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I've been pretty dissatisfied with the way the Pythagorean theorem is usually taught, mainly for two reasons:

  1. The chosen proof feels like magic and I don't feel like I have a better understanding of why the theorem is true after looking at it. I feel this way about all rearrangement proofs, the "Behold!" proof, and Euclid's I.47 for example.
  2. Even when a good proof is chosen, the theorem statement itself remains mysterious. How would one have guessed that $a^2+b^2=c^2$?

I am thus interested in having a "discovery fiction" for the Pythagorean theorem, where one goes all the way from not even knowing the theorem statement to discovering an intuitive proof. I'm not necessarily looking for something that is historically accurate, just a compelling story of how someone might discover things on their own.

I wrote a blog post giving one approach, but I am worried my audience (high school students) might find it too difficult to follow.

I'm wondering how to improve on what I have, or how to come up with some other "discovery path". I'm also interested in hearing more generally about how to teach the Pythagorean theorem to high school students (and which trade-offs one should make between, e.g. theoretical satisfyingness vs comprehensibility).

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    $\begingroup$ I think this is an interesting example because using Pythagorean Theorem doesn't require knowledge of its derivation, so there is an important question about what you mean by "teach". I'll definitely have a look at your blog post! But my first thought is that I'm not aware of a situation where a high school student will need knowledge of the theorem's proof or derivation, only cases where they will need to apply the theorem. $\endgroup$ – Carser Jan 21 at 21:37
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    $\begingroup$ @Carser I agree that students won't need to know the proof in order to solve the application problems. I'm just averse to teaching just the theorem statement, or teaching a proof without motivation, because this is the kind of thing I most disliked about learning math in high school (rote learning + grinding away at problems). $\endgroup$ – riceissa Jan 21 at 22:31
  • $\begingroup$ @Carser: Proofs, provided in the courses of algebra and geometry serve as a surrogate for Logic and Proofs course, which modern high schools don't offer. But apparently the districts' geniuses think that students don't need to learn to think and argue logically, which is why textbooks are consistently being watered down. As a former POTUS admitted, he loves poorly educated. $\endgroup$ – Rusty Core Jan 22 at 1:15
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    $\begingroup$ I think it's important for high school students to learn that proof does not have to mimic discovery (in math or in other subjects) - that the process of finding a possibly true statement does not have to mimic the process of coming to believe the statement is actually true. $\endgroup$ – Alexander Woo Jan 23 at 2:28
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    $\begingroup$ @AlexanderWoo I agree with you, but I think that the more likely danger is that students never have the experience of discovery. I am teaching a course of math majors who intend to teach high school right now and I gave them the task of giving their own definition for the translation of a set of points in the plane. They have never been asked to invent their own definition before! The results are eye opening, and suggest that they do not really understand the role of definition in mathematics. So these kinds of tasks are valuable. $\endgroup$ – Steven Gubkin Jan 23 at 18:05
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I really like the idea of a "discovery fiction" -- it gives a name to something I often try to use when teaching. Here is one suggestion.

I will try to come back and write a more elaborated version of this answer later, with diagrams and proper notation, but briefly:

(a) Don't let on that you are going to prove the Pythagorean Theorem -- don't even mention it. (Your students probably know the statement already, as it is commonly taught [without proof] in middle school, at least in the United States.) Instead, start with the fact that an altitude drawn to the hypotenuse of a right triangle divides the triangle into two smaller, similar triangles. This is a good application of similarity arguments.

(b) So we have three similar triangles (which we can informally call "big", "medium", and "small"). Using proportionality, we discover that each leg of the "big" triangle is the geometric mean of two lengths, (hypotenuse of big triangle) and (portion of hypotenuse of big triangle). Write those two geometric mean relationships down, both with and without square roots.

(c) Now "suddenly notice" that if you add the two GM equations without square roots together, you find an equation of the form (leg)^2 + (leg)^2 = (hyp)^2. Amazing!

(d) Oh, wait... That sounds familiar. Has anybody seen this before? (Wait for one of the students to say "Isn't that the Pythagorean Theorem?")

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    $\begingroup$ I was about to propose almost the same sequence! I think this proof is "the best" for high school because it reinforces ideas about similarity. $\endgroup$ – Steven Gubkin Jan 23 at 18:09
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This is a great question. Love it love it love it!!

