1. Histograms with unequal class widths
Grade 6/7
Histograms with unequal class widths
Construct and interpret histograms with unequal class widths (for grouped discrete as well as continuous data)
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2. Lesson Plan Lesson Overview Progression of Learning
Objective(s)
Construct and interpret histograms with unequal class widths (for grouped discrete as well as continuous data)
Grade
6/7
Prior Knowledge
Formula triangles
Scales
Plotting bar charts
Duration
60 minutes (variable)
Resources
Print slides:
Equipment
Ruler, Calculator
Progression of Learning
What are the students learning?
How are the students learning? (Activities & Differentiation)
Concept of a histogram
Show students slide 4 – discuss the features of a histogram and not the differences between this type of graph and others that are similar. Important to understand that the frequency is proportional to the area of the bars as this property is used in the interpretation questions. 5
How to use a frequency table to find class width and frequency density
Using slide 5 explain to students how to calculate frequency density. Give students slide 28 – which includes 3 more tables to practice finding the frequency density. 10
How to draw a histogram
Give students slide 29. Using slides 7 to 9 demonstrate how to construct a histogram from a complete frequency table. Discuss scale.
Using a histograms to calculate other values
Students need to be familiar with using a histogram – namely in these 3 ways:
To complete a table from a histogram
To complete graph and / or table which is incomplete
To calculate frequency from graph (no table given)
Give students slide 30, 31 and 31. Using slide 10 to 22 go through each step with the students. 20
Constructing and interpreting histograms in exam questions (from specimen papers)
Give students slides 33 and 34. This includes 5 exam questions related to objective. Students need to use notes from lesson to answer the questions. Ensure that all steps are shown. Relate to mark scheme to show how the marks are allocated. 15
Next Steps
Assessment
PLC/Reformed Specification/Target 6/Statistics/Histograms

3. Key Vocabulary
Frequency density Class width

4. Histograms
Histograms are a little like bar charts, but with some crucial differences:
There are no gaps between the bars
The bars can be of different widths
The frequency is proportional to the area of the bars. A bar chart frequency is proportional to the height of the bars.

5. Using the Frequency Table
Frequency Density = Freq CW
Find class width: upper - lower CW 10 5 15 FD
1.2 2 4
0.8
0.2
Height of Plants (cm)
Frequency
0 < h < 10 12
10 ≤ h < 15 10
15 ≤ h < 20 20
20 ≤ h < 25 4
25 ≤ h < 35 2
35 ≤ h < 50 3 F ÷ x FD CW

6. Using the Frequency Table - Practice
Height of Plants (cm)
Frequency
0 < h < 10 12
10 ≤ h < 15 10
15 ≤ h < 20 20
20 ≤ h < 25 4
25 ≤ h < 35 2
35 ≤ h < 50 3

7. Drawing a Histogram
This frequency table gives the weekly wage of 72 accountants The first thing we do is add 2 columns (sometimes this is already drawn in for you)
Weekly Wage
Frequency
£100 < wage < £300 10
£300 ≤ wage < £400
£400 ≤ wage < £500 20
£500 ≤ wage < £550
£550 ≤ wage < £600
£600 ≤ wage < £800 2

8. Drawing the Histogram
We look at the largest value in the frequency density column. This = 0.4, and tells us to label the vertical axis from 0 to 0.4 The horizontal axis represents our wages, which go from 0 to 800
Weekly Wage
Frequency
Class Width
Frequency density
£100 < wage < £300 10
200
10/200 =0.05
£300 ≤ wage < £400
100
10/100 = 0.1
£400 ≤ wage < £500 20
20/100 = 0.2
£500 ≤ wage < £550 50
20/50 = 0.4
£550 ≤ wage < £600
10/50 = 0.2
£600 ≤ wage < £800 2
2/200 = 0.01

9. Drawing the Histogram Frequency Density
0.4
0.3
0.2
0.1
Frequency Density
Weekly Wage

10. Using Histograms - 1
Sometimes we are asked to complete a table from a histogram
Frequency Table
Frequency Density
Weekly Wage

11. Step 1: copy the class intervals and calculate the class widths
Frequency Density
Weekly Wage
£100 < wage < £300
£300 ≤ wage < £400
£400 ≤ wage < £500
£500 ≤ wage < £550
£550 ≤ wage < £600
£600 ≤ wage < £800
Class Width
200
100 50
Weekly Wage

12. Step 2: copy the frequency density
Weekly Wage
£100 < wage < £300
£300 ≤ wage < £400
£400 ≤ wage < £500
£500 ≤ wage < £550
£550 ≤ wage < £600
£600 ≤ wage < £800
Class Width
200
100 50
Frequency density
0.05
0.1
0.2
0.4
0.01
Weekly Wage

13. Step 3: Use formula to find frequency
Frequency = Frequency Density x Class Width
Frequency Density
Weekly Wage
£100 < wage < £300
£300 ≤ wage < £400
£400 ≤ wage < £500
£500 ≤ wage < £550
£550 ≤ wage < £600
£600 ≤ wage < £800
Class Width
200
100 50
Frequency density
0.05
0.1
0.2
0.4
0.01
Frequency 10 20 2 F
Weekly Wage ÷ x FD CW

