The triangulation theorem states that every triangulation of an n-gon has exactly n-2 triangles.

But I claim that the 6-gon below can be triangulated into precisely 3 triangles.

So what do you do with this?

1. You can change the theorem, to say that there are at most n-2 triangles in a triangulation.

2. Justin’s approach:

3. Dave’s approach:

I want my students to become genuinely good at proof, and these sort of conversations emerging from counter-examples are one of the main engines of proof. The catch: in order to have practice with these thoughts, students need messy math.

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