# Lesson 5

Areas of Parallelograms

## 5.1: A Parallelogram and Its Rectangles (10 minutes)

### Warm-up

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.

Teacher Notes for IM 6–8 Accelerated
Adjust the timing of this activity to 5 minutes.

### Launch

Arrange students in groups of 2. Give students 2 minutes of quiet think time and access to geometry toolkits. Ask them to share their responses with a partner afterwards.

If using the digital activity, have students explore the applet for 3 minutes individually, and then discuss with a partner. Students will be able to see the cuts by dragging the point on the segment under the parallelogram.

### Student Facing

Elena and Tyler were finding the area of this parallelogram:

Move the slider to see how Tyler did it:

Move the slider to see how Elena did it:

How are the two strategies for finding the area of a parallelogram the same? How they are different?

### Launch

Arrange students in groups of 2. Give students 2 minutes of quiet think time and access to geometry toolkits. Ask them to share their responses with a partner afterwards.

If using the digital activity, have students explore the applet for 3 minutes individually, and then discuss with a partner. Students will be able to see the cuts by dragging the point on the segment under the parallelogram.

### Student Facing

Elena and Tyler were finding the area of this parallelogram:

Here is how Elena did it:

Here is how Tyler did it:

How are the two strategies for finding the area of a parallelogram the same? How they are different?

### Activity Synthesis

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.

## 5.2: Finding the Formula for Area of Parallelograms (15 minutes)

### Activity

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?”
Teacher Notes for IM 6–8 Accelerated

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:

### Launch

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.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. As students describe their thinking, highlight the base-height pairs on each parallelogram and record the responses in the table.
Supports accessibility for: Visual-spatial processing; Conceptual processing

### Student Facing

For each parallelogram:

• Identify a base and a corresponding height, and record their lengths in the table.
• Find the area of the parallelogram and record it in the last column of the table.
 parallelogram base (units) height (units) area (sq units) A B C D any parallelogram $$b$$ $$h$$

In the last row, write an expression for the area of any parallelogram, using $$b$$ and $$h$$ .

### Student Facing

#### Are you ready for more?

1. What happens to the area of a parallelogram if the height doubles but the base is unchanged? If the height triples? If the height is 100 times the original?
2. What happens to the area if both the base and the height double? Both triple? Both are 100 times their original lengths?

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

Representing, Conversing: MLR7 Compare and Connect. When students share their responses for the first four rows of the table, ask students to identify what is the same and what is different about how they determined the base and height for each parallelogram. Listen for and amplify student discussions that attempt to explain why their different approaches led to the same area. Then ask students if any side of a parallelogram can be used as the base. This will help students understand how the area of every parallelogram is product of the base and its corresponding height.
Design Principle(s): Support sense-making

## 5.3: More Areas of Parallelograms (25 minutes)

### Activity

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.

Teacher Notes for IM 6–8 Accelerated

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.

### Launch

Arrange students in groups of 4. Give each student access to their geometry toolkits and 5 minutes of quiet time to find the areas of the parallelograms in the first question. Then, assign each student one parallelogram (A, B, C or D). Ask each student to explain to the group, one at a time, how they found the area of the assigned parallelogram. After each student shares, check for agreement or disagreement from the rest of the group. Discuss any disagreement and come to a consensus on the correct answer before moving to the next parallelogram.

Afterwards, give students another 5–7 minutes of quiet work time to complete the rest of the activity.

For classrooms using the digital activity, arrange students in groups of 2. Ask each student to explain to their partner how they found the area of each parallelogram. When using the second applet, each student should each find one pair of quadrilaterals with equal area.

Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students describe how they calculated the area of each parallelogram, use color and annotations to scribe their thinking on a display. Ask students how they knew which measurements to use, and label each base and height accordingly.
Supports accessibility for: Visual-spatial processing; Conceptual processing

### Student Facing

1.

Calculate the area of the given figure in the applet. Then, check if your area calculation is correct by clicking the Show Area checkbox.

2. Uncheck the Area checkbox. Move one of the vertices of the parallelogram to create a new parallelogram. When you get a parallelogram that you like, sketch it and calculate the area. Then, check if your calculation is correct by using the Show Area button again.
3. Repeat this process two more times. Draw and label each parallelogram with its measurements and the area you calculated.
1. Here is Parallelogram B. What is the corresponding height for the base that is 10 cm long? Explain or show your reasoning. 2. Here are two different parallelograms with the same area.

