options: Explain volume formulas and use them to solve problems.

Prove geometric theorems.

Make geometric constructions.

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

Experiment with transformations in the plane.

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

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

Apply trigonometry to general triangles.

Analyze, compare, create, and compose shapes.

Reason with shapes and their attributes.

Understand and apply theorems about circles.

Understand similarity in terms of similarity transformations.

Apply geometric concepts in modeling situations.

Prove theorems involving similarity.

Use coordinates to prove simple geometric theorems algebraically.

Understand and apply the Pythagorean Theorem.

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

Understand congruence in terms of rigid motions.

Define trigonometric ratios and solve problems involving right triangles.

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

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

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

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

Find arc lengths and areas of sectors of circles.

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

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

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
problem_activity: Warm-up
Students examine statements deciding in each case whether the statement is always true, sometimes true, or never true. Since figures are similar if one is the result of reflecting, rotating, translating, and dilating the other, and figures are congruent only as a result of the rigid transformations, congruent figures are similar but not vice versa. Though students have not studied how angles behave under dilations, students have had multiple opportunities to see that dilations influence the size of a shape but not its angles. They also know from work in grade 7 that scaled copies have corresponding angles with the same measure, and dilations have been characterized as a process that makes scaled copies.
Launch
Display all 3 statements. Ask students to decide whether each of the statements is always, sometimes, or never true and give a signal when they have reasoning to support their decision.
Student Facing
Choose whether each of the statements is true in
all
cases, in
some
cases, or in
no
cases.
If two figures are congruent, then they are similar.
If two figures are similar, then they are congruent.
If an angle is dilated with the center of dilation at its vertex, the angle measure may change.
Student Response
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Activity Synthesis
Discuss each statement one at a time with this structure:
Poll the class on their answer choice and display the answers.
If everyone agrees on one answer, ask a few students to share their reasoning, recording it for all to see.
If there is disagreement on an answer, ask students with opposing answers to explain their reasoning to come to an agreement on an answer.


correct_option_index: 5