Lesson 8

Students calculate a scale factor given the areas of the circular base of a cone and the base of its dilation. This connects the concept of surface area dilation to cross sections, and gives practice with non-integer roots.

Monitor for pairs of students who initially consider an answer of 20.25 but come to a consensus that the scale factor is 4.5.

Arrange students in groups of 2. Provide access to scientific calculators. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

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.

Select a pair of students to explain their reasoning. If the pair considered 20.25 but moved to an answer of 4.5, ask how they knew 20.25 wasn’t correct.

Discuss with students how they can decide if 4.5 is a reasonable value for the square root of 20.25, considering the fact that 4² = 16 and 5² = 25. If time allows, ask students to calculate the scale factor for the solid’s volume (4.5³ = 91.125).

This info gap 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 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).

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 that instead apply the cube of the scale factor to the original cylinder's volume.

Here is the text of the cards for reference and planning:

Tell students they will continue to work with scale factors for dilated solids. 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 reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

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 \(k,k^2 ,\) and \(k^3\) 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 container materials and fill.

Ask students about their favorite sparkling water or juice. Tell students they’ll be playing the part of a beverage company that’s considering introducing a new product. Consider showing students several different styles of beverage cans, including mini-sizes, tall and narrow cans, and standard cans.

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, \(k\), and the scale factor for surface areas, \(k^2\).

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).