Getting Started 
Misconception/Error The student is unable to correctly complete the square. 
Examples of Student Work at this Level The student understands that the equation should be written in centerradius form and attempts to complete the square but is unable to do so correctly. The student:
 Attempts to factor .
 Reorganizes the terms, writing , but is unable to continue.
 Completes the square by adding three or six instead of 3^{2}.
 Adds nine to complete the square but rewrites as or .

Questions Eliciting Thinking Is completing the square the same as factoring?
What does it mean to complete the square? What will a quadratic expression look like when it is in this form?
What do you need to complete the square? Can you complete the square with only one variable? Why or why not?
Can you explain more about how you completed the square?
What is the equation of a circle in centerradius form?
Your initial steps are correct. Why were you unable to continue? 
Instructional Implications Review both the factored and expanded forms of a perfect square trinomial and the relationship between these two forms. Provide instruction on completing the square with expressions independent of equations of circles (e.g., explain each step in the process of rewriting as or as .
Model the process of completing the square and rewriting the given equation in centerradius form, . Explain that an advantage of centerradius form is that the coordinates of the center, (h, k), and the radius, r, can be read from the equation. Ask the student to identify the coordinates of the center and the radius of the circle whose equation is given. Provide additional examples of equations written in general form and ask the student to rewrite each equation in centerradius form and identify the coordinates of its center and its radius. 
Moving Forward 
Misconception/Error The student is unable to correctly rewrite the equation in an equivalent form. 
Examples of Student Work at this Level The student correctly completes the square by adding nine to and rewrites the trinomial as but makes an error in rewriting the equation in an equivalent form. The student:
 Adds nine to the left side of the equation but neglects to add it to the right side.
 Subtracts five from the right side of the equation but adds five to the left side.
 Inserts a yterm into the equation (or rewrite five as 5y) and attempts to complete the square of an expression containing y^{2}.
 Rewrites y^{2} as the square of y plus or minus a constant [e.g., as .
 Makes a computational error such as adding nine to negative five and writing it as 14 or 16.

Questions Eliciting Thinking If you add nine to one side of this equation, will the resulting equation be equivalent to the previous one?
If you subtract five from one side of the equation, what should you do to the other side?
If you insert a yterm into the equation, will the resulting equation be equivalent to the previous one?
Is 5y equal to five?
I think you made a computational error in your work. Can you find it? 
Instructional Implications Review properties of equality and provide specific feedback to the student concerning any errors made. Allow the student to revise his or her work and provide feedback. Then ask the student to identify the coordinates of the center and the radius of the circle.
Provide additional examples of equations written in general form and ask the student to rewrite each equation in centerradius form and identify the coordinates of its center and its radius. 
Almost There 
Misconception/Error The student is unable to correctly identify the center and radius of the circle from its equation in centerradius form. 
Examples of Student Work at this Level The student rewrites the equation in centerradius form but is unable to correctly identify the coordinates of the center and the radius. For example, the student identifies:
 The center as (3, 0), (0, 3), (3, 3), or (0, 3).
 The radius as four.

Questions Eliciting Thinking What is the centerradius form of an equation for a circle with center at (h, k) and a radius of r?
How can you identify the coordinates of the center and the radius from this equation? 
Instructional Implications Review centerradius form of the equation of a circle, . Remind the student that this form is derived using the Pythagorean Theorem and that and represent lengths. Emphasize that the coordinates of the center are the values that are subtracted from x and y in the equation. Clarify that the constant in centerradius form is the square of the radius and, again, relate this to the derivation of this form of the equation.
Provide additional examples of equations written in general form and ask the student to rewrite each equation in centerradius form and identify the coordinates of its center and its radius.
Consider implementing MFAS tasks Derive the Circle – Specific Points and Derive the Circle – General Points (GGPE.1.1) to assess the student’s understanding of the derivation of the centerradius form of the equation of a circle. 
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 correctly rewrites the equation in centerradius form, , and identifies the coordinates of the center as (3, 0) and the radius as two.

Questions Eliciting Thinking Could you use the center and radius to graph the circle?
Could you identify the center and radius of a circle directly from the general form of the equation of a circle? 
Instructional Implications Provide more challenging examples of equations of circles in general form (e.g., the coefficient of a quadratic term is different from one or the coefficient of a linear term is an odd number) and ask the student to find the center and radius.
Consider implementing MFAS task Complete the Square for CenterRadius 2 (GGPE.1.1).
Consider implementing MFAS tasks Derive the Circle – Specific Points and Derive the Circle – General Points (GGPE.1.1) to assess the student’s understanding of the derivation of the centerradius form of the equation of a circle. 