Lesson 21

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.

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.

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.)

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.

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

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. 

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?”

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:

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.

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.

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?”

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.

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

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.

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.