This Math Talk focuses on finding the hypotenuse of a right triangle. It encourages students to think about characteristics of right triangles and to rely on the Pythagorean Theorem and similarity to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students reason about the diagonals of squares.

To find the value of each hypotenuse, students need to look for and make use of structure (MP7).

The goal of this discussion is to review students’ strategies for finding the hypotenuse of a right triangle, including using the Pythagorean Theorem and similar triangles. 

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections do you see to previous problems here?”

In this activity, students compute the diagonals of some squares, measure the diagonals of others, and use repeated reasoning (MP8) to generalize a process.. From the data they collect, they reason that for a square with side length , the length of the diagonal is about . They also calculate the diagonal of a unit square exactly, using the Pythagorean Theorem, allowing them to connect the decimal approximation to the square root of 2.

Students will need a set of squares to measure. They could use various sizes of origami paper, sticky notes, or other convenient objects. If squares are not available, there is a blackline master provided with squares students can use instead.

The goal of this activity is not to simplify radicals or make explicit the connection that, for example, can be rewritten as . 

Monitor for students who:

• Use a calculator to determine that the diagonal of a unit square is about 1.4.
• Leave the diagonal of the unit square as exactly .
• Make a table to record what they learned about squares and their diagonals.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

In the previous activity, students generalized that the diagonals of squares are related to the side length of the square by a factor of about 1.4, or exactly .

In this activity, students apply their generalization about the diagonals of squares to isosceles right triangles. To find the lengths of the unlabeled sides in the second figure, students will need to generalize that a right triangle with one 45-degree angle is isosceles (because it’s half a square, because both base angles are congruent, or because it’s similar to the isosceles triangle in the first figure).

To find the unknown values in the third figure, students will have to use the ratio of diagonal length to side length to find unknown side lengths. Students may use the approximate ratio (or ) to find the side lengths, or they might use the exact ratio . Students are not expected or encouraged to rewrite answers of in another form.