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Why do this problem?

Students are often asked to calculate the average (mean) of sets of whole numbers. But what happens when the numbers vary? This problem offers students a chance to consolidate their understanding of average as a central measure, representative of the set.

It also offers a chance to rehearse the key mathematical processes of exploring, conjecturing, generalising and justifying.

Possible approach

Introduce the problem by asking the students to imagine they have an infinite supply of 3kg and 8kg weights. Can they find a combination of these weights that has an average of 4kg?

Allow some time for students to work individually or in pairs and then collect solutions. Confirm that there are many correct possibilities but that you would like to focus on the one that involves the least number of weights.

"If you had other combinations of the 3kg and 8kg weights, what other whole number averages could you make?
What's the smallest? What's the largest?
Can you make all the whole number values in between?"

Students could use a spreadsheet to explore the different combinations. Mr Judge's year 10 class, from School 21, made this spreadsheet, for example.

Allow some time for the students to work in pairs.
Collect the results on the board for future reference.
Some students may wish to comment on patterns that they notice (e.g. for all possible whole number averages, the number of 3kg and 8kg weights adds up to 5).

"What if you have a different pair of weights? What averages can you now make?"

Encourage students to work in small groups and each choose a different pair of weights (perhaps suggesting that they restrict themselves to weights less than 15kg).

"Share your results with your group. What do you notice? Do your results have anything in common?"

Draw the groups together and share ideas and conjectures. (e.g. students may notice a connection between the number of weights used and the values of those weights)

Encourage students to explain their findings.
Offering a visual image may be helpful. Can they adapt this image to explain how to work out the other averages?

Students may suggest an image like this .

"Could you make any predictions about what combinations you need to make all possible whole number averages for any pair of weights? Can you use what you notice to find, for example, the combination of 17kg and 57kg weights that have an average of 44kg......of 52kg.......of 21kg.....?"

Encourage students to test and explain their predictions.

Key questions

What's the smallest average you can make? What's the largest? How do you know?
Can you explain how to make all the whole number averages in between?

Possible extension

Given the original 3kg and 8kg weights, can you find combinations that produce averages of 4.5kg ... of 7.5kg ... of 4.2kg ...of 6.9kg ...? Convince yourself that all averages between 3kg and 8kg are possible.

What averages are possible if you are allowed a negative number of 3kg and 8kg weights?

Students could be directed to Litov's Mean Value Theorem for a suitable follow-up problem.

Possible support

You may initially wish to restrict the weights used to those which have a difference of 2kg, then 3kg, then 4kg, etc. in order to model working systematically, and to make the pattern of results more obvious.

Some students may find multilink cubes useful to support their visual images.

This interactivity might be useful for whole class or independent exploration.