Getting Started 
Misconception/Error The student is unable to apply strategies used in solving equations when rewriting formulas. 
Examples of Student Work at this Level The student writes:
 You canâ€™t solve for e.
 There arenâ€™t any numbers, so I canâ€™t solve forÂ e.
The student rewrites the formula as Â = e.

Questions Eliciting Thinking If S were a number, what is the first step you would do to solve for e? What would you do next?
Is it possible to rearrange the formula to get e by itself? What would be your first step? 
Instructional Implications Review the four basic operations (i.e., add, subtract, multiply, and divide) and give the student the opportunity to determine the inverse of each. Provide feedback as needed.
Review the reasoning that is used in solving equations and assist the student in applying it to formulas. Begin with simple threevariable formulas that require only one step to solve. Then introduce the student to twostep and finally, multistep problems.
Consider implementing MFAS tasks Literal Equations (ACED.1.4), Solving Literal Equations (ACED.1.4), and Solving Formulas for a Variable (ACED.1.4), if not done previously. 
Moving Forward 
Misconception/Error The student does not know that taking square roots is the inverse of squaring. 
Examples of Student Work at this Level The student initially divides each side of the equation by six but then stops or attempts a strategy that does not involve taking square roots.

Questions Eliciting Thinking What do you get when you square the number three? What operation can you use to â€śundoâ€ť squaring? If you divide the nine by two, will you get three? 
Instructional Implications Introduce the student to squaring and taking square roots as inverse operations. Give the student opportunities to solve equations of the form Â where c is a constant. Be sure to emphasize that roots to quadratic equations occur in conjugate pairs. Also provide the student with opportunities to solve equations of the form = c, where c is a constant. 
Making Progress 
Misconception/Error The student attempts to take the square root of each side of the equation but is unable to do so correctly. 
Examples of Student Work at this Level The student rewrites the formula as:

Questions Eliciting Thinking Can you explain what you did at each step?
Is there a difference between Â and ?Â 
Instructional Implications Provide direct instruction on solving quadratic equations by taking square roots. Begin with equations of the form Â where c is a whole number. Then introduce the student to equations in which the coefficient of x is different from one and c is a positive rational number. Require the student to explicitly show both roots. Provide the student with opportunities to solve quadratic equations in contexts in which only one root is a reasonable solution and in which both roots are reasonable solutions. Then provide the student with additional opportunities to solve formulas with a variable in the quadratic term. 
Almost There 
Misconception/Error The student is unaware that roots of quadratic equations occur in conjugate pairs. 
Examples of Student Work at this Level The student rewrites the formula as e = . When asked if there is another root of this equation, the student confidently asserts there is not.

Questions Eliciting Thinking Did you realize there is another root of this equation? Did you omit it on purpose?
Can you find a solution of the equation x^{2}Â = 9? Are there any other solutions?
What is negative three times negative three?
Would it make sense for e to be negative in the context of this problem? 
Instructional Implications Provide direct instruction on solving quadratic equations by taking square roots. Require the student to explicitly show both roots. Provide the student with opportunities to solve quadratic equations in contexts in which only one root is a reasonable solution and in which both roots are reasonable solutions. Then provide the student with additional opportunities to solve formulas with a variable in the quadratic term. 
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 formula as e = . When the student is asked why he or she did not consider both the positive and negative roots, the student replies that the length of the edge of the cube can only be positive.
The student rewrites the formula as e = Â± . When asked if the negative root is reasonable in this context, the student replies that it is not since the length of the edge of a cube cannot be negative. The student corrects his or her work. 
Questions Eliciting Thinking Why do roots of quadratic equations occur in conjugate pairs?
Can you show me an alternative, but still mathematically correct, procedure for solving this formula for e? 
Instructional Implications Ask the student to solve more complex formulas for specified variables.
Give the student formulas for which multiple methods of solution are possible. Ask the student to solve the formula in more than one way. Then ask the student to compare the solutions and decide which, if either, is more efficient. 