Getting Started 
Misconception/Error The student does not recognize the role of the discriminant in determining the type of solution. 
Examples of Student Work at this Level The studentâ€™s response indicates no understanding of the need to evaluate the discriminant in determining the type of solution.
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Questions Eliciting Thinking What kinds of solutions can a quadratic equation have?
How many solutions can a quadratic equation have?
What do the a, b, and c in the quadratic formula represent?
Can you give me an example of a rational number? An irrational number? A complex number or a number that is not a real number?
What operation gives rise to imaginary numbers? 
Instructional Implications Review the various number systems (e.g., integer, rational, irrational, real, and complex). Ask the student to identify examples of numbers whose square roots are integer, rational (but not integer), irrational, and complex. Assist the student in generalizing from the examples to statements that relate the nature of the radicand to the resulting type of number (e.g., â€śthe square root of a nonperfect square is irrationalâ€ť).
Provide instruction on the use of the quadratic formula. After the student becomes proficient with its use in solving quadratic equations with integer, rational, irrational, and complex solutions, focus instruction on the discriminant and its role in determining both the number and type of solutions. Provide opportunities for the student to analyze the discriminant to determine the number and type of solutions.
Consider using The Quadratic Formula: The Discriminant and Graphs from the Purplemath website (http://www.purplemath.com/modules/quadform3.htm). This resource provides three examples of quadratic equations solved using the quadratic formula and highlights the relationship between the discriminant and the number and type of solution. In addition, the corresponding quadratic functions are graphed and the number and type of solutions are related to the xintercepts of the graphs. 
Making Progress 
Misconception/Error The student does not understand how the discriminant is related to the number and types of solutions. 
Examples of Student Work at this Level The student understands that the expression under the radical determines the number and types of solutions. However, the student says the solutions will be complex or not real when the discriminant is:
 Positive or a nonperfect square.
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 Negative, a decimal, or not a perfect square.
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Questions Eliciting Thinking Can you give an example of a complex number? What is an example of an irrational number?
Can you describe the difference between complex and irrational numbers? 
Instructional Implications Review the various number systems (e.g., integer, rational, irrational, real, and complex). Ask the student to identify examples of numbers whose square roots are integer, rational (but not integer), irrational, and complex. Assist the student in generalizing from the examples to statements that relate the nature of the radicand to the resulting type of number (e.g., â€śthe square root of a nonperfect square is irrationalâ€ť). Provide additional examples of quadratic equations. Ask the student to examine the discriminant of each and describe the number and type of solution.
Provide additional opportunities to determine the number and type of solution of given quadratic equations. 
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 explains that the quadratic formula indicates a quadratic equation has complex or no real solutions when the discriminant is negative or when .
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Questions Eliciting Thinking What does a complex number look like? Can you describe each of its parts?
Can you describe the relationship between Â and 4ac in order for the discriminant to be negative?
What would have to be true of the discriminant in order to get irrational solutions? Rational solutions? Just one solution?
Can a quadratic equation ever have no solutions? 
Instructional Implications Provide the student with examples of three quadratic equations, one with only one real solution, one with two real solutions, and one with complex solutions. Ask the student to use the quadratic formula to solve each equation. Then ask the student to graph the corresponding quadratic functions using a graphing utility. Challenge the student to relate the number of solutions to the number of xintercepts of the graphs. 