Proving the Pythagorean Theorem

In a right triangle, where the lengths of the legs are given by a and b, and the length of the hypotenuse is given by c, a²+b²=c². 

We use this so much in math, I have no idea where I first saw it. And it's so simple that I never had trouble remembering it. (The quadratic formula, on the other hand, did not make it into my memory banks until after I had started teaching college. For the first few courses I taught, I had to have it written at the top of my notes.) So I've known and used the Pythagorean Theorem for longer than I can remember.

It comes up in beginning algebra, and for years I showed students how to use it to solve ridiculously artificial algebra problems, never once addressing the issue of proof. This seems terribly wrong to me now. Perhaps about 15 years ago, I realized I'd been 'teaching' this to students for about a decade without even knowing its proof. I tried to come up with a proof on my own and had no idea how to start. Since this was before google became a verb (or even a word), I had to search for a book that would show it. I eventually found it in a high school geometry textbook. Luckily it showed a visually simple proof that stuck with me. (There are hundreds of proofs, many of them hard to follow.)

One of the reasons Pythagoras is held in high esteem by mathematicians is his proof of this idea. It had been used long before Pythagoras and the Greeks, most famously by the Egyptians. Egyptian 'rope-pullers' surveyed the land and helped build the pyramids, using a taut circle of rope with 12 equally-spaced knots to create a 3-4-5 triangle (since 3²+4²=5² this is a right triangle, which is pretty important for building and surveying). But the first evidence we have that it was proven comes from Pythagoras. Ever since the Greeks, proof has been the basis of all mathematics. To do math without understanding why something is true really makes no sense.

Pam Sorooshian is a homeschooler who trusts kids' natural instinct for learning. So she unschooled her kids (who are now grown and doing very well). That means she never required them to learn something they weren't interested in, and never pushed her own interests on them. She has a story in Playing With Math* that really stuck with me. In a talk to other unschooling parents, she said:

Relax and let them develop conceptual understanding slowly, over time. Don't encourage them to memorize anything - the problem is that once people memorize a technique or a 'fact', they have the feeling that they 'know it' and they stop questioning it or wondering about it. Learning is stunted.

It took me decades to wonder about how we know that a²+b²=c². Now I feel that one of my main jobs as a math teacher is to get students to wonder. But my own math education left me with lots of 'knowledge' that has nothing to do with true understanding. (I wonder what else I have yet to question...) And beginning algebra students are still using textbooks that 'give' the Pythagorean Theorem with no justification. No wonder my Calc II students last year didn't know the difference between an example and a proof.

Just this morning I came across an even simpler proof of the Pythagorean Theorem than the one I have liked best over the past 10 to 15 years. I was amazed that I hadn't seen it before. (Maybe I did see it, but wasn't ready to appreciate it.)


My old favorite goes like this:

• Draw a square. 
• Put a dot on one side (not at the middle). 
• Put dots at the same place on each of the other 3 sides. 
• Connect them. 
• You now have a tilted square inside the bigger square, along with 4 triangles. At this point, you can proceed algebraically or visually.

Algebraic version:

• big square = small tilted square + 4 triangles
• (a+b)² = c² + 4*1/2*ab
• a²+2ab+b² = c² + 2ab
• a²+b² = c²



www.cut-the-knot.org/pythag

Visual version:

• Move the triangles around.
• What was c² is now a²+b².
• Also check out Vi Hart's video showing a paper-folding proof (with a bit of ripping). It's pretty similar to this one.

To me, that seemed as simple as it gets. Until I saw this:

from The Step to Rationality, by R. N. Shepard

This is an even more visual proof, although it might take a few geometric remarks to make it clear. In any right triangle, the two acute (less than 90 degrees) angles add up to 90 degrees. Is that enough to see that the original triangle, triangle A, and triangle B are all similar? (Similar means they have exactly the same shape, though they may be different sizes.) Which makes the 'houses with asymmetrical roofs' also all similar. Since the big 'house' has an 'attic' equal in size to the two other 'attics', its 'room' must also be equal in area to the two other 'rooms'.^# Wow!

Added note (6-9-13): I've been asked to clarify why the big house must be equal in size to the two smaller ones added together. Since all three houses are similar (exact same shape, different sizes), the size of the room is some given multiple of the size of the attic. More properly, area(square) = k*area(triangle), where k is the same for all three figures. The square attached to triangle A (whose area we will say is also A) has area kA, similarly for the square attached to triangle B. kA+kB=k(A+B), which is the area of the square attached to the triangle labeled A+B. But kA = a², and kB = b². So k(A+B) = a²+b². And it also equals c², giving us what we sought, a²+b² = c².

