Lesson 5

In this warm-up, students compare and contrast two ways of decomposing and rearranging a parallelogram on a grid such that its area can be found. It serves a few purposes: to reinforce the work done in the previous lesson; to allow students to practice communicating their observations; and to shed light on the features of a parallelogram that are useful for finding area—a base and a corresponding height.

The flow of key ideas—to be uncovered during discussion and gradually throughout the lesson—is as follows:

• There are multiple ways to decompose a parallelogram (with one cut) and rearrange it into a rectangle whose area we can determine.
• The cut can be made in different places, but to compose a rectangle, the cut has to be at a right angle to two opposite sides of the parallelogram.
• The length of one side of this newly composed rectangle is the same as the length of one side of the parallelogram. We use the term base to refer to this side.
• The length of the other side of the rectangle is the length of the cut we made to the parallelogram. We call this segment a height that corresponds to the chosen base.
• We use these two lengths to determine the area of the rectangle, and thus also the area of the parallelogram.

As students work and discuss, identify those who make comparisons in terms of the first two points so they could share later. Be sure to leave enough time to discuss the first four points as a class.

Ask a few students to share what was the same and what was different about the methods they observed.

Highlight the following points on how Elena and Tyler's approaches are the same, though do not expect students to use the language. Instead, rely on pointing and gesturing to make clear what is meant. If any of these are not mentioned by the students, share them.

• The rectangles are identical; they have the same side lengths. (Label the side lengths of the rectangles.)
• The cuts were made in different places, but the length of the cuts was the same. (Label the lengths along the vertical cuts.)
• The horizontal sides of the parallelogram have the same length as the horizontal sides of the rectangle. (Point out how both segments have the same length.)
• The length of each cut is also the distance between the two horizontal sides of the parallelogram. It is also the vertical side length of the rectangle. (Point out how that distance stays the same across the horizontal length of the parallelogram.)

Begin to connect the observations to the terms base and height. For example, explain:

• “The two measurements that we see here have special names. The length of one side of the parallelogram—which is also the length of one side of the rectangle—is called a base. The length of the vertical cut segment—which is also the length of the vertical side of the rectangle—is called a height that corresponds to that base.”
• “Here, the side of the parallelogram that is 7 units long is also called a base. In other words, the word base is used for both the segment and the measurement.”

Tell students that we will explore bases and heights of a parallelogram in this lesson.

In previous lessons, students reasoned about the area of parallelograms by decomposing, rearranging, and enclosing them and by using what they know about the area of rectangles. They also identified base-height pairs in parallelograms. Here, they use what they learned to find the area of new parallelograms, generalize the process, and write an expression for finding the area of any parallelogram.

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 our answer?”

In the launch for this activity, before giving students time to complete the first four rows of the table, display these images and the corresponding text. Then, ask students what they notice and wonder. Give students 3–5 minutes to share their thoughts. Highlight the idea that the base is one of the sides of the parallelogram and a corresponding height is a segment perpendicular to the line containing the base extending to the line containing the opposite side of the parallelogram.

The dashed segments do represent the height for the given bases in these parallelograms:

Keep students in groups of 2. Give students access to their geometry toolkits and 5–7 minutes of quiet think time to complete the first four rows of the table. Ask them to be prepared to share their reasoning. If time is limited, consider splitting up the work: have one partner work independently on parallelograms A and C, and the other partner on B and D. Encourage students to use their work from earlier activities (on bases and heights) as a reference.

Ask students to pause after completing the first four rows and to share their responses with their partner. Then, they should discuss how to write the expression for the area of any parallelogram. Students should notice that the area of every parallelogram is the product of a base and its corresponding height.

Finding a height segment outside of the parallelogram may still be a rather unfamiliar idea to students. Have examples from the “The Right Height?” section visible so they can serve as a reference in finding heights.

Students may say that the base of Parallelogram D cannot be determined because, as displayed, it does not have a horizontal side. Remind students that in an earlier activity we learned that any side of a parallelogram could be a base. Ask students to see if there is a side whose length can be determined. 

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 all answers for the first four rows are shared, discuss the following questions, displayed for all to see:

• “How did you determine 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 be the same as the base of the parallelogram. The height of the parallelogram is also the height of the rectangle—both are perpendicular to the base.)
• “Do you think this expression will always work?”

Be sure everyone has the correct expression for finding the area of a parallelogram by the end of the discussion. The second discussion question is meant to elicit connections to the parallelogram’s related rectangle as they decomposed and rearranged to find the area. The third question (about whether the expression will always work) is not meant to be proven here, so speculation on students’ part is expected at this point. It is intended to prompt students to think of other differently-shaped parallelograms beyond the four shown here.

This activity allows students to practice finding and reasoning about the area of various parallelograms—on and off a grid. Students need to make sense of the measurements and relationships in the given figures, identify an appropriate pair of base-height measurements to use, and 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 clearly explain how they found the area of each of the parallelograms.

Adjust the timing of this activity to 15 minutes.

Tell students to skip the question asking about the height of Parallelogram B that corresponds to the base that is 10 cm long. If time allows, include this question as part of the activity synthesis.

Some students may continue to use visual reasoning strategies (decomposition, rearranging, enclosing, and subtracting) to find the area of parallelograms. This is fine at this stage, but to help them gradually transition toward abstract reasoning, encourage them to try solving one problem both ways—using visual reasoning and their generalization about bases and heights from an earlier lesson. They can start with one method and use the other to check their work.

Use whole-class discussion to draw out three important points:

1. We need base and height information to help us 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 the base to the opposite side. Other measurements may not be as useful.
2. A parallelogram generally has two pairs of base and height. Both pairs produce the same area (it's the same parallelogram), so the product of pair of numbers should equal the product of the other pair.
3. Two parallelograms with different pairs of base and height can have the same area, as long as their products are equal. So a 3-by-6 rectangle and a parallelogram with base 1 and height 18 will have the same area because \(3 \boldcdot 6 = 1 \boldcdot 18\).

To highlight the first point, ask how students decided which measurements to use when calculating area.

• “When multiple measurements are shown, how did you know which of the measurements would help you find area?”
• “Which pieces of information in parallelograms B and C were not needed? Why not?”

To highlight the second point, select 1–2 previously identified students to share how they went about finding the missing height in the second question. Emphasize that the product \(8 \boldcdot 15\) 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 with equal area and an explanation of how they know the areas are equal. If not made explicit in students' explanations, stress that the base-height pairs must have the same product.