Unit 7, Lesson 12
Radian Sense
Geometry
12.1
Warm-up
Instructional Routines
Which Three Go Together?Materials
NoneActivity Narrative
This Warm-up prompts students to compare four angle measurements. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the figures for all to see. In the figures, and 1 represent angles measured in radians. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three figures that go together and can explain why. Next, tell students to share their response with their group and then work together to find as many sets of three as they can.
Activity
NoneWhich three go together? Why do they go together?
A
B
C
D
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose is for students to consider the ideas of others and notice new similarities and differences between the figures.
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any geometric terminology they use, such as “arc length” or “radian,” and to clarify their reasoning as needed. Consider asking:
- “How do you know _____?”
- “What do you mean by _____?”
- “Can you say that in another way?”
Student Lesson Summary
We can divide circles into congruent sectors to get a sense for the size of an angle measured in radians.
Suppose we want to draw an angle that measures radians. We know that radians is equivalent to 180 degrees. If we divide a sector with a central angle of radians into thirds, we can shade in two of the sectors to create an angle measuring radians.
Another way to understand the size of an angle measured in radians is to create a double number line with degrees on one line and radians on the other. On the double number line shown here, the degree measures are aligned with their equivalent radian measures. For example, radians is equivalent to 180.
Suppose we need to know the size of an angle that measures radians. The left half of the double number line represents radians. Divide the left half of the top and bottom number lines into fourths, then count out 3 of them on the radians line to land on . On the top line, each interval we drew represents 45 degrees because . If we count 3 of those intervals, we find that radians is equivalent to 135 degrees because .