If students are struggling, ask them what shapes they see. (rectangles and right triangles) If students are still stuck, ask them how they would find or if there were only a right triangle. (by using the Pythagorean Theorem)

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).

If students are struggling to organize their thinking, suggest that they make a table. Help students brainstorm categories that would be effective for organizing their measurements and calculations, for example, “side length,” “diagonal length,” and “diagonal length divided by side length.”

The purpose of this discussion is ensure students understand that the ratio of the side length of any square to its diagonal is , or about .

Ask students what patterns they noticed and what conjectures they made. Invite students who organized their thinking using a table to display their work for all to see. If no students made a table, create one as a class, displayed for all to see. Include students who approximated the diagonal length of the unit square using a calculator, and students who left it as . Leave the table visible for all to see during the next activity as well.

If this conjecture is not mentioned by students, point it out in the table, and then ask students to explain:

• The diagonal length of a square seems to be the side length multiplied by 1.4 each time. (All squares are similar to the unit square by a scale factor of the side length, so it makes sense that all the diagonal lengths are multiples of the diagonal length of the unit square.)

Ask students if they agree that both of these things are true:

• The diagonal of a square with side length is about 1.4 times .
• The diagonal of a square with side length is exactly .

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.

If students are struggling, encourage them to analyze the three triangles, look for patterns, and identify the triangles as isosceles right triangles. Students can then use the patterns from the previous activity. 

The goal of this discussion is for students to consider given solutions and contrast degree of accuracy with degree of efficiency.

Make sure all students understand that the three triangles are isosceles right triangles and  each represents half of a square. Students will then be prepared to connect their reasoning from earlier activities to this activity.

Display this list of solutions for triangle :

• , or 19.6

Invite students to determine which answers are most accurate. (The first two methods are equally accurate.) Ask students which answers are most efficient. (The second and fourth are very efficient since they were found using scale factors. The fourth answer may be best for estimating.)