Lesson 21

Cylinders, Cones, and Spheres

21.1: Sphere Arguments (5 minutes)

Warm-up

The purpose of this warm-up is to catch errors students are making in calculating the volume of a sphere. Monitor for students who use the formula for a cylinder or cone, who use \(r^2\) instead of \(r^3\), or who forget to include \(\pi\) as a factor in the computation.

Launch

Arrange students in groups of 2. Tell students that if they have a hard time visualizing this sphere, they can sketch it. Give students 1–2 minutes of quiet work time followed by time to discuss their responses with their partner.

Student Facing

Four students each calculated the volume of a sphere with a radius of 9 centimeters and they got four different answers.

  • Han thinks it is 108 cubic centimeters.
  • Jada got \(108\pi\) cubic centimeters.
  • Tyler calculated 972 cubic centimeters.
  • Mai says it is \(972\pi\) cubic centimeters.

Do you agree with any of them? Explain your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

For each answer, ask students to indicate whether or not they agree. Display the number of students who agree with each answer all to see. Invite someone who agreed with \(972\pi\) to explain their reasoning. Ask students if they think they know what the other students did incorrectly to get their answers. (To get 108, Han and Jada likely used \(r^2\) instead of \(r^3\), and Tyler probably didn’t realize that multiplying \(\frac43 r^3\) by \(\pi\) is necessary.)

21.2: Sphere’s Radius (5 minutes)

Optional activity

The purpose of this activity is for students to think about how to find the radius of a sphere when its volume is known. Students can examine the structure of the equation for volume and reason about a number that makes the equation true. They can also notice that \(\pi\) is a factor on each side of the equation and divide each side by \(\pi\). Both strategies simplify the solution process and minimize the need for rounding. Watch for students who substitute a value for \(\pi\) to each side of the equation, who use the structure of the equation to reason about the solution, or who solve another way, so strategies can be shared and compared in the whole-class discussion.

Launch

Allow students 3–4 minutes work time followed by a whole-class discussion. 

Writing, Speaking: MLR1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their response to the question, “What is the value of \(r\) for this sphere? Explain how you know.” Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them to strengthen their ideas and clarify their language (e.g., “What did you do first?”, “Why did you…?”, and “How did you deal with \(r^3\)?”, etc.). Students can borrow ideas and language from each partner to strengthen their final version.
Design Principle(s): Optimize output (for explanation)

Student Facing

A sphere. A dashed line is drawn from the center of the sphere to the edge of the sphere and is labeled "r."

The volume of this sphere with radius \(r\) is \(V=288\pi\). This statement is true:

\(\displaystyle 288\pi =\frac43 r^3 \pi.\) What is the value of \(r\) for this sphere? Explain how you know.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Students who substituted a value for \(\pi\) and solved the resulting equation might have rounded along the way, making the value for the radius slightly less than 6 while the actual value is exactly 6. This is a good opportunity to talk about the effects of rounding and how to minimize the error that rounding introduces. 

Activity Synthesis

The purpose of the discussion is to examine how students reasoned through each step in solving for the unknown radius. Ask previously identified students to share their responses.

Consider asking students the following questions to help clarify the different approaches students took:

  • \(\pi\) appears on both sides of the volume equation. Did you deal with this as a first step or later in the solution process?”
  • “How did you deal with the fraction in the equation?”
  • “If the final step in your solution was solving for \(r\) when \(r^3=216\), how did you solve? If you found that \(r^3\) was a different number, how did you solve?”
Engagement: Develop Effort and Persistence. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. All group members should be prepared to share if invited. 
Supports accessibility for: Attention; Social-emotional skills

21.3: Info Gap: Unknown Dimensions (20 minutes)

Activity

In this info gap activity, students determine and request the information needed to answer questions related to volume equations of cylinders, cones, and spheres.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Infogap cards.

Launch

Tell students that they will continue to refresh their skills of working with proportional relationships. Explain the Info Gap structure and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for the second problem and instruct them to switch roles.

Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity. 
Supports accessibility for: Memory; Organization
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary solve problems involving the volume of shapes. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?”
Design Principle(s): Cultivate Conversation

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to be able to answer the question.

  2. Ask your partner for the specific information that you need.

  3. Explain how you are using the information to solve the problem.

    Continue to ask questions until you have enough information to solve the problem.

  4. Share the problem card and solve the problem independently.

  5. Read the data card and discuss your reasoning.

If your teacher gives you the data card:

  1. Silently read your card.

  2. Ask your partner “What specific information do you need?” and wait for them to ask for information.

    If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

  3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

  4. Read the problem card and solve the problem independently.

  5. Share the data card and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Students may have a difficulty making sense of the relationships between the dimensions of the two figures. Encourage these students to sketch the figures and label them carefully.

Activity Synthesis

Select several groups to share their answers and reasoning for each of the problems. If any students made sketches with labeled dimensions, display these for all to see. In particular, contrast students who used volume formulas versus those who remembered that the volume of a cone is half the volume of a sphere with the same dimensions (radius and height).

Consider asking these discussion questions:

  • For students who had a Problem Card:
    • “How did you decide what information to ask for? How did the information on your card help?”
    • “How easy or difficult was it to explain why you needed the information you were asking for?”
    • “Give an example of a question that you asked, the clue you received, and how you made use of it.”
    • “How many questions did it take for you to be able to solve the problem? What were those questions?”
    • “Was anyone able to solve the problem with a different set of questions?”
  • For students who had a Data Card:
    • “When you asked your partner why they needed a specific piece of information, what kind of explanations did you consider acceptable?”
    • “Were you able to tell from their questions what volume question they were trying to answer? If so, how? If not, why might that be?”