The following is just what occurs to me off the top of my head.

Show a graph paper grid with a dot at the origin and a dot at (3,4). Say we want to find the distance between the dots. We could guess that it would be 3+4=7. Well, that would be right if these were city blocks, but it's not the right answer as the crow flies. Roughly what do you think the distance is as the crow flies?

OK, let's measure it. Could someone draw it on the board at a scale where the coordinates are 30 cm and 40 cm? Thanks. What do you get when you measure it? 49 cm? Can you estimate the millimeters? I know it's rough. OK, you got 49.3 cm.

So now we know that the formula x+y isn't right. Actually, our answer looks really close to 50 cm. We don't know who the first person was to notice this, but supposedly it was already known to the ancient Egyptians. (Picture of rope with knots.) Yes, the conjecture is right, the true answer really is 5, but the Egyptians never knew why.

Here's one thing we can tell for sure about the mystery formula that gives the distance in terms of x and y. Suppose I draw a circle with a radius of 5, centered on the origin. It passes through our point from before. Now look at the top of the circle. It looks really flat there. If you were to zoom in on that point under an infinitely powerful microscope, it would look perfectly flat. You wouldn't even be able to see that it was curved, just like you can't tell that the earth is round when you're standing on the earth.

This is a hint that the relationship is nonlinear, meaning that it isn't just about ordinary addition and subtraction, like x+y or 2x+3y or something like that. I'm going to give you a preview of the actual answer, and then we'll come back and prove it. People probably did this originally as a guess and check (or guess and measure), and may never have been able to give an actual proof until hundreds of years later.

So the actual expression relating the distance d to x and y is $d^2=x^2+y^2$. This is called nonlinear because it has the exponents in it. When you square a big number, it gets really big. When you square a small number, it gets really small by comparison. What do you get when you square 1000? What do you get when you square 0.1?

This makes sense in terms of the close-up flatness idea. At a point like this (point to somewhere close to the top of the circle), x is really small. The square of a small number is really small, so $x^2$ is so small, we hardly care about it. So $d^2\approx y^2$. Can anyone solve this equation for y?

[...continue from here, lead in to formal proof...]

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I really like the idea of discovery fictions because they capture an essential component of how mathematics is best understood and communicated. Almost all ideas in mathematics follow simply and inexorably from previous ideas, and understanding any mathematical discipline consists almost entirely of figuring out how you could have developed these ideas on your own.

That being said, there are some statements in mathematics that are much simpler than any of their explanations. Such a statement is true because it's true, and no explanation will ever be entirely satisfying, because none of the explanations reflect the beauty and power of the statement itself. These are the kinds of statements that people bring up when they argue that mathematics is "discovered" instead of invented. They often become famous theorems, and sometimes many different proofs are known.

The Pythagorean theorem is a paradigmatic example of such a statement. Most statements in geometry are best understood using their explanations, but the Pythagorean theorem is more like a diamond that mathematicians unearthed when we started digging into the subject. There is nothing expected about it, and no matter how many proofs of it we find there is no way to completely dispel the mystery of why it was there.

The real history of the Pythagorean theorem agrees with this assessment. The history is that the statement itself was discovered several times by several different cultures. The Egyptians seem to have been aware of the (3,4,5) triple in 2000 BCE (apparently used as a way of making right angles for building construction), and by 1000 BCE both the Babylonians and the Chinese were aware of many different Pythagorean triples as well as the general rule. However, it wasn't until 500 years later (around 500 BCE) that the first demonstration of the theorem was given by Pythagoras. This is perhaps the first significant example in history of the now familiar mathematical process of forming a conjecture based on empirical evidence, and then later finding a proof of the conjecture.

So my argument is that, unlike almost every other statement in secondary-school mathematics, there isn't a good discovery fiction for the Pythagorean theorem, and in fact any such fiction would be highly misleading. Instead, the Pythogorean theorem is shining example of mathematics behaving like a science, with a process of discovery based on first gathering empirical evidence, then guessing the general rule, and finally verifying it, with the difference between mathematics and science being that the verification comes in the form of a proof instead of an experiment. This method of discovering mathematical truth is very powerful and important, and there are very few genuine examples of it in secondary mathematics, so I would advocate for presenting the Pythagorean theorem from that point of view rather than straining to make the theorem seem like an "intuitive" or "obvious" part of the development of geometry.

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