Getting Started 
Misconception/Error The student draws conclusions without formally investigating the formula. 
Examples of Student Work at this Level The student:
 Assumes that both the radius of the base and the height must double in order for the volume to double.
 Uses the structure of the cone to informally reason about the effects on the volume.

Questions Eliciting Thinking How did you determine your answers? Did you look at the formula?
Why do you think the volume of a cone increases when the height is doubled? How could you determine by what factor it will increase?
What does it mean to double a value? In math, what is the same as doubling? 
Instructional Implications Explain to the student that it is difficult to know exactly how the volume is affected in each case without using the formula to investigate. Guide the student to use the formula to determine how doubling the height changes the volume of the cone. For example, ask the student to substitute 2h for h in the formula and rewrite the resulting expression in the form . Assist the student in interpreting this result as indicating that the volume is now twice what is was originally. To verify, suggest that the student use the formula to calculate the volume for an arbitrary value of h and then again for twice that value. Then ask the student to use the formula in a similar manner to investigate the questions posed in parts B and C. Provide assistance and feedback as needed.
Provide examples of other formulas and expressions such as A = b + c, A = bc, and . Ask the student to describe each formula using terms such as sum, product, or square (e.g., A is the sum of b and c, A is the product of b and c, A is the product of b and the square of c) to call the studentâ€™s attention to the structure of the formula. Ask the student to determine the effect on A of multiplying c by two in each instance and then to determine the effect on A of multiplying b by two in each instance. Ask the student to compare the effects on the third formula of doubling c versus doubling b and to relate the differences to the structure of the formula. 
Moving Forward 
Misconception/Error The student explains using the formula in at least some instances but draws incorrect conclusions. 
Examples of Student Work at this Level The student:
 Uses the formula to investigate the question in A but assumes the volume will also double in part B when the radius is doubled. In addition, the student does not use the formula or the answers to A and B to determine an answer to C.
 References the formulas but provides incomplete and unclear explanations. In addition, the studentâ€™s conclusion for part B is incorrect.

Questions Eliciting Thinking You substituted numerical values into the formula to investigate what happens when the height is doubled. Do you think you can do that for parts B and C also?
Is doubling the radius of a cone going to have the same effect on the volume of the cone as doubling the height did? Why or why not?
How can you show, with the formula, what will happen to the volume when the height is doubled? When the radius is doubled? When both are doubled? 
Instructional Implications Guide the student to use the formula in all instances to determine the effects on the volume. Encourage the student to algebraically show the effect by substituting 2h for h in the formula and then rewriting the resulting expression in the form . Allow the student to revise his or her responses to the questions.
Provide opportunities for the student to explore and summarize the changes to the perimeter, area, or volume of other two or threedimensional shapes such as circles, rectangles, prisms, spheres, and pyramids when a dimension is doubled. 
Almost There 
Misconception/Error The student is able to determine how the volume will change when the dimensions change but is unable to fully and completely justify his or her responses. 
Examples of Student Work at this Level The student states correct conclusions but:
 Provides no explanation or justification.
 Provides incomplete or imprecise justifications.

Questions Eliciting Thinking How do you know that doubling the height will affect the volume of the cone in that way?
Can you explain to me how you determined your answers?
Can you show, using the formula, what will happen to the volume when the height is doubled? When the radius is doubled? When both are doubled? 
Instructional Implications Ask the student to provide algebraic justifications for each of the three questions and provide feedback that will enable the student to better understand the nature of a mathematical justification.
Provide opportunities for the student to explore changes to the perimeter, area, or volume of other two or threedimensional shapes such as circles, rectangles, prisms, spheres, and pyramids when a dimension is increased by a given factor. Ask the student to justify algebraically the effects of doing so. 
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 uses the formula to investigate each question:
 For question A, the student substitutes 2h for h into the volume formula and shows that . The student concludes the volume is doubled.
 For question B, the student substitutes 2r for r into the volume formula and shows that . The student concludes the volume is increased by a factor of four.
 For question C, the student rewrites the formula as Â to show that . The student concludes the volume is increased by a factor of eight.

Questions Eliciting Thinking How did you know the volume would double when the height doubled? Did you need to try an example or use the formula?
What would happen to the area of the base if the radius is doubled?
Can you describe the effect on the volume when the height is multiplied by some number n? When the radius is multiplied by some number n? 
Instructional Implications Challenge the student to determine the effect of altering parts of more complex expressions. For example, ask the student to determine the effect on F of doubling r in the formula .
Consider implementing MFAS tasks Dot Expressions (ASSE.1.1) and Interpreting Basic Tax (ASSE.1.1). 