8.1: Going Backwards (5 minutes)
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.
The image shows a cone that has a base with area \(36\pi\) square centimeters. The cone has been dilated using the top vertex as a center. The area of the dilated cone’s base is \(729\pi\) square centimeters.
What was the scale factor of the dilation?
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).
8.2: Info Gap: Originals and Dilations (20 minutes)
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.
Design Principle(s): Cultivate Conversation
Supports accessibility for: Memory; Organization
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
- Silently read the information on your card.
- Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
- Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
- Read the problem card, and solve the problem independently.
- Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
- Silently read your card and think about what information you need to answer the question.
- Ask your partner for the specific information that you need.
- Explain to your partner how you are using the information to solve the problem.
- When you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.
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.
8.3: Jumbo Can (15 minutes)
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.
Design Principle(s): Support sense-making
Supports accessibility for: Memory; Organization
A beverage company manufactures and fills juice cans. They spend $0.04 on materials for each can, and fill each can with $0.27 worth of juice.
The marketing team wants to make a jumbo version of the can that’s a dilated version of the original. They can spend at most $0.16 on materials for the new can. There’s no restriction on how much they can spend on the juice to fill each can. The team wants to make the new can as large as possible given their budget.
- By what factor will the height of the can increase? Explain your reasoning.
- By what factor will the radius of the can increase? Explain your reasoning.
- Create drawings of the original and jumbo cans.
- What geometric solid do the cans resemble? What are some possible differences between the geometric solid and the actual can?
- What will be the total cost for materials and juice fill for the jumbo can? Explain or show your reasoning.
- Describe any other factors that might cause the total cost to be different from your answer.
Are you ready for more?
As of 2019, the Burj Khalifa, located in Dubai, was the tallest building in the world. Suppose a scale model of the Burj Khalifa (without antennae) is 30 inches tall.
- To what scale is this model? You will need to use the internet or another resource to find the actual height of the building.
- How tall would a model of the Eiffel tower be at this scale?
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).
The main idea of the lesson is that if we know the factor by which the volumes, surface areas, or lengths change when a solid is dilated, it’s possible to find the factor by which the remaining values change. Here are some questions for discussion:
- “Suppose you know the volumes of an original solid and its dilation. How can you find the factor by which the surface area changed?” (Divide the dilated volume by the original volume to find the factor by which the volume changed. Then take the cube root of that to find the scale factor of dilation. Finally, square that value to get the surface area scale factor.)
- “Suppose you know the surface areas of an original solid and its dilation. How can you find the factor by which the volume changed?” (Divide the dilated volume by the original volume to find the factor by which the surface area changed. Then take the square root of that to find the scale factor of dilation. Finally, cube that value to get the surface area scale factor.)
- “What are some real-world applications for these concepts?” (One example is designing any kind of packaging including shampoo, cereal, and coffee—we often need to understand how a change in volume for these products affects the packaging materials and product dimensions. Other examples include designing spaces that need a certain volume like car trunks, cargo containers, and tanker trucks, and engineering objects that are expensive to paint, such as airplanes.)
8.4: Cool-down - Dog Food Bags (5 minutes)
Student Lesson Summary
Suppose a solid is dilated. If we know the factor by which the surface area or volume scale changed, we can work backwards to find the scale factor of dilation. Then we can use that information to solve problems.
A company sells 10 inch by 10 inch by 14 inch 5-gallon aquariums, but a museum wants to buy a 135-gallon aquarium with the same shape. The company needs to know the dimensions of the new tank and by what factor the surface area will change.
Gallons are a measure of volume. So, the volume of the tank increases by a factor of \(135\div 5=27\). To find the scale factor for the dimensions of the tank, calculate the cube root of 27, or 3. This tells us that the height, length, and width of the tank will each be multiplied by 3. Next, we can square the scale factor of 3 to find that the tank’s surface area will increase by a factor of 32 = 9.
|height (inches)||10||\(10\boldcdot 3=30\)|
|length (inches)||14||\(14\boldcdot 3=42\)|
|width (inches)||10||\(10\boldcdot 3=30\)|
|surface area (square inches)||760||\(760\boldcdot 9=6,\!840\)|
|volume (gallons)||5||\(5\boldcdot 27=135\)|