If students start their proof by substituting expressions into , tell them great job working backwards! To write a proof they need to reorganize their ideas to start with things they know are true and end the proof with .

Invite students to share their thinking about how Elena’s and Diego’s discoveries can be used to prove the Pythagorean Theorem. Continue discussing until students have described all the key ideas:

• Adding the expressions equal to and together
• Figuring out why =
• Figuring out why

Ask students if is true for all types of triangles. If students aren’t sure, display an example of an altitude in an acute triangle. Once students are convinced that  works only for right triangles, ask students what aspect of the proof worked only for right triangles. (The altitude forms three similar triangles only if the biggest triangle is a right triangle.)

Tell students that proofs cannot rely on the result of the theorem in the justification. Ask students, “Did you use the equation in your proof?” (No, we used only ideas of similarity, which rely on transformations, dilations, and ratios.)

Students may be familiar with this proof from middle school. Revisiting this proof helps students connect a familiar diagram and proof to the similarity proof. It also gives students an opportunity to compare and contrast different types of proof and to think about why mathematicians might bother to re-prove something that has already been proven.

When students articulate their reasoning, they have an opportunity to attend to precision in the language that they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

Encourage students to look for . Students have scissors and tracing paper in their geometry toolkits, and they may want to trace, cut out, and reassemble the pieces.

If students continue to struggle, ask them to find the areas of the smaller parts of each figure and see how they could be related. Point to the displayed colored image and ask how those parts could be related.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their proof. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.

If time allows, display these prompts for feedback:

• “ makes sense, but what do you mean when you say ?”
• “Can you describe that another way?”
• “How do you know ? What else do you know is true?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, invite students to talk about the differences between this proof and the one in the previous activity, including which one made more sense to them, which one they preferred writing, or which one felt most convincing. Ask students why they think mathematicians might continue to try to come up with different ways to prove a theorem as old as the Pythagorean Theorem.