Sectors of Circles

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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".

Patterns with Sectors of Circles

50 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).

  1. Review formulas for Circumference and Area with the whole class. 
  2. 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
  3. 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.

Homework: Triangles and Rectangles

5 minutes

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.