A typical proof for zero/negative exponents that isn’t mindless

(Park Math, Book 5, p. 46)

What I like about this isn’t the proof — it’s the same old proof that is in every textbook in the world. But what this sequence of problems shows me is that any old proof can be transformed into something that really deeply engages student thinking. I admire the way this argument just puts the weird idea in front of kids, and asks them to come up with a meaning. In the class discussion that follows, kids will figure out what makes the most sense. Since math makes sense, enough careful thought will bring students to where the mathematical community has been for a while.

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