21.4: The Right Fit (10 minutes)

Activity

In this activity, students once again consider different figures with given dimensions, this time comparing their capacity to contain a certain amount of water. The goal is for students to not only apply the correct volume formulas, but to slow down and think about how the dimensions of the figures compare and how those measurements affect the volume of the figures.

Launch

Arrange students in groups of 2. Allow 5 minutes of quiet work time, then a partner discussion followed by a whole-class discussion.

Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Speaking: MLR8 Discussion Supports. Provide sentence frames to support students as they work with their partner and explain their reasoning. For example, “I think ____ , because _____ .”, or “I (agree/disagree) because ______ .”, or “The volume is (greater than/less than) _____ , which means _____ .”
Design Principle(s): Support sense-making; Optimize output for (explanation)

Student Facing

A right circular cylinder witha diameter of 3 centimeters and a height of 8 centimeters.

A cylinder with diameter 3 centimeters and height 8 centimeters is filled with water. Decide which figures described here, if any, could hold all of the water from the cylinder. Explain your reasoning.

  1. Cone with a height of 8 centimeters and a radius of 3 centimeters.
  2. Cylinder with a diameter of 6 centimeters and height of 2 centimeters.
  3. Rectangular prism with a length of 3 centimeters, width of 4 centimeters, and height of 8 centimeters.
  4. Sphere with a radius of 2 centimeters.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Are you ready for more?

A thirsty crow wants to raise the level of water in a cylindrical container so that it can reach the water with its beak.

  • The container has diameter of 2 inches and a height of 9 inches.
  • The water level is currently at 6 inches.
  • The crow can reach the water if it is 1 inch from the top of the container.

In order to raise the water level, the crow puts spherical pebbles in the container. If the pebbles are approximately \(\frac12\) inch in diameter, what is the fewest number of pebbles the crow needs to drop into the container in order to reach the water?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Students might think that doubling a dimension doubles the volume. Help students recall and reason that changes in the radius, because it is squared or cubed when calculating volume, have a greater affect.

Activity Synthesis

The purpose of this discussion is to compare volumes of different figures by computation and also by considering the effect that different dimensions have on volume.

Ask students if they made any predictions about the volumes before directly computing them and, if yes, how they were able to predict. Students might have reasoned, for example, that the second cylinder had double the radius of the first, which would make the volume 4 times as great, but the height was only \(\frac14\) as great so the volume would be the same. Or they might have seen that the cone would have a greater volume since the radius was double and the height the same, making the volume (if it were another cylinder) 4 times as great, so the factor of \(\frac13\) for the cone didn’t bring the volume down below the volume of the cylinder.

Lesson Synthesis

Lesson Synthesis

In this unit, students have learned how to find the volume of cylinders, cones, and spheres, how to find an unknown dimension when the volume and another dimension are known, and how to reason about the effects of different dimensions on volume. Assign groups of 2–3 students one of the questions shown here and provide them with the tools to make a visual display explaining their response. Encourage students to make their displays as though they are explaining the answer to the question to someone who is not in the class and to make up values for dimensions to use to illustrate their ideas. Suggest sketches of figures where appropriate.

  • “Describe some relationships between the volumes of cylinders, cones, and spheres.”
  • “How do we find a missing dimension when we know the volume and another dimension of a cylinder, cone, or sphere (or just the volume in the case of the sphere)?”
  • “What happens to the volume of a cylinder or cone when its height is doubled? Tripled?”
  • “What happens to the volume of a sphere when its height is doubled? Tripled?”
  • “What happens to the volume of a cylinder, cone, or sphere when its radius is doubled? Tripled?”
  • “What happens to the volume of a cylinder or cone when its height is doubled and its radius is halved?”
  • “What happens to the volume of a cylinder or cone when its radius is doubled and its height is halved?”

21.5: Cool-down - New Four Spheres (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Lesson Summary

Student Facing

The formula

\(\displaystyle V=\frac43 \pi r^3\)

gives the volume of a sphere with radius \(r\). We can use the formula to find the volume of a sphere with a known radius. For example, if the radius of a sphere is 6 units, then the volume would be

\(\displaystyle \frac{4}{3} \pi (6)^3 = 288\pi\)

or approximately \(904\) cubic units. We can also use the formula to find the radius of a sphere if we only know its volume. For example, if we know the volume of a sphere is \(36 \pi\) cubic units but we don't know the radius, then this equation is true: 

\(\displaystyle 36\pi=\frac43\pi r^3\)

That means that \(r^3 = 27\), so the radius \(r\) has to be 3 units in order for both sides of the equation to have the same value.

Many common objects, from water bottles to buildings to balloons, are similar in shape to rectangular prisms, cylinders, cones, and spheres—or even combinations of these shapes! Using the volume formulas for these shapes allows us to compare the volume of different types of objects, sometimes with surprising results.

For example, a cube-shaped box with side length 3 centimeters holds less than a sphere with radius 2 centimeters because the volume of the cube is 27 cubic centimeters (\(3^3 = 27\)), and the volume of the sphere is around 33.51 cubic centimeters (\(\frac43\pi \boldcdot  2^3 \approx 33.51\)).