14. Using Histograms - 2 Sometimes the graph and / or table is incomplete
Frequency Table
Mark
Frequency
0 < mark ≤ 30 6
30 < mark ≤ 50
50 < mark ≤ 60 32
60 < mark ≤ 80 20
80 < mark ≤ 100
Frequency Density

15. Step 1: Add class width and calculate frequency density (where possible)
Frequency Table
Mark
Frequency
0 < mark ≤ 30 6
30 < mark ≤ 50
50 < mark ≤ 60 32
60 < mark ≤ 80 20
80 < mark ≤ 100 CW 30
 20 10 20 FD
0.2
3.2 1
Frequency Density

16. Step 2: Look for a match frequency table and histogram to find the scale
3.5 3
Mark
Frequency
0 < mark ≤ 30 6
30 < mark ≤ 50
50 < mark ≤ 60 32
60 < mark ≤ 80 20
80 < mark ≤ 100 CW 30
 20 10 20 FD
0.2
3.2 1
2.5
Frequency Density 2
1.5 1
0.5

17. Step 3: Use the scale to find the missing frequency densities – which will then mean can find frequency
Frequency Table
3.5 3
Mark
Frequency
0 < mark ≤ 30 6
30 < mark ≤ 50
50 < mark ≤ 60 32
60 < mark ≤ 80 20
80 < mark ≤ 100 CW 30
 20 10 20 FD
0.2
3.2 1
2.5
Frequency Density 10
0.5 2 4
0.2
1.5 1
0.5
Frequency = Frequency Density x Class Width

18. Step 4: Use the scale to plot the remaining frequency densities
Frequency Table
3.5 3
Mark
Frequency
0 < mark ≤ 30 6
30 < mark ≤ 50
50 < mark ≤ 60 32
60 < mark ≤ 80 20
80 < mark ≤ 100 CW 30
 20 10 20 FD
0.2
3.2 1
2.5
Frequency Density 20 1 2 4
0.2
1.5 1
0.5
Frequency = Frequency Density x Class Width

19. Using Histograms - 3
Sometimes need to calculate frequency from graph (no table given)
The histogram shows the ages, in years, of members of a chess club. There are 22 members with ages in the range 40 ≤ age < 65. Work out the number of members with ages in the range 25 ≤ age < 40

20. Step 1: Find area of the bar we know the frequency for
Sometimes need to calculate frequency from graph (no table given)
The histogram shows the ages, in years, of members of a chess club. There are 22 members with ages in the range 40 ≤ age < 65. Work out the number of members with ages in the range 25 ≤ age < 40
Area = 25 x 11
= 275

21. Step 2: Use area to find how many squares represent 1 frequency
Sometimes need to calculate frequency from graph (no table given)
The histogram shows the ages, in years, of members of a chess club. There are 22 members with ages in the range 40 ≤ age < 65. Work out the number of members with ages in the range 25 ≤ age < 40
275 ÷ 22 = 12.5
Means that every 12.5 squares represents a frequency of 1
Area = 25 x 11
= 275

22. Step 3: Compare with area of bar where trying to find frequency
Sometimes need to calculate frequency from graph (no table given)
The histogram shows the ages, in years, of members of a chess club. There are 22 members with ages in the range 40 ≤ age < 65. Work out the number of members with ages in the range 25 ≤ age < 40
Area = 15 x 30
= 450
275 ÷ 22 = 12.5
Means that every 12.5 squares represents a frequency of 1
Area = 25 x 11
= 275
To find frequency 450 ÷ 12.5 = 36

23. Exam Question – Specimen Papers

24. Exam Question – Specimen Papers
[2 marks]

25. Exam Question – Specimen Papers

26. Exam Question – Specimen Papers

27. Exam Question – Specimen Papers
[3]
[4]
[1]

28. Using the Frequency Table
Height of Plants (cm)
Frequency
0 < h < 10 12
10 ≤ h < 15 10
15 ≤ h < 20 20
20 ≤ h < 25 4
25 ≤ h < 35 2
35 ≤ h < 50 3
Student Sheet 1

29. Drawing a Histogram Student Sheet 2 Weekly Wage Frequency10
£300 ≤ wage < £400
£400 ≤ wage < £500 20
£500 ≤ wage < £550
£550 ≤ wage < £600
£600 ≤ wage < £800 2
Student Sheet 2

30. Using Histograms - 1
Sometimes we are asked to complete a table from a histogram
Frequency Table
Frequency Density
Weekly Wage
Student Sheet 3

31. Using Histograms - 2 Sometimes the graph and / or table is incomplete
Frequency Table
Mark
Frequency
0 < mark ≤ 30 6
30 < mark ≤ 50
50 < mark ≤ 60 32
60 < mark ≤ 80 20
80 < mark ≤ 100
Frequency Density
Student Sheet 4

32. Using Histograms - 3
Sometimes need to calculate frequency from graph (no table given)
The histogram shows the ages, in years, of members of a chess club. There are 22 members with ages in the range 40 ≤ age < 65. Work out the number of members with ages in the range 25 ≤ age < 40
Student Sheet 5

33. Exam Questions – Specimen Papers - 1
[2 marks]
Student Sheet 6

34. Exam Questions – Specimen Papers - 2
[3]
[4]
[1]
Student Sheet 7