Compare participation when the groups have different sizes

model. In Group A, 3/4 of 16 learners choose drawing. In Group B, 2/3 of 24 learners choose drawing. Which group has more drawing participants?

1. Read exactly what is being compared. More participants asks for a count of learners; a larger fraction asks for the proportion of a specified whole.
2. Find one quarter of Group A's 16 learners: 4. Three such groups give 12 drawing participants. Record both 3/4 and 12 instead of replacing one quantity with the other.
3. Find one third of Group B's 24 learners: 8. Two such groups give 16 participants, so Group B has four more people choosing drawing.
4. Check the counts against their group totals and compare the fractions separately if needed. Three quarters exceeds two thirds, yet twelve is below sixteen because the reference groups have different sizes.

Worked example: Group B has more participants: 16 compared with Group A's 12. Group A has the larger participation fraction, but that fraction refers to a smaller group.

p1. Support: 1/4 of a group of 20 learners chooses drawing. Enter the participant count.

Answer: 5

Teacher response: Twenty learners split into four equal groups of five.

p2. Core: 2/3 of a group of 18 learners participates. Enter the participant count.

Answer: 12

Teacher response: One third is six learners, so two thirds is twelve.

p3. Challenge: Group C has 1/2 of 30 learners participating; Group D has 3/4 of 20 participating. Enter the difference in participant counts, and explain why their fractions differ.

Answer: 0

Teacher response: Both have 15 participants. The same count is one half of 30 but three quarters of 20.

e1. Independent check: Group E has 3/4 of 24 learners participating; Group F has 2/3 of 21. How many more participants are in E? Enter the count and record both totals.

Answer: 4

Look for:

• Keeps each fraction attached to its own reference whole.
• Uses counts to answer a count question and does not infer equal group sizes.

Teacher response: E has 18 participants and F has 14; their difference is four learners.

transfer. A report says one half of one group and one half of another participate. What additional information is needed to compare participant counts?

Answer: The size of each reference group is needed. Equal fractions produce equal participant counts only when the group totals match; half of 10 is five while half of 20 is ten.

teach-1. 3 minutes: Ask whether the question seeks a fraction or a number of people. Circle the reference whole in each statement.

teach-2. 4 minutes: Complete the worked comparison table using equal groups, then state the count and proportion conclusions separately.

teach-3. 5 minutes: Use the support and core items to practise counts before discussing the challenge's equal-count result.

teach-4. 3 minutes: Complete the independent two-group record and explain which quantity justifies the answer.

misconception. Three quarters is larger than two thirds, so Group A must have more people choosing drawing.

Teacher response: Ask what whole each fraction uses. The fraction comparison alone answers a proportion question; multiply each fraction by its own group size before comparing head counts.

support. Draw one box per learner in two small groups and shade the requested fraction before using multiplication.

extend. Create two different-sized groups with equal participant counts and explain how each fraction depends on its own group total.