Knowing the Pythagorean Theorem

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Knowing the Pythagorean Theorem

June 10, 2013 Cathy O'Neil, mathbabe

This guest post is by Sue VanHattum, who blogs at Math Mama Writes. She teaches math at Contra Costa College, a community college in the Bay Area, and is working on a book titled Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, which will be published soon.

Here’s 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 $c$ , we have $a^2+b^2=c^2.$

Do you remember when you first learned about it? Do you remember when you first proved it?

I have no idea when or where I first saw it. It feels like something I’ve always ‘known’. I put known in quotes because in math we prove things, and I used the Pythagoeran Theorem for way too many years, as a student and as a math teacher, before I ever thought about proving it. (It’s certainly possible I worked through a proof in my high school geometry class, but my memory kind of sucks and I have no memory of it.)

It’s used in beginning algebra classes as part of terrible ‘pseudo-problems’ like this:

Two cars start from the same intersection with one traveling southbound while the other travels eastbound going 10 mph faster. If after two hours they are 10 times the square root of 24 [miles] apart, how fast was each car traveling?

After years of working through these problems with students, I finally realized I’d never shown them a proof (this seems terribly wrong to me now). I tried to prove it, and didn’t really have any idea how to get started.

This was 10 to 15 years ago, before Google became a verb, so I searched for it in a book. 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.

There is something wrong with an education system that teaches us ‘facts’ like this one and knocks the desire for deep understanding out of us. Pam Sorooshian, an unschooling advocate, said in a talk to other unschooling parents:

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.

She sure got my number! I thought I knew it for all those years, and it took me decades to realize that I didn’t really know it. This is especially ironic – the reason it bears Pythagoras’ name is because the Pythagoreans were the first to prove it (that we know of).

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 $3^2+4^2=5^2$ this is a right triangle, giving them the right angle that’s so important for building and surveying.

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.

Nowadays I feel that one of my main jobs as a math teacher is to get students to wonder and to question. 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. Well, perhaps I had seen it but never took it in before, not being ready to appreciate it. I’ll talk about it below.

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
$(a+b)^2 = c^2 + 4 \cdot 1/2 \cdot ab$
$a^2+2ab+b^2 = c^2 + 2ab$
$a^2+b^2 = c^2$

Visual version:

Move the triangles around.

What was $c^2$ is now $a^2+b^2.$
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:

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! (I got this language from Alexander Bogomolny’s blog post about it, which also tells a story about young Einstein discovering this proof.

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) = $k \cdot$ 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 $k \cdot A$ , similarly for the square attached to triangle $B$ . But note that $kA + kB = k (A+B)$ which is the area of the square attached to the triangle labeled $A + B$ . But $kA = a^2$ , and $kB= b^2$ , so $k(A+B) = a^2 + b^2,$ and it also equals $c^2$ giving us what we sought: $a^2 + b^2 = c^2$

I stumbled on the article in which this appeared (The Step to Rationality, by R. N. Shepard) while trying to find an answer to a question I have about centroids. I haven’t answered my centroid question yet, but I sure was sending out some google love when I found this.

What I love about this proof is that the triangle stay central in our thoughts throughout, and the focus stays on area, which is what this is really about. It’s all about self-similarity, and that’s what makes it so beautiful.

I think that, even though this proof is simpler in terms of steps than my old favorite, it’s a bit harder to see conceptually. So I may stick with the first one when explaining to students. What do you think?

Categories: guest post

Comments (31)

peacockg

June 10, 2013 at 7:45 am

Thank you. What a perfect complement to Cathy’s recent “proxy” posts. It’s not the formula, but what’s underneath that really matters for understanding.

When we unschool the means-variance optimizers that Wall Street deploys so effectively on the investing public, It’s not just the two digits to the right of the decimal point that seem laughable.

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- Abe Kohen
  
  June 10, 2013 at 8:12 am
  
  You’ve got me scratching my bald head trying to figure out what the dumbing down of high school math has to do with Wall Street.
  
