Teaching the Pythagorean Theorem Proof through Discovery

Using a discovery lab to introduce students to the Pythagorean Theorem was an amazing way to kick off my Pythagorean Theorem unit. I’m excited to share with you exactly how we got hands-on with the Pythagorean Theorem proof and how it helped my students really understand this geometry concept. I’ll also share with you how I used foldable notes to introduce this unit to my students.

Why discovery labs?

Having students begin a unit with a discovery lab activity gets them involved in creating the mental model of what’s happening mathematically right from the start. They turn from passive recipients of information to more involved learners. As I’ve shared before, this is an easy to implement way to get students to use math talk, search for and identify patterns, draw conclusions, and make mathematical arguments, all before the formal instruction. And the Pythagorean Theorem discovery lab directly addressed our first learning goal for the unit (see how I used “I Can” statements to plan the learning and help students track their learning here.)

Calling these activities “labs” reminds students that they are expected to experiment and try things in a low stress environment. It encourages students to ask questions, make predictions, and test their ideas. And with my students I’ve seen it provide a great shared experience we can refer back to throughout the whole unit of study. Students have more context for the formal instruction when we get to it, and they ultimately understand it in a deeper way than I’ve seen before. The Pythagorean Theorem in so many ways is especially perfect for this kind of lesson because it’s based in understanding a proof. Given the right direction, students can come to the same conclusions as Pythagoras. That’s pretty cool!

For several years I’ve seen all over Pinterest different ways people model the mathematical argument of the Pythagorean Theorem. The use of square numbers represented with boxes for the numbers (as seen below) is a physical way of showing what the equation a² + b² = c² means. In this discovery lab, I wanted to use that same proof, but also have students look at examples and non-examples of the Pythagorean Theorem with right triangles so they had a richer understanding of why it works with just specific numbers.

Step 1- Building background

To get students ready to dive in and play around with the Pythagorean theorem, I started with vocabulary terms. We quickly defined the following terms:

Right angle
Leg (as it relates to triangles, of course!)
Hypotenuse
Square number

Then, I asked students to look at a right triangle with the accompanying squares (see below). They recorded their thoughts and observations about the shape.

My students knew absolutely nothing about the Pythagorean Theorem. They were truly starting at ground zero. But I wanted them to get warmed up looking at shapes and making any observations they could.

Step 2- Trials and observations

Once we’d defined some terms, we got right into hands-on exploration. Students worked with a partner. Each partnership got a work mat with a printed right triangle and a clearly labeled place for each leg and the hypotenuse. Students also cut out visual representations of 3, 4, 5, and 6 all squared. I gave students different numbers for the side lengths and asked them to build that triangle. Once they modeled it on their work mat, they sketched what it looked like and determined whether or not they created a right triangle.

Students had to be precise in how they lined up pieces on the work mat. Then, it became clear which right triangles worked, and which ones didn’t. (Last year, my attempt at this discovery lab was a major flop. Mostly due to the fact that one of the squares was missing a row. Oops. Precision really matters here. But with that fix made, and emphasizing to students to carefully have the corners meeting, this year went sooooo much better.)

This set had only one combination that works: 3-4-5. The other two combinations don’t. Once students had completed building all three triangles for set 1, I asked them to explain for each of them if they saw a relationship between the squares of the side lengths? I wanted them to start looking for a pattern related to the squares, nudging them towards recognizing the same pattern Pythagoras did.

After students completed the first set, they repeated the process. With the second set, students built triangles with the numbers 5-10-12-13. (Tip: I copied each set on different colors to help students keep them separate) As in the first set, one combination worked, while the other two did not. Students started observing out loud that not all combinations of numbers work, or make a right triangle.

Again, students sketched the models they built and then reported whether or not they made a right triangle. Then, they answered the question, “Is there a relationship between the squares of the side lengths?”

Finding the pattern of the Pythagorean Theorem

The conversations were good, and students realized that only certain number combinations make a right triangle. However, they weren’t seeing any patterns between the numbers, their squares, and what makes a right triangle. So I decided to drop some specific hints.

First, I wrote on the board 9, 16, and 25 (from the first set’s 3-4-5 triangle). I asked students if they could see a pattern now. They didn’t. But one student started laying the squares from our trials on top of each other. As I saw her doing that, I just thought, yes! I asked the rest of the students to also start stacking them on top of each other and see if they noticed anything. One by one, each partnership started to get it. They would lay one over the other, and then count what was left. As they counted what was left over, the light bulbs started going off.

The first student who realized it quietly came to me and whispered the answer. “I think that the two numbers squared is the same as the third number.” When I told her she had it, her eyes lit up. It was especially awesome to see because she has traditionally struggled in math and doesn’t always have a lot of confidence. After she figured it out, other partnerships found the pattern too. As each of them told me what they were thinking, I told them they were right on. Then, I challenged them to test the other triangle we made from set 2 to see if it followed the same pattern.

Step 3- Drawing conclusions about the Pythagorean Theorem proof

To wrap up the discovery lab, I gave students three reflection prompts to respond to:

Write a rule about the lengths of the sides of a triangle.
Write a formula for this rule.
Explain what the Pythagorean Theorem is. How could the Pythagorean Theorem help you find the third side of a right triangle?

For the rule, I asked students to describe what they figured out. Then, I asked them to write a formula. And I got blank stares. So, we worked together to take their rules and write them as a formula. Finally, students reflected on what they had discovered about the Pythagorean Theorem.

The rest of this unit went far faster than I expected. While discovery labs are typically a great way to start a unit before getting into the formal instruction, what I noticed with this unit is that we flew through the note taking. Students also showed proficiency very quickly in our practice with this concept. I think a big part of that was the solid understanding they created themselves through this activity.

Taking notes in interactive notebooks

With the discovery lab behind us, we moved ahead with formal instruction. Students added this foldable to their math interactive notebooks. We simply added tape to one side to attach it to our notebook. On the front of the foldable, students labeled the parts of the triangle to review the key vocabulary.

Then, we walked through a worked problem with Pythagorean Theorem together on the back. We annotated the problem to make sure students knew exactly what was happening.

Under the flap of the triangle, students wrote notes as we modeled a few more problems. I wanted to make sure that students saw how to solve for a leg of a right triangle, as well as the hypotenuse, using the Pythagorean Theorem.

Reflecting on discovery labs and foldable notes

Using discovery labs and math interactive notebooks has revolutionized my classroom over the past two years. Together they make a powerful way to combine student centered learning and meaningful class notes that students will actually look back at and use. They get students more involved in the learning process, with relatively small adjustments for me as a teacher.

The foldable notes provided my students a simple, clear way to organize the new information about using the Pythagorean Theorem. This was a great resource for students. They referred back to them whenever they had a question. I don’t remember them looking back in a text book in the same way in past years 🙂 There’s something about students creating their own reference that’s really powerful.

Try it out!

Want everything I used in my classroom for this unit? The entire Pythagorean Theorem discovery lab (student pages, work mat, and prepared squares) can be found here.