Getting Started 
Misconception/Error The student is unable to use the assumptions to derive each of the three formulas. 
Examples of Student Work at this Level The student is unable to correctly derive any of the formulas. The student:
 Attempts some algebraic manipulations of the assumptions but is unable to derive any of the formulas.
 Uses the formula in its own derivation.
 Does not start with any of the given assumptions.
 Simply writes the formulas without any derivations (some of which may contain errors).
 Selects the correct assumptions for the derivations of the area and circumference of a circle, but does not attempt to derive either formula.
 Correctly derives the formula for the area of a circle but is unable to derive the other two formulas.

Questions Eliciting Thinking What formulas are you being asked to derive?
What assumptions can you make? What statements can you use to derive these formulas?
Do you think you will need to use all three assumptions to derive each formula? How can you determine which assumptions lead to which formulas?
Once you have derived the area of a circle formula, can you use it to derive the other formulas? 
Instructional Implications Have the student circle or highlight the definition of and the two previously established statements (e.g., the assumptions) so that what is available to use in the derivations is clear. Then have the student write the area of a circle formula in the margin so that he or she has a clear understanding of what is to be derived. Ask the student to compare the area of a circle formula to the assumptions and the definition of . Guide the student to use the definition of (e.g., = A(1)) as the basis for a substitution into the second assumption (e.g., ) to derive the formula .
Explain to the student that since the area of a circle formula has now been established, it can be used in the derivation of the remaining formulas (if it is needed). Ask the student if he or she can now use the area of a circle formula and the first assumption (e.g., ) to derive the circumference of a circle formula. Then challenge the student to use the definition of the diameter of a circle (e.g., d = 2r) and the circumference of a circle formula to derive the formula for in terms of C and d.
Acknowledge that this sequence of derivations relies on a definition of that is probably novel but allows for the derivation of the previously encountered area and circumference formulas as well as the formula that establishes the relationship between the circumference and the diameter of a circle. 
Making Progress 
Misconception/Error The student’s work contains some errors or omissions. 
Examples of Student Work at this Level The student:
 Displays sufficient work for at least two of the formulas but work is not displayed for one of the formulas.
 Derives the three formulas but one formula contains a mistake and/or is not written in reduced form.

Questions Eliciting Thinking You correctly wrote the formula for question number (one, two, or three) but did not show how it is derived from one of the assumptions or the other formulas. Can you clarify how you derived that formula?
Your formulas are generally correct but one formula contains a mistake. Can you identify your mistake?
Your formulas are generally correct but one formula could be simplified. Can you review your formulas and rewrite any in simpler form, if possible? 
Instructional Implications Group the students at this level in pairs. Ask each student to verbally describe to his or her partner how each formula is derived. Tell the partners to edit (refine) each original response until a clear and concise answer for deriving each formula is agreed upon. Next, return to each student the MFAS tasks Area and Circumference  1 (GGMD.1.1) and Area and Circumference  2 (GGMD.1.1). Ask the partners to repeat the process of verbally describing the derivation of each formula and refining the notation in the written responses until both partners agree on clear and concise answers for each question on the first two tasks. Provide feedback as necessary. 
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 is able to use the definition of and the two assumptions to clearly and concisely derive each of the three formulas. The student writes:
 Since = A(1) and , then .
 Since and , then so that and C = 2r.
 Since d = 2r and C = 2r, then C = d so that .

Questions Eliciting Thinking How does the definition of as the area of the unit circle compare to your previous understanding of ?
Can you interpret the formula ? What does this formula indicate about the relationship between C, d, and ? 
Instructional Implications Ask the student to develop informal arguments for other formulas such as the formulas for the volume of a cylinder, pyramid, and cone. 