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  - peacockg
    
    June 10, 2013 at 8:54 am
    
    It’s not the dumbing down of math in school. It’s thinking about the underlying reasoning of things. Cathy has written two or three posts recently about “proxies” and what happens over time when people just memorize or use a formula. It struck me that simply learning the “fact” of the Pythagorean theorem and not having to learn or think about the proof was similar in nature.
    
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    - Abe Kohen
      
      June 10, 2013 at 9:03 am
      
      Having taught Geometry in an inner city high school, I have found that some students have a real hard time doing proofs, while for others it is second nature. Given the time constraints it is to the benefit of the less mathematically inclined students to learn how to use a theorem rather than learning how to prove it. It’s not about proxies it’s about applied math vs theoretical math. Yet the smarter students get shortchanged by switching to applied only. That’s why providing alternatives is important.
      
      My two cents.
      
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    - peacockg
      
      June 10, 2013 at 9:07 am
      
      I totally agree with you on that. My mother was a math teacher.
      
      I guess my complaint was not so much with teaching children and young adults math, but with the adults who should know better relying upon “proxies” that over time “degrade” in the manner that Cathy has described.
      
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    - suevanhattum
      
      June 10, 2013 at 11:27 am
      
      You might want to check out Henri Piccioto’s Math Ed Page for some activities that will work for both low and high level students.
      
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    - Michelle
      
      June 10, 2013 at 10:37 pm
      
      If what you mean is that some students are better off simply learning the statement of the Pythagorean Theorema and how to apply it: I totally disagree. Why should anyone bother knowing the fact of the Pythagorean Theorem? Why in the world is that particular fact (or any particular fact) valuable to them? It’s not! Almost no particular fact you teach in high school has any relevance to their later lives, unless you’re talking about future scientists and engineers.
      
      So what is valuable? Reasoning, proof, logical argument, making connections, seeking understanding, curiosity, reasoning about pictures like this… all the stuff in the post is valuable. Way more valuable than the Pythagorean Theorem itself. And it’s worth the time, and it’s worth pushing students who struggle with it.
      
      There is some value in learning to understand the statement of and how to use a “black box” tool, and maybe that’s what you meant. In that case, I think that’s valuable across the board for all students. But it’s not the only thing that they should get out of math class.
      
      And the Pythagorean Theorem is one place where you can actually get at the proof without too much struggle. There are literally hundreds of proofs out there, some of them lovely and transparent like the two here and some much harder. President Garfield, before taking office, created his own proof of the Pythagorean Theorem. (Though, in fairness, it’s essentially the first one in this post with the picture cut in half. But still… a US President! Created a proof!) This is not the piece of content where we should focus on “let’s try to understand and apply this tool without worrying about the why’s.”
      
      Disclaimer (?): I’m a mathematician, teacher educator, and former high school teacher (among many other levels).
      
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Abe Kohen

June 10, 2013 at 8:09 am

As far as I can recall proving the Pythagorean theorem was part of the standard for NY State Regents Tenth Year Mathematics (aka Geometry) back when I was in high school in the 60s. Just checked a copy of “Geometry” 3rd Edition by Harry Lewis (c) 1973, page 452 has a proof and refers to someone collecting “close to 300 of these proofs, one of which was developed by President Garfield.” (Don’t know if that Harry Lewis is at all related to the Harry R. Lewis of Harvard.) I don’t know what that says about Contra Costa College, except for bringing back memories of driving from Berkeley to Hilltop Mall in Richmond.

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- suevanhattum
  
  June 10, 2013 at 11:21 am
  
  I was talking about an algebra course. I doubt that our algebra is any less thought-provoking than your 60’s algebra course. I can’t imagine how this post has anything to do with math allegedly being ‘dumbed down’.
  
  I probably did learn a proof in my 70’s high school geometry class, and then forgot it. But a better memory would not have helped me to understand math deeply.
  
  I’m not sure you’re getting the point of this article, Abe.
  
