072e. Pie Charts and Divided Bar Graphs

Learning Intentions

• To understand that pie charts and divided bar graphs are used to represent proportions of a total
• draw and interpret a pie chart
• draw and interpret a divided bar graph

Pre-requisite Summary

• Understand that a proportion compares a part to a whole
• Know that fractions, decimals and percentages can represent proportions
• Be able to find a fraction or percentage of a total
• Understand that a full circle represents a whole, or $100\mathrm{\%}$
• Know that a full angle around a point is ${360}^{\circ }$
• Be able to label categories and frequencies clearly
• Understand that equal totals are important when comparing proportions in divided bar graphs
• Be able to use a ruler and protractor accurately when constructing graphs

Worked Examples

Worked Example 1

a) Explain why pie charts and divided bar graphs are useful for showing proportions of a total.
b) State what the whole represents in a pie chart.
c) State what the whole represents in a divided bar graph.

Worked Example 2

A class survey gives:

• Sport: $50\mathrm{\%}$
• Music: $30\mathrm{\%}$
• Art: $20\mathrm{\%}$
  a) Draw a pie chart.
  b) Label each sector.
  c) State which category has the greatest proportion.

Worked Example 3

A group of $40$ students travel to school by:

• bus: $10$
• car: $20$
• walk: $10$
  a) Convert each category to a proportion of the total.
  b) Draw a divided bar graph.
  c) Interpret the graph.

Worked Example 4

A pie chart shows the proportions of favourite fruits:

• apples: ${\displaystyle \frac{1}{2}}$
• bananas: ${\displaystyle \frac{1}{4}}$
• oranges: ${\displaystyle \frac{1}{4}}$
  a) Convert each fraction to an angle.
  b) Draw the pie chart.
  c) Explain how the sector sizes show the proportions.

Worked Example 5

A divided bar graph shows the proportions of water use in a house:

• bathroom: $40\mathrm{\%}$
• kitchen: $35\mathrm{\%}$
• garden: $25\mathrm{\%}$
  a) Draw the divided bar graph.
  b) Label each section.
  c) State which section is the smallest.

Worked Example 6

A pie chart or divided bar graph is given.
a) Read the largest category.
b) Read the smallest category.
c) Estimate a proportion for one category.

Problems

Problem 1

a) Explain why pie charts and divided bar graphs are useful for showing proportions of a total.
b) State what the whole represents in a pie chart.
c) State what the whole represents in a divided bar graph.

Problem 2

A class survey gives:

• Reading: $40\mathrm{\%}$
• Gaming: $35\mathrm{\%}$
• Drawing: $25\mathrm{\%}$
  a) Draw a pie chart.
  b) Label each sector.
  c) State which category has the greatest proportion.

Problem 3

A group of $50$ students choose lunch options:

• sandwich: $20$
• pasta: $15$
• sushi: $10$
• salad: $5$
  a) Convert each category to a proportion of the total.
  b) Draw a divided bar graph.
  c) Interpret the graph.

Problem 4

A pie chart shows the proportions of pets owned:

• dogs: ${\displaystyle \frac{1}{2}}$
• cats: ${\displaystyle \frac{3}{10}}$
• fish: ${\displaystyle \frac{1}{5}}$
  a) Convert each fraction to an angle.
  b) Draw the pie chart.
  c) Explain how the sector sizes show the proportions.

Problem 5

A divided bar graph shows the proportions of time spent after school:

• homework: $30\mathrm{\%}$
• sport: $45\mathrm{\%}$
• screen time: $25\mathrm{\%}$
  a) Draw the divided bar graph.
  b) Label each section.
  c) State which section is the largest.

Problem 6

A pie chart or divided bar graph is given.
a) Read the largest category.
b) Read the smallest category.
c) Estimate a proportion for one category.

