We started a new section in our Geometry book this week and it starts with some properties of triangles that require slightly more difficult proofs that we’ve seen so far. My son is struggling with the difficulty and I have a lot of sympathy – learning techniques of proof was absolutely one of the most difficult things for me.
Today we were exploring non-right triangles and, in particular, when a triangle’s sides might satisfy 
It took about 30 minutes to explore the idea this morning. Even with that long exploration, reviewing the idea again in our movie took a long time. Initially we just talked through about the problem and the initial ideas in the proof.
I broke this video after we write down three algebraic relationships in the triangle.
Once we had those relationships, the idea was to see how we could apply them to our problem. I think there are two parts of learning math that are particularly hard here. The first is understanding the connection between the geometry and the algebra. The second is understanding the relationships between the numbers / sides implied by the algebra.
Again, I have a lot of sympathy for my son as learning more difficult proofs really is hard. A line from an old Indigo Girls song comes to mind – “the hardest to learn was the least complicated.”
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