options: Apply trigonometry to general triangles.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Understand and apply theorems about circles.

Solve real-world and mathematical problems involving area, surface area, and volume.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Prove theorems involving similarity.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Understand similarity in terms of similarity transformations.

Define trigonometric ratios and solve problems involving right triangles.

Use coordinates to prove simple geometric theorems algebraically.

Analyze, compare, create, and compose shapes.

Visualize relationships between two-dimensional and three-dimensional objects.

Find arc lengths and areas of sectors of circles.

Reason with shapes and their attributes.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Translate between the geometric description and the equation for a conic section.

Understand and apply the Pythagorean Theorem.

Apply geometric concepts in modeling situations.

Experiment with transformations in the plane.

Understand congruence in terms of rigid motions.

Make geometric constructions.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Prove geometric theorems.

Explain volume formulas and use them to solve problems.

Classify two-dimensional figures into categories based on their properties.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
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
Teachers with a valid work email address can
click here to register or sign in
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: 8

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