In this activity, students find the missing dimensions of cylinders when given the volume and the other dimension. A volume equation representing the cylinder is given for each problem to help students focus on solving for the unknown value instead of setting up the equations themselves, which will come later.

In this partner activity, students take turns sharing their initial ideas and first drafts. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). 

The purpose of this discussion is to compare the different strategies used to calculate the unknown values. Invite previously selected students to share their strategies. If one of these strategies is not brought up, share it with the class:

• Guess and check: Plug in numbers for to find a value that makes the statements true. Since the solutions for these problems are small whole numbers, this strategy works well. In other situations, this strategy may be less efficient.
• Divide each side of the equation by the same value to solve for the missing variable: For example, divide each side of by the common factor, . It’s important to remember that is a number that can be multiplied and divided like any other factor.
• Use the structure of the equation to reason about the missing variable: For example, is double , so the missing value must be 2. For the second cylinder, , so the radius must be 3 since .

Conclude the discussion by inviting students to share which strategy they liked best. It is important to note that while all three of these strategies work for the cylinders here, the numbers will not always lead to guess and check being efficient, especially if the value of the volume is approximated instead of written in terms of .

The purpose of this activity is for students to use the structure of the volume formula for cylinders to find unknown dimensions of a cylinder given other dimensions. The students are given the image of a generic cylinder 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 unknown 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 students to share their answers for a row along with any patterns they noticed while filling out the table, such as seeing that since rows f and g have the same radius, the area of the base must also be the same.

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

• Rows a and c have the same radius, which means they have the same base area. The height in row c is double the height in row a, and the volume in row c is double the volume in row a. The height and volume increase proportionally.
• Rows d and g have the same height, but the radius for row g is double the radius for row d, and the volume for row g is quadruple the volume for row d. 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 .

Students will have more opportunities to consider this aspect of volume formulas in future lessons.