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
This warm-up sets the stage for this lesson. Students are presented with the basic question of whether baths or showers use more water and they brainstorm information that might help them investigate the question.
Launch
"Some people take showers, some people take baths. There is disagreement over which one takes more water. What do you think?" Ask students to think for just a minute about whether they think a shower or a bath uses more water. (This is just to record their first instinct—they should not spend any time researching or calculating right now.) Poll the class and record the total for each category for all to see: "think a bath takes more water" and "think a shower takes more water."
Student Facing
Some people say that it uses more water to take a bath than a shower. Others disagree.
What information would you collect in order to answer the question?
Estimate some reasonable values for the things you suggest.
Student Response
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Activity Synthesis
Invite students to share their responses. Record and display their responses for all to see. If these quantities do not come up in conversation, ask students to discuss the ideas and provide reasonable estimates:
Time spent in the shower
Volume of the bath tub
Rate of water coming out of the shower head


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