Lesson 14

The purpose of this warm-up is for students to get familiar and comfortable with the diagram of an altitude drawn to the hypotenuse of a right triangle with side lengths labeled with variables, which will be useful when students write equations using these variables and manipulate them in a later activity. By engaging with this explicit prompt to take a step back and become familiar with a context and the mathematics that might be involved, students are making sense of problems (MP1).

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. Don’t feel the need to wrap anything up, the next activity has students exploring this diagram further.

All the work students have done to make sense of the altitude to the hypotenuse in right triangles pays off in this activity, as students prove the familiar Pythagorean Theorem. In middle school, students explored and proved the Pythagorean Theorem visually, but now they can also prove it using similarity. Because manipulating the equivalent ratios involved in this proof is challenging, and it’s hard to see where it’s going when you’re in the middle of calculations, we introduce the proof through a dialogue between two characters, but students are the ones to come up with the insight of how we can use the expressions involving \(a^2\) and \(b^2\) to prove that \(a^2 + b^2 = c^2\).

If students start their proof by substituting expressions into \(a^2+b^2=c^2\) 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 \(a^2+b^2=c^2\).

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 \(a^2\) and \(b^2\) together
• figuring out why \(xc + yc\) = \((x+y)c\)
• figuring out why \((x+y)c = c^2\).

Ask students if \(a^2 + b^2 = c^2\) 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 \(a^2 + b^2 = c^2\) only works for right triangles, ask students what aspect of the proof only worked for right triangles. (The altitude only forms three similar triangles if the biggest triangle is a right triangle.)

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 think about why mathematicians might bother to re-prove something that has already been proved.

Use the Notice and Wonder routine to help students make sense of the context and images they will be using. Ask students what they notice and wonder about the images. If not mentioned by students, ask where \(c\) shows up in these images.

Encourage students to look for \(a^2\). Students have scissors and tracing paper in their geometry toolkits, they may want to trace, cut out, and reassemble the pieces.

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.