Exercises

Understanding and Fluency

1. State whether each graph type shows proportions of a whole:
   a) pie chart
   b) divided bar graph
   c) line graph

2. For each graph, state what represents the whole:
   a) the full circle in a pie chart
   b) the full bar in a divided bar graph
   c) one sector in a pie chart

3. Convert each percentage to an angle for a pie chart:
   a) $50\mathrm{\%}$
   b) $25\mathrm{\%}$
   c) $10\mathrm{\%}$

4. Convert each percentage to an angle for a pie chart:
   a) $40\mathrm{\%}$
   b) $30\mathrm{\%}$
   c) $20\mathrm{\%}$

5. Draw a pie chart for:
   a) red $=50\mathrm{\%}$
   b) blue $=30\mathrm{\%}$
   c) green $=20\mathrm{\%}$

6. Draw a pie chart for:
   a) football $=40\mathrm{\%}$
   b) basketball $=35\mathrm{\%}$
   c) tennis $=25\mathrm{\%}$

7. Draw a divided bar graph for:
   a) apples $=60\mathrm{\%}$
   b) bananas $=25\mathrm{\%}$
   c) oranges $=15\mathrm{\%}$

8. Draw a divided bar graph for:
   a) walking $=20\mathrm{\%}$
   b) bus $=50\mathrm{\%}$
   c) car $=30\mathrm{\%}$

9. Interpret the graph by stating:
   a) the largest category
   b) the smallest category
   c) one category that is about one-quarter of the total

10. Interpret the graph by estimating:
    a) a category that is about half the total
    b) two categories that together make about $60\mathrm{\%}$
    c) the difference in proportion between the largest and smallest categories

Reasoning

11. Explain why a pie chart is suitable for showing parts of a whole.

12. A student draws a pie chart but the sector angles add to ${390}^{\circ }$. Explain the mistake.

13. Explain why the full bar in a divided bar graph must represent the same total each time.

14. A student says the tallest section in a divided bar graph means the largest proportion. Explain why this is incorrect.

15. Explain why larger sectors in a pie chart represent larger proportions.

16. A student labels a pie chart but does not include a title. Explain why this makes interpretation harder.

Problem-solving

17. A survey of favourite pets gives: dogs $=18$, cats $=12$, fish $=6$, birds $=4$.
    a) Find the total.
    b) Draw a divided bar graph.
    c) State which pet is chosen by the greatest proportion.

18. A class of $40$ students chooses transport to school:
    bus $=16$, car $=12$, walk $=8$, bike $=4$.
    a) Convert each category to a fraction or percentage of the total.
    b) Draw a pie chart.
    c) Interpret the chart.

19. A daily time-use survey gives: sleep $=8$ h, school $=6$ h, homework $=2$ h, leisure $=8$ h.
    a) Express each part as a proportion of $24$ h.
    b) Draw a divided bar graph.
    c) State which two categories are equal.

20. A pie chart shows spending in a week:
    food $=50\mathrm{\%}$,
    travel $=20\mathrm{\%}$,
    entertainment $=15\mathrm{\%}$,
    savings $=15\mathrm{\%}$.
    a) Draw the pie chart.
    b) State the angle for food.
    c) Identify the two equal categories.

21. A sports club survey gives: soccer $=45\mathrm{\%}$, netball $=25\mathrm{\%}$, swimming $=20\mathrm{\%}$, tennis $=10\mathrm{\%}$.
    a) Draw a divided bar graph.
    b) State the most popular sport.
    c) Estimate how much more popular soccer is than tennis.

22. A pie chart is used to show favourite fruits in a class: apples $={\displaystyle \frac{1}{2}}$, bananas $={\displaystyle \frac{1}{4}}$, grapes $={\displaystyle \frac{1}{8}}$, oranges $={\displaystyle \frac{1}{8}}$.
    a) Draw the chart.
    b) State the angle for bananas.
    c) Explain why apples take the largest sector.

Potential Misunderstandings

• Students may think pie charts and divided bar graphs show frequencies directly rather than proportions of a total
• Students may forget that the whole pie chart must add to ${360}^{\circ }$
• Students may forget that the full divided bar must represent the entire total, or $100\mathrm{\%}$
• Students may convert percentages to angles incorrectly
• Students may draw sectors that do not match the calculated proportions
• Students may compare the height of sections in a divided bar graph instead of the length of each section
• Students may label categories unclearly or forget to include a title
• Students may think the largest category in a pie chart must always be more than half, which is not true