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  - Abe Kohen
    
    June 10, 2013 at 11:39 am
    
    Are you talking about teaching algebra in community college or in high school?
    
    The dumbed down curriculum got rid of the 9th grade algebra, 10th grade geometry, 11th grade trigonometry, and 12th grade calculus or pre-calculus sequence and replaced it with a hodge-podge of Math A, Math B (at least in NY State). Dan L makes some very good points about whether you need to know a proof BEFORE using the results.
    
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    - suevanhattum
      
      June 10, 2013 at 12:07 pm
      
      The course I reference in the post is a beginning algebra course at the community college. That’s pretty much equivalent to a high school Algebra I, except that we do it in one semester. So many students come to community college at this level, and our goal is to help them truly understand math, so they can benefit from it.
      
      Mixing the subjects together does not have to be a dumbing down. Phillips Exeter Academy has math 1, 2, 3 and 4, and shares their wonderful materials. I believe they are successful with both low and high level students.
      
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moosesnsquirrels

June 10, 2013 at 8:25 am

My wife was a Montessori student and teacher, and has often pointed out to me that Montessori’s directed discovery approach to teaching mathematics and science is far better, at least in the elementary grades, than the standard textbook approach that took over the public schools around a century ago. We both find Montessori’s “guiding hand” preferable to the wholly undirected unschooling philosophy of “they’ll get to it some time” and the highly structured and contrived textbook method that emphasizes passive memorizing.

Eventually, I believe there is too much material to cover to avoid more standards texts. But I emphasizing the idea of actively proving proving mathematical statements early in education makes the use of textbooks later on much more effective.

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- suevanhattum
  
  June 10, 2013 at 12:23 pm
  
  I think I probably agree with you. Although I personally lean toward unschooling, I suspect it works best with very well-educated parents. Montessori is a good way to engage kids who might not start out with an interest in the math. I quoted Pam Sorooshian not because of her unschooling philosophy, but because I was intrigued by the idea – do we think we know things that we’ve memorized, and not wonder about them as much as we might?
  
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Steve Stein

June 10, 2013 at 9:28 am

I like the first one better. (BTW, that’s the original Chinese proof, I think, from the Zhou bi suan jing). But you should teach BOTH to your students. And, for good measure, teach Euclid’s proof.

It’s a good way to teach the concept of a proof. Which is more convincing?

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Jon Doe

June 10, 2013 at 10:23 am

It might be a stretch but if more people really understood the tools they are using it is possible that they would not put as much faith in Black–Scholes as they did. Remember the phrase picking up nickels in front of a steam roller?

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- Abe Kohen
  
  June 10, 2013 at 11:53 am
  
  A better question is why Long Term Capital, with Myron Scholes on board, failed to recognize the inherent risks of finite capital and margin calls in an illiquid market. No amount of deriving BSM from scratch (as was done in my advanced finance class in B-school at Berkeley) would have helped Merriwether and his young turks from Harvard and MIT via Salomon Brothers. Mark Rubinstein said that B,S and M sought help to find a closed form solution from their colleagues in engineering who enlightened them about the heat conduction equation.
  
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Dan L

June 10, 2013 at 10:33 am

Yes, concepts are important. Proofs are important. Intellectual curiosity is *very* important. But I think you run into trouble when you start making blanket statements that memorization should not come before conceptual understanding. It doesn’t bother me at all that most students (including myself) learn the Pythagorean Theorem before learning why it’s true. There are various reasons why I believe this, and they apply as well to high-level college math courses as they do to elementary mathematics:

First, a lot of mathematical learning and problem-solving skill comes from working out examples. So for example, your “pseudo-problem” is not such a bad problem, because it allows students to practice their *algebra* skills (even though it teaches them almost nothing about triangles).

