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

Understand and apply theorems about circles.

Reason with shapes and their attributes.

Understand and apply the Pythagorean Theorem.

Use coordinates to prove simple geometric theorems algebraically.

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

Apply trigonometry to general triangles.

Make geometric constructions.

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

Explain volume formulas and use them to solve problems.

Understand congruence in terms of rigid motions.

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

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

Understand similarity in terms of similarity transformations.

Apply geometric concepts in modeling situations.

Find arc lengths and areas of sectors of circles.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

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

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

Define trigonometric ratios and solve problems involving right triangles.

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

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

Prove geometric theorems.

Analyze, compare, create, and compose shapes.

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

Prove theorems involving similarity.

Experiment with transformations in the plane.
problem_activity: Activity
The purpose of this activity is for students to compare the possible ways of constructing equilateral triangles. Look for student work that illustrates a variety of methods and sizes of triangles.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Encourage students to construct the equilateral triangles using different methods rather than repeating the same method twice.
Student Facing
Use straightedge and compass moves to construct at least 2 equilateral triangles of different sizes.
Student Response
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Student Facing
Are you ready for more?
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Examine the figure carefully. What different shapes is it composed of? Be specific.
Figure out how to construct the figure with a compass and straightedge.
Then, cut it out, and see if you can fold it up into a container like this.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students get stuck, ask them to refer to the warm-up to see whether they can spot equilateral triangles. Then encourage them to do the compass and straightedge moves required to make those equilateral triangles.
Activity Synthesis
Display several student responses for all to see. Ask students to explain their methods for producing equilateral triangles and justify how they know each triangle is equilateral. If not mentioned by students, conjecture that, in some of these constructions, the entire construction remains the same even when rotated
\(\frac13\)
of a full turn (120 degrees) around the center. This means that each side can be rotated onto the other sides, and each angle can be rotated onto the other angles. This plants ideas about symmetry that will be explored in later lessons.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to help students prepare for the whole-class discussion. Arrange students in groups of 2. Invite students to take turns explaining their method for constructing equilateral triangles of different sizes. This will provide all students with additional opportunities to clarify their thinking and to produce language.
Design Principle(s): Support sense-making; Cultivate conversation
Action and Expression: Develop Expression and Communication.
Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy and ideas. For example, “First, I _____ because _____. Then, I....” and “I know _____ because….”
Supports accessibility for: Language; Organization


correct_option_index: 7