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 cone from the formula of the pyramid. Instead the student:
 Describes a concept unrelated to the task (e.g., the volume of the cone is one third the volume of a cylinder).
 Writes an incorrect formula or only part of the formula for the volume of a cone.
 Attempts to write an explanation of the volume of a cone formula.

Questions Eliciting Thinking What does it mean to use one formula to derive another?
What are you told about the pyramid and the cone 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 cone from the formula for the volume of a pyramid, he or she must begin with the formula for the volume of a pyramid, , and show how it can be transformed into a formula for the volume of a cone. Explain a general strategy for the derivation:
 Establish that the volume of the cone is equal to the volume of the pyramid.
 Substitute the expression for the area of the base of the cone for the expression that represents the area of the base of the pyramid in the formula for the volume of the pyramid.
 Recognize that this formula now can be used to find the volume of the cone.
Assist the student in understanding the logic of the derivation. Then guide the student to write the details of each step.
Consider implementing MFAS task Volume of a Cylinder (GGMD.1.1). 
Moving Forward 
Misconception/Error The student does not clearly 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 cone is the same as the volume of the pyramid. However, the student substitutes for in the volume formula for the pyramid, and asserts that the volume formula for the cone is .

Questions Eliciting Thinking Why are you justified in making the substitution of for in the volume formula for the pyramid, ?
If you substitute for in the volume formula for the pyramid, , haven’t you just rewritten the volume formula for the pyramid? How does this formula become a volume formula for the cone?
What else would you need to know about the pyramid and the cone in order to use the volume formula for the pyramid to derive the volume formula for the cone?
Why do you suppose it was given that the pyramid and the cone 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?
What do you know about 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 pyramid, results in a formula that still describes the volume of the pyramid and indicates nothing about the volume of the cone. In order to claim the new volume formula applies to the cone, it needs to be established that the volume of the cone is the same as the volume of the pyramid. Review Cavalieri’s Principle and guide the student to apply it in order to conclude that the volumes of the two solids are equal. 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 cylinder from the formula for the volume of a prism (if not done previously).
Consider implementing MFAS task Volume of a Cylinder (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 (l·w) in the volume formula for the pyramid, , resulting in , the volume formula for the cone. However, the student omits an important detail or uses imprecise wording in writing the derivation. For example, the student:
 States that the volumes of the solids are the same but does not justify this assertion.
 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 cone from the formula for volume of the pyramid.

Questions Eliciting Thinking How do you know the volumes are the same? What do you know about 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 cone?
Given that the volumes are equal, can you show me, algebraically, how you derived a formula for the volume of the cone from the formula for the volume of a pyramid? 
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 cylinder from the formula for the volume of a prism (if not done previously).
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 pyramid and the cone have the same height and the same crosssectional area at any given height above the base, their volumes are equal (by Cavalieri’s Principle). Therefore, the area of the base of the cone, , can be substituted for the area of the base of the pyramid, , in the volume formula for the pyramid, , resulting in , the volume formula for the cone. 
Questions Eliciting Thinking Could Cavalieri’s Principle be used to derive the volume of a cylinder formula? If yes, to what solid would you compare the cylinder to in the derivation? If not, why not?
Suppose the dimensions of the pyramid are measured in centimeters and the dimensions of the cone are measured in inches. Would this make a difference in the derivation of the volume formula? Why or why not? 
Instructional Implications Provide the student with an example of a square pyramid and cone in which the edge of the square base is equal to the diameter of the circular base. Have the student determine the ratio of the area of the bases, the ratio of the crosssectional areas, and the ratio of the volumes of the cone and the pyramid.
Consider implementing other MFAS tasks such as Area and Circumference1 (GGMD.1.1), Area and Circumference2 (GGMD.1.1), Area and Circumference3 (GGMD.1.1), Volume of a Pyramid (GGMD.1.1), or Volume of a Cylinder (GGMD.1.1). 