Unit 7, Lesson 12
Arcs and Sectors
Integrated Math 2
12.1
Warm-up
Materials
NoneActivity Narrative
This Math Talk focuses on calculating areas of sectors and finding arc lengths for simple fractions of circles. It encourages students to think about area and circumference formulas and to rely on the structure of a circle to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students develop general methods for calculating sector areas and arc lengths.
To combine fraction operations with area and circumference formulas, students need to look for and make use of structure (MP7).
Launch
If necessary, provide students with formulas for the area and circumference of a circle.
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneEvaluate mentally.
- Find the area of the shaded portion of the circle.
- Find the length of the highlighted portion of the circle’s circumference.
- Find the area of the shaded portion of the circle.
- Find the length of the highlighted portion of the circle’s circumference.
Activity Synthesis
The goal of this discussion is to review students’ strategies for evaluating arc lengths and sector areas.
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:
Advances: Speaking, Representing
- “First, I because .”
- “I noticed , so I .”
Advances: Speaking, Representing
Lesson Synthesis
In this lesson, students solved problems involving arc length and sector area. Here are some questions for discussion:
- “Compare and contrast the processes for finding the area of a sector and calculating the length of an arc.” (Each calculation involves finding the fraction of a circle represented by a sector. For arc length, the fraction is multiplied by the circle’s circumference. For sector area, the fraction is multiplied by the circle’s area.)
- “What are some examples in the real world of arcs or sectors?” (A slice of pizza, the area wiped by a windshield wiper, and the wedges of a dartboard are shaped like sectors. Arc length can describe the path of an object in circular motion, like the orbit of a satellite around the earth or the path of someone riding a Ferris wheel. It’s also the distance around the circular portion of a racetrack.)
Finally, add the following theorem to the class reference chart, and ask students to add it to their reference charts. Tell students that this wording is just a suggestion. They should add the method that they used in this lesson.
To calculate the area of a sector or the length of an arc, first find the fraction of the circle represented by the central angle of the arc or sector. Multiply this fraction by the circle’s area or circumference. (Theorem)
- arc length: units
- sector area: square units
Student Lesson Summary
A sector of a circle is the region enclosed by two radii. To find the area of a sector, start by calculating the area of the whole circle. Divide the measure of the central angle of the sector by 360 to find the fraction of the circle represented by the sector. Then multiply this fraction by the circle’s total area. We can use a similar process to find the length of the arc lying on the boundary of the sector.
The circle in the image has a total area of square centimeters, and its circumference is centimeters. To find the area of the sector with a 225-degree central angle, divide 225 by 360 to get , or 0.625. Multiply this by to find that the area of the sector is square centimeters. The length of the arc defined by the sector is because .