Until recently, I'd always introduced Pythagoras's Theorem in the same way: by showing a picture of a cartoon man in a toga and writing a squared + b squared = c squared on the board.
Not very inspiring! (Although I did manage to write the squared symbol properly, something I haven't figured out on here yet)
THEN a colleague of mine told me that she never even mentioned the letters a, b and c, she just showed them this picture of Perigal's dissection, and by discussing it, her students arrived at their own method.
So I tried it with 2 of my classes recently and it went down a storm. They came up with the fact that the area of the two smaller sqaures added together to make the larger square, then I added numerical values for the lengths of the shorter sides and set them to work figuring out the length of the hypotenuse. Each group managed to figure it out for themselves and then apply the same system to simple questions involving both the hypotenuse and a shorter side.
It's a shame to deprive anyone of algebra, so the next lesson I asked both classes if they could see how the formula a squared + b squared = c squared would apply to Pythagoras's Theorem and they picked it up pretty quickly.
I think starting from the visual basis for the theorem helps students to think it through logically. In the past, I've always seen furrowed brows and cross expressions when I tried to show weaker students how to rearrange the formula to find a shorter side. Introducing Pythagoras using the picture first helped students to understand it more intuitively. In fact, it was a student who usually complains at the merest whiff of algebra, who suggested that the formula for a shorter side would be c squared - b squared = a squared.
Once again - not rocket science. But pretty useful.