In this activity, students determine the total area of five shapes. How students determine the area of the shapes is left open-ended on purpose (MP1). 

Monitor for groups who calculate the area of each shape individually, or fit the shapes into the squares and then calculate the area. It is possible to fit all five shapes into the two smaller squares or into the one larger square. 

These shapes are part of a transformations-based proof of the Pythagorean Theorem that students will see in a future lesson. The five cut-out shapes in this activity can be reserved at the end of the activity for future use.

In the digital version of the activity, students use an applet to determine the total area of five shapes. The applet allows students to manipulate the shapes to fit them inside of squares with known areas. This activity works best when each student has access to the applet because preparation of paper manipulatives may be time-consuming. If students don't have individual access, displaying the applet for all to see would be helpful during the Activity Synthesis.

The purpose of this discussion is to prime student thinking about strategies for calculating area, particularly the use of rearranging and composition, in preparation for future activities. Ask groups previously identified to share their strategies. 

If not brought up during the discussion, ask students how they know the total area is 25 square units. Make sure they understand that if the shapes fit in a square perfectly without overlaps or gaps, then the combined area of the shapes must be the same as the area of the square. 

The purpose of this activity is for students to find the areas of squares whose side lengths are not easily determined by inspection. The squares are represented in an increasingly abstract way: first showing all of the square units, next showing only the grid units along the edges, and finally providing labels only for the side lengths. This sequence provides students with practice decomposing and rearranging shapes, which will help them understand and explain a proof of the Pythagorean Theorem in a later lesson.

After students find the area of the square in the first problem, pause for a class discussion about strategies used. In particular, students should see that the side length and area of the larger square (with sides that lie along grid lines surrounding the shaded “tilted” square) can easily be found by counting (MP7).

Monitor for students who find the area of the first shaded square by:

• Composing two triangles to make a rectangle and then subtracting the area of two rectangles from the larger surrounding square.

• Using the formula for the area of a triangle and then subtracting the area of four triangles from the larger surrounding square.

Make these strategies explicit to help students who are only able to count unit squares make a successful transition to the third square. 

The goal of this discussion is to emphasize the strategy of decomposing the larger surrounding square and subtracting the area of the smaller triangles, either by rearranging the triangles into rectangles or by finding the areas of the triangles using the triangle area formula. 

Select 2–3 students to explain how they found the area of the third square. Here are some questions for discussion:

• “Why did you choose that particular strategy?”

• “Did anyone solve the problem the same way but would explain it differently?”