Extending Great Squares

Explore one of these five pictures.

Polygonals

Polygonals

Polygonal numbers are those that are arranged in shapes as they enlarge (starting with $1$). Here are examples of the $5$th polygonal numbers for the first $7$ shapes Triangle ($3$) through to Nonegon ($9$).

If we were to just look at the heagons we'd have a form of hexagonal numbers that grow  $1,6, 15, 28, 45$.

There is another group that are called "Centred Polygonals"  that look like these;

and for example, the centred hexagon numbers go $1, 7, 19, 37, 61$.

Now it's over to you!

This investigation invites you to explore these sets of numbers and explore relationships within ordinary polygonal numbers and/or centred polygonal numbers.
You could also explore relationships between ordinary polygonal numbers and the centred polygonal numbers.

For example, you could explore which different polygonals (both centred and ordinary) have the same number occuring in the series?

KEY QUESTIONS:

What is the relationship between ordinary triangular polygonal numbers and others?

Can you re-arrange the dots from one polygonal to make another, and then generalise.?

Why do this problem?

This activity is specially designed for the highest-attaining pupils that you ever come across. It may act as a further extension to 3D Stacks as polygonal numbers can occur in that investigation. It's an activity that is intended to give opportunities for those pupils to explore more deeply using their intuition and flair in the areas of both spatial awareness and number relationships and patterns.

Possible approach

As this is designed for the highest attaining, it might be presented as on the website or in a one-to-one situation, encouraging discussion between adult and pupil. The pupils may need access to a spreadsheet once many number results are being acquired.

Key questions

Tell me about what you have found?
Can you describe the ways that you arrived at these numbers?
How did you construct this on the spreadsheet you used?

Possible extension

If your pupils have investigated this very thoroughly they might like to look at Steps to the Podium and seek to find connections.