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.Â
 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?
Could you complete the square if the leading coefficient was one instead of four?
What constant could you add to the expression x to complete the square? 
Instructional Implications Review the concept of a perfect square trinomial and show the student that a perfect square trinomial results from squaring a binomial. Have the student square several binomials (including ones in which the coefficient of x is different from one) and assist the student in understanding the relationship between the constant in the binomial and the constant and coefficients in the resulting trinomial. Guide the student to observe the features of the terms 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., ) and ask the student to identify a constant so that the polynomial can be written as the product of a coefficient and a perfect square trinomial [e.g., as ]. Then have the student rewrite the perfect square trinomial as the square of a binomial [e.g., as ].
Review with the student the process of solving a quadratic equation of the form 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. Encourage the student to show all work in a neat and organized manner to avoid errors.
If necessary, review how to multiply, divide, and square fractions. Suggest to the student that computations in this context may be easier if fractions are not changed to mixed numbers.
Consider using MFAS tasks Complete the Square  1Â (AREI.2.4) or Complete the Square  2Â (AREI.2.4) if not previously used. 
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 divides all terms by two and determines that adding will complete the square, but the student:
 Is unable to factor the perfect square trinomial or factors incorrectly.
 Makes errors computing with fractions.
 Does not add to both sides of the equation.

Questions Eliciting Thinking How did you factor the trinomial? Did you check your factoring?
After you found that was the value needed to complete the square, what did you do?
I see that you added to this side of the equation to complete the square. What did you add to the other side of the equation? How did you determine this value? 
Instructional Implications Review with the student the process of completing the square to solve a quadratic equation. Encourage the student to be mindful of any value factored from the terms of the trinomial and to take this factor into account when determining the appropriate value to add to the other side 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. Encourage the student to show all work in a neat and organized manner to avoid errors.
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.
If the student factored out two on the leftÂ side of the equation, show the student an alternative solution of dividing all terms by two. The student may prefer this approach. 
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:
 Makes an error taking the square root.
 Only finds the positive square root.
 Does not complete solving.
 Makes a computation error.

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 the work. If the student found only one of the solutions, use a simplified example such as , to remind the student that there are two values which result in four when squared. Clearly identify the values and ask the student to revise his or her work by applying this result.
Remind the student that a fraction can be a perfect square.Â Provide additional examples of perfect square fractions and ask the student to find the square root of each.
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 adds three to both sides of the equation and then divides all terms by two resulting in the equationÂ .Â The student correctly completes the square by addingÂ to both sides of the equation and rewrites the equation as x = . The student then factors the left side of the equation as . The student takes the square root of each side of the equation and rewrites it as x + = Â± . The student then solves theÂ resulting two equations and determines the solution set is {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?
How can you use graphing technology to check your solutions? 
Instructional Implications Ask the student to prepare completed problems that contain errors to share with classmates. Ask the student to explain what type of incorrect thinking would lead to each error. Challenge the studentâ€™s classmates to find and correct the errors. Allow the student to provide feedback.
Also consider using NCTM lessons Two Squares Are Equal http://www.illustrativemathematics.org/illustrations/618 or Completing the Square http://www.illustrativemathematics.org/illustrations/1690. 