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## 'Caterpillars' printed from http://nrich.maths.org/

Well, here are the
caterpillars!

Our caterpillars have numbers on
each little part - numbers $1, 2, 3, 4, \ldots$ up to $16$.

You can see their pale blue head, and their body bending at
right angles so that each part is lying in a square.

There are two explorations to go with these caterpillars:

1. You could look carefully at the shapes of the caterpillars and
let them turn in other ways. Here are a few to start you off:

When you've discovered many new
ones, with the shapes - and therefore the numbers - all showing
differently, you could compare them. What is the same and what is
different? Can you explain why?

You could try to put the
caterpillars in shapes that are not squares!

OR

2.

a) Choose one of the caterpillars and, using the numbers and the
way that they are arranged, explore the patterns and relationships
you can find.

b) Then let that caterpillar grow nine more parts so that it
becomes a $25$ caterpillar with the shape bending in just the same
way. Explore those patterns and relations.

c) Finally, compare the two different groups of things you've
discovered in a) and in b).

Do tell us about all the things you find out.