Second, it’s often good to first understand *why* a theorem is important or interesting. That’s why, even in pure math courses, it’s sometimes nice to discuss applications of a theorem *before* discussing its proof. In fact, this is probably why many people who are “taught” the proof of the Pythagorean Thm will never remember it. Because they have no context for caring about it. It’s just one more lesson in an endless parade of lessons, just like the lesson on the Louisiana Purchase. However, if one first uses the theorem and deals with squares and triangles for years and does tons of SAT problems and whatnot, and *then* one sees the proof, it has the chance to be memorable and interesting.

Third (although this is not so relevant to the Pythagorean Thm), practicing something *without* conceptual understanding often leads to better conceptual understanding through practice and repetition. I think of this as proof by example. Solving simultaneous linear equations is a decent example of this. You can learn to do it without having a real sense of *why* the method works, and the more you do it, the more likely you are to understand why it works. (The algorithms for multi-digit addition/subtraction/multiplication/division–the standard ones and the nonstandard ones–might also fit this description.)

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suevanhattum

June 10, 2013 at 11:40 am

I go back and forth on what you’re saying, Dan. But I think the math circle philosophy that “What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises.” (G. C. Lichtenberg) is valuable here.

I think the old books by W.W. Sawyer do a great job of staying centered on understanding. I’m relearning abstract algebra right now, and having a blast.

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josh

June 10, 2013 at 12:51 pm

Sue,

Thanks very much. Great article and beautiful proof.

I like the last proof. It does require some explanation but doing so involves other useful concepts and so is worthwhile.

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Nitpicker

June 10, 2013 at 1:09 pm

I have a problem with your first proof. As I see it, you handwave the part of the proof where you state that your internal quadrilateral is actually a square. Maybe it is a rectangle or even a rhombus that just looks like a square. It’s not hard to show that all four sides are equal (they are the long side of 4 identical triangles) and that all four angles are equal and “square” (all equal because they finish the lines of the outer square, “square” because they must sum to 360 and are all equal). But I think you should include those details in your proof. Better yet, when you present the proof; leave it out; and tell students there is a step missing; and challenge them to find it.

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- suevanhattum
  
  June 10, 2013 at 1:26 pm
  
  Good catch, Nitpicker. (And perfect handle, eh?) I do discuss that issue with my students, although I’ve never tried your idea of challenging them to find “the missing step”. Perhaps because it’s too directive. I’m more likely to say, “Hmm, how can we be sure that’s a square?”
  
  Each time we write a proof, we decide how much to say. Even after adding the step you point out, some mathematicians would want to add more detail, depending on their perspective. A proof is a communication tailored to a particular audience. I thought it best to leave that out, but am not surprised to see a disagreement.
  
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  - Cathy O'Neil, mathbabe
    
    June 10, 2013 at 1:27 pm
    
    Nicely said, Sue.
    
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  - Nitpicker
    
    June 10, 2013 at 1:46 pm
    
    I thought a bit more after I posted the above and I agree completely with your comment about proofs being tailered to an audience. However, I think that at least early on students should be enouraged to be as complete as possible in their proofs. Furthermore when proofs are presented to them, they should either be complete or it should be carefully pointed out where the “handwaving” is taking place.
    
    I wonder if there is a collection of incomplete or just plain wrong proofs for simple (and possibly true) theorems which could be used to teach students to notice when “handwaving” is occurring. It seems to me that being able to detect “handwaving” in an argument of any sort is a very important part of critical thinking.
    
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    - suevanhattum
      
      June 10, 2013 at 2:07 pm
      
      If anyone has a collection like that, I’d like to use it with my Discrete math students.
      
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    - Michelle
      
      June 10, 2013 at 10:42 pm
      
      I always create my own collections of these based on student responses to questions. These are sometimes proofs and sometimes just solutions to problems with less formal justifications. But I will anonymize them and then bring them in, asking the class to read them critically. They do a pretty good job of finding big flaws and missing steps, but have a lot of trouble with handwaving and subtle flaws (like where someone says “clearly it’s a square”… they are very inclined to just believe that rather than question it.)
      
