Pythagorean Theorem Proofs
Written by: Dr. Julian Fleron
For advanced mathematics majors and practicing mathematicians we think of proofs as tight, streamlined, efficient arguments. Proof 1 is a bit wordier than a typical proof — but one sees that has nicely used natural language to provide all of the requisite details for a complete proof — with a closing emphasizing the essential role of the picture in the proof. In Proof 2 and Proof 3 you notice some of what we refer to in our paper as the narrative structure coming through: "We cut a triangle while the papers was folded in quarters..." and "Using the 5 pieces you can..." In Proof 4 the author clearly emphasizes that we really have a Proof Without Words, we just need to fill in some of the details, which she does graphic novel style.
The real surprise was the significant number of students (perhaps 20 of my 64 students) who successfully completed the construction in class but somehow felt that it was not sufficient when they went to write up their final proof. So they modified and adapted their approaches — many with the help of the Internet I am guessing — into proofs that involved algebra. I had purposely avoided algebra as it is a weakness of many of our students in this course. Despite this, and despite the fact that they had made sense of a complete construction/picture proof of the Pythagorean theorem, they felt the inclusion of algebra legitimized their proof more. Their work shows this and several told me this explicitly when I ask them, despite the fact that several even admitted it made their final proofs less intuitive! Proof 5 is one example of this, where the student essentially includes two different proofs to cover her bases. And Proof 6 entirely abandons the approach in favor of an approach typically attributed to Chou Pei Suan Ching from 200 BCE which uses the same four congruent triangles constructed at the outset but a different construction. This construction is also most naturally completed visually, but, as expected, the student chose to do it algebraically.