Toni is selling caps for the basketball club.

The picture shows the number of caps that Toni sold during the first three weeks.

How many hats must Toni sell in week 4 so that the average number of caps that she has sold per week is 7?

This problem involves students in working with the average or mean, an important concept in statistics, giving the ‘centre’ of the distribution of numbers. When incomplete data are given, as in this problem, the necessary information can be reconstructed from that data.

Averages are calculated by an equation of the form 30 ÷ 2 = 15, although the nature of the unknown information can vary. When the problem is expressed by an equation such as ? ÷ 4 = 7, students learn that the unknown can be found by using the inverse operation of multiplication: The inverse operation shows 4 x 7 = 28.

Copymaster of the problem (Māori)

Copymaster of the problem (English)

A cap to introduce the problem

Counters

### Problem

Toni is selling caps for the basketball club. The picture shows the number of caps that Toni sold during the first three weeks.

How many hats must Toni sell in week 4 so that the average number of caps that she has sold per week is 7?

### Teaching Sequence

- Pose the problem to the class.
- As the students work on the problem, individually or in pairs, ask questions that focus on their understanding of average. Students should understand that the average number represents an equal redistribution (of the caps).

*What is the average number of caps sold?*

What does this mean?

How would you describe average to a friend who had forgotten what it was? - Ask the students to consider how they can write an equation to express the problem, and how they will record their solution.
- Share written records.

### Extension

Aria is also selling caps. She averages sales of 8 caps per week and sells 32 caps altogether. How many weeks does she sell caps?

If Aria sells a different number each week, and doesn't actually sell 8 caps in any of these weeks, show what her weekly sales table might look like.

Students can write their own average problem for others to solve.

#### Other Contexts

Other fundraising contexts, games and scores

### Solution

The average is the number of caps sold altogether divided by the number of weeks (4). The given average is 7. The equation is ? ÷ 4 = 7

The inverse operation shows 4 x 7 = 28, so 28 caps are sold over the 4 weeks giving an average of 7 per week.

Toni sells 9 in the first week, 6 in the second and 3 in the third. This is a total of 18. She must sell 28 to keep the average of 7. So Toni must sell 10 caps in the last week.

#### Extension Solution

The total caps sold = 32 . The average is 8 caps per week. So the number of weeks is 32 ÷ 8 = 4

Responses will vary. One example is: 9 caps in week 1, 7 in week 2, 6 in week 3, 10 in week 4.