Some students may struggle to find the square root of 20.25. Remind students that their calculators can find square roots, and prompt them to use an estimate to check the reasonableness of the calculator output.

This activity gives students an opportunity to determine and request the information needed to infer characteristics of original and dilated solids based on one-, two-, and three-dimensional scale factors.

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 to ask increasingly more precise questions until they get the information they need (MP6).

Monitor for pairs that complete Problem Card 2 by using the radius and the height of the dilated cylinder in the volume formula, and for other pairs who instead apply the cube of the scale factor to the original cylinder's volume.

The goal is to make sure students understand how scaling a three-dimensional figure impacts length, surface area, and volume. After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Select groups that found Problem Card 2’s answer using the volume formula, and other groups that applied the cubed scale factor to the original cylinder’s volume.

Here are some questions for discussion:

• “What was the easiest part about this activity? What was the most difficult part?”
• “Was all the data on each card used? If not, which pieces of data weren’t used?”
• “How did you determine the scale factor for lengths in the first problem?”
• “How did you find the volume of the dilated cylinder on the second card? Did you use the volume formula, or did you use another method?”

Highlight for students the scale factors of and for lengths, surface areas, and volumes, respectively.

In this activity, students are building skills that will help them in mathematical modeling (MP4). They recognize that a geometric solid can be a mathematical model of a real-life object, and have an opportunity to consider the accuracy of that model. They’re prompted to connect surface area and volume to the real-life context of metal to create the container and liquid to fill it.

Some students may double the height of the can, but not the radius, in their drawings. Prompt them to verify that their dilated can has the same proportions as their original.

Some students may identify the scale factor as 2 or as 16. Remind them of the relationship between the scale factor for dimensions, , and the scale factor for surface areas, .

The goal is for students to understand that the cylinder is an inexact mathematical model for the real-life can. The model can give insight into the real-world situation. Ask students to share their thoughts on factors that affect the final cost. Invite them to consider whether the original proportions of the can matter (they don’t matter, because the scale factors are the same regardless of the actual shape of the can).