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.