# Lesson 15

Putting All the Solids Together

## 15.1: Math Talk: Volumes (5 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for calculating volumes of solids. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to do more complex volume calculations.

In this activity, students have an opportunity to notice and make use of structure (MP7). To successfully evaluate volumes mentally, students must carefully consider the order in which to perform the required arithmetical operations. The structure of the number system, in particular, prime factorization, helps direct efficient choices.

### Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

### Student Facing

Evaluate the volume of each solid mentally.

### Anticipated Misconceptions

When calculating the cone’s volume, students may multiply the radius measurement by $$\frac 13$$ before squaring. Remind them of the order of operations convention that says to evaluate exponents prior to performing multiplication.

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{.5in}}$$’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

## 15.2: Missing Measurements (20 minutes)

### Activity

Students practice finding volumes of pyramids and prisms in problems that require the Pythagorean Theorem or trigonometry. When students articulate their strategies in advance of their calculations, they are making sense of a problem (MP1).

### Launch

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to increase awareness of the language used to talk about the features of cones and oblique rectangular prisms. Before revealing the questions in this activity, display the image of the right cone and oblique rectangular prism. Ask students to write down possible mathematical questions that could be asked about the solids. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the radius, height, and volume of the cone or prism. For example, “What is the radius of the cone?”, “What is the height of the rectangular prism?”, and “What is the volume of the cone?”
Design Principle(s): Maximize meta-awareness; Support sense-making
Representation: Internalize Comprehension. Activate background knowledge about the Pythagorean theorem and trigonometric ratios.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

1. Answer the questions for each of the two solids shown.
1. Which measurement that you need to calculate the volume isn’t given?
2. How can you find the value of the missing measurement?
3. What volume formula applies?
4. Calculate the volume of the solid, rounding to the nearest tenth if necessary.
2. Calculate the volume of each solid, rounding to the nearest tenth if necessary.

### Student Facing

#### Are you ready for more?

For a sphere with radius $$r$$, its volume is $$\frac43 \pi r^3$$ and its surface area is $$4 \pi r^2$$. Here is a half-sphere bowl pressed out of a piece of sheet metal with area 1 square foot. What is the volume of the bowl?

### Activity Synthesis

Invite students to share how they knew which measurements needed to be calculated and how they chose calculation strategies. When was the Pythagorean Theorem helpful, and when did they need to use trigonometry? Ask students to describe the easiest and most difficult aspects of this task.

## 15.3: Spinning into Three Dimensions (10 minutes)

### 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.

1. Draw the solid that would be traced out. Label the dimensions of the solid.
2. Find the volume of the solid. Round your answer to the nearest tenth if needed.

### 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

## Lesson Synthesis

### Lesson Synthesis

The goal is to discuss strategies for selecting formulas and determining all the necessary dimensions. Here are some questions for discussion:

• “What dimensions are needed to calculate the volume of a cone?” (The radius and height are needed.)
• “What formulas are used when calculating the volume of a triangular pyramid?” (The expression $$\frac12 bh$$ allows you to find the area of the triangular base. Then use $$\frac13 Bh$$ for the pyramid volume.)
• “What is the difference between the $$b$$ and $$h$$ in the triangle area formula, and the $$B$$ and the $$h$$ in the pyramid volume formula?” (In the triangle area formula, the $$b$$ and $$h$$ are the base and height of the triangle. In the pyramid volume formula, $$B$$ is the area of the base and $$h$$ is the height of the pyramid.)
• “What formulas are used when calculating the volume of a cylinder?” (The formula $$\pi r^2$$ allows you to find the area of the base. Then the volume formula is $$Bh$$.)

## Student Lesson Summary

### Student Facing

Before computing volume, it’s important to select the right formula and find all the dimensions represented in the formula. For example, consider a company that makes two chew toys for dogs. One toy is in the shape of a cylinder with radius 9 cm and height 2.5 cm. The other looks like the cone in the image. The company wants to know which toy uses more material. The toys are solid, not hollow.

To calculate the cylinder toy’s volume, use the expression $$Bh$$. The radius measures 9 cm, so the area of the base, $$B$$, is $$81\pi$$ cm2. The volume is $$202.5\pi$$, or approximately 636 cm3, because $$81\pi \boldcdot 2.5 = 202.5\pi$$.
For the cone, the height is unknown. A right triangle is formed by the radius 6 cm and the height $$h$$, with hypotenuse 16 cm. By the Pythagorean Theorem, $$6^2+h^2=16^2$$. Solving, we get $$h=\sqrt{220}$$.
Since this is a cone, use the expression $$\frac13 Bh$$. The area of the base, $$B$$, is $$36\pi$$ cm2. The volume is approximately 559 cm3 because $$\frac13 \boldcdot 36 \pi \boldcdot \sqrt{220} \approx 559$$. The cylinder-shaped toy uses more material.