The purpose of this activity is for students to think about the relationships between the squares of the side lengths of triangles as a lead up to the Pythagorean Theorem at the end of this lesson. 

Students record both the side lengths and areas of squares and look for patterns (MP8). Note that some side lengths are intentionally positioned so that students won’t be able to easily draw squares. In these cases, the segments are congruent to others whose lengths are already known or could be calculated. Some side lengths lie along gridlines.

The purpose of this discussion is to allow students to communicate what they noticed using precise mathematical language. Students may notice that for Triangles E and Q and that they are also right triangles. If so, ask students if any of the other triangles are right triangles. (They are not.) If students do not see these patterns yet, do not give them away. Instead, tell them that they are going to look at more triangles to find a pattern.

The goal of this activity is to reach a formal statement of the Pythagorean Theorem. Students will work with a proof of the theorem in a later lesson, so the focus here is on building a basic understanding of what the theorem says and that it is not true for all triangles.

Invite selected students to share their strategies for determining the missing side lengths. Make sure the class comes to an agreement for which triangles . If not brought up in students’ explanations, bring to their attention that it works for the two right triangles, Triangles J and K, but not for Triangle L. Then tell students that the Pythagorean Theorem says:

If , , and are the sides of a right triangle, where and are legs and  is the hypotenuse, then .