options: Use coordinates to prove simple geometric theorems algebraically.

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

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

Reason with shapes and their attributes.

Analyze, compare, create, and compose shapes.

Prove geometric theorems.

Apply geometric concepts in modeling situations.

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

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

Prove theorems involving similarity.

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

Understand congruence in terms of rigid motions.

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

Define trigonometric ratios and solve problems involving right triangles.

Understand and apply theorems about circles.

Find arc lengths and areas of sectors of circles.

Understand and apply the Pythagorean Theorem.

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

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

Apply trigonometry to general triangles.

Understand similarity in terms of similarity transformations.

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

Explain volume formulas and use them to solve problems.

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

Make geometric constructions.

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

Experiment with transformations in the plane.
problem_activity: Activity
In previous grades, students learned how to calculate the volume of a cylinder. In this activity, they revisit that process. Students are prompted to think about volume in 1-unit layers. This will be helpful in upcoming lessons when students learn about Cavalieri’s Principle, or the idea that if two solids have equal-area cross sections at all heights, the solids have equal volumes. Cavalieri’s Principle will be used to derive a formula for the volume of a pyramid.
Note that in this activity, “volume of a container” is used as shorthand for “the volume of the region enclosed by the container.”
Monitor for students who calculate the volume of the water in the prism then work backward to find the height of the water in the cylinder, and for those who use deductive logic (for example, reasoning that because the two bases have equal area, the height of the water must be equal in both solids) to arrive at the same answer.
Students reason abstractly and quantitatively (MP2) when they compare the prism and cylinder volumes.
Launch
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to increase awareness of the language used to talk about the volumes of solids. Before revealing the questions in this activity, display the image of both solids along with the statement: “Here are two solids. All measurements are in centimeters,” and ask students to write down possible mathematical questions that could be asked about the image. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the volume of both solids. For example, “Does the cylinder have the same volume as the prism?”
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because _____. Then, I….”, “I noticed _____ so I….”, and “I tried _____, and what happened was….”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Here are two containers. All measurements are in centimeters.
[IMG]
Suppose the prism contains water that reaches a height of 1 cm.
Draw a representation of this situation.
The water is poured from the prism into the cylinder. What is the height of the water in the cylinder? Explain your reasoning.
Suppose the prism contained water that reached a height of 3 cm instead of 1 cm. If the water were poured into the empty cylinder, what would the height of the water in the cylinder be?
Student Response
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Anticipated Misconceptions
Students may be unsure how to work with the prism measurement
\(4\pi\)
cm. Invite them to write out the area calculation as
\(9\boldcdot4\pi\)
. Alternatively, they can use an approximation of
\(\pi\)
to get rounded answers.
If students struggle to find the height of the water in the cylinder, ask them what they already know about the cylinder, and to consider what additional information they would need to find the height. Prompt them to use their responses to the first question to help them answer this one.
Activity Synthesis
The goal is to make connections between methods of calculating volumes for prisms and cylinders. Ask previously identified students to share their reasoning, starting with those who did volume calculations and worked backward, and ending with those who used deductive reasoning. Connect these methods by asking these questions:
“Describe what volume formulas you used. Do they apply to just one of the containers, or to both?” (Sample responses: The expression
\(\ell wh\)
applies to the prism. The expression
\(\pi r^2 h\)
applies to the cylinder. The expression
\(BH\)
applies to both.)
“Consider the volume formula
\(V=Bh\)
. How does this formula help explain why the heights were the same for the same volume of water in each container?” (The area of the base,
\(B\)
, is the same for each container:
\(36\pi\)
square centimeters. The volume formula can be rewritten
\(h=V\div B\)
, so if the containers have the same volume of water in each and have bases with the same area, the height will be the same.)
Invite students to describe how the volumes of the two containers compare (they are equal), and challenge them to describe another solid that has the same volume as these two.


correct_option_index: 22