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

Apply trigonometry to general triangles.

Use coordinates to prove simple geometric theorems algebraically.

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

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

Prove geometric theorems.

Reason with shapes and their attributes.

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

Experiment with transformations in the plane.

Prove theorems involving similarity.

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.

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

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

Define trigonometric ratios and solve problems involving right triangles.

Understand and apply the Pythagorean Theorem.

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

Understand congruence in terms of rigid motions.

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

Apply geometric concepts in modeling situations.

Understand and apply theorems about circles.

Make geometric constructions.

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

Understand similarity in terms of similarity transformations.

Find arc lengths and areas of sectors of circles.

Explain volume formulas and use them to solve problems.

Analyze, compare, create, and compose shapes.
problem_activity: Activity
The purpose of this activity is for students to practice solving problems that involve volumes of pyramids and cones. Monitor for various problem-solving strategies—in particular, for the problem in which students find the radius or height of a cone given a particular volume. Some students may substitute values into a formula and solve the resulting equation, and others may follow a process of deductive reasoning.
Launch
Provide students access to scientific calculators. Arrange students in groups of 2. Ask students to discuss their thinking with their partner, and if they disagree, work to reach an agreement.
Representation: Internalize Comprehension.
Activate or supply background knowledge about the formulas for the area of a triangle, square, and circle. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Calculate the volume of each solid. Round your answers to the nearest tenth if necessary.
A
[IMG]
B
[IMG]
C
[IMG]
D: height 12 cm; area of base 32 cm
2
[IMG]
A particular cone has radius
\(r\)
and height
\(h\)
.
Write an expression for the volume of this cone in terms of
\(r\)
and
\(h\)
.
What is the height of a cone whose volume is
\(16\pi\)
cubic units and whose radius is 3 units?
What is the radius of a cone whose volume is
\(16\pi\)
cubic units and whose height is 3 units?
Student Response
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Student Facing
Are you ready for more?
The Pyramid of Giza is 455 feet tall. The base is square with a 756-foot side length. How many Olympic-size swimming pool volumes of water can fit inside the Pyramid of Giza? Explain or show your reasoning.
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
For the pyramid with a triangular base, students may forget that the triangle area formula involves dividing by 2, resulting in a base area of 12 square units. Prompt them to draw and label the base prior to calculating its area.
Activity Synthesis
Ask students to share their strategies for volume calculations. For example, for the cone, did they calculate the area of the base first, and substitute it into the formula
\(V=\frac13 Bh\)
? Or, perhaps they wrote out
\(V=\frac13 \pi r^2 h\)
and performed the calculations all at once.
Select previously identified students to share their thought process on the “working backward” cone problem. If possible, find one student who solved an equation algebraically and another who used backward reasoning rather than formally solving an equation.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to enhance the whole-class discussion about strategies for volume calculation. After students calculate the volume of each solid and determine the radius or height of each cone, invite them to create a visual display of their work for one of the solids. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their strategies. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast strategies for volume calculation.
Design Principle(s): Cultivate conversation


correct_option_index: 25