Extra Credit you may want to work on (Updated)

below the fold are some possibilities, most of which I have mentioned in class already.

Extra credit possibilities: you may want to do one or more of these, if they interest you.

EC1) Verifying that this proof of the Pythagorean Theorem works. (It is the first proof given in the Math is Fun page on the Pythagoras Theorem, as they refer to it.) The thing that you need to verify is that all of the side that are fitted together in the final picture actually match up in length. For convenience, I am providing you with a picture of the big square being cut up and the two smaller squares after the pieces have been rearranged: (from this blog)

The proof in Math is Fun's gif starts out by looking at the big square which lies along the hypotenuse of the yellow triangle in this picture, then they cut up that big square and rearrange the pieces. What needs to be checked is that everything adds up in the two smaller squares when you do this.

The lengths that have to be checked are where the hypotenuse of the blue triangle meets the pink trapezoid in the small square, where the side of the blue triangle and one of the sides of the pink trapezoid appear to add up to a (the side of the small square), where the sides of the red triangle and the blue triangle apparently add up to b (the side of the middle-sized square), and where the side of the blue triangle and the side of the green trapezoid appear to add up to b (the side of the middle-sized square). I think it's clear that the hypotenuse of the red triangle and the side of the green trapezoid that it meets in the middle-sized square are both the same, namely they have length c.

Here's a sort of idea how to proceed: you can prove that  the sides of the red triangle and the blue triangle add up to b (the side of the middle-sized square), because the red triangle and the yellow triangle are exactly the same size and shape (by the way this was constructed), so the blue triangle must be exactly the right size and shape to fill in the lower right corner of the square formed by the yellow triangle and the blue trapezoid. But that square has sides of length b, so it works.

So for each thing that has to be checked, you look back to where the side came from in the original big square, figure out how its length is related to the sides of the yellow right triangle a, b, and c (or to some other thing you know about), and then see if they add up to what they should. [I found this rather challenging, but it helped to give all the lengths their own letters. I'll try to post a picture like that, but so far I have not succeeded in getting one that is readable. But do check back!]

EC2) Research one of the area paradoxes which are based on squares rather than triangles, and find out (or figure out) where the extra area comes from or goes to. There may be a connection to Fibonacci numbers as well!

Here is one for example: you can easily find others by searching for "area paradox".

EC3) Here is a pdf with some excerpts from a comical novel which happened to have some math jokes in it. (Most of the jokes in this series of novels are literary, or anyway non-mathematical.) See how many of the in-jokes you can find as instructed in this pdf.