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grwww

June 10, 2013 at 3:31 pm

If you can work in the unit circle, then all will be good. Try to find a good explanation about how the “squares of the ratios of the two sides of the right triangle to the hypotenuse, always add up to 1”, which just happens to be how sin and cos relate to all of this. For example, if you use the 3, 4, 5 triangle, then (3/5)^2 + (4/5)^2 = 1. You can also show how that works in relation to 3^2 + 4^2 = 5^2 by dividing by 5^2 on both sides etc.

Once you have given them several working examples, then you can go for the throat, by adding in a 30, 60, 90 right triangle, and then showing that sin(30)^2 + cos(30)^2 = 1 as well as sin(60)^2 + cos(60)^2 = 1, and then let them see that sin(x) = cos(90-x) for all 0 <= x <= 90, and thus complete the picture of how a and b are just points of reference and not always the long side or the short side etc.

Math wizardry is something that requires multiple perspectives and simple proofs as you've alluded to. I know, that for me, the unit circle and the Pythagorean theorem were always not related to me in teaching practices in enough ways and enough times for me to grasp the simplicity of it and the power of what I needed to know.

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Michelle

June 10, 2013 at 10:44 pm

Random aside: I think the Egyptian rope pullers were really using the converse of the Pythagorean Theorem: If the sides of a triangle satisfy a^2 + b^2 = c^2, then you have a right triangle on your hands. To prove that, you invoke SSS congruence plus the Pythagorean Theorem itself.

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- suevanhattum
  
  June 10, 2013 at 10:47 pm
  
  Thanks, Michelle, I did miss that subtle difference.
  
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kaleberg

June 11, 2013 at 1:19 am

I first encountered the Pythagorean theorem in 2nd or 3rd grade while reading a big blue, but rather thin, hardcover book called “Mathematics”. It had the usual squares on the sides illustration and a picture of a 3-4-5 rope triangle with some pyramids in the background. In 9th grade math, I saw a geometric construction proofs which worked by translating and rotating tiles. In 10th grade I did a NY State Regents proof, I believe using similar triangles and some ratio formula, but I can’t swear to it. In other words, there was a 7 or 8 year gap. If I understand correctly, the proof I did is actually full of holes since Euclid made all sorts of crackpot assumptions about 2D space, so it is possible that I still haven’t learned how to prove the Pythagorean theorem. (I do remember President Garfield’s proof. Apparently, there was a time in the 19th century that proving the Pythagorean Theorem was a popular thing to do in certain circles.)

Personally, I have no problem with teaching people how to do things without first teaching them how and why. I can’t imagine teaching a one or two year the rules of grammar so that they could understand and utter English sentences, especially if they have no command of language whatever at that point. On the other hand, the current practice of never teaching grammar has caused real problems for many of the high school students I have been tutoring. I recently explained the parts of speech to one young lady who responded that it would have been nice to have been taught about nouns and verbs and adverbs some time before her senior year.

I think the proper approach is to teach how to do things and provide a framework of facts. Rote memorization may suck, but there are a huge number of tasks for which it is absolutely essential. It is also very important to teach how and why things work and how they are derived, but you don’t want to do it to soon, or expect the child to do all the work of thousands of years of collective human knowledge development.

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Kids will be kids

June 13, 2013 at 1:52 pm

This is just my two cents. I just finished Geometry as a middle school student, but it is the same exact course that people in high school have to take. We did only did two proofs the entire year on proving the similarity or congruence of triangles. We never once tried proving the Pythagorean theorem. Yeah, we didn’t prove it in algebra either…..

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josh

June 14, 2013 at 8:34 am

47, 18, 76, 123, 11, 29, 7

If I asked you to memorize these seven numbers, I suspect it would take some time to do so and you would not retain them very long.

On the other hand, if you ordered them an noticed that each is the sum of the previous two, you would get them much more quickly and confidently.

7, 11, 18, 29, 47, 76, 123

Actually, you would only memorize the first two and the rule.

The difference between memorizing formulas and seeing proofs is the same.

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