Lesson 3: Reasoning to Find Area
About this lesson
This lesson is the third of three lessons that use the following principles for reasoning about figures to find area:
 If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
 If a figure is composed from pieces that don't overlap, the sum of the areas of the pieces is the area of the figure. If a given figure is decomposed into pieces, then the area of the given figure is the sum of the areas of the pieces.
Following these principles, students can use several strategies to find the area of a figure. They can:
 Decompose it into shapes whose areas they can calculate.
 Decompose and rearrange it into shapes whose areas they can calculate.
 Consider it as a shape with one or more missing pieces, calculate the area of the shape, then subtract the areas of the missing pieces.
 Enclose it with a figure whose area they can calculate, consider the result as a region with missing pieces, and find its area using the previous strategy.
Use of these strategies involves looking for and making use of structure (MP7); explaining them involves constructing logical arguments (MP3). For now, rectangles are the only shapes whose areas students know how to calculate, but the strategies will become more powerful as students’ repertoires grow. This lesson includes one figure for which the “enclosing” strategy is appropriate, however, that strategy is not the main focus of the lesson and is not included in the list of strategies at the end.
Note that these materials use the “dot” notation (for example \(2 \cdot 3\)) to represent multiplication instead of the “cross” notation (for example \(2 \times 3\)). This is because students will be writing many algebraic expressions and equations in this course, sometimes involving the letter \(x\) used as a variable. This notation will be new for many students, and they will need explicit guidance in using it.
Lesson overview
 3.1 Warmup: Comparing Regions (5 minutes)

3.2 Activity: On the Grid (20 minutes)
 Includes "Are you Ready for More?" extension problem
 3.3 Activity: Off the Grid (15 minutes)
 Lesson Synthesis
 3.4 Cooldown: Maritime Flag (5 minutes)
Learning goals:
 Compare and contrast (orally) different strategies for calculating the area of a polygon.
 Find the area of a polygon by decomposing, rearranging, subtracting or enclosing shapes, and explain (orally and in writing) the solution method.
 Include appropriate units (in spoken and written language) when stating the area of a polygon.
Learning goals (student facing):
 Let’s decompose and rearrange shapes to find their areas.
Learning targets (student facing):
 I can use different reasoning strategies to find the area of shapes.
Required materials:
 copies of blackline master
 geometry toolkits
Required preparation:
 Make sure students have access to items in their geometry toolkits: tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles.
 For the warmup activity, prepare several copies of the pair of figures on the blackline master, in case students propose cutting them out to compare the areas.
Glossary:
 Access the complete Grade 6 glossary.
Standards:

This lesson builds on the standard:CCSS.3.MD.C.7.dMS.3.MD.7dMO.3.GM.C.14
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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