these data.Use the information in your table to draw a cumulative frequency diagram.Estimate how many members failed to catch a fish weighing more than 8 kg.Estimate how many members caught fish weighing more than 22 kg.

a)b)c)d)

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3.4 Equal interval histogramsKey wordsContinuous scaleHistogramClass intervals

ResourcesGraph paper

LinksAutograph activity: www.heinemann.co.uk/hotlinks (code 4554T)

ActiveTeach resourcesMS PowerPoint presentation 3.4 Equal interval histogramsChapter 3 game

Follow upStudent Book sections 3.4–3.5 and Exercise 3D

Extra practiceWorksheet 3.4 (answers on page 183)

Student BookSections 3.4–3.5

StarterTell students you are going to collect data from the class to use later in the lesson.

Ask each student to measure their hand span and record the results on slide 1 of PowerPoint 3.4.

Main teachingRemind students of the bar charts they have drawn in the past. Display slide 2 of PowerPoint 3.4 and identify particular features:

The height represents the frequency.

There are gaps between bars.

The bars represent whole numbers.

Move on to slide 3. What are the differences between this histogram and the bar chart on the first slide? Reveal the differences when students have offered plenty of suggestions:

No gaps.

Continuous scale on axis.

The area of each rectangle represents the frequency of a class interval.

Discuss each of these points and explain that diagrams such as this are called histograms.

Display the frequency table on slide 4 and discuss how to draw a histogram of these data.

Reveal the histogram on slide 5 and explain how it has been drawn. What is the shape of the distribution? Display page 91 of the Student Book to support a discussion about the different shapes formed by distributions.

Tell students they are going to draw a histogram using the hand span data gathered at the beginning of the lesson. Display the data on slide 1 and remind them of the importance of axis labels, scales and titles.

Invite students to compare their histograms to look for mistakes.

What is the shape of the distribution?

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ObjectivesUse a continuous scaleDraw a histogram with equal class intervals

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The table shows the distance (d) employees of a small company travel to work.1.

Draw a histogram for these data.Draw a frequency polygon for these data.Describe the shape of the distribution.

The table below shows the yield in kilograms of 100 fruit trees in two orchards.

a)b)c)

2.

i) Draw a histogram to represent the data for Hill Top Orchard.Draw a frequency polygon for these data.

Describe the shape of the distribution for Hill Top Orchard.i) Draw a histogram to represent the data for Valley Orchard.

Draw a frequency polygon for these data.Describe the shape of the distribution for Valley Orchard.Write down the name of the orchard that produced the most fruit.

The table shows the time taken by some employees to complete a task.

a)ii)

b)c)

ii)d)e)

3.

Draw a histogram for these data.Draw a frequency polygon for these data.Describe the shape of the distribution.

a)b)c)

Worksheet 3.4Equal class intervals andthe shape of distributions

Student Book sections 3.4–3.5

Distance, d (km) Frequency0 < d ≤ 3 33 < d ≤ 6 66 < d ≤ 9 10

9 < d ≤ 12 4

Yield, y (kg) Hill Top Orchard Valley Orchard5 < y ≤ 10 5 18

10 < y ≤ 15 10 4315 < y ≤ 20 25 2420 < y ≤ 25 42 1225 < y ≤ 30 18 3

Time, t (min) Frequency0 < t ≤ 5 5

5 < t ≤ 10 810 < t ≤ 15 1215 < t ≤ 20 3

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3.6 Unequal interval histograms (Higher)Key wordsHistogramUnequal class intervalsFrequency density

ResourcesGraph paper

LinksAutograph activity: www.heinemann.co.uk/hotlinks (code 4554T)

Active Teach resourcesMS PowerPoint presentation 3.6 Unequal interval histograms (Higher)Chapter 3 game

Follow upStudent Book sections 3.6–3.7 and Exercise 3E (Higher)

Extra practiceWorksheet 3.6 Higher (answers on page 183)

Student BookSections 3.6–3.7

Starter Discuss how blood pressure is measured. Two measurements are taken: systolic and diastolic pressure. A normal reading is in the region of 120/80 (120 over 80). The higher a person’s blood pressure, the more likely they are to have health problems.