1. Explain why their areas are equal.
2. Drag points to create two new parallelograms that are not identical copies of each other but that have the same area as each other. Sketch your parallelograms and explain or show how you know their areas are equal. Then, click on the Check button to see if the two areas are indeed equal.

### Launch

Arrange students in groups of 4. Give each student access to their geometry toolkits and 5 minutes of quiet time to find the areas of the parallelograms in the first question. Then, assign each student one parallelogram (A, B, C or D). Ask each student to explain to the group, one at a time, how they found the area of the assigned parallelogram. After each student shares, check for agreement or disagreement from the rest of the group. Discuss any disagreement and come to a consensus on the correct answer before moving to the next parallelogram.

Afterwards, give students another 5–7 minutes of quiet work time to complete the rest of the activity.

For classrooms using the digital activity, arrange students in groups of 2. Ask each student to explain to their partner how they found the area of each parallelogram. When using the second applet, each student should each find one pair of quadrilaterals with equal area.

Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students describe how they calculated the area of each parallelogram, use color and annotations to scribe their thinking on a display. Ask students how they knew which measurements to use, and label each base and height accordingly.
Supports accessibility for: Visual-spatial processing; Conceptual processing

### Student Facing

2. In Parallelogram B, what is the corresponding height for the base that is 10 cm long? Explain or show your reasoning.
3. Two different parallelograms P and Q both have an area of 20 square units. Neither of the parallelograms are rectangles.

On the grid, draw two parallelograms that could be P and Q.

### Student Facing

#### Are you ready for more?

Here is a parallelogram composed of smaller parallelograms. The shaded region is composed of four identical parallelograms. All lengths are in inches.

What is the area of the unshaded parallelogram in the middle? Explain or show your reasoning.

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion when students explain how they created two parallelograms with equal area. For each explanation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. If students are not able to restate, they should ask for clarification. Call students' attention to any words or phrases that helped to clarify the original statement, such as area, product, base, or height. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making; Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

In this lesson, we identified a base and a corresponding height in a parallelogram, and then wrote an algebraic expression for finding the area of any parallelogram.

• “How do you decide the base of a parallelogram?” (Any side can be a base. Sometimes one side is preferable over another because its length is known or easy to know.)
• “Once we have chosen a base, how can we identify a height that corresponds to it?” (Identify a perpendicular segment that connects that base and the opposite side. Find the length of that segment.)
• “Do parallelograms that have the same base and height always look the same?” (No.) “Can you show an example?”
• “In how many ways can we identify a base and a height for a given parallelogram?” (There are two possible bases. For each base, many possible segments can represent the corresponding height.)
• “What is the relationship between the base and height of a parallelogram and its area?” (The area is the product of base and height.)

## Student Lesson Summary

### Student Facing

In this lesson, we learned about 2 important parts of parallelograms, the base and the height.

• We can choose any of the four sides of a parallelogram as the base. Both the side (the segment) and its length (the measurement) are called the base.
• If we draw any perpendicular segment from a point on the base to the opposite side of the parallelogram, that segment will always have the same length. We call that value the height. There are infinitely many segments that can represent the height!

Any pair of base and corresponding height can help us find the area of a parallelogram, but some base-height pairs are more easily identified than others.

We often use letters to stand for numbers. If $$b$$ is the length of a base of a parallelogram (in units), and $$h$$ is the length of the corresponding height (in units), then the area of the parallelogram (in square units) is the product of these two numbers, $$b \boldcdot h$$. Notice that we write the multiplication symbol with a small dot instead of a $$\times$$ symbol. This is so that we don’t get confused about whether $$\times$$ means multiply, or whether the letter $$x$$ is standing in for a number.

When a parallelogram is drawn on a grid and has horizontal sides, we can use a horizontal side as the base. When it has vertical sides, we can use a vertical side as the base. The grid can help us find (or estimate) the lengths of the base and of the corresponding height.

In this parallelogram, the corresponding height for the side that is 10 units long is not given, but the height for the side that is 8 units long is given. This base-height pair can help us find that the area is 64 square units since $$8 \boldcdot 8 = 64$$.