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

Explain volume formulas and use them to solve problems.

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

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

Understand similarity in terms of similarity transformations.

Understand and apply the Pythagorean Theorem.

Analyze, compare, create, and compose shapes.

Use coordinates to prove simple geometric theorems algebraically.

Make geometric constructions.

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

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

Define trigonometric ratios and solve problems involving right triangles.

Apply geometric concepts in modeling situations.

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

Understand and apply theorems about circles.

Prove geometric theorems.

Experiment with transformations in the plane.

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

Find arc lengths and areas of sectors of circles.

Reason with shapes and their attributes.

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

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

Understand congruence in terms of rigid motions.

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.

Prove theorems involving similarity.

Apply trigonometry to general triangles.
problem_activity: Activity
Students combine their experience with solids of rotation and their understanding of cylinder volume.
No specific directions are given in regard to specifying how students should express their volume answer. Monitor for students who leave their answers in terms of
\(\pi\)
and those who found an approximate decimal answer.
Launch
Action and Expression: Develop Expression and Communication.
Maintain a display of important terms and vocabulary. During the launch, take time to review the following terms from previous lessons that students will need to access for this activity: axis of rotation, solid of rotation, volume of a cylinder.
Supports accessibility for: Memory; Language
Student Facing
Suppose each two-dimensional figure is rotated around the vertical axis shown. Each small square in the grid represents 1 square centimeter.
A
[IMG]
B
[IMG]
For each solid:
Either sketch
or
describe in words the three-dimensional solid that would form.
Find the solid’s volume.
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Given a cylinder with radius
\(r\)
and height
\(h\)
, write an expression for the volume if each of these changes were made.
The height is tripled to
\(3h\)
and the radius remains
\(r\)
.
The radius is tripled to
\(3r\)
and the height remains
\(h\)
.
Given a cube of side length
\(s\)
, write an expression for the volume if the side length were tripled to
\(3s\)
.
Which change affected the shape’s volume the most? The least? Explain or show your reasoning.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may struggle to visualize the rotated figures. For the L shape, encourage them to divide it into smaller pieces and consider each independently. For the rectangle, remind them of the doughnut cross section they drew in a previous lesson.
Activity Synthesis
In the discussion, explore different aspects of the calculations students made. Invite one student who left the answer in terms of
\(\pi\)
and another who found a decimal answer to share their results. Remind students that the former is called an exact answer, while the latter is called an approximate answer because the value of
\(\pi\)
needs to be approximated and rounded to find a decimal answer. Emphasize that the choice of expression depends on the situation at hand.
If time permits, ask students would would happen if the first figure were rotated around the horizontal axis formed by the bottom edge of the figure. Challenge them to decide if the volume would be the same as with the rotation around the given vertical axis. (The volume of this figure would actually be
\(99\pi\)
cubic centimeters.)
Speaking: MLR8 Discussion Supports.
As students share their sketches or descriptions of the three-dimensional solid that would form, press for details by asking how they know that the first figure would form two cylinders after a rotation around the vertical axis. Also, ask how they know that the cylinder on top will have a radius of 2 cm and the cylinder at the bottom will have a radius of 5 cm. This will help students justify their reasoning for the measurements of the cylinders in their sketches or descriptions.
Design Principle(s): Support sense-making; Optimize output (for justification)


correct_option_index: 1

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