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Geometry toolkits
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. This work allows students to practice communicating their observations and prompts them to notice features of a parallelogram that are useful for finding area—a base and its corresponding height.
The flow of key ideas—to be uncovered during discussion and gradually throughout the lesson—is as follows:
As students work and discuss, identify those who recognize that both Elena and Tyler decomposed the parallelogram by making a cut that is perpendicular to one side and then rearranged the pieces into a rectangle. Ask them to share their observations later. Be sure to leave enough time to discuss the first four key ideas as a class.
In the digital version of the warm-up, students use applets to animate the moves that Elena and Tyler made (decomposing and rearranging) to find the area of the parallelogram.
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 afterward.
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?
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.
Begin to connect the observations to the terms “base” and “height.” For example, explain:
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,
After sharing, tell students that they will have the opportunity to suggest additions to the teacher section during the Cool-down.
To Gather
Geometry toolkits
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:
The Narrative refers to previous lessons about base-height pairs which are not included in this course. To 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, display these images and corresponding text for all to see during the Launch. Then ask students what they notice and wonder.
In the first four drawings, each dashed segment represents a height that corresponds to the given base.
In the next four drawings, each dashed segment does not represent a height that corresponds to the given base.
Students may not need 4–5 minutes of partner work time to complete the table.
In Building Student Thinking, the referenced activity “The Right Height?” is not included in this course. If necessary, refer to the examples and non-examples added to the 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.
For each parallelogram:
| 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
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:
Be sure everyone has the correct expression for finding the area of a parallelogram by the end of the discussion.
Any pair of a base and a corresponding height can help us find the area of a parallelogram.
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!
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.
Two parallelograms drawn on two grids. First parallelogram, 2 horizontal sides each 8 units long, 2 slanted sides that rise 2 vertical units over 4 horizontal units. Bottom horizontal side labeled, b. A 2-unit perpendicular segment labeled, h, connects the horizontal sides. Second parallelogram, 2 vertical sides each 6 units long, 2 slanted sides that rise 4 vertical units over 4 horizontal units. The left vertical side is labeled, b. A 4-unit perpendicular segment labeled, h, connects one vertex of the vertical side to a point on the other vertical side.
When a parallelogram is not drawn on a grid, we can still find its area if we know a base and a corresponding height.
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 often use letters to stand for numbers. If
Notice that we write the multiplication symbol with a small dot instead of a
Parallelograms that have the same base and the same height will have the same area; the product of the base and height will be equal. Here are 4 different parallelograms with the same pair of base-height measurements.
Help us improve by sharing suggestions or reporting issues.
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.
Arrange students in groups of 4. Give each student access to their geometry toolkit 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 the students to take turns explaining to the group how they found the area of their assigned parallelogram. Explain that while one group member explains, the others should listen and make sure they agree. If they don’t agree, they should discuss their thinking and work to reach an agreement before moving to the next parallelogram.
Afterward, give students another 5–7 minutes of quiet work time to complete the rest of the activity.
Here are two different parallelograms with the same area. Explain why their areas are equal.
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. Explain how you know.
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 using their generalization about bases and heights from an earlier lesson. They can start with one method and use the other to check their work.