Pupils explore which numbers can be written as the difference of two square numbers; a nice way to practise calculating with square numbers, but also encourages them to pattern-spot at the same time.
All the odd numbers can be written as the sum of two consecutive square numbers. Some pupils notice this very quickly, but some will need to try numbers for themselves first. It helps to ask pupils to look for patterns in answers they already have, then ask them to think about whether this pattern could apply to other numbers. It’s a little trickier for pupils to work out the other group – the multiples of 4.
It’s highly unlikely that proofs of these will be applicable when first introducing square numbers to a group. However, the activity could be revisited in the context of lessons on proof or expanding quadratics.