options: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Activity
In this activity students build off the specific calculations from the previous lesson to generalize the perimeter of a polygon inscribed in a circle of radius 1. The relatively unstructured presentation of this activity is purposeful. Students work with their groups to determine what information they need, how to calculate in the specific cases, and how they can express those repeated procedures in a generalized formula (MP8). Monitor for groups who have generalized any part of the process.
Digital
Launch
Encourage students to refer to the examples from the previous lesson as they work to generalize.
Action and Expression: Internalize Executive Functions.
Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Look for students who are struggling to begin and review examples from the previous lesson. Record their thinking on a display and keep the work visible as students continue to work.
Supports accessibility for: Organization; Attention
Student Facing
The applet shows a regular
\(n\)
-sided polygon inscribed in a circle.
Come up with a general formula for the perimeter of the polygon in terms of
\(n\)
.  Explain or show your reasoning.
Student Response
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for free access to Student Response.
Print
Launch
Encourage students to refer to the examples from the previous lesson as they work to generalize.
Action and Expression: Internalize Executive Functions.
Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Look for students who are struggling to begin and review examples from the previous lesson. Record their thinking on a display and keep the work visible as students continue to work.
Supports accessibility for: Organization; Attention
Student Facing
Here is one part of a regular
\(n\)
-sided polygon inscribed in a circle of radius 1.
Come up with a general formula for the perimeter of the polygon in terms of
\(n\)
.  Explain or show your reasoning.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students are struggling invite them to go back to the problems from the previous lesson to generalize the process. (Draw in the altitude. Find the measure of the central angle. Find the length of the opposite leg.) Suggest students generalize each step before trying to write a single formula.
Activity Synthesis
Invite previously selected groups to share one step at a time:
generalize the angle measure
generalize the segment length
extend to the whole perimeter
Invite another student to summarize by explaining where each piece of
\(P=2n \boldcdot \sin \left( \frac{360}{2n} \right)\)
appears in the diagram.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite students to create a visual display showing their general formula for the perimeter of the polygon in terms of
\(n\)
and their reasoning. Allow students time to quietly circulate and analyze the formulas and reasoning in at least 2 other displays in the room. Give students quiet think time to consider what is the same and what is different. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations that highlight what information each group used, any calculations they completed, and how they expressed those procedures in a generalized formula.
Design Principle(s): Optimize output (for generalization); Cultivate conversation


correct_option_index: 2