Getting Started 
Misconception/Error The student is unable to correctly complete the square. 
Examples of Student Work at this Level The student attempts to complete the square but is unable to do so correctly. For example, the student:
 Is unable to find the constant to complete the square.
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 Finds the constant needed to complete the square but is unable to correctly rewrite the trinomial as the square of a binomial.
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Questions Eliciting Thinking How can you find the constant to complete the square?
What does it mean to complete the square? What will a quadratic function look like when it is in this form?
How could you check to see if your new equation is equivalent to the original? 
Instructional Implications Model the process of completing the square using the equation given on the worksheet. Indicate that after the square has been completed, the equation is in vertex form [i.e., y = ]. Explain that the advantage of vertex form is the coordinates of the vertex (h, k) can be read from the equation. Provide instruction on completing the square starting with quadratic polynomials of the form + bx + c where a = 1 and b is an even integer. Eventually, introduce quadratic polynomials in which b is an odd integer, a 1 (including negative values of a), or both. Guide the student to check his or her work by putting the quadratic back into standard form and comparing to the original quadratic given.
Be sure the student understands how to determine the signs of the coordinates of the vertex when a quadratic polynomial function is written in vertex form [i.e., y = ].
Provide additional opportunities to write quadratic functions in vertex form by completing the square. 
Making Progress 
Misconception/Error The student is unable to identify or interpret the coordinates of the vertex. 
Examples of Student Work at this Level The student correctly completes the square and rewrites the function as A(x) = + 625 or A(x) â€“ 625 = . However, the student:
 Incorrectly identifies the coordinates of the vertex.
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 Interprets the coordinates of the vertex incorrectly or incompletely.Â
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Questions Eliciting Thinking Do you know what the vertex form of a quadratic function looks like in general? Can you identify the coordinates of the vertex in general?
What is the independent variable in this problem? What is the dependent variable in this problem? What are their units of measure?
If you graphed this function, would it be opening up or opening down? Where is the vertex located on the graph? Is it a maximum or a minimum? 
Instructional Implications If needed, remind the student that vertex form of a quadratic function isÂ Â and when the equation is in this form, the coordinates of the vertex are (h, k). Provide a number of quadratic functions written in vertex form and ask the student to identify the coordinates of the vertex. Have the student use a graphing utility to graph each equation and check the location of the vertex.
Be sure the student understands that the vertex is a minimum when a parabola is opening upward and a maximum when a parabola is opening downward. Ask the student to review the meaning (and unit of measure) of the independent and dependent variables. Then, guide the student to write the vertex as an ordered pair including units [e.g., (25 feet, 625 square feet)]. Ask the student to determine if the vertex is a maximum or a minimum. Then, model interpreting the coordinates of the vertex by saying, â€śThe area of the rectangular enclosure attains a maximum value of 625 square feet when the length of a side is 25 feet.â€ť Emphasize that the maximum area occurs when the length of each side of the enclosure is 25 feet.
Provide additional opportunities to write quadratic functions in vertex form by completing the square and to interpret the coordinates of the vertex in 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 correctly completes the square and writes the function as A(x) = + 625 or A(x) â€“ 625 = . The student identifies the coordinates of the vertex as (25, 625) and explains that the maximum area of the enclosure, 625 square feet, is attained when the length of each side of the enclosure is 25 feet. 
Questions Eliciting Thinking Can you factor  + 50x? How are the expressions x and (50 â€“ x) related to the problem? Why is the equation written as A(x) = x(50 â€“ x)?
How would the equation change if there were 120 feet of fencing available?
Is the graph symmetric? Where is the vertex in relationship to the line of symmetry of the graph? 
Instructional Implications Ask the student to consider the relationship between the equation and the length of fencing, 100 feet, given in the problem and to explain the meaning of x, (50 â€“ x), and x(50 â€“ x). Ask the student to evaluate 50 â€“ x for x = 25 and explain the significance of this value to the problem.
Change the problem so there is 120 feet of fencing available and the enclosure is adjacent to a garage so only three sides require fencing. Challenge the student to write an equation to model the area of the enclosure and find the length of a side that maximizes the area of the enclosure. 