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Unit 1

Area of Triangles

Lesson 8

Area and Surface Area

Expressions and Equations

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Let’s use what we know about parallelograms to find the area of triangles.

Unit 1 ● Lesson 8

Learning

Goal

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Composing Parallelograms

Unit 1 ● Lesson 8 ● Activity 1

What do you notice? What do you wonder?

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Warm-up: Notice and Wonder

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Composing Parallelograms

Unit 1 ● Lesson 8 ● Activity 1

Han made a copy of Triangle M and composed three different parallelograms using the original M and the copy.

For each parallelogram Han composed, identify a base and a corresponding height, and write the measurements on the drawing.
Find the area of each parallelogram Han composed. Show your reasoning.

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Warm-up

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Composing Parallelograms

Unit 1 ● Lesson 8 ● Activity 1

Why do all the parallelograms have the same area even though they all have different shapes?
What do you notice about the bases and heights of the parallelograms?
How are the base-height measurements related to the right triangle?
Can we find the area of the triangle? How?

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Warm-up

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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More Triangles

Unit 1 ● Lesson 8 ● Activity 2

Find the areas of at least two of these triangles. Show your reasoning.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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More Triangles

Unit 1 ● Lesson 8 ● Activity 2

As you listen to others present their strategies, think about these questions.

Did anyone else reason the same way?
Did anyone else draw the same diagram but think about the problem differently?
Can this strategy be used on another triangle in this set? Which one?
Is there a triangle for which this strategy would not be helpful? Which one, and why not?

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Ask one student to identify the base, height, and area of each parallelogram, as well as how they reasoned about the area. If not already brought up by students in their explanations, discuss the following questions:

“Why do all parallelograms have the same area even though they all have different shapes?” (They are composed of the same parts—two copies of the same right triangles. They have the same pair of numbers for their base and height. They all call be decomposed and rearranged into a 6-by-4 rectangle.)
“What do you notice about the bases and heights of the parallelograms?” (They are the same pair of numbers.)
“How are the base-height measurements related to the right triangle?” (They are the lengths of two sides of the right triangles.)
“Can we find the area of the triangle? How?” (Yes, the area of the triangle is 12 square units because it is half of the area of every parallelogram, which is 24 square units.)

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Decomposing a Parallelogram

Unit 1 ● Lesson 8 ● Activity 3

Your teacher will give you two copies of a parallelogram. Glue or tape one copy of your parallelogram and find its area. Show your reasoning.
Decompose the second copy of your parallelogram by cutting along the dotted lines. Take only the small triangle and the trapezoid, and rearrange these two pieces into a different parallelogram. Glue or tape the newly composed parallelogram on your paper.
Find the area of the new parallelogram you composed. Show your reasoning.
What do you notice about the relationship between the area of this new parallelogram and the original one?
How do you think the area of the large triangle compares to that of the new parallelogram: Is it larger, the same, or smaller? Why is that?
Glue or tape the remaining large triangle to your paper. Use any part of your work to help you find its area. Show your reasoning.

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Decomposing a Parallelogram

Unit 1 ● Lesson 8 ● Activity 3

How many possible parallelograms can be created from each set of trapezoid and triangle?
Do they all yield the same area? Why or why not?
How does the area of the new parallelogram relate to the area of the original parallelogram?” (It is half the area of the original.) Why do you think that is?
Can the area of the large triangle be determined? How?

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Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Area of Triangles

Unit 1 ● Lesson 8

What can we say about the area of a triangle and that of a parallelogram with the same height?
In the second activity, we cut a triangle along a line that goes through the midpoints of two sides and rearranged the pieces into a parallelogram. What did we notice about the area and the height of the resulting parallelogram?
How might we start finding the area of any triangle, in general?

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Lesson Synthesis

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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Unit 1 ● Lesson 8

I can use what I know about parallelograms to reason about the area of triangles.

Learning Targets

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An Area of 14

Unit 1 ● Lesson 8 ● Activity 4

Elena, Lin, and Noah all found the area of Triangle Q to be 14 square units but reasoned about it differently, as shown in the diagrams. Explain at least one student’s way of thinking and why his or her answer is correct.

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Cool-down

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

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This slide deck is copyright 2020 by Kendall Hunt Publishing, https://im.kendallhunt.com/, and is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0), https://creativecommons.org/licenses/by-nc/4.0/.

All curriculum excerpts are under the following licenses:

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

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