The purpose of this activity is for students to work through an area-based algebraic proof of the Pythagorean Theorem (MP1). Figure G was first encountered by students in an earlier unit on transformations. 

While there are many proofs of the Pythagorean Theorem similar to the one in this activity, they often rely on , which is beyond the scope of grade 8. For this proof, students reason about the areas of the two squares with the same dimensions. Each square is divided into smaller regions in different ways. Using the equality of the total area of each square, they uncover the Pythagorean Theorem. The extension uses this same division to solve a challenging area composition and decomposition problem.

The goal of this discussion is to make sure students understand this area-based algebraic proof of the Pythagorean Theorem. Display the image from the Task Statement for all to see.

First of two squares of the same area. This square is divided into the following: A square with side lengths “a”. Two rectangles with side lengths “a” and “b”. A square with side lengths “b”.

Second of two squares of the same area. This square is divided into the following: Four identical triangles on each corner of the square with sides labeled “a” and “b”. A square in the center with unlabeled side lengths.

Select 2–3 groups to share their work for the third question. Make sure the last group presenting concludes with or something close enough that the class can get there with a little prompting. For example, if groups are stuck with the equation looking something like , encourage them to try and combine like terms and remove any quantities both sides have in common.

Here are some questions for discussion:

• “How do you see regions from one figure matching regions in the other figure?” (The two small squares in Figure F match the one large square in Figure G.)

• “How do the rectangles and triangles match?” (The area of the two rectangles is the same as the area of four of the triangles, since two of the triangles together make a rectangle that is wide and long.)

If needed, show students an image with the diagonals added in, such as the one shown here, to help make the connection between the two figures clearer.

Second of two squares of the same area. This square is divided into the following: Four identical triangles on each corner of the square with sides labeled “a” and “b”. A square in the center with unlabeled side lengths.

Note how these figures can be made for any right triangle with legs and and hypotenuse .

The purpose of this activity is to practice using the Pythagorean Theorem. Prior to this lesson, students could only find the length of a segment by using a grid. The Pythagorean Theorem now makes it possible to find the length of any segment that is a side of a right triangle.

The goal of this discussion is to make sure students understand how to use the Pythagorean Theorem when finding a leg versus when finding the hypotenuse. Students will have more practice using the theorem in future lessons. 

Before students share their methods for finding the side length of a right triangle, present an incorrect solution based on a common error you observe in the class. For example, “I know that , , and , so when I use the Pythagorean Theorem, I get the equation .

This equation simplifies to .

When I solve for , I get .” 

Ask students to identify the error and revise the original statement. Remind students that in the Pythagorean Theorem, and represent the squares of the legs of the right triangle, and represents the square of the hypotenuse.

Invite a few students to share their reasoning with the class for each unknown side length. As students share, record their steps for all to see, clearly showing the initial setup with .

In this optional activity, students explore a transformations-based proof of the Pythagorean Theorem. Since this proof is not one students are expected to derive on their own, the focus is on understanding why the steps are possible from a transformations perspective.  

In the digital version of the activity, students use an applet to compare the areas of two smaller squares to the area of one larger square. The applet allows students to decompose the two smaller squares into smaller pieces and quickly rearrange them. This activity works best when each student has access to the applet and when having students cut shapes from the handout may take more time than is available. If students don't have individual access, displaying the applet for all to see would be helpful during the synthesis.

The goal of this discussion is for students to see that the sum of the areas of the 2 smaller squares is congruent to the area of the larger square. Invite several groups to share their answers to the last question.