Getting Started 
Misconception/Error The student does not select an appropriate formula or have an appropriate strategy for finding the volume of the cone. 
Examples of Student Work at this Level The student:
 Selects and attempts to use an incorrect formula.
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 Attempts to determine the volume by multiplying values given in the problem without considering a formula or general strategy for finding volumes of cones.

Questions Eliciting Thinking What is the difference between area and volume? What were you asked to calculate in this problem?
Do you know a general method for finding the volume of a cone? Can you remember a formula for calculating the volume of a cone?
Is the formula you used the correct formula? What could you do to fix it? How can you tell that itâ€™s not correct? 
Instructional Implications Ensure that the student is familiar with cylinders and cones as well as terms used to describe their parts and dimensions such as base, lateral surface, height, slant height, diameter, and radius. If necessary, review the formula for finding the area of a circle and be sure the student understands how to apply it. Explain that the volume of a prism or cylinder can be determined by multiplying the area of a base by the height. Similarly, the volume of a pyramid or cone can be found by multiplying the product of the base area and height by .
Explain or demonstrate why the volume of a cone is onethird the volume of a cylinder with the same base radius and height. Emphasize the general formulas for finding the volumes of prisms and cones. Explain to the student that the general formulas along with some basic area formulas are all that is needed to calculate volumes of prisms, cylinders, pyramids, and cones.
Provide the student with the general formula for finding the volume of a cone, V= Bh, and show how the specific formula, V= , can be easily derived from it. Clearly identify the meaning of the variables in each formula and explain why the two formulas are equivalent. Address any misconceptions about the meaning of pi (e.g., clarify that is not a variable). Be sure the student can locate the base, base radius or diameter, and height on a threedimensional model and on a drawing of a cone.
Provide specific examples of cones and ask the student to identify a relevant formula and calculate the volume. Provide feedback. 
Moving Forward 
Misconception/Error The student does not substitute correct values into the formula for the volume of a cone. 
Examples of Student Work at this Level The student substitutes incorrect values for variables in the formula. For example, the student substitutes:

Questions Eliciting Thinking What do the variables in your formula represent?
How are the radius and diameter related? Can you identify the radius of the cone? 
Instructional Implications Review the terms used to describe the parts and dimensions of cylinders and cones such as base, lateral surface, height, slant height, and radius. Be sure the student can locate the base, base radius or diameter, and height on a model and in the drawing of a cone. Guide the student through the process of identifying the variables in the formula for the volume of a cone and describing what each represents. Provide additional opportunities to find the volume of cones in both mathematical and realworld problems. Provide feedback.
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Almost There 
Misconception/Error The studentâ€™s work contains a minor mathematical error. 
Examples of Student Work at this Level The student selects an appropriate formula, substitutes correct values into the formula, and calculates an answer but the student:
 Uses incorrect units or no units.
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 Writes units incorrectly.
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 Makes a multiplication error when evaluating the formula.
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 Rounds incorrectly or is uncertain of an appropriate decimal place to round the volume.
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Questions Eliciting Thinking What is the unit of measure for volume? Did you include a unit of measure in your answer?
I think you may have made a mistake in your calculations. Can you check your work to see if you can find it?
Is your answer appropriate to the question? How can you decide the appropriate place value for rounding? 
Instructional Implications Provide specific feedback, and allow the student to revise his or her work. If needed, assist the student in choosing an appropriate decimal place to round the volume.
If needed, provide the student with handson activities to better establish the concept of a cubic unit. Ask the student to explain the difference among a centimeter, a square centimeter and a cubic centimeter and why each is used to measure length, area, and volume, respectively. Remind the student to always include a unit of measure in length, area, and volume problems. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student identifies an appropriate formula, V = or V = Bh (where B is the area of the base). The student substitutes correct values into the formula and determines a final answer of 368.43 (or appropriately rounded to a different place) if using 3.14 for .
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Questions Eliciting Thinking What is a cubic centimeter? How does the formula V = determine how many cubic centimeters there are in a cone?
What approximation did you use for ? Would your answer change if you used a different approximation?
Can you write your answer in terms of ? 
Instructional Implications Provide opportunities to solve mathematical and realworld problems by calculating volumes of cylinders, cones, and spheres. Include some figures that are composites of these solids.
Ask the student to develop a model in which the volume of a cone can be approximated with stacked cylinders. Ask the student to consider how his or her model could be improved (e.g., by increasing the number and decreasing the height of the cylinders).
Introduce examples of conical frustums and ask the student to calculate their volumes. Challenge the student to derive the formula for the volume of a conical frustum using similar triangles.
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