This post by Michael Pershan has an excellent way to give students opportunities to play around with exponents and problem solve with them.

The basic idea is, give students a bunch of dots in an array, and ask them to find ways to change the picture so that it represents a power.

A related, and more generic, question is:

How do I change this diagram/image/equation/formula so that it represents <something else>?

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*Related*

Related but slightly different question I’ve tried to get at with some number talks on this topic — how does changing the diagram help us learn more about it?