##
* *Reflection: Trust and Respect
Strategies for Decomposing 2-D Figures - Section 1: Warm-Up: Area Decomposition

This is one of my favorite Warm-Ups of the school year because there are so many ways students can attack the problem. The whole-class conversation is rich because there is no one right answer. This is the kind of problem that forces students to see the answer is the least important part of the problem since the goal is to have a strategy for how to think about decomposing a 2-D shape, not to calculate an area.

I made sure to circulate the room as students worked on the warm-up to get a sense of the strategies that emerged. Making time to observe students working gave me a sense of who I might call on to share out in the whole class discussion. By inviting several students to share their work under the document camera and pausing to check in—“who else thought of this problem this way?”—I validated common approaches while showing the class there were still other approaches in the class.

Something I decided to do this year was to ask students “what surprised you in talking about this warm-up?” This was the best part of the discussion because students were able to assign status to their peers, praising them for a novel approach they hadn’t before considered and telling them how their strategy created connections and made something “click” for them.

*Breaking it Down*

*Trust and Respect: Breaking it Down*

# Strategies for Decomposing 2-D Figures

Lesson 4 of 14

## Objective: Students will be able to decompose 2-D figures to strategically calculate area.

## Big Idea: Through small group and whole class discussion, students will be able to make connections between multiple ways of viewing composite areas and shaded areas of circles.

*60 minutes*

#### Warm-Up: Area Decomposition

*20 min*

I want students to focus on sense making, particularly as students consider sector area and ratios of similarity in 2-D and 3-D figures later in this unit. I give this warm-up so students can develop a variety of strategies and make connections between the strategies.

When I debrief the warm-up with students, I call on the Recorder/Reporter from each group to share out a method that emerged from his/her group. After the Recorder/Reporter shares, I open up the problem to student volunteers who want to add another strategy (which I record in a different color) or connect the strategies that have been shared, which requires them to construct viable arguments and critique the reasoning of others (**MP3**). I have found that having the expectation that the Recorder/Reporter will share out encourages a smooth and lively discussion.

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#### Notes: Sector Area

*10 min*

I give brief notes on sector area, encouraging students to first consider how much of the circle we must deal with, then writing out a formula. Since proportional reasoning is sometimes confusing for students, I make sure we try out at least a couple of practice problems as a whole class to model how to think about and apply the formula when solving problems.

#### Resources

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#### Homework Review

*15 min*

Since the Area Application homework featured complex problems, I want to make sure we have time to make sense of our work. In small groups, students compare their work, with the goal of making sure everyone understands. As a whole class, I facilitate a discussion around Problem #3, which requires students to think about how changing the side length of similar figures changes their areas—this is often a difficult concept for students to grasp, which is why I encourage students to come to the board to write out their work, draw models, explain their thinking, and get feedback from their peers.

#### Resources

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#### Check for Understanding

*10 min*

At this point in the unit, I want to assess my students’ understanding of regular polygon area and circle area. I collect students' work, which they do on scratch paper, as students exit the classroom.

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#### Homework: More Circles

*5 min*

In the More Circles homework assignment, students must strategically decompose figures so that they can find the area of shaded regions of circles.

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- UNIT 1: Creating Classroom Culture to Develop the Math Practices
- UNIT 2: Introducing Geometry
- UNIT 3: Transformations
- UNIT 4: Discovering and Proving Angle Relationships
- UNIT 5: Constructions
- UNIT 6: Midterm Exam Review
- UNIT 7: Discovering and Proving Triangle Properties
- UNIT 8: Discovering and Proving Polygon Properties
- UNIT 9: Discovering and Proving Circles Properties
- UNIT 10: Geometric Measurement and Dimension
- UNIT 11: The Pythagorean Theorem
- UNIT 12: Triangle Similarity and Trigonometric Ratios
- UNIT 13: Final Exam Review

- LESSON 1: Sectors of Circles
- LESSON 2: Making Sense of Area Formulas for Triangles, Parallelograms, Trapezoids, and Kites
- LESSON 3: Making Sense of Area Formulas for Regular Polygons and Circles
- LESSON 4: Strategies for Decomposing 2-D Figures
- LESSON 5: Sector Area Application: The Grazing Goat
- LESSON 6: Surface Area and Area Differentiation
- LESSON 7: Extreme Couponing: Pizza Edition
- LESSON 8: Area "Quest"
- LESSON 9: Introduction to Volume: Origami Boxes
- LESSON 10: Origami Boxes Gallery Walk
- LESSON 11: Volume Formulas, Cavalieri's Principle, and 2-D Cross-Sections
- LESSON 12: Real World Volume Context Problems
- LESSON 13: Ratios of Similarity and 3D Solids Generated by Revolving 2D Figures
- LESSON 14: Volume "Quest"