This activity elicits different strategies for reasoning about finding the area of regions:

• Decomposing
• Decomposing and rearranging
• Subtracting
• Enclosing and subtracting

The figures are on a grid, which reinforces the meaning of area and supports students in quantifying the square units. Students may start by counting squares, as they had done in earlier grades, but the figures have been chosen to encourage other approaches. 

Monitor for students who use at least two different strategies for finding the area of each figure (one strategy as shown in the student responses and at least one other). Here are some approaches students might take for each figure:

• Figure A can be easily decomposed into rectangles.
• Figure B can be decomposed into rectangles. Or, more efficiently, it can be seen as a square with a missing piece. The area of the unshaded inner square can be subtracted from the area of the larger square.
• Figure C can also be seen as having a missing piece, but subtracting the area of the unshaded shape does not work because the side lengths of the inner square are unknown. Instead, the shaded triangles can be decomposed and rearranged into rectangles.
• Figure D can be decomposed and rearranged into rectangles. It can also be viewed as the inner square of Figure C.

The purpose of this discussion is to highlight a few key strategies for finding area. Invite previously selected students to share their work. Before sharing begins, ask students to notice similarities and differences in the strategies and be ready to explain them. Sequence the discussion of the strategies for each figure as shown here, starting with decomposing.

• Figure A: 
  • decomposing
  • enclosing and subtracting
• Figure B: 
  • decomposing
  • subtracting
• Figure C:
  • decomposing and rearranging
  • subtracting (after finding the area of the inner square, which is Figure D)
• Figure D:
  • decomposing and rearranging
  • enclosing and subtracting

As students share their strategies, record and display their moves on each figure for all to see (or ask the presenters to do so). 

After each student shares, name the strategy and ask if anyone else reasoned the same way. If one of these strategies does not appear in students’ work, illustrate it for the class.

Connect the different responses to the learning goals by asking questions such as:

• “How are the strategies used to find the areas of figures A and D alike?” (We can find the areas by decomposing the figures.)
•  “How are the strategies used to find the areas of figures A and D different?” (After decomposing A, we can find the area of each piece as they are. After decomposing D, we need to rearrange the pieces before finding their areas.)
• “After decomposing figures C and D, what shapes were the pieces rearranged into?” (rectangles) “Why might that be?” (We know how to find the area of rectangles.)

Highlight that there are multiple ways to find the area of each figure and that decomposing, with or without rearranging, is generally a useful strategy.

In this activity, students apply the strategies they have learned to find the areas of figures, but now the figures are not on a grid.

• Figure E can easily be decomposed and rearranged into a rectangle.
• Figure F can be decomposed and rearranged into rectangles (as was done with Figure C in the previous activity). Students cannot use the strategy of subtracting the area of the inner square from that of the outer square because the side lengths of the inner square are unknown.

As students discuss their approaches in groups, support students in naming the strategies and by asking clarifying questions. Monitor for students who observed that the same strategies for reasoning about area can be applied both on and off the grid. 

For Figure F, students may estimate the side lengths of the inner square so that its area could be subtracted from that of the outer square. They may struggle to see how the triangles could be rearranged. Suggest that they use tracing paper to help them in their thinking.

Students might not be familiar with the symbols that indicate right angles and might think these symbols indicate square units. Remind them that those symbols indicate 90 degree angles.

Briefly discuss the question, “Which strategies are similar to the ones you used in an earlier activity to find the areas of figures on a grid?”

Select 1–2 previously identified students to share their observations—that in both activities they decomposed and rearranged figures to find their area. (Some students may have also enclosed Figure A and subtracted the areas of unshaded triangles after rearranging them into a rectangle.) 

Emphasize that these strategies for finding area can be used whether the measurements are indicated by a grid or given directly (without a grid).

If students did not include appropriate area units for figures E and F, remind them that the side lengths of these figures are given in centimeters, so their areas are in square centimeters.

If time permits, ask students: 

• “For Figure F, can we use the strategy of subtracting the area of the inner square to find the area of the shaded region? Why or why not?” (No, because we don’t know the side lengths of the inner square or its area.) 
• Display or sketch Figure F with the sides of the inner square labeled “5 cm.”

  

  “Can we use subtraction to find the area of the shaded region now? Why or why not? (Yes, because we know the side lengths of the outer and inner squares. We can find the area of the smaller square, 25 square centimeters, and subtract it from the area of the larger square, 49 square centimeters, which gives 24 square centimeters.)