### Why do this problem?

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.

### Possible approach

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.
"OK so this is too easy for you - 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?"

Allow some time for individual work and then ask:
"Can you convince yourself you have them all?"
"Can you convince a friend?"
"Can you convince the rest of the class?"

It would be useful to give some time for students to organise their set of solutions for sharing with the whole class.
Invite early finishers to list their set on the board in a way that makes their system explicit.
You may end up with several different orderings.
You could either:
invite each pair of writers to explain their logic to the class, OR
invite the class to work out what the reasoning is behind each ordering and ask the writers to confirm, OR
ask students to just list the first few sets of numbers and ask the class to predict which sets will follow.

"If I had also told you that the range of these five numbers was 10, how many solutions would there have been?"
Could you add one number to these five and still have the same mean, mode, median and range?

Finish the activity by asking the students to make up a similar question including mean, mode, median and range, for their partner. Can they find a question which has a unique solution?

### Key questions

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

### Possible extension

"You should have found quite a few solutions...
Could we have chosen a mean, mode and median for which there was only one solution?
Could we have chosen a mean, mode and median for which there were no solutions?
Would your answers be different if you were allowed to include negative numbers?"

The Final Challenge from the problem also makes a good follow-up question:

How many sets of five positive whole numbers are there with the following property?
Mean = Median = Mode = Range = a single digit number

### Possible support

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.