Getting Started 
Misconception/Error The student does not select a solution strategy based on the initial form of the quadratic equation. 
Examples of Student Work at this Level The student:
 Chooses the same method for three or more equations without relating the choice of method to the form of the equation.
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 Consistently chooses methods that are not the most efficient and provides insufficient justifications.
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Questions Eliciting Thinking Why did you choose taking square roots (or another specific strategy) as your solution method for this equation?
How can you tell if a quadratic is factorable? Is every quadratic factorable?
Can every method be used to solve every quadratic equation?
When is completing the square the most efficient method? 
Instructional Implications If needed, provide instruction on solving quadratic equations using each method. Throughout instruction, relate each method to features of the equation. For example, explain to the student that taking square roots is a good solution method when the quadratic contains only the quadratic term and a constant and is particularly good if the constant is a perfect square.
After the student is proficient using each solution method, focus instruction on selecting the most efficient solution method based on the initial form of the quadratic equation. Ask the student to make a fourcolumn chart. The columns should contain the (1) name of each method, (2) examples of quadratic equations suited to each method, (3) values of a, b, and c for each example, and (4) stepbystep solutions. Provide the student with additional examples of quadratic equations and ask the student to identify and justify an appropriate solution method. 
Moving Forward 
Misconception/Error The student does not recognize when a quadratic is factorable. 
Examples of Student Work at this Level The student suggests using factoring to solve Â or .
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Questions Eliciting Thinking Is every quadratic factorable?
How can you tell if a quadratic is factorable?
When might factoring be a better choice than the quadratic formula? 
Instructional Implications Provide an example of a quadratic equation that is not factorable such as . Guide the student to list all factor pairs for the constant along with their sums. To convince the student that the quadratic is not factorable, compare the sums to the coefficient of the linear term. Then provide instruction on examining the discriminant to determine when a quadratic is factorable. Finally, discuss with the student what solution methods can be used to solve quadratic equations that are not factorable. 
Almost There 
Misconception/Error The student selects a less efficient strategy in one instance. 
Examples of Student Work at this Level The student:
 Suggests using completing the square to solve Â but makes reasonable choices in all other cases.
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 Suggests using the quadratic formula to solveÂ Â but makes reasonable choices in all other cases.

Questions Eliciting Thinking You suggested completing the square to solve . Did you consider using the quadratic formula? Why or why not?
When might you want to use the quadratic formula instead of completing the square?
When is factoring a good choice of strategy? 
Instructional Implications Present the student with a variety of quadratic equations and discuss all possible methods of solution for each. Then engage the student in a discussion of which method might be best suited to each type of equation. Model justifying a choice of solution method based on the form of the equation.
Ask the student to create an example of a quadratic equation suited to each solution strategy and to explain/justify the equation with regard to the strategy. Provide feedback as necessary. 
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 selects an appropriate solution strategy and provides adequate justification. The student says to solve:
 Â by taking square roots because there are only two terms (the quadratic term and a constant) and each is a perfect square.Â
 Â using the quadratic formula because Â is not factorable,Â and completing the squareÂ would be cumbersome because of the coefficients.
 Â by factoring because Â is easily factored into (x+2)(x+3).
 Â by completing the square because it is written in the form, Â where the leading coefficient, a = 1. Also, Â is not factorable. The quadratic formula can also be used but it may not be the most efficient method because of the leading coefficient and the form of the quadratic.
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Questions Eliciting Thinking You suggested using the quadratic formula to solve . Did you consider using completing the square? Why or why not?
Are there any methods that can be used to solve all types of quadratic equations? 
Instructional Implications Provide the student with a variety of quadratic equations and ask him or her to examine the discriminant of the quadratic formula and relate it to the number and types of solutions.
Challenge the student to derive the quadratic formula using completing the square. 