Basketball caps

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Purpose

This problem solving activity has a number focus.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
Student Activity

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

A table recording how many hats Toni sold. Week 1: 9, Week 2: 6, Week 3: 3, Week 4: unknown.

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

Specific Learning Outcomes
  • Apply the concept of average to solve a problem.
  • Devise and use problem solving strategies (draw a picture, use equipment, think).
Description of Mathematics

This problem involves students working with the average or mean. This is an important concept in statistics and gives 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 using an equation like 30 ÷ 2 = 15 (i.e. total sum of the numbers given divided by the count). In this problem the total sum will be the total sum of the caps, and the count will be the number of weeks. In a problem like this, 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.

Required Resource Materials
Activity

The Problem

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

A table recording how many hats Toni sold. Week 1: 9, Week 2: 6, Week 3: 3, Week 4: unknown.

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

  1. Use the cap to introduce the problem to the class.
  2. 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?
    How would you find the average number if we knew the missing number of caps? 
    How can we find the average without knowing the missing number?
  3. Ask the students to consider how they can write an equation to express the problem, and how they will record their solution.
  4. 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.

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. 

Solution to the Extension 

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.

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Level Three