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Main teachingShow the blood pressure frequency table on slide 1 of PowerPoint 3.6. Explain that it shows the frequencies of blood pressure for a group of men.

Discuss the width of the class intervals. Why do you think unequal class intervals have been used? Complete the class-width column on slide 2.

Remind students of the rules for histograms:

No gaps.

Continuous scale.

The area of each rectangle represents the frequency of a class interval.

Discuss how a histogram might be drawn. Explain that area must be used to represent the frequency and define frequency density as shown on slide 3.

Ensure students understand that the frequency density is the height of the bar and therefore the vertical axis is labelled frequency density, not frequency.

Ask students to calculate the frequency densities for the table on slide 4 and then reveal the correct answers.

If time allows, provide students with graph paper to draw the histogram for these data or complete one as a class on the whiteboard.

A completed histogram can be displayed on slide 5 if needed.

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ObjectivesUnderstand why frequency density is usedDraw a histogram with unequal class intervals

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An athletics club gave intensive training to improve the ability of its members to throw the javelin. The improvements are recorded in this table.

1.

Draw a histogram to display these data.Estimate how many people improved their throw by between 7 and 10 metres.

The histogram shows the length of some nails found in a carpenter’s toolbox.

a)b)

2.

Estimate how many nails are between 5 mm and 15 mm long.Design and complete a frequency table for these data.

In a TV game show contestants tried to beat the clock to complete a puzzle.The times taken are shown in the table.

a)b)

3.

Draw a histogram to display these data.Estimate how many people took more than 35 seconds to complete the puzzle.

a)b)

Worksheet 3.6HigherUnequal interval histograms

Student Book sections 3.6–3.7

Distance improvement, d (m) Frequency0 < d ≤ 1 31 < d ≤ 3 43 < d ≤ 5 6

5 < d ≤ 10 510 < d ≤ 20 5

10 20 30Length of nail (mm)

40

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nsity

0

0.5

1.0

1.5

2.0

0

2.5

50

Time, t (seconds) Frequency0 < t ≤ 15 3

15 < t ≤ 25 525 < t ≤ 30 1530 < t ≤ 40 640 < t ≤ 60 2

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3.9 Choropleth mapsKey wordsChoropleth map

Resources4 × 4 square grids on paper

LinksPlayground choropleth map: www.heinemann.co.uk/hotlinks (code 4554T)

ActiveTeach resourcesMS PowerPoint presentation 3.9 Choropleth mapsChapter 3 game

Follow upStudent Book section 3.8Student Book section 3.9 and Exercise 3FStudent Book section 3.10, Exercise 3G and Worksheet 3.9B

Extra practiceWorksheet 3.9A (answers on page 184)

Student BookSections 3.8–3.10

Starter Display slide 1 of PowerPoint 3.9 and discuss the map. Tell students it is called a choropleth map and they use shading to give data about the regions shown on the map. Explain that this map shows the woodland cover county by county for Scotland.

Ask students for other examples of choropleth maps they may have seen.

What do all choropleth maps need? (A key to explain what the colours mean)

Which of the lettered areas has the most trees? (G)

Which has the least? (A)

What percentage of Area E is covered by trees? (15–17.9%)

Suggest a reason why there are few trees in areas A and C. (E.g. climate/terrain not suitable for trees)

Main teachingMove on to slide 2 and discuss each of the listed choropleth map features.

A scientist is looking into the spread of buttercups and divides a rectangular field into square sections. The grid on slide 3 shows how many buttercups were found in each square.

Ask students to use the information on the grid and the key to complete a choropleth map. Use a 4 × 4 square grid.

Compare their answers with the map given on slide 4.

Describe the spread of buttercups in the field.

Most in top left-hand corner and least in the bottom right-hand corner.

Ask students for pros and cons of choropleth maps and reveal the key points at the bottom of slide 4.

