Geometry.

PROOF 1

I really liked this proof - probably because it was “in motion” and it was easier for me to see what the proof was trying to say. The proof I like starts at 2:12. I like this proof because it is both visual and kinesthetic. You can give these pieces to students and they can prove the theorem to themselves. You can “touch” the proof.

https://www.youtube.com/watch?v=YompsDlE dtc

PROOF 2

Area of the large square: �(a +b)(a + b) = a2 + 2ab + b2 Area of the 4 right triangles: �4 x (1/2 ab) = 2ab Area of the inside square with sides c is c2 a2 + 2ab + b2 = 2ab + c2 a2 + 2ab + b2 = 2ab + c2

PROOF 3

I find this proof to be the most elegant, because it requires minimal explanation. Maybe that makes me lazy, or maybe that just makes me efficient! This proof relies primarily on visuals and a little bit of algebra to show the Pythagorean theorem. Next year, I'll be teaching high school algebra, and could imagine my students having a "wow" moment when seeing how easily the Pythagorean theorem can be proven. --Jordan

The one I usually end up sharing with my students is Proof#4 – this amazing YouTube video helps a lot. I love it that it is hands-on, interactive, and uses algebra and my students get to paste it later in their interactive notebooks.

--- Shalini

I would say that Proof #3/#4 are the ones that I always remember and naturally make sense to me. To me, an "elegant" proof is one that is simplistic and accessible, or one that takes an approach which shows a truth and connection in an unexpected way. Proofs #3/#4 are elegant proofs in the first sense.

---Molly

I like these proofs because they offer a visual starting point and an algebraic conclusion. Although many (most? all?) Pythagorean Theorem proofs will or can end with the algebraic statement a^2 + b^2 = c^2, these two seem to have a more substantial algebraic solving component. I am attracted to the notion that there are fun examples of algebra and geometry not really being different fields, but rather two different possible representations of the same abstract concept.

PROOF 4

I like that it incorporates an origami approach, with no cutting or rearranging required. It took me a couple tries before I was able to follow the whole approach from beginning to end, but when I finally was successful it was super cool to see the Pythagorean theorem this way!

PROOF 5

Proof by Indian mathematician Bhaskar (2nd cent AD)!

Blue triangles, congruent, yellow squares congruent.

Compute area two different ways!

This first one I liked because it uses simple shapes like the ones that we have used in class- without complicating the Pythagorean Theorem too much this one makes sense.

PROOF 6 Drop perpendicular CE to hypotenuse. Triangles ABC and CBE similar. Triangles ABC and ACE similar.

BUT, I found the second particularly interesting as it uses similar triangles - a topic that we have been studying more recently. I like that this uses similarity rather than area and still does not over complicate things. AA similarity is one that I think students understand easily and the

algebra is quite simple as well.

-- Ali

Since we have been looking at similar triangles wanted to share this proof. It is both simple in its proof of similarity and in algebra. The students have an easy time understanding AA similarity and the three triangles are easy to see. Then using substitution the Pythagorean identity becomes clear.

-- Alice

Thanks Becky! 7 Rotate triangles so that the angles between the c sides is right! What is the area of the weird quadrilateral?

Proof 51 is my favorite, and I would argue the best proof for a very straight-forwarded reason. I teach middle school in a very low income area. Most of my students and their parents are convinced that they are bad at math. Many of my students openly are waiting until they turn 16 to drop out of school. They don't

believe they are capable of understanding academics. Enter this proof. I KNOW that my 7th graders could understand and interact with this proof. The glimpses of hope that keep us going, are the moments when they grasp something challenging and explain it to someone. I actually plan on showing them this after the high stakes testing in May. I think as someone who loves math, it is easy to make it inaccessible for others if we aren't careful. While we should ardently pursue rigor and depth, I think any opportunity we have to give others and entry point to the beauty of mathematics should be taken!

PROOF 8

For

students just making their way into geometry (grade 6 to 8) I might prefer this really fun and basic picture proof. At the end of the day the fun of the proofs, for me, is helping someone else understand, so I always see them through the lens of how my students would access them.

PROOF 9

AD is the angle bisector of angle A DE is perpendicular to AB.�AB = c, BC = a, and AC = b.�Let CD = DE = x.

Then BD = a - x and BE = c - b. Triangles ABC and DBE are similar.

This is my favorite of the proofs for four reasons; it makes students construct an angle bisector and a perpendicular, it contains a wealth of congruence and similarity, it makes us flip the shapes around, so we are not stuck in the single perspective, and it highlights the intersection of algebra and constructions in proofs. It does not highlight the most mathematical gymnastics, but it has so many great parts to have students practice essential skills and increase their geometric flexibility while being completely accessible to a 10th grade second semester student. It is challenging, simple, and compactly precise.

PROOF 10

Uses the super cool Power of a Point Theorem!

The one that stuck with me was Proof #22 – just because we had covered Circles and it uses the Power of Points theorem (quiet a rare proof—I haven’t seen that one before!!)

PROOF 11

This one is from 30th Nov, 2017!

Area of top is a^2 + b^2 Area of parallelogram is c^2

Take a look at this proof, Pythagorean Theorem from a Rearrangement of a Parallelogram. Does it make you think of the first day of class. The missing square puzzle that makes us think about how shapes can be

rearranged. I saw this proof and immediately questioned what was going on and wondered if it was really a Pythagorean proof ? Well, it is and I love how simple and elegant it is. Just by adding the dotted line and understanding that it is also length c when it is constructed by a 90 degree angle from side c is awesome. The fact that this proof uses information we just learned about parallelograms should push this to the top as best proof!

Thanks Kelsey! 12

I chose to show that the Pythagorean theorem cannot only be proven with squares as it most commonly is, but with any shape! I chose this because when I came for Harvard Extension School orientation night and met with Andy Engelward he showed us this proof.

As a math student and teacher, I had never thought of this nor did I ever see it in middle school, high school, or even college as a math major. �This proof could really become fun and creative if you were to do this in a math classroom and allow your students to use

whichever shape they want and prove that it works along the way.

PROOF 13

There were so many to choose from I decided that 32 was simple enough for a middle/high school student to appreciate without "standing on their head". I also saw that it was incorporated in some later proofs.

Calculate the area of triangle ADE in two different ways!

PROOF 14 Left as a puzzle for you!

Pythagorean Theorem “in nature”?! Do these tiles hide a proof of the pythagorean theorem? Used by 9th cent. Arabic mathematicians Al-Nayinzi and Thabit ibn Qurra to prove it! https://www.youtube.com/watch?v=