In this activity, students find the area of more parallelograms, generalize the process, and write an expression for finding the area of any parallelogram. To do so, they apply what they learned in previous lessons about base-height pairs in parallelograms and about strategies for reasoning about area.

As students discuss their work, monitor conversations for any disagreements between partners. Support them by asking clarifying questions:

• “How did you choose a base? How can you be sure that is the height?”
• “How did you find the area? Why did you choose that strategy for this parallelogram?”
• “Is there another way to find the area and to check your answer?”

Display the parallelograms and the table for all to see. Select a few students to share the correct answers for each parallelogram. As students share, highlight the base-height pairs on each parallelogram and record the responses in the table. Although only one base-height pair is named for each parallelogram, reiterate that there is another pair. Show the second pair on the diagram or ask students to point it out.

After the first four rows of the table are completed, discuss the expression in the last row. Ask students:

• “How did you figure out the expression for the area for any parallelogram?” (The areas of Parallelograms A–D are each the product of base and height.)
• “Suppose you decompose a parallelogram with a cut and rearrange it into a rectangle. Does this expression for finding area still work? Why or why not?” (Yes. One side of the rectangle will have the same length as the base of the parallelogram. The height of the parallelogram is also the height of the rectangle—both are perpendicular to the base.)

Be sure everyone has the correct expression for finding the area of a parallelogram by the end of the discussion.

This activity allows students to practice finding and reasoning about the area of various parallelograms—on and off a grid. Students make sense of the measurements and relationships in the given figures and identify an appropriate pair of base-height measurements to use (the length of a side and that of a segment that is perpendicular to that side). Students learn to recognize that two parallelograms with the same base-height measurements (or with different base-height measurements but the same product) have the same area.

As they work individually, notice how students determine base-height pairs to use. As they work in groups, listen to their discussions and identify those who can explain how they found the area of each of the parallelograms.

In the digital version of the activity, students use an applet to create two parallelograms with the same area.

Use whole-class discussion to highlight three important points:

• We need base and height information to calculate the area of a parallelogram, so we generally look for the length of one side and the length of a perpendicular segment that connects that side to the opposite side. Other measurements may not be as useful.
• A parallelogram has two pairs of base and corresponding height. Both pairs produce the same area, so the product of one pair of numbers should equal the product of the other pair.
• Two parallelograms with different pairs of base and corresponding height can have the same area, as long as their products are equal. A 3-by-6 rectangle and a parallelogram with a base of 1 and a height of 18 will have the same area because .

To highlight the first point, consider asking:

• “The parallelograms in the first question show multiple measurements. How did you know which ones would help you find the area?”
• “Which pieces of information in Parallelograms B and C were not needed? Why not?”

To highlight the second point, select 1 or 2 students to share how they found the missing height in the second question. Emphasize that the product of 815 and that of 10 and the unknown h must be equal because both give us the area of the same parallelogram.

To highlight the last point, invite a few students to share their pair of parallelograms and how they know that the areas are equal. If not made explicit in students' explanations, stress that the base-height pairs must have the same product. Consider displaying the applet for all to see and using it to facilitate this discussion.