Getting Started 
Misconception/Error The student does not demonstrate an understanding of completing the square. 
Examples of Student Work at this Level The student:
 Attempts to solve the equation using a different method such as factoring.
 Is unable to correctly identify a constant term to create a perfect square trinomial.

Questions Eliciting Thinking What does it mean to complete the square?
What is a perfect square trinomial? Can you give me an example? What constant could you add to the expression + 18x to make it a perfect square trinomial? 
Instructional Implications Review the concept of a perfect square trinomialÂ and show the student that a perfect square trinomial results from squaring a binomial. Have the student square a number of binomials (initially in which the coefficient of x is one) and assist the student in understanding the relationship between the constants in the binomial and the constants and coefficients in the resulting trinomial. Guide the student to observe the features of the constants and coefficients of a trinomial that indicate it is a perfect square. Provide the student with a number of examples of the quadratic and linear terms of a trinomial (e.g., + 10x) and ask the student to identify a constant so the trinomial is a perfect square. Then have the student rewrite each trinomial as the square of a binomial. When the student has mastered the simpler cases (e.g., the coefficient in the quadratic term is one and the coefficient in the linear term is even), transition the student to more challenging cases.
Review the process of solving a quadratic equation of the form = c by taking the square root of each side. Guide the student through several examples emphasizing the reasons for each step in the process. Give the student additional equations to solve by first completing the square.Â 
Moving Forward 
Misconception/Error The student completes the square but makes errors when rewriting the equation. 
Examples of Student Work at this Level The student identifies 81 as the constant that will complete the square but the student:
 Does not use the Addition Property of Equality and only adds 81 to one side of the equation.
 Disregards the constant, (i.e.,  45 on the right side of the equation) and does not add it to 81.
 Does not add 81 to 45 correctly and is unsure how to continue.
 Does not correctly factor the perfect square trinomial.

Questions Eliciting Thinking If you add 81 to one side of the equation, what must you do to the other side of the equation? Why must you add 81 to both sides of the equation?
Did you simplify the right side of the equation correctly? What happened to 45?
How did you factor the left side of the equation? How can you check that your factoring is correct? 
Instructional Implications Review with the student the process of completing the square to solve a quadratic equation. Stress the necessity of adding the constant to both sides of the equation. Guide the student through several examples emphasizing the reasons for each step in the process. Give the student additional equations to solve by completing the square.
If the student struggles to factor the perfect square trinomial, remind him or her that the constant in the binomial can be found in two different ways: by taking half of the coefficient of the linear term or by taking the square root of the constant. Ask the student to find the constant term in each way and check that they are the same. 
Almost There 
Misconception/Error The student makes an error when solving the equation. 
Examples of Student Work at this Level The student correctly completes the square, factors the perfect square trinomial, and rewrites the equation as . However, the student makes an error in solving this equation. For example, the student:
 Only writes the positive root when taking the square root of 36.
 Neglects to take the square root of 36.
 Adds nine to both sides of the equation instead of subtracting nine.
 Makes a computational error, for example, adds six to 9 and writes the sum as three.

Questions Eliciting Thinking What type of equation is this? How many solutions can a quadratic equation have? How many solutions should this equation have?
Did you take the square root of both sides of the equation?
Can you simplify your answer any further?
There is an error in the last step. Can you find it? 
Instructional Implications Provide feedback on any errors made and allow the student to revise his or her work. If the student found only one of the solutions, use a simplified example such as = 4, to remind the student that there are two values that result in four when squared. Clearly identify the values and ask the student to revise the work by applying this result.
Provide the student with a completed problem that contains errors. Have the student identify and correct the errors. 
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 completes the square by adding 81 to both sides of the equation and rewrites the equation as = 36. The student then takes the square root of each side of the equation rewriting it as x + 9 = Â± 6. The student then solves the two resulting equations and determines the solution set is {15, 3}.

Questions Eliciting Thinking How can you check your solutions?
Do you think it would have been easier to solve this equation using another method? Why?
How can you use the process of completing the square to write a quadratic equation in two variables in vertex form? 
Instructional Implications Challenge the student to solve, by completing the square, quadratic equations for which the coefficient of the quadratic term is different from one or the coefficient of the linear term is an odd number.
Consider implementing MFAS tasks Completing the Square  2 (AREI.2.4) and Complete the Square  3 (AREI.2.4).
Also consider using NCTM activities Two Squares Are Equal http://www.illustrativemathematics.org/illustrations/618 or Complete the Square http://www.illustrativemathematics.org/illustrations/1690. 