Getting Started 
Misconception/Error The student does not know the formula for the volume of a cylinder. 
Examples of Student Work at this Level The student cannot correctly identify a formula for finding the volume of a cylinder. The student writes an incorrect expression or equation and imprecisely describes the meaning of the variables.

Questions Eliciting Thinking What are the parts of a cylinder? If you were to create a net of a cylinder, what twodimensional shapes would you draw?
What is a variable? Is a variable? Why or why not?
What terms describe the dimensions of the cylinder? 
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, and radius. If necessary, review the formula for finding the area of a circle and be sure the student understands how to apply it. Remind the student that the volumes of prisms and cylinders can be found by multiplying the area of their bases by their heights. Similarly, the volume of a pyramid or cone can be found by multiplying the product of the base area and height by onethird. Emphasize the general formulas for finding the volumes of prism and pyramids. Explain to the student that the general formulas along with some basic area formulas is 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 cylinder, 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 (or how pi is spelled). Be sure the student can locate the base, base radius or diameter, and height on a model and in a drawing of a cylinder.
Provide specific examples of cylinders and ask the student to identify a relevant formula and calculate the volume. Provide feedback. 
Making Progress 
Misconception/Error The student does not understand the variables in the formula. 
Examples of Student Work at this Level The student correctly identifies a formula for finding the volume of a cylinder but:
 Does not precisely explain the meaning of each variable and does not correctly label the diagram.

Questions Eliciting Thinking Can you identify any parts of a cylinder? How does the formula you wrote correspond to the diagram?
What is ? Is it a variable? Why or why not?
What is the difference between the base and the area of the base? 
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. Provide the student with both the general formula for finding the volume of a cylinder, V=Bh, and the specific formula, V=, and 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 (or how pi is spelled). The student should understand that pi is a specific value and is not a variable. Be sure the student can locate the base, base radius or diameter, and height on a model and in a drawing of a cylinder.
Provide specific examples of cylinders and ask the student to calculate the volume. Provide feedback. 
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 writes:
 V=Bh or V=.
 V is volume, B is the area of the base, r is the radius of the base, and h is the height of the cylinder.
 The student correctly labels the dimensions on the diagram.

Questions Eliciting Thinking Can you explain in general how to find the volume of any solid?
Can you explain how the two different volume formulas, V=Bh and V=, are related? Why do they result in the same answer? 
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.
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