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• Multiplying the radian measure by the radius, which in this case would rst involve solving for the radius.
Here are some questions for discussion:
• “What fraction of the entire circle is an angle of ?” (It is of a circle because an entire circle measures radians.)
• “When might one strategy be easier than another?” (It might be easier to nd the fraction of the whole and multiply the circumference if the circumference is given rather than the radius. It might be easier to multiply the radian measure by the radius if the radius is given instead, or if the radian measure is not easily recognized as a divided into equal parts.)
• “Which way of thinking about arc length do you prefer? Why?” (Possible response: Thinking in terms of fractions of the circumference connects to previous understanding about fractions.)
3.3 Radians and Degrees
15 minutes
The purpose of this activity is for students to explore a context that provides some physical intuition about how large 1 degree of rotation is and to practice converting between degrees and radians. Degrees will remain an important part of students’ understanding of angle measure, so it will be important to maintain that understanding and integrate it with new experiences with radian measure.
Launch
If desired, display an analog clock and explain the roles of the second hand and minute hand. It’s important to show an example of a clock whose minute hand moves when the second hand moves. Pause after question 3 to discuss that the small movement of the minute hand in 10 seconds represents of a full rotation around the circle. This is what we call 1 degree.
Student Task Statement
An angle that traces out an arc that goes all the way around a circle (the circumference) measures radians because the circumference of a circle is times the radius.
1. There are 60 seconds in a minute and 60 minutes in an hour. How many seconds are there in an hour?
2. On an analog clock, what radian angle measure does the minute hand trace out in 1 minute? Give your answer in terms of .
3. On an analog clock, what radian angle measure does the minute hand trace out in 10 seconds? Give your answer in terms of . The de nition of 1 degree is radians.
Unit 7
Lesson 3: Measure That Arc
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