standard_description: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
problem_activity: Activity
This activity gives students opportunities to find areas of regions using a variety of strategies. When working with a grid, students may start by counting squares, as they had done in earlier grades. However, the figures have been chosen to elicit the strategies listed in the Lesson Narrative.
Figure A can be easily decomposed into rectangles.
Figure B can be decomposed into rectangles. Or, more efficiently, it can be seen as a square with a missing piece, and the area of the inner unshaded square can be subtracted from the area of the larger square.
Figure C can also be seen as having a missing piece, but subtracting the area of the unshaded shape does not work because the side lengths of the inner square are unknown. Instead, the shaded triangles can be decomposed and rearranged into rectangles.
Figure D can be decomposed and rearranged into rectangles. It can also be viewed as the inner square of Figure C.
As students work, identify students who use these strategies and can illustrate or explain them clearly. Ask them to share later. Look for at least two strategies being used for each figure (one strategy as shown in the Student Response and at least one other).
Launch
Tell students that they will find the areas of various figures on a grid. To encourage students to use a more grade-appropriate strategy for finding areas, show them a strategy from earlier grades. As a class, find the area of Figure A by counting the squares one by one aloud. Confirm that there are 24 square units, and then ask students to think about other ways to find the area of Figure A and other figures besides counting each square.
Arrange students in groups of 2. Ask one partner to start with Figures A and C, and the other with B and D. Give students 4–5 minutes of quiet think time, and provide access to their geometry toolkits. Then, give them a few minutes to share their responses with their partner. Emphasize that as one partner explains, the other should listen carefully and see if they agree or disagree with the answer and explanations.
Student Facing
Each grid square is 1 square unit. Find the area, in square units, of each shaded region without counting every square. Be prepared to explain your reasoning.
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Student Response
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Student Facing
Are you ready for more?
Rearrange the triangles from Figure C so they fit inside Figure D. Draw and color a diagram of your work.
Student Response
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Anticipated Misconceptions
Some students may count both complete and partial grid squares instead of looking for ways to decompose and rearrange larger shapes. Ask them if they can find a way to to find the area by decomposing and rearranging larger pieces. The discussion at the end, during which everyone sees a variety of strategies, is especially important for these students.
Activity Synthesis
Discussion should center around how different strategies—decomposing, rearranging, subtracting, and enclosing—are used to find area. For each figure, select two students with different strategies to share their work, if possible. Sequence students’ presentations so that, for each figure, a subtracting strategy comes last, as that is typically the most challenging.
Before sharing begins, explain to students that they should notice similarities and differences in the strategies shared and be ready to explain them. As students share their strategies, consider recording the moves on each figure for all to see. After each person shares, name the strategy, and poll the class to see if anyone else reasoned the same way. If one of these strategies does not appear in students’ work, illustrate it for the class.
Decomposing (A and B)
Decomposing and rearranging (C and D)
Subtracting (B)
Enclosing, then subtracting (D)
If time permits, give partners a minute to talk to their partner about the similarities and differences they saw in the strategies used to find the areas of the four figures. Consider displaying sentence starters such as:
The strategies used to find the areas of figures ____ and ____ are alike in that. . .
The strategies used to find the areas of figures ____ and ____ are different in that. . .
Representation: Internalize Comprehension.
Use color and annotations to highlight distinctions between strategies. Create a display that includes multiple copies of each figure. As students describe their strategies, use color and annotation to scribe their thinking so that it is visible for all students. Be sure to label each figure with the strategy described (decomposing, rearranging, subtracting, or enclosing).
Supports accessibility for: Visual-spatial processing; Conceptual processing
Speaking: Math Language Routine 3 Clarify, Critique, Correct.
This is the first time Math Language Routine 3 is suggested as a support in this course. In this routine, students are given an incorrect or incomplete piece of mathematical work. This may be in the form of a written statement, drawing, problem-solving steps, or another mathematical representation. Pairs of students analyze, reflect on, and improve the written work by correcting errors and clarifying meaning. Typical prompts are “Is anything unclear?” or “Are there any reasoning errors?” The purpose of this routine is to engage students in analyzing mathematical thinking that is not their own, and to solidify their knowledge through communicating about conceptual errors and ambiguities in language.
Design Principle(s): Maximize meta-awareness
How It Happens:
In the class discussion for Figure C of this activity, provide the following “draft” explanation:
“I cut a square in the middle, then I saw that it was a bunch of triangles, so then I figured those out to get my answer.”
Prompt students to identify the ambiguity of this response. Ask students, “What do you think this person is trying to say? What is unclear? Did the author use any of the strategies we’ve been using to find area?”
Give students 1 minute of individual time to respond to the questions in writing, and then 3 minutes to discuss with a partner.
As pairs discuss, provide these sentence frames for scaffolding: “I think the author is trying to use the strategy _ because.…”, “I think what the author meant by ‘figured it out’ was….”, and “The part that is most unclear to me is … because.…”. Encourage the listener to press for detail by asking follow-up questions to clarify the intended meaning of the statement. Allow each partner to take a turn as the speaker and listener.
Invite students to improve the “draft” response using the target vocabulary and structures.
The targeted vocabulary includes the names of the strategies (decomposing, rearranging, subtracting, enclosing, or any combination of them), and other terms from this lesson such as “alike,” “different,” “area,” and “polygon.”
The targeted structures of the response should include an explanation of each step, order/time transition words (first, next, then, etc.), and/or reasons for decisions made during steps.
Here is one example of an improved response:
“First, I cut out the square in the middle of the shaded regions. Next, I noticed that all the shaded regions were a bunch of right triangles. Then, I rearranged the triangles into rectangles. Lastly, I figured out the area of each rectangle and added them together to get my answer.”
Ask each pair of students to contribute their improved response to a poster, the whiteboard, or digital projection. Call on 2–3 pairs of students to present their response to the whole class, and invite the class to make comparisons among the responses shared and their own responses.
Listen for responses that include the strategy of decomposing and then rearranging the triangles. Emphasize that this is one way of figuring out the area of the shaded region.
Share one more improved response, discuss to reach a general understanding, and then move on to Figure D.


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