Getting Started 
Misconception/Error The student is unable to draw a right triangle that can be used to derive the equation of the circle. 
Examples of Student Work at this Level The student may draw axes on the grid, graph the center, and sketch the circle. However, the student is unable to draw a right triangle whose hypotenuse is a radius of the circle.

Questions Eliciting Thinking What kind of triangle were you trying to sketch?
Can you draw a radius whose endpoints are the center and a point on the circle? Can you form a right triangle with this radius as the hypotenuse? 
Instructional Implications Ask the student to draw axes on the grid and then sketch the given circle [with a radius of six whose center is at (5, 7)]. Explain that the equation of the circle will show how any point, P(x, y), on the circle is related to the center of the circle, (5, 7). Guide the student to draw a radius of the circle and label its endpoint on the circle P(x, y). Then assist the student in drawing a right triangle so that the radius is the hypotenuse. Guide the student to identify the lengths of the legs and describe them as and . Ask the student to use the Pythagorean Theorem to relate the lengths of the sides of the right triangle. Explain that the resulting equation, , is the equation of the circle in centerradius form. Ask the student to use this strategy to derive the equation of a circle with a different center and radius. 
Moving Forward 
Misconception/Error The student is unable find representations for the lengths of the legs of the right triangle. 
Examples of Student Work at this Level The student draws axes on the grid, graphs the center, sketches the circle, and draws a right triangle whose hypotenuse is the radius of the circle. The student may represent the coordinates of the endpoint of the hypotenuse on the circle as (x, y). However, the student is unable to represent the lengths of the legs of the right triangle in terms of the coordinates of the endpoints of the hypotenuse. The student represents the lengths as a and b (or incorrectly as x and y).

Questions Eliciting Thinking Suppose you represented the coordinates of this point (indicate the endpoint of the radius on the circle) as P(x, y). How could you represent the coordinates of the vertex of the right angle?
Can you think of a way to use these coordinates to represent the lengths of the legs? 
Instructional Implications Ask the student to label the endpoint of a radius on the circle P(x, y). Guide the student to identify the coordinates of the vertex of the right angle in terms of the coordinates of the endpoints of the hypotenuse of the right triangle. Review how to find lengths of vertical and horizontal segments in the coordinate plane. Then ask the student to determine the lengths of the legs and describe them as and . Explain the rationale for using the absolute value symbols. Then ask the student to use the Pythagorean Theorem to relate the lengths of the sides of the right triangle. Explain that the resulting equation, , is the equation of the circle in centerradius form. Ask the student to use this strategy to derive the equation of a circle with a different center and radius. 
Almost There 
Misconception/Error The student makes a minor error or writes the equation in an unconventional form. 
Examples of Student Work at this Level The student:
 Makes a sign error in the equation writing it as .
 Writes the equation as .

Questions Eliciting Thinking I think you made a small error in your equation. Can you see if you can find it?
What do you think is the relationship between and ? Are they equal or unequal? 
Instructional Implications Provide feedback to the student regarding any errors made and allow the student to revise his or her work.
If the student wrote the equation in a correct form but different from , explain that equations of circles are usually written in the form . Show the student that his or her equation is equivalent to . However, the conventional form is . Explain that the form is called centerradius form since the coordinates of the center and the radius can be easily identified from the equation.
Ask the student to use the approach taken in this specific case to derive the equation of a circle of radius r whose center is given by (h, k).
Consider implementing MFAS task Derive the Circle – General Points (GGPE.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 draws axes on the grid and sketches a circle of radius six whose center is given by C(5, 7). The student picks a point on the circle described as P(x, y), and draws right triangle CPA so that is the hypotenuse of the right triangle (see below).
The student explains that AC = and AP = . The student then uses the Pythagorean Theorem to write an equation that relates the lengths of the sides of the right triangle: . The student may rewrite the equation as . 
Questions Eliciting Thinking Could you have drawn the right triangle in a different way? Would that change anything about the equation?
Could you use this approach to write the equation of any circle?
How would your equation be written if the center had been at C(8, 4) with a radius of nine? 
Instructional Implications Introduce the student to the general form of the equation of a circle and ask the student to rewrite the equation in general form.
Ask the student to use the approach taken in this specific case to derive the equation of a circle of radius r whose center is given by (h, k).
Consider implementing MFAS task Derive the Circle – General Points (GGPE.1.1). 