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?

What is this problem about?
This problem takes a fresh look at the notion of average or mean. The usual way to ask a question about averages is to give the numbers and the number of periods and get the children to divide one number by the other. This is the way that cricket averages, for instance, are calculated. As we have said elsewhere though, children will learn the concept better if they are given the standard question from a different angle.
Averages are calculated by an equation of the form . If caps and weeks are given we can find the average. But we can vary things. If we give average and weeks, we can find the number of caps sold. And if we give average and caps, we can find the number of weeks. You could take this last idea to produce an extension of the current problem.

There are two reasons for doing this problem. First it deals with averages or means. These are an important concept in statistics as they give some idea of the ‘centre’ of the distribution of numbers. The second reason is that people often have to deal with incomplete data. Sometimes the necessary information can be reconstructed from the data. The data in this problem is incomplete and the necessary information, the average, can be found from the data.

Achievement Objectives
Statistics (Level 3)
- use their own language to talk about the distinctive features in their own and others’ data displays.
Mathematical Processes
- devise and use problem solving strategies (draw a picture, use equipment, think).

Resources
- A cap to introduce the problem
- Counters
- Copymaster of the problem (English)

Specific learning outcomes
The children will be able to:
- apply the concept of average to solve a problem

1. Pose the problem to the class using the cap as a prop.
2. As the children work on the problem, individually or in pairs, ask questions that focus on their understanding of average. It is important that children understand that the average number represents an equal redistribution (of the caps in this case), rather than just being able to state the rule.
   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?
3. Ask the children to record their solutions to the problem.
4. Share written records.

Extension
Children write their own average problem for others to solve.

Other Contexts
This problem can be put into any number of contexts. For instance, Toni can be selling anything anywhere. On the other hand, weeks could be games and caps could be goals in a netball game. And the caps could also be money earned in different weeks at work.

Solution
There are a number of ways of tackling this problem. We’ll first do it using the definition of average. So, the average is the number of caps sold altogether divided by the number of weeks. This is something divided by 4. But we know that the average is to be 7. Since something divided by 4 = 7, then the something must be 28. So Toni has to sell 28 caps over the 4 weeks.

Now add up all the caps she has sold. There were 9 in the first week, 6 in the second and 3 in the third. This is a total of 18. But she has to sell 28 to keep the average of 7. So Toni has to sell another 10 caps in the last week. The answer is therefore 10 caps.

But the problem can be done with equipment. Let’s take the original table and then spread out the stars representing the caps into 7 per week. We can then see how many caps have to be added to get an average of 7. The caps we add will be shown by plus signs.

Now rearrange to make at most 7 stars per week.

Now redistribute the extra two stars from week 1.

Now to get 7 per week we need to add in some pluses.

Since there are 10 pluses, Toni needs to sell 10 caps in the fourth week to average sales of 7 per week.