Monday, 16 November 2015
Is this a Proof of the Pythagorean Theorem, and to Whom ?
The Image above is from The Puzzle Universe, A History of Mathematics in 315 Puzzles, by Ivan Moscovich, showing the bottom third of a page about the Pythagorea Theorem. I have mentioned this book recently with the advance notice I wanted to return to some of these beautiful problems as time permits. This morning, I decided that time permits. In looking over these beautiful proofs, I realized how our mathematical experience allows us to see these proofs differently from what students see.
The image shows five different proofs of the Pythagorean Theorem, On the left (1) a dissection proof from the Chinese classic from about 200 BC, the "Chou Pei Suan Ching." For me, this is the proof of the Pythagorean theorem that is most understandable to students. I have seen this made into a puzzle block with the four right triangles able to slide back and forth to reveal the two smaller squares (as shown on bottom left) and then slid to positions showing the one larger square. I think one of the reasons this sits so well with students is that the Algebra provides support. They look at the whole container and see it has sides of a+b, so it's area is (a+b)2 , and the individual pieces are two smaller squares with areas a2 and b2, and four rght triangles each with area 1/2 a*b. The same container when pieces are moved contains the same four triangles, and the square on their hypotenuse, c2. The elimination of the four triangles provides the equality they know and seek. If you gave them a different shaped triangle and asked them to create a similar proof for that triangle, most could surely do so. As I will try to illustrate as I go on, I believe they do NOT "see" the other proofs in as clear a way. However in it's original form, I don't think any of this becomes clear. In my experience, very few students look at the overlapping squares in the original, and see the simple proof below and to the right. Here is a somewhat different image of the Chinese proof as given in Eli Maor's history of the Pythagorean theorem. In fact, the earliest mention of the clearer version, is given in Elisha Scott Loomis' classic The Pythagorean Proposition, published in 1927, but written almost 20 years before. In the 1939 revision, Loomis (pp. 49-50) mentions that the proof "was devised by Maurice Laisnez, a high school boy, in the Junior-Senior High School of South Bend, Ind., and sent to me, May 16, 1939, by his class teacher, Wilson Thornton."
The Second proof is credited to Leonardo Da Vinci. I doubt that many teachers can quickly walk through this proof on first site, and without being told, I wonder how many students would even guess that it was intended to be a proof of the Pythagorean theorem. For those who want to explore this proof, which does have a certain geometric beauty on reflection, there is a demonstration at Wolfram alpha.
The third proof, the five white and red images in the black box are credited to New York mathematician Hermann Baravalle in 1945. This appears to me to be very like the "Brides Chair" proof by Euclid in Elements Book I, Prop 47. They seem close enough to me that if I wanted a student to struggle through one of them, I would return to the classic and help them mount the obstacles of it.
If you are not familiar with Euclid's original proof, see David Joyce's beautiful page on the Elements.
The fourth proof shown, Moscovich calls, "The simplest proof." I will heartily disagree, and think for high school students, this may be one of the two hardest. This is proof #230 in The Pythagorean Proposition, and Loomis credits it to nineteen year old Stanley Jashemski of Youngstown Ohio, and calls him, "A young man of superior intellect." In his book on the history of the Pythagorean theorem, Eli Maor claims to have independently rediscovered the proof, and gave it the name, "The Folding Bag."
Why is it so difficult for students? I propose a simple test, show the image to students and ask, "What do you see?" I imagine the most common response will be something like, "some triangles." And there is the rub, triangles. Students have been shaped in the culture and their classroom that the Pythagorean theorem is about "Squares." So maybe they know that it doesn't have to be squares, but can they explain what CAN be in place of squares? Is "right triangles" enough of an answer? Can they be any three right triangles, or must they be similar? Are these similar? Explain!
And is this just for one right triangle, or will it work for ANY right triangle... again, Explain!
With a little introduction, and some challenging questions, this can be a powerful proof that changes student's perceptions of the Pythagorean Theorem and proofs.
The fifth image is the beautiful dissection proof of Henry Perigal, who was so proud of it that he had it placed on his tombstone. The proof is usually presented as a (somewhat easy) puzzle with the five pieces in the two smaller squares and asking for them to be reassembled into the square on the hypotenuse. So if the student achieves this, what has she done? Ask them? Did you prove the Pythagorean Theorem? If they believe they did, ask them to tell you in their own words what the Pythagorean Theorem says. What we have done here is prove that there is one triangle, in this case a right triangle, for which it is true that the sum of the areas of the squares on the legs is equal to the area of the square on the hypotenuse. What we have not done is prove this is true for any right triangle (and we haven't even brought up the idea of "only if".
To prove it for any right triangle, we must be able to provide a method to show that such a dissection is possible for any given right triangle. Can the student do that? (Can the teacher?) There are two ways to generalize the proof, one that was the approach that Perigal used, and another introduced after his death that produces unlimited approaches to dissections of a right triangle. I will show Perigal's method below, and provide a nice link to where both methods, as well as some more history about the long lived Perigal.
To perform Perigal's dissection, draw the triangle with the hypotenuse horizontally beneath the two legs (hypotenuse actually comes from the Greek for "to stretch below") so this is the "authentic" way to draw right triangles. Now complete the three squares on the sides.
Find the center point of the square on the larger leg, and draw a line through it both vertically and horizontally. Now from the midpoint of the hypotenuse, draw a line parallel to the shorter leg extending to the other side of the square on the hypotenuse.
If you are really good at Geogebra or another interactive geometry tool, you can actually show the translations for each piece of the smaller triangles to the appropriate position in the square on the hypotenuse.
The other, more modern method is called a Pythagorean Tesselation and you can see an example, as well as the promised history notes about Perigal and another explanation of the method I have just shown if it is still unclear. The material is at the Plus Math site.
And it this has stimulated your interest in the beautiful book from which it comes, the link below will take you to the Amazon page for it.