This Math Talk focuses on finding the unknown value in an equation. It encourages students to think about the fraction and to rely on the structure of the equations to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students are solving for the unknown length of the radius or height of a cone given its volume.
To solve each equation mentally, students need to look for and make use of structure (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Solve each equation mentally.
Student Response
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Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed , so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
16.2
Activity
Optional
Standards Alignment
Building On
Addressing
8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
The purpose of this activity is for students to calculate the radius of the cone given the volume and height. This activity is similar to an activity in a previous lesson in which students calculated the radius of a cylinder given the volume and height. The difference here is that solving for the unknown takes an additional step since there is also a in the equation.
Launch
Arrange students in groups of 2. Give students 2–3 minutes of quiet work time followed by time to share their strategies with their partner. Follow with a whole-class discussion.
MLR8 Discussion Supports. Display sentence frames for students to use when they share their strategies with their partners. Examples: “The radius is because ” or “To find the radius, first I . Then, I .” or “I used the by .” Advances: Speaking, Conversing
Activity Synthesis
The purpose of this discussion is for students to share strategies they used to calculate the radius. Invite 1–3 students to share their thinking. Ask students, “Which do you think is more challenging: calculating the radius of a cone or a cylinder when the heights and volumes of the shapes are known? Why?” (I think they are the same: I just have to work with when calculating the radius of the cone, which adds an extra step.)
16.3
Activity
Standards Alignment
Building On
Addressing
8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
The purpose of this activity is for students to use the structure of the volume formula for cones to calculate missing dimensions of a cone given other dimensions. Students are given the image of a generic cone 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 missing 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).
Launch
Give students 6–8 minutes of quiet work time, followed by a whole-class discussion. As they work, select students who are using the information given in some rows to find missing information in other rows to share during the Activity Synthesis.
Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, ask students to complete two lines of the table at a time and then assess for accuracy and comprehension. Supports accessibility for: Organization, Attention
Activity
None
Each row of the table has some information about a particular cone. Complete the table with the missing dimensions.
diameter (units)
radius (units)
area of the base (square units)
height (units)
volume of cone (cubic units)
a
4
3
b
8
c
d
20
e
6
f
3
Activity Synthesis
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 2–3 previously selected students to share the strategies they used to fill in the missing information. Ask students:
“Which information, in your opinion, was the hardest to calculate?”
“If you had to pick two pieces of information given in the table which would you want? Why?”
If not brought up by students, highlight the following rows:
Rows a and e have the same radius, which means they have the same base area. The height in row e is double the height in row a, and the volume in row e is double the volume in row a. The height and volume increase proportionally.
Rows a and b have the same height, but the radius for row b is double the radius for row a, and the volume for row b is quadruple the volume for row a. 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 .
If time allows, select a few rows of the table, and ask students how they might find the volume of a cylinder with the same radius and height as the cone. (Multiply by 3.)
Conclude the discussion by making sure students understand that when working with the volume formula for either a cylinder or cone, if they know two out of three values for the radius, height, and volume, they can always calculate the third.
16.4
Activity
Standards Alignment
Building On
Addressing
8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
The purpose of this activity is for students to reason about the volume of popcorn that cone- and cylinder-shaped popcorn cups hold and the price they pay for it. Students start by picking the popcorn container they would buy (without doing any calculations) and then work with a partner to answer the question of which container is a better value. The fact that one container (the cone) has a wider diameter and looks like it has more coming out of the top might sway students to think that the cone is a better deal. However, the difference in the volume amounts should support the fact that even with a taller height and a wider diameter, the cone still holds less volume than the cylinder.
After having time to complete some calculations to support or reject their initial choice, groups have opportunities to explain their reasoning and critique the reasoning of others during the discussion (MP3).
Launch
Arrange students in groups of 2. Give students 1 minute of quiet think time to decide which popcorn container they would purchase. Students are not asked to use any written calculations to determine this answer. Survey the class, and display the results for all to see. Keep this information displayed to refer back to during the Activity Synthesis.
Give students 3–4 minutes to work with their partner to determine which container is a better value. As they work, select groups reasoning about the choice using both volume formulas and calculating the cost per unit to share during the whole-class discussion.
MLR5 Co-Craft Questions. Keep books or devices closed. Display only the problem stem and the image, without revealing the question. Give students 2–3 minutes to write a list of mathematical questions that could be asked about this situation, before comparing their questions with a partner. Invite each group to contribute one written question to a whole-class display. Ask the class to make comparisons among the shared questions and their own. Reveal the intended question for this task, and invite additional connections. Advances: Reading, Writing
Activity
None
A movie theater offers two containers:
Which container is the better value? Use 3.14 as an approximation for .
Student Response
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Building on Student Thinking
If students try to use just the image to reason about which is a better deal without noticing the images are not in the same scale, consider asking:
“How did you choose the cone or cylinder as the better deal?”
“How could the measurements given help you make your choice?”
Activity Synthesis
Survey the class again for which container they would purchase. Display the new results next to the original results for all to see during the discussion.
Invite previously selected groups to share their arguments for which container has a better value. Encourage groups to share details of their calculations, and record these for all to see in order to mark any similarities and differences between the groups’ arguments. After each group shares, ask the class if they agree or disagree with any of the statements made, and give students opportunities to justify their reasoning.
Here are some questions to choose from to help students think deeper about the situation:
“Do you think your volume calculations overestimate or underestimate the amount of popcorn each container can hold?” (I think the calculations underestimate the amount because the popcorn piles higher than the lip of the container.)
“Why do you think movie theaters charged more for the cone?” (It may look like it has more volume to some people since it has a larger diameter and height.)
“Do you think a lot of people would buy the cone over the cylinder?” (Yes. The 50-cent difference is not a lot, and since it is taller and wider, people might think the cone is bigger. Although the cylinder is a better value, there may be other considerations, like how easy the cup is to hold or put in the cup holder in the seat or whether you have time in line to calculate the value.)
Lesson Synthesis
The purpose of this discussion is to for students to engage with the volume equation and notice that since there are two variables, and , that are unknown, there are many possible solutions.
Display the image of the popcorn cone from the activity "Popcorn Deals," including the dimensions of the cone. Ask students, “What size of cylinder cup would you need to have the same volume as the cone?”
Working in groups of 2, tell partners to determine the height and radius of a possible cylinder of equivalent volume and to make a sketch with labels on the dimensions. Display sketches, and invite students to share the strategies they used to find their cylinders. Highlight that there are many possible cylinders that would meet the requirements of having the same volume as the cone.
Student Lesson Summary
As we saw with cylinders, the volume of a cone depends on the radius of the base and the height :
If we know the radius and height, we can find the volume. If we know the volume and one of the dimensions (either radius or height), we can find the other dimension.
For example, imagine a cone with a volume of cm3, a height of 3 cm, and an unknown radius . From the volume formula, we know:
Looking at the structure of the equation, we can see that , so the radius must be 8 cm.
Now imagine a different cone with a volume of cm3, a radius of 3 cm, and an unknown height . Using the formula for the volume of the cone, we know:
So, the height must be 6 cm. Can you see why?
Standards Alignment
Building On
6.EE.B.5
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
The volume of a cone with radius is given by the formula .
This cone has a height 3 units and radius units.
The volume of this cone is given by the equation cubic units.
This statement is true:
What does the radius of this cone have to be? Explain how you know.
Student Response
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Building on Student Thinking
Students might struggle with the while solving for the unknown radius length. Encourage students to think about the strategies they used in the Warm-up to work with the .
