Personal Learning Path #5

Background

I am working from the New York State Teacher Certification Examinations (NYSTCE) standards, Field 005: Mathematics. This week, we will be discussing a topic from Competency 0005 - Geometry and Measurement. Within this Competency, I have chosen the following performance indicator: "Understands the Pythagorean theorem and its converse (including proofs) and applies the theorem to solve problems in two and three dimensions and in the coordinate plane."

Here are the NY State Common Core Learning Standards for Mathematics which align with my selected topic:

My goal is to provide students with an geometric proof of the Pythagorean Theorem. The theorem is often taught using algebra alone despite it being one of the most fundamental geometric theorems. To be clear, I have not invented this geometric proof, nor did I create the activity I am working through here. I first discovered it from Eddie Woo's video, though I imagine that he discovered the activity elsewhere. Right now, I am trying to condense the activity into one manageable set of instructions for both the teacher and the students. The goal is that you could print out the instructions and cut-out shapes in this blog post and be ready to go. The activity allows teachers do demonstrate the proof in front of the class while also allowing students to conduct the proof for themselves.

For reference, here is the Pythagorean Theorem:

The Activity

I have decided to record a video of myself doing this activity, since I think it is much easier to explain in video form.

Here is a link to the video. The actual proof and a quick walkthrough of the activity is included in the video.

And here is the worksheet you can give to your students:

  

Potential modification #1: If you don't feel comfortable letting your students cut their own shapes, here is a document you can print and hand to your student with pre-cut shapes:

Potential modification #2: You could simply demonstrate the proof yourself if you are pressed for time.

Potential modification #3: You could create a quick worksheet with questions for students to work through as they complete the geometric proof. Here are some sample questions:

• After Step 4: What is the area of the square inside the triangles? How do you know it is a square?
• After Step 4: What is the area of the square outside the triangles? How do you know it is a square?
• After Step 5: What is the area of the square outside the triangles? How do you know it is a square? Is it the same square from Step 4?

STRONGLY RECOMMENDED - Potential modification #4: You could extend this activity by allowing students to cut out various right triangles. Have them measure any two sides, and let them predict the length of the third side using the Pythagorean Theorem. Have them record their measurements and predictions before they measure the third side to check their prediction. Have them repeat this process with several right triangles.

Reflection

The goal of this post was to condense this common geometric proof of the Pythagorean Theorem in a way that teachers can easily implement in their classroom. I'm happy with how this activity went, and I will definitely use it if I ever teach a geometry class of my own. There are many potential modifications, and I think some variation of this activity could work in any geometry class. I also enjoyed creating the video for this activity, though there are some small tweaks I wish I could make in my video. In retrospect, it would have been useful to actually draw the larger square the first time so that when the triangles are rearranged, it is clear that the square is the same size (a^2 + b^2). After working on this activity for so long, I feel like I have gotten better and better at explaining the proof to the point that my recorded video no longer seems like the most efficient and clear way I could have explained it. One other thing I would like to do is create a worksheet for students to work through in conjunction with this proof. The questions would be similar to the ones in "potential modifications #3 and #4." Should I ever implement this activity in my future classes, I will be sure to create such a worksheet. Nevertheless, I am mostly happy with this blog post, and I am glad to have created these resources for such an important topic.

References

• Eddie Woo's "Visual Proof of Pythagoras' Theorem: https://www.youtube.com/watch?v=tTHhBE5lYTg
• Pythagorean Theorem image: https://medium.com/swlh/why-the-pythagorean-theorem-is-true-1d4c8a508510