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Activity Synthesis

Select previously identified students to share what was the same and what was different about Elena’s and Tyler’s methods.

If not already mentioned by students, highlight the following points on how Elena’s and Tyler's approaches are the same, though do not expect students to use the language as written here. Clarify each point by gesturing, pointing, and annotating the images.

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 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.)

<p>6 parallelograms on a grid</p>

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.

Math Community
After the Warm-up, display the Math Community Chart with the “doing math” actions added to the teacher section for all to see. Give students 1 minute to review. Then share 2–3 key points from the teacher section and your reasoning for adding them. For example,

If “questioning vs. telling,” a shared reason could focus on your belief that students are capable mathematical thinkers and your desire to understand how students are making meaning of the mathematics.
If “listening,” a shared reason could be that sometimes you want to sit quietly with a group just to listen and hear student thinking and not because you think the group needs help or is off-track.

After sharing, tell students that they will have the opportunity to suggest additions to the teacher section during the Cool-down.

5.2

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR8: Discussion Supports

Materials

None

Activity Narrative

In this activity, students further develop their understanding of bases and heights of parallelograms by studying examples and non-examples and by analyzing statements. The goal is for students to see that in a parallelogram:

The term "base" refers to the length of one side and "height" to the length of a perpendicular segment between that side and the opposite side.
Any side of a parallelogram can be a base.
There are always two base-height pairs for a given parallelogram.

In the digital version of Are You Ready for More?, students use an applet to create a dynamic parallelogram with a height displayed. The applet allows students to see placements of height segments in a variety of parallelograms and when any side is chosen as a base. Consider allowing students to use the applet to check their responses to the last question (about whether the bases and heights in parallelograms A–E are correctly labeled) and to gain additional insights about base-height pairs.

Launch

Display the examples and non-examples of bases and heights for all to see. Read aloud the first paragraph of the activity and the description of each set of images. Give students a minute to observe the images. Then tell students to use the examples and non-examples to determine what is true about bases and heights in a parallelogram.

Arrange students in groups of 2. Give students 4–5 minutes to complete the first question with their partner. Ask them to pause for a class discussion after the first question. Select a student or a group to make a case for whether each statement is true or false. If one or more students disagree, ask them to explain their reasoning and discuss to reach a consensus. Before moving on to the next question, be sure students record the verified true statements so that they can be used as a reference later.

Give students 3 minutes of quiet time to answer the second question and another 2–3 minutes to share their responses with a partner. Ask them to focus partner conversations on the following questions, displayed for all to see:

How do you know the parallelogram is labeled correctly or incorrectly?
Is there another way a base and height could be labeled on this parallelogram?

After answering the questions, students with digital access can explore the applet ggbm.at/UnfbrN96 and use it to verify their responses and further their understanding of bases and heights.

MLR8 Discussion Supports. Pair gestures with verbal directions to clarify the meaning of any unfamiliar terms such as “dashed,” “horizontal,” “opposite,” or “perpendicular.” Advances: Listening, Representing

Activity

None

Here are some drawings of parallelograms. In each drawing, one side is labeled “base.”

In the first four drawings, each dashed segment represents a height that corresponds to the given base.

2 parallelograms with perpendicular heights provided

2 parallelograms with perpendicular heights provided<br>

In the next four drawings, each dashed segment does not represent a height that corresponds to the given base.

2 parallelograms where heights are not perpendicular to bases

2 parallelograms where heights are not perpendicular to bases

Select all the statements that are true about bases and heights in a parallelogram.
1. Only a horizontal side of a parallelogram can be a base.
2. Any side of a parallelogram can be a base.
3. A height can be drawn at any angle to the side chosen as the base.
4. A base and its corresponding height must be perpendicular to each other.
5. A height can only be drawn inside a parallelogram.
6. A height can be drawn outside of the parallelogram, as long as it is drawn at a 90-degree angle to the base.
7. A base cannot be extended to meet a height.
Five students labeled a base and a corresponding height for each of these parallelograms. Are all drawings correctly labeled? Explain how you know.

A

B

C

D

E

Activity Synthesis

Ask the class to give a quick agree-or-disagree signal on whether each figure in the last question is labeled correctly. After getting the responses for each figure, ask a student to explain how they know it is correct or incorrect.

If a parallelogram is incorrectly labeled, ask students where a correct height could be. If it is correctly labeled, ask students if there is another base and height that could be labeled on this parallelogram.

Before moving forward in this lesson, be sure that students understand which parallelograms are labeled correctly and emphasize the following points:

We can choose any side of a parallelogram as a base.
To find the height that corresponds to that base, draw a segment that joins the base and its opposite side. That segment has to be perpendicular to both the base and the opposite side.

Consider using the applet ggbm.at/UnfbrN96 to further illustrate possible base-height pairs and reinforce students' understanding of them.

Representation: Internalize Comprehension. Invite students to identify which details were most important in finding base-height pairs to solve the problem. Display the sentence frame, “The next time I need to find the base and height of a parallelogram, I will look for . . .“
Supports accessibility for: Language

5.3

Activity

Standards Alignment

Building On

Addressing

Instructional Routines

None

Materials

To Gather

Geometry toolkits

Activity Narrative

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?”

Launch

Keep students in groups of 2. Give students access to their geometry toolkits and 4–5 minutes of partner work time to complete the table. Ask them to be prepared to share their reasoning. Encourage students to use their work from earlier activities (on bases and heights) as a reference.

Activity

None

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.

Four parallelograms A--D.

parallelogram	base (units)	height (units)	area (sq units)
A
B
C
D
any parallelogram

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

Student Response

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Building on Student Thinking

Finding a height segment outside of the parallelogram may still be unfamiliar to students. Have examples from the “The Right Height?” activity 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 and that rotating our paper can help us see this. 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 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.

Student Lesson Summary

We can choose any side of a parallelogram as the base. Both the side selected (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!

Here are two copies of the same parallelogram.

2 copies of the same parallelogram.

On the left, the side that is the base is 6 units long. Its corresponding height is 4 units.

On the right, the side that is the base is 5 units long. Its corresponding height is 4.8 units.

For both, three different segments are shown to represent the height. We could draw in many more!

No matter which side is chosen as the base, the area of the parallelogram is the product of that base and its corresponding height. We can check this:

and

We can see why this is true by decomposing and rearranging the parallelograms into rectangles.

2 parallelograms, On left, base = 6, height = 4. On right, base = 5, height = 4 and 8 tenths.

Notice that the side lengths of each rectangle are the base and height of the parallelogram. Even though the two rectangles have different side lengths, the products of the side lengths are equal, so they have the same area! And both rectangles have the same area as does the parallelogram.

We often use letters to stand for numbers. If is a base of a parallelogram (in units), and is the corresponding height (in units), then the area of the parallelogram (in square units) is the product of these two numbers:

Notice that we write the multiplication symbol with a small dot instead of a symbol. This is so that we don’t get confused about whether means multiply, or whether the letter is standing in for a number.