Possibly the most famous theorem in all of mathematics. Over 300 distinct proofs of this theorem exist, including one discovered in 1876 by future president James Garfield. [Unfortunately, his mathematical prowess did not protect him from the assassin’s bullet.]

I introduced the Pythagorean Theorem today in class. Here’s how I’ve taught it before:

- State the theorem. For every right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs (a.k.a. for a right triangle, ). Almost every student has seen it before, either in middle school, in algebra, or in engineering.
- Do some practice questions involving solving for the unknown hypotenuse.
- Do some practice questions involving solving for an unknown leg.
- Do some practice questions involving Pythagorean whole number triples, where you have to figure out the third number without knowing whether it’s a leg or hypotenuse.
- Introduce the Pythagorean Theorem converse and do some practice questions of figuring out whether a triangle is acute, right, obtuse, or nonexistent.
- Have students research a proof of the Pythagorean Theorem, come to an understanding of it by reading it and working out the pictures and/or algebra, then write up a mini-report. Here‘s the project description.

Anything vital (or just extremely cool) that I’m missing? How do you teach the Pythagorean Theorem?

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I usually do a graphical proof. I give each student a set of four identical right triangles in which the legs are labeled

aandb, and the hypotenuse is labeledc, and a square whose sides arec–a. The first task is to arrange the shapes so that you have areas ofa2 +b2, and then rearrange the shapes so that you have an area ofc2. Since the two shapes have the same area, they are equal.(b-a)for the square, correct?Yes, that’s one of my favorite proofs! I like that it can be seen geometrically, as compared to some proofs that seem at first to students like mere algebraic manipulation.

Thanks for the comment!

I really want to do this with my pre-AP classes next semester. I have a few questions, as I’ve never actually done a research project with a class.

1. How long total do they have to complete the project from the time it’s assigned?

2. Do you have resources on hand for the students to look in? And even if you don’t, can you point me to some that I may look at myself?

3. Is there any way to keep every kid from doing the same proof?

4. Any other advice you can give for doing this would be appreciated.

Thanks!

I started doing this proof research project, adapted from another teacher at my school Mr. Beck, when the school began pushing for a research project in every class. I am lucky to teach in a computer lab, so the internet is the main resource students choose (though I have had students choose from my library of math books instead). One site in particular, used by many students, has 88 different proofs (though students usually reject #1 as looking too complicated and rarely get past #5 before settling on one they like). This is probably the most frequent site because it is a top hit when googling “pythagorean theorem proof”. I actually did get this book as a gift last year, and while I haven’t read it all yet, it may be another good resource.

This year I gave my students three class periods (ninety minutes each, minus passing time and warm-up activity time) to do their research, settle on a proof, and write up the report. Other years I have attempted to do two class periods spaced a week apart (while doing more practice with the P.T. and then moving on to trigonometry of the right triangle in between) so that students have homework time to come to understand and work on aspects of the proof/report. This didn’t work so well, as most of the students didn’t do any work in between the two days.

I do ask the students to check in with me about which proof they are using, after exploring different options. I try to get them to vary it up some, at least not to pick the same proof as their neighbors. This also helps me catch some who think a statement or demonstration of the Pythagorean Theorem is the same as a proof 🙂 .

Some students I encourage to do the example first, to help them understand the proof’s steps, while others are sophisticated enough to dive right in to the geometry or algebra of the proof itself. I often ask the students to talk through the proof (in their own words) with me, before writing it down, when they don’t seem to know how to summarize the proof without plagiarizing.

Hope some of this helps!

i am very like to learn about this problem and i hope you can help me

As I mentioned, the Pythagorean Theorem is one of the most famous theorems in math, and one of my favorites too. Let me know how I can help!

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