standard_description: 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.
problem_activity: Activity
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 approximations and exact values of
\(\pi\)
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. Identify students who use this strategy, and ask them to share during the discussion.
Launch
Give students 6–8 minutes of quiet work time, followed by a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. 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
Student Facing
[IMG]
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) |
|------------------|-----------------|---------------------------------|----------------|------------------------------|
|                  |        4        |                                 |       3        |                              |
|                  | \(\frac{1}{3}\) |                                 |       6        |                              |
|                  |                 |           \(144\pi\)            |  \(\frac14\)   |                              |
|        20        |                 |                                 |                |          \(200\pi\)          |
|                  |                 |                                 |       12       |          \(64\pi\)           |
|                  |                 |                                 |       3        |             3.14             |
Student Response
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Student Facing
Are you ready for more?
A
frustum
is the result of taking a cone and slicing off a smaller cone using a cut parallel to the base.
[IMG]
Find a formula for the volume of a frustum, including deciding which quantities you are going to include in your formula.
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Students might try to quickly fill in the missing dimensions without the proper calculations. Encourage students to use the volume of a cone equation and the given dimensions to figure out the unknown dimensions.
Activity Synthesis
Select previously identified 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 information would you want? Why?”
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)
Make sure students understand that when working with the volume formula for either a cylinder or cone, if they know two out of three for radius, height and volume, they can always calculate the third.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each strategy that is shared, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This will help students to produce and make sense of the language needed to communicate their own ideas when calculating missing dimensions of a cone.
Design Principle(s): Support sense-making; Optimize output (for explanation)


aligns: no