Proving the Pythagorean Theorem

As I type these first sentences, I am watching my students in full math mode. I do not know whether to cry, to jump for joy, to laugh out loud, or high five all the students as quickly as I can.

Cliff is explaining his group’s proof to his group members. Bianca and Ellen are struggling through understanding the algebra behind their proof. Stephen and Eloise are creating a detailed slide show presentation to explain their proof. Jennifer, Clay, and Thomas are moving around triangles they cut out from paper to get a grasp of their proof. Only Veronica seems completely distracted and uninvolved. (Not their real names, but I assure you they are real people.)

Next week, each of these groups are tasked with presenting a proof of the Pythagorean Theorem. This a class of tenth graders all of whom have run into a-squared plus b-squared equals c-squared at some point in their math lives. Last year, I did a project based on a MARS task on proofs of the Pythagorean Theorem. I liked it pretty well, but this year I decided to make it a lot more open ended and give my students a chance to practice being mathematicians.

Each group has to find at least one proof of the Pythagorean Theorem and then present it and teach it to their classmates. (See my last post on presentations.) How they teach the proof is completely up to them. They can do a computer slide show, they can project up sheets of graph paper. They can demonstrate how manipulating triangles and squares can justify the Theorem. They can show a clip from a Khan Academy video or make their own.

This project is another attempt at a “low floor, high ceiling” assignment. Jo Boaler discusses such assignments in her book Mathematical Mindsets. The idea is that an assignment is accessible to every student. Every student should have some way in to the problem and have the ability to produce something of quality that adequately addresses the prompt. At the same time, the assignment should have enough freedom so that students who are inspired or sufficiently motivated can extend the problem in different ways to make it more challenging and to grow more. I went so far as to suggest possible extensions of this assignment such as proving the converse of the Pythagorean Theorem, presenting real-life applications of the Theorem, discussing Pythagorean Triples, and researching the history of their proof. (President James Garfield anyone?)

One group has already started making a video in which they ask a people at our school what they know about the Theorem. They clearly are taking great pride in making a humorous and well-produced video.

I am very curious to see what Alex and his group do with the project. At times this year, Alex has expressed frustration with the difficulty of the work. He appears to want more challenging material. However, up to now, he has not shown the willingness to take risks and extend assignments. I have been gently nudging him, explaining the opportunity before him.

I think I am most pleased by the work I saw Cliff doing today. Cliff has been one of my struggles this year. He is often disinterested and shows little effort. He rushes through work without much thought and sees the course as a punch-the-clock type of class. But today, he was as engaged as anybody and working hard to explain a challenging proof to his group. His struggle is clearly increasing his understanding and I am thrilled to finally see Cliff leave his comfort zone and lose himself in the mathematics.

Presentations are next week, but those seem perfunctory almost at this point. Just watching my students work this week is all the evidence I really need to know how hard they have worked and how much they have learned.