Getting Started 
Misconception/Error The student is unable to show or explain why the area of equals area of . 
Examples of Student Work at this Level The student:
 Attempts to use the formula , but does not address the similarity of AB'C' and ABC and never establishes the relationship between the heights of the triangles.
 Attempts to use the formula , but assumes that and represent the base and height of ABC (and and represent the base and height of AB'C').
 Assumes the statement is true by the Substitution Property.
 Attempts an explanation in terms of the scale factor but, never establishes that AB'C' ABC or references the specific relationship between the areas of the triangles in terms of the scale factor.

Questions Eliciting Thinking How do you know that the height of AB'C' is r times the corresponding height of ABC?
How are the height and the base of a triangle related? If is the base of ABC, can be its height?
You mentioned the scale factor. Can you elaborate on how the scale factor relates to areas of similar triangles? 
Instructional Implications Review or establish the general relationship between the areas of similar triangles (e.g., if two triangles are similar then the ratio of their areas is the square of the ratio of any two corresponding sides). Show the student how this theorem can be applied to conclude that the Area of AB'C' = Area of ABC. Make sure the student understands that in order to apply this theorem, it must first be determined that AB'C' ABC. If necessary, review the SAS Similarity Theorem and assist the student in using the theorem to show that AB'C' ABC. Then explain that since it was given that , then the Area of AB'C' = Area of ABC.
Provide opportunities for the student to use this theorem (e.g., if two triangles are similar then the ratio of their areas is the square of the ratio of any two corresponding sides) to solve problems involving areas of similar figures. 
Moving Forward 
Misconception/Error The student does not to show that the Area of = Area of . 
Examples of Student Work at this Level The student shows (or explains) that the Area of AB'C' = Area of ABC but, is unable to show that the Area of = Area of . The student:
 Does not realize that the area of is n times the area of ABC and the area of is n times the area of AB'C'.
 Attempts to calculate the areas of the polygons using the formula .

Questions Eliciting Thinking What is the second question asking you to show or explain? What do and represent? What does n represent?
What is the relationship between n and ? What are the relationships among the Area of AB'C', n, and ?
If the variable n represents the number of sides of the ngon, what does n tell you about the number of congruent isosceles triangles that compose the ngon?
Do you know the apothem of either polygon? If you do not know the apothem, can you use the formula ? 
Instructional Implications Assist the student with recognizing that n is both the number of sides of the regular polygon (ngon) and the number of congruent isosceles triangles that compose the regular polygon (ngon). Ask the student to decompose the ngon into n congruent triangles by sketching the remaining triangles in the diagram. Guide the student to observe that the Area of = Area of ABC and the Area of = Area of AB'C'. Assist the student, in writing this explanation mathematically [e.g., Area of = n(Area of AB'C' ) = n( Area of ABC) = Area of ABC = Area of ].
Provide additional opportunities to decompose regular ngons into n congruent triangles in order to find their areas. Use this as an opportunity to develop the formula . 
Almost There 
Misconception/Error The student provides an incorrect or incomplete explanation of A(r) = A(1). 
Examples of Student Work at this Level The student shows that the Area of AB'C' = Area of ABC, and the Area of = Area of , but does not provide a complete explanation of A(r) = A(1). The student:
 Does not observe the convergence relationship between the regular ngons and their circumscribed circles.
 Has difficulty verbalizing the convergence relationship between the regular ngons and their circumscribed circles.
 Uses unconventional notation that renders parts of the explanation unclear or ambiguous.

Questions Eliciting Thinking As n increases, what happens to the ngon in relation to the circle?
What happens to the relationship between the radius of the circle and the apothem of the ngon as n increases?
You seem to have an overall understanding of the progression from Area of AB'C' to A(r) = A(1) using Area of = Area of , but some of your work or notation is unclear and/or inaccurate. Can you explain what you meant by (refer to a part of the student’s work that is unclear)? 
Instructional Implications Explain the convergence relationship between the regular ngons and their circumscribed circles as n increases. Assist the student in adopting and using language such as, “As n increases, the area of the ngon approaches the area of the circle.” Ask the student to also consider the relationship between the apothem of the ngon and the radius of the circle as n increases. Explain that as n increases, the length of the apothem approaches the length of the radius. Explain how this convergence relationship can be used to reason from the relationship between the areas of the ngons to the relationship between the areas of their circumscribed circles.
After the student corrects mistakes and/or refines explanations, consider implementing the final MFAS task in this sequence Area and Circumference  3 (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:
 Since and , and is a shared angle, ABC AB'C' (by the SAS Similarity Theorem). Consequently,
 Area of AB'C' = Area of ABC.
 The Area of = n(Area of AB'C' ) = n( Area of ABC) = n· Area of ABC = Area of .
 As n increases, the length of the apothem of approaches the radius of circle A of radius r, and the area of approaches the area of circle A of radius r. Likewise, as n increases, the length of the apothem of approaches the radius of the unit circle (r = 1), and the area of approaches the area of the unit circle. Given that Area of = Area of , then A(r) = A(1).

Questions Eliciting Thinking If is defined as the area of a unit circle A(1), then what is A(r), ? Explain. 
Instructional Implications Ask the student to demonstrate that when .
Implement the final MFAS task in this sequence Area and Circumference  3 (GGMD.1.1). 