The purpose of this activity is for students to calculate the radius of the cone given the volume and height. This activity is similar to an activity in a previous lesson in which students calculated the radius of a cylinder given the volume and height. The difference here is that solving for the unknown takes an additional step since there is also a in the equation.

The purpose of this discussion is for students to share strategies they used to calculate the radius. Invite 1–3 students to share their thinking. Ask students, “Which do you think is more challenging: calculating the radius of a cone or a cylinder when the heights and volumes of the shapes are known? Why?” (I think they are the same: I just have to work with when calculating the radius of the cone, which adds an extra step.)

The purpose of this activity is for students to use the structure of the volume formula for cones to calculate missing dimensions of a cone given other dimensions. Students are given the image of a generic cone with marked dimensions for the radius, diameter, and height to help their reasoning about the different rows in the table.

While completing the table, students work with exact values of as well as statements that require reasoning about squared values. The final row of the table asks students to find missing dimensions given an expression representing volume that uses letters to represent the height and the radius. This requires students to manipulate expressions consisting only of variables representing dimensions.

Encourage students to make use of work done in some rows to help find missing information in other rows. By paying attention to what rows have values in common, students can use the structure of the table and their knowledge of the volume formula to calculate related values more efficiently (MP7).

The purpose of this discussion is to make visible the different strategies students used to calculate the values in the table and to highlight some key relationships between radius, height, and volume.

Display the table from the Task Statement for all to see. Invite 2–3 previously selected students to share the strategies they used to fill in the missing information. Ask students:

• “Which information, in your opinion, was the hardest to calculate?”
• “If you had to pick two pieces of information given in the table which would you want? Why?”

If not brought up by students, highlight the following rows:

• Rows a and e have the same radius, which means they have the same base area. The height in row e is double the height in row a, and the volume in row e is double the volume in row a. The height and volume increase proportionally.
• Rows a and b have the same height, but the radius for row b is double the radius for row a, and the volume for row b is quadruple the volume for row a. The radius and volume do not increase proportionally.

Give students 30 seconds of quiet think time, then invite 1–2 students to share why they think this is happening. (The volume formula for a cylinder is  . Doubling the value of  doubles the volume , but doubling the value of quadruples the volume because the radius is squared and .

If time allows, select a few rows of the table, and ask students how they might find the volume of a cylinder with the same radius and height as the cone. (Multiply by 3.)

Conclude the discussion by making sure students understand that when working with the volume formula for either a cylinder or cone, if they know two out of three values for the radius, height, and volume, they can always calculate the third.

The purpose of this activity is for students to reason about the volume of popcorn that cone- and cylinder-shaped popcorn cups hold and the price they pay for it. Students start by picking the popcorn container they would buy (without doing any calculations) and then work with a partner to answer the question of which container is a better value. The fact that one container (the cone) has a wider diameter and looks like it has more coming out of the top might sway students to think that the cone is a better deal. However, the difference in the volume amounts should support the fact that even with a taller height and a wider diameter, the cone still holds less volume than the cylinder.

After having time to complete some calculations to support or reject their initial choice, groups have opportunities to explain their reasoning and critique the reasoning of others during the discussion (MP3).

If students try to use just the image to reason about which is a better deal without noticing the images are not in the same scale, consider asking:

• “How did you choose the cone or cylinder as the better deal?”
• “How could the measurements given help you make your choice?”

Survey the class again for which container they would purchase. Display the new results next to the original results for all to see during the discussion.

Invite previously selected groups to share their arguments for which container has a better value. Encourage groups to share details of their calculations, and record these for all to see in order to mark any similarities and differences between the groups’ arguments. After each group shares, ask the class if they agree or disagree with any of the statements made, and give students opportunities to justify their reasoning.
Here are some questions to choose from to help students think deeper about the situation:

• “Do you think your volume calculations overestimate or underestimate the amount of popcorn each container can hold?” (I think the calculations underestimate the amount because the popcorn piles higher than the lip of the container.)
• “Why do you think movie theaters charged more for the cone?” (It may look like it has more volume to some people since it has a larger diameter and height.)
• “Do you think a lot of people would buy the cone over the cylinder?” (Yes. The 50-cent difference is not a lot, and since it is taller and wider, people might think the cone is bigger. Although the cylinder is a better value, there may be other considerations, like how easy the cup is to hold or put in the cup holder in the seat or whether you have time in line to calculate the value.)