standard_description: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
problem_activity: Warm-up
The purpose of this warm-up is to engage students' intuition about likelihood of events. The following activities in this lesson continue to develop more formal ways of thinking about likelihood leading to the definition of probability in the next lesson.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet work time and time to share their response with a partner. Follow with a whole-class discussion.
Student Facing
Which is more likely to happen?
When reaching into a dark closet and pulling out one shoe from a pile of 20 pairs of shoes, you pull out a left shoe.
When listening to a playlist—which has 5 songs on it—in shuffle mode, the first song on the playlist plays first.
Student Response
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Anticipated Misconceptions
Students may think that it is required to pull out a specific shoe rather than any left shoe. Ask students to visualize the problem and determine how many left shoes are in the closet.
Activity Synthesis
The purpose of the discussion is to help students recognize their own intuition about the likelihood of an event even when prior outcomes are not available.
Have partners share responses with the class, and ask at least one student for each option for their reasoning. Give students time to discuss their reasoning until they come to an agreement.
It may be helpful to reiterate that the outcomes from these actions are not certain. It is certainly possible to do both things and draw out a right shoe and listen to the first song on the list first, but it is not very likely based on the situations.


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