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Why do this problem?

This activity can supplement work being done with triangular numbers. It also lends itself to being opened out for further investigation.

Possible approach

It would be best to introduce this challenge in as practical way as possible to start with and then question the pupils to find out if they understand the idea of triangular numbers.

Key questions

So, what is a triangular number?
How have you found out how many are in your triangular stacks?

Possible extension

As is often the case with straightforward challenges it is good to explore taking the patterns further.
So we could start by:

a] looking at what numbers are in the following layers;
b] looking at what the totals become as each new layer is added;
c] looking at the digital roots

For the first simple stacking we'd have:
For the second stacking as a square based pyramid we'd have:
Each of these sequences can be explored, and, depending on the pupils' experience, encourage them to explain WHY the things they notice have occurred.

Possible support

Pupils may need their own small cubes to represent the cans. Some will need to have assitance when exploring the square-based shapes.