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ObjectivesExtract information from a choropleth mapRepresent data using a choropleth map

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The population pyramid below shows the age distribution of a small town in England.1.

Write down the percentage of the males who are aged over 60.Write down the percentage of the females who are aged over 60.Comment on your answers to parts a and b.

Third world countries are very poor and people often get little food. Diseases are common.

What difference would you expect to see if a similar population pyramid was drawn for a third world country?

A university is carrying out research into the distribution of a type of worm. They take a small plot and divide it into squares.The number of worms in each square is shown in the following diagram.

a)b)c)

d)

2.

Use the information in the diagram to complete a choropleth map for these data.Describe how the worms are spread across the plot.

a)b)

Worksheet 3.9APopulation pyramids andchoropleth maps

Student Book sections 3.8–3.9

70 < a ≤ 80

a > 80

60 < a ≤ 70

50 < a ≤ 60

40 < a ≤ 50

30 < a ≤ 40

20 < a ≤ 30

10 < a ≤ 20

0 < a ≤ 10

0510152025 2520151050Percentage Percentage

Population percentage by age groupFemales Males

10 10 5 3 1

10 7 4 2 1

8 5 4 2 1

6 5 3 2 1

2 3 2 1 0

1 1 1 0 0

0–2 worms3–5 worms6–8 worms9–11 worms

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This pie chart was drawn to show the amount of time given to different types of sport by a cable TV channel.

1.

Explain why this is not a good way to display the data.

The Tourist Board at town A has produced the following chart to show the average daily hours of sunshine in 3 resorts.

2.

Explain why the diagram could be misleading.Discuss why Town A might want to misrepresent the results.

Tom looks at the bar chart showing the length of some rivers.

a)b)

3.

He says the Yangtze is less than half as long as the Nile.Explain why the bar chart is misleading.Draw a diagram to display the lengths of the rivers to so that Tom can easily make comparisons.

a)b)

Worksheet 3.9B

Misleading diagrams

Student Book sections 3.10

Sports coverage

Outdoor sports 43%Combat sports 18%

Indoor sports 15%

Motor sports 15%

Town A2 hours

Town B8 hours

Town C4 hours

Yangtz

e6000

6100

6200

6300

6400

6500

6600

River length6700

Amazon Nile

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The time taken by some students to complete a puzzle is given in the table below.1.

Quick Test 3A

Complete the table.

The stem and leaf diagram shows the length of some blades of grass in a scientific experiment.

2.

How many blades of grass were measured in the experiment?Write down the length of the shortest blade of grass.Work out the range of the lengths of the blades of grass.

Write down the missing word in each of the statements below.A frequency polygon often joins the _____________ of the top of the bars of a histogram with straight lines.A population pyramid is used to compare aspects of the _____________ .

The vertical scale gives the _____________ group. The bars usually have _____________ widths. One side of the pyramid shows __________ and the other shows _________ . The horizontal scale is usually given as a _____________ .

A choropleth map is used to classify regions using ____________. A key shows what each _____________ represents.

a)b)c)

3.a)

b)

c)

Time, t (minutes) Frequency Cumulative frequency0 < t ≤ 5 5

5 < t ≤ 10 1010 < t ≤ 15 1415 < t ≤ 20 6

8 2 4 6 69 3 5 6 8 9

10 2 4 7Key 8 6 means 8.6 cm

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A nurse is drawing up a frequency table to record the heights of some patients. Which of the following would be suitable class intervals to use in the table?Give a reason for each answer.A 120–130 cm, 130–140 cm etc.B 120 < h ≤ 129, 130 < h ≤ 140 etc.C 120 < h ≤ 130, 130 < h ≤ 140 etc.

The cumulative frequency curve shows the times taken by a group of students to complete an exercise.

1.

2.

How many students were in the group?Estimate how many students took longer than 30 seconds.Estimate how long it took before 50% of the students had completed the exercise.

Fill in the blanks for these statements about a histogram.When there are unequal intervals in a frequency table then the _____________ of each bar is proportional to the frequency. The height of the bar is often called the _____________ .

