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 is a factor on each side of the equation, so the equation is still true if rewritten without on each side. Both strategies simplify the solution process and minimize the need for rounding. Watch for students who substitute a value for to each side of the equation, who use the structure of the equation to reason about the solution, or who solve another way so that these strategies can be shared and compared in the whole-class discussion.

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:

• “ 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 when , how did you solve? If you found that was a different number, how did you solve?”

In this activity, students calculate the volumes of spheres but do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.

The Information 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).

If any partners finish quickly, perhaps because they recall that the volume of a cone is half the volume of a sphere with matching dimensions, encourage them to collaborate to find a different way to solve the problem.

After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “How did you decide what information to ask for first? How did the information on your card help?” (My card said the cone and the sphere have the same dimensions, so I started by asking for those dimensions so I could use the volume formula.)
• “Give an example of a question that you asked, the clue you received, and how you made use of it.” (I asked for the volume of the cone and was told it was cm³. I used that value, the fact that the height of the cone is double the radius, and the volume formula for a cone to figure out the radius of the cone. Since the radius of the sphere is the same, I could then calculate the volume of the sphere.)
• “When you asked your partner why they needed a specific piece of information, what kind of explanations did you consider acceptable?”

As students respond, highlight any student sketches that include labeled dimensions, and display these for all to see. In particular, contrast students who used volume formulas to determine the value of the radius with those who remembered that the volume of a cone is half the volume of a sphere with the same dimensions (radius and height).

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Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm ‘’ in action today, and it really helped me/my group because .”

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.

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 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 was the same, making the volume (if it were another cylinder) 4 times as great, so the factor of for the cone didn’t bring the volume down below the volume of the cylinder.