options: Understand and apply the Pythagorean Theorem.

Define trigonometric ratios and solve problems involving right triangles.

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

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

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

Use coordinates to prove simple geometric theorems algebraically.

Prove theorems involving similarity.

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

Understand similarity in terms of similarity transformations.

Analyze, compare, create, and compose shapes.

Explain volume formulas and use them to solve problems.

Find arc lengths and areas of sectors of circles.

Apply geometric concepts in modeling situations.

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

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

Understand congruence in terms of rigid motions.

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

Experiment with transformations in the plane.

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

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

Understand and apply theorems about circles.

Apply trigonometry to general triangles.

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

Prove geometric theorems.

Reason with shapes and their attributes.

Make geometric constructions.

Classify two-dimensional figures into categories based on their properties.
problem_activity: Warm-up
In this warm-up, students reason about the volume and dimensions of a cylinder based on information given about a sphere of the same height. The purpose of this warm-up is for students to recognize when they do not have enough information to reach a single answer. While they can determine the height of the cylinder, the radius is completely undefined, which means the volume of the cylinder could be anything.
Launch
Give students quiet work time followed by a whole-class discussion.
Student Facing
A cylinder and sphere have the same height.
If the sphere has a volume of
\(36\pi\)
cubic units, what is the height of the cylinder?
What is a possible volume for the cylinder? Be prepared to explain your reasoning.
Student Response
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Anticipated Misconceptions
Some students may not understand why the second problem talks about “possible” volumes if they assume the radius of the cylinder is the same as the sphere. Ask these students to explain how they found the radius of the cylinder to help them notice that the problem never gives that information.
Activity Synthesis
Ask students to share how they calculated the height of the cylinder. If any students made a sketch, display these for all to see.
Select at least 5 students to give a possible volume for the cylinder and record these for all to see. Ask students,
“Why do we not know what the volume of the cylinder is?” (We don’t know the radius, only the height, so the volume could be anything.)
“Is knowing the height of a sphere enough information to determine the volume?” (Yes. The volume of a sphere is based on the radius, which is half the height.)
Tell students that in the next activity, they will investigate how changes to the radius of a sphere changes the volume of the sphere.


correct_option_index: 13