This problem offers the students an opportunity to consolidate what they are expected to know about mean, mode and median whilst also challenging them to work systematically, and justify their reasoning.

Start by writing five numbers on the board: 5, 3, 6, 3, 3 and ask for the mean, median and mode of this set. Resolve any disagreements.

"You know how to answer questions like this, but what if I turn the question round? What if I had told you that the mean, mode and median of five positive whole numbers were

mean: 4

mode: 3

median: 3

Would you have been able to tell me the five numbers?"

"Are there any other sets of five numbers that fit these conditions?"

Collect a few suggestions and then ask:

"There seem to be quite a few - can you find some more?"

"Can you find them all?"

"Can you find them all?"

Allow some time for students to work on the problem on their own, before inviting them to work with a partner.

Circulate around the class. Observe which students are working randomly and which are working more systematically. Prompt students who are working randomly to consider breaking up the problem into smaller parts:

"What if one of the numbers was 1? What if you had no 1s? …"

Or: "Can you have just one 3? What about two 3s? Three 3s? …"

When appropriate ask each pair:

"Can you convince yourselves you have found all the solutions?"

"Do you think you could convince the rest of the class?"

Students who find it difficult to work systematically could record each solution on a separate slip of paper and then arrange them into groups.

Circulate around the class. Observe which students are working randomly and which are working more systematically. Prompt students who are working randomly to consider breaking up the problem into smaller parts:

"What if one of the numbers was 1? What if you had no 1s? …"

Or: "Can you have just one 3? What about two 3s? Three 3s? …"

When appropriate ask each pair:

"Can you convince yourselves you have found all the solutions?"

"Do you think you could convince the rest of the class?"

Students who find it difficult to work systematically could record each solution on a separate slip of paper and then arrange them into groups.

It would be useful to give students some time to rewrite their set of solutions in a way that makes it easy to convince others that they haven’t missed any possibilities.

Invite some pairs to list their set on the board, or on a large sheet of paper, in a way that makes it clear that all the possibilities are included.

If you end up with several different orderings, you could:

- invite each pair to explain their logic to the class
- invite the class to work out what the reasoning is behind each ordering
- ask students to just list the first few sets of numbers and ask the class to predict which sets will follow.

Which piece of information is the most useful to start with?

What process allows you to be confident that you will have found all the results by the end?

Possible extension

Unequal Averages follows on from this problem.

Students who find it difficult to work systematically may be helped by recording each solution on a separate slip of paper and rearranging them into 'families'. If you are using an interactive white board you could model this by recording solutions as they are produced and rearranging them.