Fill in the missing values in the table.

a)b)c)

3.

4.

Comment on the skew for this distribution.5.

Quick Test 3BHigher

10 20 30Time (s)

40

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0 50 60

Time, t (s) Frequency Class width Frequency density20 < t ≤ 40 80 2040 < t ≤ 50 10 750 < t ≤ 60 60 6

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The Chapter Test is available in two formats: as it appears here and exam-style write-on format on the CD-ROM in the back of this Teacher Guide.

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The table shows the distribution of males and females in a small town.1.

Draw a population pyramid for these data.(4)

The table shows the maximum temperature recorded each month in Oxford over a period of two years.

2.

Copy and complete the frequency table.a)

Chapter 3 TestFoundation

Age, a (years) Males Females0 < a ≤ 10 8 7

10 < a ≤ 20 12 1020 < a ≤ 30 13 1130 < a ≤ 40 22 1940 < a ≤ 50 17 1550 < a ≤ 60 11 1360 < a ≤ 70 9 1170 < a ≤ 80 6 8

80 < a 2 6

10.4 17.5 19.2 10.3 18.7 17.6

9.7 20.5 15.1 10.5 20.1 14.3

12.0 20.6 11.3 10.6 21.7 9.9

17.8 21.7 8.3 13.1 20.8 6.5

Maximum temperature, t (°C) Frequency5 < t ≤ 88 < t ≤ 11

11 < t ≤ 1414 < t ≤ 1717 < t ≤ 2020 < t ≤ 23

Use the information in your table to draw a histogram.Draw a frequency polygon to display these data.

(6)

b)c)

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The choropleth map shows the spread of bacteria in a rectangular container during a laboratory experiment.

3.

Which rectangle contains the greatest concentration of the bacteria?The following day all of the squares were found to contain an additional 10 bacteria.

Draw a choropleth map to show the spread of the bacteria on this day.(3)

The cumulative frequency diagram shows the age of members of a club.

a)

b)

4.

How many members does the club have?Estimate the age of the club’s oldest member?Estimate the age of the club’s youngest member?Estimate how many members are between the ages of 30 and 50.

(5)

a)b)c)d)

KeyMore than 20 bacteria observed

11–20 bacteria observed

1–10 bacteria observed

No bacteria observed

A B C D

1

2

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4

5

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40

50

60

70

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The stem and leaf diagram shows the weight of each egg in a box of 15 eggs, to the nearest gram.

5.

4 6 7 75 4 5 7 96 2 2 3 6 7 87 1 2

Key 4 6 means 46 g

How heavy is the heaviest egg?How many eggs weigh less than 60 g?Draw a grouped frequency table for these data.Construct a pie chart for these data.

(8)Total 26

a)b)c)d)

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The Chapter Test is available in two formats: as it appears here and exam-style write-on format on the CD-ROM in the back of this Teacher Guide.

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The pie chart shows the diet of some people in the USA. 1.

How many take-off distances were measured?Draw a histogram to display these data.Use your histogram to estimate how many aircraft took off in less than 390 m.Draw a frequency polygon for these data.Describe the skewness of your histogram.

(8)

a)b)c)d)e)

Chapter 3 TestHigher

America diet

Dairy products 7%Starchy food 6%

Cereals 12%

Sugar 8%

Vegetables 31%

Meat and fish 36%

What makes the pie chart misleading?(2)

An airport measures the take-off distances of some aircraft. The distances are given in the table.

2.

Take-off distance, d (m) Frequency320 < d ≤ 350 15350 < d ≤ 370 14370 < d ≤ 380 19380 < d ≤ 400 35400 < d ≤ 450 3

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Over how many months were the data collected?Estimate the number of months the rainfall exceeded 200 mm.Construct a frequency table for these data.Use your frequency table to draw a histogram for these data.

(5)Total 15

a)b)c)d)

The cumulative frequency diagram shows the monthly rainfall for a town.3.

60 80 100Rainfall (mm)

12040 140 160 180 200 220 240

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20

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