Getting Started 
Misconception/Error The student does not understand what it means to derive a formula. 
Examples of Student Work at this Level The student makes no attempt to derive the formula of the cylinder from the formula of the prism. Instead the student:
 Says the two objects are “similar and equal” and writes a formula for the surface area of the prism.
 Attempts to equate the length of the radius of the cylinder to the length of the base of the prism.
 Writes an unrelated formula and does not explain.

Questions Eliciting Thinking What does it mean to use one formula to derive another?
What are you told about the prism and cylinder in this problem?
Where do you think you might start? 
Instructional Implications Explain to the student that in order to derive the formula for the volume of a cylinder from the formula for the volume of a prism, he or she must begin with the formula for the volume of a prism, , and show how it can be transformed into a formula for the volume of a cylinder. Explain a general strategy for the derivation:
 Establish that the volume of the cylinder is equal to the volume of the prism.
 Substitute the expression for the area of the base of the cylinder for the expression that represents the area of the base of the prism in the formula for the volume of the prism.
 Recognize that this formula now can be used to find the volume of the cylinder.
Assist the student in understanding the logic of the derivation. Then guide the student to write the details of each step.
Ask the student to apply similar reasoning to derive a formula for the volume of a cone from the formula for the volume of a pyramid.
Consider implementing MFAS task Volume of a Cone (GGMD.1.1). 
Moving Forward 
Misconception/Error The student does not establish that the volumes of the two solids are the same. 
Examples of Student Work at this Level The student does not establish (using Cavalieri’s Principle) that the volume of the cylinder is the same as the volume of the prism. However, the student substitutes for in the volume formula for the prism, V = lwh and asserts that the volume formula for the cylinder is .

Questions Eliciting Thinking Why are you justified in making the substitution of for in the volume formula for the prism, V = lwh?
If you substitute for in the volume formula for the prism, V = lwh, haven’t you just rewritten the volume formula for the prism? How does this formula become a volume formula for the cylinder?
What else would you need to know about the prism and the cylinder in order to use the volume formula for the prism to derive the volume formula for the cylinder?
Why do you suppose it was given that the prism and the cylinder have the same height and the same crosssectional area at any given height above the base?
If both solids have the same height and crosssectional area at every level, what might that tell you about the bases of the figures? About the volume of each solid?
Do you remember Cavalieri’s Principle? How might it apply in this situation? 
Instructional Implications Assist the student in recognizing the need to establish that the volumes of the two solids are equal. Explain that if the volumes are not equal, then substituting for in the volume formula for the prism results in a formula that still describes the volume of the prism and indicates nothing about the volume of the cylinder. In order to claim the new volume formula applies to the cylinder, it needs to be established that the volume of the cylinder is the same as the volume of the prism. Review Cavalieri’s Principle and guide the student to apply it in order to conclude that the volumes of the two solids are the same. Ask the student to revise his or her derivation. Then, ask the student to apply similar reasoning to derive a formula for the volume of a cone from the formula for the volume of a pyramid.
Consider implementing MFAS task Volume of a Cone (GGMD.1.1). 
Almost There 
Misconception/Error The student’s derivation is incomplete or imprecisely worded. 
Examples of Student Work at this Level The student states or establishes that the volumes of the two solids are equal and substitutes for in the volume formula for the prism, V = lwh, resulting in , a volume formula for the cylinder. However, the student omits an important detail or uses imprecise wording in writing the derivation. For example, the student:
 Says, “Since the areas are the same, the volumes are the same” but is not clear what is meant by “the areas” and does not justify the equality of the volumes (by citing Cavalieri’s Principle).
 Uses Cavalieri’s Principle to establish that the volumes of the solids are the same but does not clearly derive the formula for the volume of the cylinder from the formula for the volume of the prism.

Questions Eliciting Thinking What did you mean by “the areas”?
How do you know the volumes are the same? Do you remember Cavalieri’s Principle? How might it apply in this situation?
How does knowing the two solids have the same volume help you find the formula for the cylinder?
Given that the volumes are equal, can you show me, algebraically, how you derived a formula for the volume of the cylinder from the formula for the volume of a prism? 
Instructional Implications Provide feedback to the student concerning any omissions or use of imprecise language. Allow the student to revise his or her response. Show the student a complete and convincing derivation, so the student may compare his or her derivation to it. Then, ask the student to apply similar reasoning to derive a formula for the volume of a cone from the formula for the volume of a pyramid.
Consider implementing other MFAS tasks for GGMD.1.1. 
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 explains that since the prism and the cylinder have the same height and the same crosssectional area at any given height above the base, their volumes are equal (by Cavalieri’s Principle). So the area of the base of the cylinder, , can be substituted for the area of the base of the prism, , in the volume formula for the prism, V = lwh, resulting in , a volume formula for the cylinder. 
Questions Eliciting Thinking Suppose the dimensions of the prism are measured in inches and the dimensions of the cylinder are measured in centimeters? Would this make a difference in the derivation of the volume formula?
Can you explain why the volume formula of a prism can be given as V = lwh? 
Instructional Implications Challenge the student to use Cavalieri’s Principle to derive a formula for the volume of a cone.
Consider implementing the other MFAS tasks for this standard, including Area and Circumference1 (GGMD.1.1), Area and Circumference2 (GGMD.1.1), Area and Circumference3 (GGMD.1.1). Ensure the student has an understanding of how the volume of a prism and pyramid relate by implementing MFAS task The Volume of a Pyramid (GGMD.1.1) before asking the student to attempt MFAS task Volume of a Cone (GGMD.1.1). 