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# Sectors of Circles

Lesson 1 of 14

## Objective: Students will be able to use proportional reasoning to solve problems about sector area and arc length.

## Big Idea: In a group-worthy task, students use clues about sector area, arc length, and sector perimeter to solve their "dominos".

*55 minutes*

Since this is the first day of our new unit on Measurement and Dimensionality, I want to make sure to challenge students by having them engage in a task that requires them to look for patterns, find relationships between sectors, angles, and arcs, and get a sense of how these change in relation to the size of the circle—this task requires students to recall their understanding that all circles are similar and to get a sense of the proportionality of perimeters and areas of circles of different radii. Additionally, this group-worthy task encourages students to use multiple intelligences as they use a table, detect patterns, and apply formulas.

In Part 1 of the Sectors of Circles task, students complete a table featuring sectors of different sizes, writing out their areas and perimeters, using formulas and/or seeing relationships between the sizes of the sectors. When I check in with students, I often choose a student to explain how the group thought about the problem and ask them to choose one or two sectors through which to explain their methods and ideas.

Part 2 is the main focus for the lesson—this is the part of the task where students apply the knowledge they gained from completing and making sense of the table in Part 1. In Part 2, groups must solve each “domino” by using the given clues and making sense of the relationships between the sectors and arcs of the three different-sized circles (**MP1**). I offer little assistance in this task to encourage groups to depend on themselves, especially because they themselves can verify if their answers are correct (if all of their answers are correct, all of their “dominos” will connect to form a loop).

- Review formulas for Circumference and Area with the whole class.
- Review the definition of perimeter broadly, then, in the context of the perimeter of a sector, where the perimeter of a sector equals the arc length + two of its radii
- I give a brief overview of Part II and tell students how they can check their answer (all of the dominos, if answered correctly, should loop back around to the beginning)

We will debrief Part 2 early in tomorrow's lesson.

Resource Citation*: *I want to acknowledge the Geometry Team at Mission High School in San Francisco, California for sharing Sectors of Circles.

#### Resources

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I give this homework assignment to support my formative assessment strategy and to ensure students can find the areas and perimeters of squares and rectangles and that they can make sense of triangle area.

*expand content*

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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"