Getting Started 
Misconception/Error The student does not understand how to solve for a variable within an exponential expression. 
Examples of Student Work at this Level The student:
 Ignores the exponent and solves for x as if the equation had read .
 Attempts to square Â before solving for x.
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Questions Eliciting Thinking Did you realize thatÂ Â is raised to the second power? How does that affect the process of solving for x?
If you squareÂ , how many terms now contain a factor of x? 
Instructional Implications Review solving equations of the form , such as ,Â orÂ Â by taking the square root of each side. Provide equations for the student to solve.
Remind the student that the purpose of this task is to solve the equation for x in order to complete the derivation of the quadratic formula. Explain why squaring the expressionÂ Â is not a good strategy. Assist the student in understanding that a good first step is to â€śundoâ€ť the squaring by taking the square root of each side of the equation. Use a simpler example such x^{2} = 7 to illustrate. Remind the student that there are two roots to consider and show the student the conventional notation used in representing the roots (i.e., the â€śplus or minusâ€ť symbol). Ask the student to complete the process of solving for x.
If the student does not represent x in the conventional form of the quadratic formula, ask the student to convert to the final form . 
Moving Forward 
Misconception/Error The student makes an error in rewriting the expression on the right side of the equation. 
Examples of Student Work at this Level The student understands the need to take the square root of each side of the equation in order to solve for x. However, the student:
 Rewrites Â as Â .
Â
 Only takes the square root of the numerator on the right side of the equation (and is unable to continue to solve).
Â

Questions Eliciting Thinking Is the square root of Â equal to 3 + 4? In general, can the square root of a sum be written as the sum of the square roots?
What are you actually taking the square root of on this side of the equation? 
Instructional Implications Assist the student in understanding that the square root of the right side of the equation is written . If needed, explain that, in general, Â does not equal . Use a specific example to illustrate. Guide the student to rewrite the right side of the equation asÂ . Ask the student to continue to solve for x. Provide assistance as needed. 
Almost There 
Misconception/Error The student makes a minor error. 
Examples of Student Work at this Level The student:
 Omits the â€śplus or minusâ€ť symbol and writesÂ .
 Makes an error rewriting the expression in the conventional form of the quadratic formula. The student makes an error when rewriting Â with a common denominator.
 Makes a sign error.
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 Writes some step of the work incorrectly but is able to solve the equation for x.
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 Does not completely solveÂ for x.
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Questions Eliciting Thinking How many solutions can a quadratic equation have?
What can be used as a common denominator when subtractingÂ ? How would you rewrite the numerators? 
Instructional Implications Provide feedback to the student concerning any errors made. If needed, assist the student in rewriting the expression in the conventional form of the quadratic formula.
Ask the student to completely derive the quadratic formula by solving the equation Â for x and writing the result in the conventional form. 
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 solves the equation for x and, upon request, can write the final result asÂ .
Â 
Questions Eliciting Thinking What does the symbol Â± mean?
How many solutions can a quadratic equation have?
How could you check to see if you solved correctly for x?
How will this help you solve a quadratic equation? 
Instructional Implications Challenge the student to show that the expressionÂ Â satisfies the equation .
Ask the student to completely derive the quadratic formula by solving the equation Â for x and writing the result in the conventional form. 