Practice for high attainers, a pile of vector resources and a bit of mathematical colouring 

This investigation is great for getting pupils to “spot” square number patterns – it’s particularly useful as a little starter a couple of weeks after completing the topic to remind them of the sequence of square numbers. Pupils could also make the patterns from multi-link cubes and then demonstrate how the top and bottom patterns can be turned into squares.

I’ve also used this as a time-filler activity with half a class of Year 10s and asked them to come up with the nth term rule for the top and bottom patterns (n² and (n – 1)² respectively), then combine this to get the nth term rule for the complete pattern (gives 2n² - 2n + 1 when expanded and simplified). They then used this to predict and check the next pattern in the sequence.

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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.

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This is one of my absolute favourite investigations to use when teaching square numbers. It can fit quite nicely into 15/20 minutes as a simple activity, or extend to a whole lesson with the class working together.

To find out if a number is happy:

1. Pick any two-digit number (e.g. 23).
2. Square and add the digits (4 + 9 = 13).
3. Repeat (1 + 9 = 10).
4. And again (1 + 0 = 1).

A number is happy if it eventually gets to 1. There are 20 happy numbers between 1 and 100; asking pupils to find two or three happy numbers works well as a quick activity, possibly towards the end of a lesson introducing square numbers.

However, the most interesting part is what happens with numbers that aren’t happy. Every number that isn’t happy eventually “spirals” into a repeating loop:

4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4

By extending the time allowed, pupils quickly discover this loop for themselves. Some then realise that they can rule out reverse digits (e.g. if 42 isn’t happy, then 24 won’t be either). It’s quite powerful to ask pupils to work in pairs or groups, as this allows lots of opportunity for planning and discussion.

I’ve previously used the board to map this “spiral” out for pupils, asking them to come up and add any numbers that they can. I also spent one very rainy Sunday afternoon transferring this map to a PowerPoint slide with colour coding (cause I'm a loser)... I think it looks very satisfying!

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