Lesson 5: Bases and Heights of Parallelograms
About this lesson
Students begin this lesson by comparing two strategies for finding the area of a parallelogram. This comparison sets the stage both for formally defining the terms base and height and for writing a general formula for the area of a parallelogram. Being able to correctly identify a baseheight pair for a parallelogram requires looking for and making use of structure (MP7).
The terms base and height are potentially confusing because they are sometimes used to refer to particular line segments, and sometimes to the length of a line segment or the distance between parallel lines. Furthermore, there are always two baseheight pairs for any parallelogram, so asking for the base and the height is not, technically, a wellposed question. Instead, asking for a base and its corresponding height is more appropriate. As students clarify their intended meaning when using these terms, they are attending to precision of language (MP6). In these materials, the words “base” and “height” mean the numbers unless it is clear from the context that it means a segment and that there is no potential confusion.
By the end of the lesson, students both look for a pattern they can generalize to the formula for the area of a rectangle (MP8) and make arguments that explain why this works for all parallelograms (MP3).
Lesson overview

5.1 Warmup: A Parallelogram and Its Rectangles (10 minutes)
 There is a digital applet in this activity.

5.2 Activity: The Right Height? (20 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.

5.3 Activity: Finding the Formula for Area of Parallelograms (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 5.4 Cooldown: Parallelograms S and T (5 minutes)
Learning goals:
 Comprehend the terms “base” and “height” to refer to one side of a parallelogram and the perpendicular distance between that side and the opposite side.
 Generalize (orally) a process for finding the area of a parallelogram, using the length of a base and the corresponding height.
 Identify a base and the corresponding height for a parallelogram, and understand that there are two different baseheight pairs for any parallelogram.
Learning goals (student facing):
 Let’s investigate the area of parallelograms some more.
Learning targets (student facing):
 I can identify pairs of base and height of a parallelogram.
 I can write and explain the formula for the area of a parallelogram.
 I know what the terms "base" and "height" refer to in a parallelogram.
Required materials:
 geometry toolkits
Glossary:
 base (of a parallelogram or triangle)  We can choose any side of a parallelogram or triangle to be the shape’s base. Sometimes we use the word base to refer to the length of this side.

height (of a parallelogram or triangle)  The height is the shortest distance from the base of the shape to the opposite side (for a parallelogram) or opposite vertex (for a triangle).
We can show the height in more than one place, but it will always be perpendicular to the chosen base.
 Access the complete Grade 6 glossary.
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 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/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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