Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
The purpose of this Warm-up is to prepare students for the Information Gap activity that follows. First, students are given a problem with incomplete information. They are prompted to brainstorm what they need to know to solve a problem that involves scale factor. Next, they practice asking for information, explaining the rationale for their request, and persevering if their initial questions are unproductive (MP1). Once students have enough information, they solve the problem.
Launch
Display the first two paragraphs of the Task Statement for all to see. Ask students to solve the problem. When they recognize that not enough information is given, display the second prompt and ask what they need to know to be able to solve the problem. Display the sentence frame “Can you tell me ?” for all to see and invite students to use it to frame their information request. Give students 2 minutes of quiet think time.
Activity
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Student Task Statement
Activity Synthesis
Tell students that the problem is a part of an Information Gap routine. In the routine, one person has a problem with incomplete information, and another person has data that can help with solving it. Explain that it is the job of the person with the problem to think about what is needed to answer the question, and then request it from the person with information.
Tell students they will try to solve the problem this way as a class to learn the routine. In this round, the students have the problem, and the teacher has the information needed to solve the problem.
Ask students, “What specific information do you need to find out the height and radius of the original cylinder?”
Select students to ask their questions. Encourage students to use the format of “Can you tell me ?” Respond to each question with, “Why do you need to know ?”
Once students justify their question, only answer questions if they can be answered using these data:
The dilated cylinder’s volume is in3.
The original cylinder’s volume is in3.
The radius of the original prism is 2 in.
The height of the dilated prism is 15 in.
The original prism’s surface area is in2.
The dilated prism’s surface area is in2.
If students ask for information that is not on the data card, respond with, “I don’t have that information.”
When students think they have enough information, give them 2 minutes to solve the problem. (a cylinder with labeled dimensions: height 5 cm and radius 2 cm)
Tell students they will work in small groups and use the routine to solve problems in the next activity.
Math Community
After the Warm-up, display the Math Community Chart with the “Doing Math” actions added to the teacher section for all to see. Give students 1 minute to review. Then share 2–3 key points from the teacher section and your reasoning for adding them. For example,
If “questioning vs. telling,” a shared reason could focus on your belief that students are capable mathematical thinkers and your desire to understand how students are making meaning of the mathematics.
If “listening,” a shared reason could be that sometimes you want to sit quietly with a group just to listen and hear student thinking and not because you think the group needs help or is off-track.
After sharing, tell students that they will have the opportunity to suggest additions to the teacher section during the Cool-down.
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 surface area by the original surface area 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 volume 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.)
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 backward 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 . 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.
original aquarium
dilated aquarium
height (inches)
10
length (inches)
14
width (inches)
10
surface area (square inches)
760
volume (gallons)
5
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This activity gives students an opportunity to determine and request the information needed to find measurements of original and dilated solids, using their knowledge of one-, two-, and three-dimensional scale factors.
This is the first Information Gap activity in the course. In this activity, students represent sequences in different ways but do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.
The Information Gap routine facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, thereby creating a need for communication. This routine supports language development by providing students with opportunities to ask for and share information, and to justify their reasoning within conversation. 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 who instead apply the cube of the scale factor to the original cylinder’s volume.
This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.
Launch
Display for all to see the graphic that illustrates a framework for the Information Gap routine.
Explain that in an Information Gap routine students work with a partner. One partner gets a problem card with a question that doesn’t have enough given information, and the other partner gets a data card with information relevant to the problem card.
The person with the problem card asks questions like “Can you tell me ?” and is expected to explain what they will do with the information. If that person asks for information that is not on the data card (including the answer!) and gives their reason, then the person with the data card must respond with, “I don’t have that information.” The person with the data card should just be providing information, not making assumptions. Note that it is okay to help a stuck partner by saying something like “I don’t have the height of the original cylinder. I only have information about the dimensions of the prism.”
Once the partner with the problem card has enough information, both partners look at the problem card and solve the problem independently.
Arrange students in groups of 2 or 4. If students are new to the Information Gap routine, allowing them to work in groups of 2 for each role supports communication and understanding. In each group, distribute a problem card to one student (or group) and a data card to the other student (or group). After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.
Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Information Gap graphic visible throughout the activity or provide students with a physical copy. Supports accessibility for: Memory, Organization
Activity
None
Student Task Statement
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 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. “Can you tell me _____?”
Explain to your partner how you are using the information to solve the problem. “I need to know _____ because _____.”
Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.
If your teacher gives you the data card:
Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know _____?”
Listen to your partner’s reasoning, and ask clarifying questions. Give only information that is on your card. Do not figure out anything for your partner!
These steps may be repeated.
Once your partner has enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
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.
None
Activity Narrative
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.
Launch
Ask students what types of beverages they and their family drink from cans (sparkling water, energy drinks, coconut water, juice, tea). 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.
Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students understand that the materials cost of the can is related to the can’s surface area and the fill cost is related to the can’s volume. Supports accessibility for: Memory, Organization
Activity
None
Student Task Statement
A beverage company manufactures and fills juice cans. The company spends $0.04 on materials for each can, and fills 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.
Student Response
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Building on Student Thinking
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, .
Activity Synthesis
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).
Standards Alignment
Building On
Addressing
HSN-Q.A.1
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
