options: 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.

Understand and apply the Pythagorean Theorem.

Understand congruence in terms of rigid motions.

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

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

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

Understand and apply theorems about circles.

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

Analyze, compare, create, and compose shapes.

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

Experiment with transformations in the plane.

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

Apply geometric concepts in modeling situations.

Prove geometric theorems.

Reason with shapes and their attributes.

Understand similarity in terms of similarity transformations.

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

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

Prove theorems involving similarity.

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

Use coordinates to prove simple geometric theorems algebraically.

Make geometric constructions.

Explain volume formulas and use them to solve problems.

Apply trigonometry to general triangles.

Define trigonometric ratios and solve problems involving right triangles.

Find arc lengths and areas of sectors of circles.
problem_activity: Activity
This task combines concepts of decomposition, cylinder and cone volume formulas, and solids of rotation.
If students choose to use 3D graphing technology, reviewing the Axis of Rotation lesson may be helpful. Making this technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Student Facing
Suppose this two-dimensional figure is rotated 360 degrees using the vertical axis shown. Each small square on the grid represents 1 square inch.
[IMG]
Draw the solid that would be traced out. Label the dimensions of the solid.
Find the volume of the solid. Round your answer to the nearest tenth if needed.
Student Response
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Anticipated Misconceptions
Students may struggle to visualize and draw the solid of rotation. Suggest that they divide the two-dimensional figure into two pieces horizontally and think about what each would look like rotated using the vertical axis. If necessary, they can draw the two resulting solids (cone and cylinder) separately.
Activity Synthesis
Ask students to compare and contrast the cylinder and the cone. They each have the same radius measurement, but their heights are different.
Invite students to summarize the kinds of solids that can be traced out through rotating a two-dimensional figure: spheres, cones, and cylinders can be created through spinning half-circles, triangles, and rectangles, but pryamids and prisms can’t be created through rotation. In general, a solid created through rotation will have circular cross sections.
Speaking: MLR8 Discussion Supports.
As students share the solids that can be formed by rotating a two-dimensional figure, press for details by asking students to elaborate or give an example. Also, ask students how they know that prisms and pyramids cannot be created through rotation. This will help students justify their reasoning for the solids that can and cannot be formed by rotating a two-dimensional figure.
Design Principle(s): Support sense-making; Optimize output (for justification)
Representation: Internalize Comprehension.
Provide examples of actual three-dimensional models of cylinders, cones, spheres, and prisms for students to view or manipulate. Ask students which solids can be formed by rotating the two-dimensional figure around the vertical axis.
Supports accessibility for: Visual-spatial processing; Conceptual processing


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