I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in university. We will be doing a lesson on visual proofs of the Pythagorean theorem.
There are at least three that are easy and fun to do using some paper triangles. They'll be encouraged to vote on their favorite. But these students are bound to ask why do it three ways? Why do it 118 ways?
ThisThis post, this post, and others do a fine job discussing what makes two proofs different and how many proofs a theorem can have, but what I want to know is this: why do we care? Is it just a sense of nerdy joy in finding a different argument for the same theorem?