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) If you have any questions regarding these resources or come across any errors, please contact helpful-report@pixl.org.uk

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: 28 - 34 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

Key Vocabulary Frequency density Class width

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.

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

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

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

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

Drawing the Histogram Frequency Density 0.4 0.3 0.2 0.1 Frequency Density 0 100 200 300 400 500 600 700 800 Weekly Wage

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

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

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

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

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

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

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

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

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

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

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

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

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

Exam Question – Specimen Papers

Exam Question – Specimen Papers [2 marks]

Exam Question – Specimen Papers

Exam Question – Specimen Papers

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

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

Drawing a Histogram Student Sheet 2 Weekly Wage Frequency 10 £300 ≤ wage < £400 £400 ≤ wage < £500 20 £500 ≤ wage < £550 £550 ≤ wage < £600 £600 ≤ wage < £800 2 Student Sheet 2

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

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

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

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

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