Getting Started 
Misconception/Error The student does not understand how to use the process of completing the square to write the function in vertex form. 
Examples of Student Work at this Level The student:
 Writes the expression in standard form.
 Factors x from the first two terms of the expression [i.e., writes x(x  400)+ 40100].
 Rewrites the expression incorrectly as .
 Factors 100 from the last two terms of the expression.

Questions Eliciting Thinking What is vertex form?
In the following, , what do h and k represent?
What process can be used to write this equation in vertex form?
How do you complete the square of a quadratic expression? 
Instructional Implications Allow the student to use graphing technology to explore the horizontal and vertical shifts in the graphs of the following functions: , , , , , , and . Ask the student to locate and record the coordinates of the vertex of each graph and compare the coordinates to values in the equation. Guide the student to observe that (h, k) represents the coordinates of the vertex in functions of the form . Explain to the student that because the vertex can be determined directly from the equation, it is said to be in vertex form.
Explain that a quadratic expression in standard form can be written in vertex form using the process of completing the square. Carefully guide the student through the process of completing the square. Begin with quadratic expressions of the form . Then, introduce quadratics in which the coefficient of x is other than one. Provide many examples and opportunities to practice completing the square.
Provide a variety of quadratic functions and allow the student to use a graphing utility to explore key features of each graph (e.g., the orientation of the graph, the location of the vertex, whether the graph has a maximum or minimum, and its intercepts). Provide examples of quadratic functions in realworld contexts and have the student find and interpret the meaning of key features of the graphs. 
Moving Forward 
Misconception/Error The student makes errors in completing the square and writing it as the square of a binomial. 
Examples of Student Work at this Level The student understands that the process of completing the square must be used to rewrite the function in vertex form. However, the student:
 Squares 400 first then divides by two and writes .
 Finds the square root of 400, divides by two, and then squares the quotient, writing the function as .
 Initially tries to solve for x but writes the function as .
 Completes the square but cannot represent it correctly as the square of a binomial, for example, writing the function as .
 Compensates for adding to complete the square by subtracting 200, writing the function as .
 Rewrites the expression by completing the square but does not write the perfect square trinomial as the square of a binomial or does not complete the process of writing the function in vertex form.

Questions Eliciting Thinking What strategy did you use to rewrite the expression in vertex form?
What are the steps in completing the square of an expression?
I think you made a mistake in your work. Can you identify your mistake and correct it?
What does vertex form look like? Is your function in vertex form? 
Instructional Implications Carefully guide the student through the process of completing the square. Begin with quadratic expressions of the form . Then, introduce quadratics in which the coefficient of x is other than one. Ask the student to reevaluate his or her work on this task to find and correct any errors. Then, ask the student to complete the remaining questions and provide feedback.
Provide additional opportunities to practice completing the square. 
Almost There 
Misconception/Error The student does not identify the coordinate of the vertex and/or does not describe it correctly. 
Examples of Student Work at this Level The student:
 Identifies the vertex as the maximum because â€śthe aâ€ť is positive.
 Identifies the vertex as (200,100).
 Is unable to interpret the vertex in the context of the problem.Â

Questions Eliciting Thinking How do you know if the vertex is a maximum or minimum?
In the following, , what do h and k represent? What does a represent? How does the value of a affect the graph?
Can you write the vertex as an ordered pair? What do the coordinates represent in this problem? 
Instructional Implications Remind the student that (h, k) represents the coordinates of the vertex in functions of the form . Provide examples of functions in vertex form and ask the student to determine the orientation of the graph and the location of the vertex. Allow the student to use a graphing utility to graph each function and check his or her responses.
Provide additional examples of quadratic functions in realworld contexts and have the student find and interpret the meaning of key features of the graph (e.g., the orientation of the graph, the location of the vertex, whether the graph has a maximum or minimum, and its intercepts) in the problem context. 
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 rewrites the expression in vertex form as . The student describes the vertex as a minimum value because a is positive and the graph opens upward. The student writes the coordinates of the vertex as (200,100). The student explains this means the minimum cost of $100 occurs when the company builds 200 parts each week. 
Questions Eliciting Thinking Is there a way to identify the vertex when the quadratic expression in is standard form?
Can you find the yintercept and the zeros of a quadratic written in vertex form? What would these values tell you? 
Instructional Implications Introduce the student to the centerradius form of the equation of a circle, and explain this form of the equation reveals the coordinates of the center, (h, k), and the radius, r. Challenge the student to use completing the square to write the equation of a circle given in general form, for example, , in centerradius form. 