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 for students to reason about the various measurements of a circle. In each of the pictures, students are given a piece of information about the circle and asked to reason about the other measurements of the circle. Some of these measurements can be found based on the given information while others, without a calculator, would require an estimate.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet think time followed by 2 minutes to discuss their estimates with a partner.
Student Facing
Here are some pictures of circular objects, with measurement tools shown. The measurement tool on each picture reads as follows:
Wagon wheel: 3 feet
Plane propeller: 24 inches
Sliced Orange: 20 centimeters
[IMG]
For each picture, which measurement is shown?
Based on this information, what measurement(s) could you estimate for each picture?
Student Response
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Activity Synthesis
For each picture, ask selected students to share the measurements shown in the pictures. Ask students to explain the strategies or calculations they would use for computing or estimating other measurements (for example, the radius and circumference of the wheel). Record and display student responses on the actual pictures. Refer to MLR 2 (Collect and Display) to highlight student responses and language about what they know and what they estimated.


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