I stumbled on the article in which this appeared (The Step to Rationality, by R. N. Shepard) while searching on 'thought experiment weight times distance must equal to balance'. I'm working on a handout for my Calc II students to explain centroid (since the Briggs textbook leaves this topic out). I was wondering if we need experimental evidence to show that the two sides of a teeter-totter will balance only when the weights times distances from the fulcrum are equal on the two sides. I thought maybe we could come up with a thought experiment that would convince us it must be true. I wasn't having any bright ideas, and turned to google. It hasn't solved my centroid question yet, but I love what I discovered.

I think that, even though this proof is simpler in terms of steps, it's a bit harder to see conceptually. So I may stick with my old favorite when explaining to students. Or maybe there's a way to test out which one is more helpful for a deep understanding of both the notion of proof and this theorem in particular.

What do you think? 





____________
*Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is a collection of great writing about math education (often outside the classroom), from over 30 authors. I've been working on for 4 years now; it will be available within 2 to 4 months. 
^#I got this language (of houses, attics, and rooms) from a similar description of this proof which I found on Cut-the-Knot.

Maths and Stats by Email (CSIRO)

About two and a half years ago, I signed up for this online newsletter. I have enjoyed most of the issues quite a bit. There's no link to head you over to the latest issue. They don't do it that way. (I have no idea why not.) You pretty much have to sign up for it if you want it.

This month's issue is typical. A simple topic, explained well for young people:

Take a sheet of A4 paper and measure its sides. A4 is 210 millimetres wide and 297 millimetres long. It’s probably the most common size of paper and it’s used in most countries. However, A4 side lengths aren’t simple numbers like 200 or 300 millimetres. So why don’t we use something easier to measure?

If you take a sheet of paper and cut it halfway down the longer side, you end up with two new pieces of paper. These pieces of paper each have half the area of the original sheet, but they are the same proportions as the original sheet! There’s only one type of rectangle that has this ability. Because these half sheets have the same proportions as A4, they also have a name – A5.  If you cut an A5 sheet in half, you get two pieces of A6 paper, with the same proportions as A5 and A4. All these paper sizes are part of a set called the A series.

This pattern also works if you want to go bigger instead of smaller.  If you take two sheets of A4 paper and stick the long sides together, you’ll end up with a sheet of paper that has the same proportions as A4, but is twice as big. This size is called A3. You can use the same process to make A3 sheets into A2, and even A2 sheets into A1 paper.

So why is A4 paper called A4? A4 is half an A3, or one quarter of A2, but more importantly, it’s one sixteenth of A0. A0 has an area of one square metre (but it isn’t a square), and every other paper size in the A series is based on A0. We use A4 for writing on because it is a lot more convenient than trying to write on a square metre sheet of paper!

After the introductory article, they always have a 'try this' activity. This month's is on tangrams (and the activity relates tangrams to the paper sizes described).

Let me know if you decide to sign up.

Seeking Math-Poets for a Reading (JMM, 1-11-13)

JoAnne Growney just posted this on her blog (Intersections - Poetry with Mathematics):

Call for Readers:
     The Journal of Humanistic Mathematics will host a reading of poetry-with-mathematics at the annual Joint Mathematics Meetings (JMM) on Friday, January 11, 5-6:30 PM in Room 1B, Upper Level, San Diego Convention Center.  If you wish to attend the reading and participate, please send,  by December 1, 2012 (via e-mail, to Gizem Karaali (gizem.karaali@pomona.edu)) up to 3 poems that involve mathematics (in content or structure, or both) -- no more than 3 pages -- and a 25 word bio.

There's more at her post, check it out.

I'll be co-hosting this event, and would love to meet you there. (I hope to read a poem or two of my own. I'd better get my submission in.)

Good Text for Discrete Math?

I'll be teaching Discrete Math next semester, for the first time ever. (Kind of surprising I waited this long, since many of the topics are dear to my heart and the whole course looks like playing with math to me.)

We've been using Discrete Mathematics and its Applications, by Kenneth Rosen. Cheapest used I see on Amazon is $92. My colleague likes a book by Washburn, Marlowe and Ryan, but it was published in 2000, and I don't see any newer editions online. It looks like it's probably out of print.

I have a few days to pick a textbook. Does anyone know a good one? Are there any open source texts? (I see one they use at UCSD.)


Edited to include the goodies:

• Exeter has a Discrete Math problem set. I'll probably use many of their problems as supplementary material.
• Discrete Math with Ducks, by sarah-marie belcastro, is about $60, and looks pretty good. I've chosen this for my course.
• John Golden pointed out another interesting book, Applied Combinatorics, by Keller and Trotter. Mitch Keller just changed it to a Creative Commons copyright. Thanks, Mitch!