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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:
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.
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, base = 6 units. Corresponding height = 4 units. On the right, base = 5 units. Corresponding height = 4.8 units. For both, 3 different segments are shown to represent the height.
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.
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
Notice that we write the multiplication symbol with a small dot instead of a
Help us improve by sharing suggestions or reporting issues.
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:
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.
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.
In the next four drawings, each dashed segment does not represent a height that corresponds to the given base.
Select all the statements that are true about bases and heights in a parallelogram.
Students might say that Parallelogram E is correctly labeled because the labeled sides remind them of the labeled length and width of a rectangle. Ask students to revisit the true statements about base-height pairs and see if those conditions are met in Parallelogram E.
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.
Building Toward