Pythagorean triangles and differences of squares

This is the second short (in words) post for today.

Problem #7 from the 2008 AMC 10 gave my son a little trouble this morning, but talking about it led to a fun conversation about geometry and algebra. Here’s the problem:

Problem #7 from the 2008 AMC 10 A

After talking about the solution to this problem, I scrapped the actual lesson for today and talked a bit about some special Pythagorean triangles. Then I gave my son this problem as a challenge:

 

He was able to solve the problem in the last video by finding a pattern in the side lengths. Next I challenged him to find a different pattern. This part was a little bit of a struggle, but he did eventually find a different pattern that connects the side lengths:

 

Finally, I wanted to use the pattern that we found in the second video to find some new triangles. We found the next couple of triangles in the pattern and that showed him, I think, that this new pattern could be pretty useful.

 

This was an interesting little project for me. I guess there’s no way to know what patterns that people will find easy to see and what ones that they will find hard to see, but I do think looking for patterns that you don’t see initially is an important skill in problem solving. I’ll be on the lookout for similar project connecting geometry, algebra, and patterns in the future.

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Comments

2 Comments so far. Leave a comment below.
  1. I had a go at this, more by accident than design. here’s the link to my post:
    https://howardat58.wordpress.com/2014/11/14/pythgoras-triples-345-a-calculator/

  2. It might be instructive to see if he can prove that his earlier pattern produces right triangles. Should be two available proofs: a direct proof and a proof by induction.

    The 2x + 1 = n^2 observations helps you see that n needs to be odd.

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