About Average

Why do this problem?

This problem improves fluency in working with the numerical definitions of the mean, median, mode and range. It also develops understanding of how each measure is affected by individual numbers in a sample, as well as students’ reasoning. If appropriate, you could also use this problem to encourage students to represent the concepts algebraically and set up equations.
 

Possible approach

This problem should be used once students have already seen the mean, median mode and range. As a starter, you could give students some sets of numbers and ask them to find the mean, median, mode and range. Then, ask them to find three numbers with mean 3 and mode 2. Is there more than one way of doing it?

After the first example, invite students to share their strategies.

Students could then work in pairs or small groups to find numbers for each set of conditions. You could encourage them to record their strategies, and whether they used the same strategy for all of the questions. At the end of the activity, students could share their strategies and conclusions in a whole class discussion.

Note that number 5 is only possible if you allow 0 as one of the numbers, and in that case there are several possible solutions. There are two possible solutios for number 6 if 0 is allowed.

Manipulatives could be used to introduce this work. Number cards could be used to allow students to visually see the problem.
 

Key questions

What is the difference between the averages i.e. How is the mode different to the mean?

Is there only one answer?

Is there a method that always works?

Of the mean, median, mode and range, which should you focus on first as you choose your numbers, and why?
 

Possible support

You could begin by giving students some of the numbers, for example 2, ___, 5 (choose the third number so that the mean is 3 and the mode is 2).

Alternatively, you could show them helpful examples: you could give them some sets of three numbers with mode equal to 2, such as 2, 2, 3 and 1, 2, 2 and ask them to find the mean and the mode, before asking them to find three numbers with mode 2 and mean 3.


Possible extension

Ask students to create their own conditions that can be presented to their peers.

Ask students to investigate when it is possible to find numbers that meet their conditions, and when it is impossible. Can they create impossible conditions that look possible?