Getting Started 
Misconception/Error The student is unable to apply strategies used in solving equations to rewrite formulas. 
Examples of Student Work at this Level The student:
 Solves for the wrong variable.
 Manipulates symbols and variables without any mathematical justification.
 Subtracts b from every term in the equation.
 Subtracts b twice from the same side of the equation.
 Correctly subtracts b from both sides of the equation but is unable to solve further for v.

Questions Eliciting Thinking What is the problem asking you to do? If all of the other variables were numbers, what would you do first? Second?
Is it possible to rewrite the formula to get v by itself? What would be your first step?
When solving equations, whatever is done to one side of the equation, must be done to the other side to keep the equation â€śbalanced.â€ť Â Does your work adhere to that rule? 
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.
Use manipulatives (e.g., integer chips) to model the equation 10 = 6 + 4. Demonstrate that subtracting 2 from every term does not keep the equation equal, or â€śbalanced.â€ť Help the student apply this understanding to the equation 10 = 6 +Â v, and then to the equation T =Â b + v. Finally, ask the student to rework the original equation on a separate sheet of paper.
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. Remind the student to use inverse operations when solving for the variable and correct notation when showing work. Provide feedback as needed.
Consider implementing MFAS tasks Literal Equations (ACED.1.4), Solving Literal Equations (ACED.1.4), Solving Formulas for a Variable (ACED.1.4) or Surface Area of a Cube (ACED.1.4), if not done previously. 
Moving Forward 
Misconception/Error The student makes errors in using the Distributive Property when solving. 
Examples of Student Work at this Level The student:
 Does not distribute the term 2d to both terms T and b, writing .
 Does not distribute the term 2d to every term in the equation, writing .

Questions Eliciting Thinking How do you multiply a binomial by a monomial?
Can you begin by first subtracting b from both sides of the equation? Why or why not?
What operation does the fraction bar represent? What is the inverse operation? 
Instructional Implications Provide feedback to the student regarding his or her error. Review the Distributive Property and how to correctly apply it when multiplying a part of an equation that contains more than one term. Explain the difference between 2dÂ·T â€“ b and 2d(T â€“ b).
Have the student solve theÂ equation Â for v.Â Â Then have the student solve the equationÂ Â for v. Finally, ask the student to rework the original equation on a separate sheet of paper.
Give the student additional multistep literal equations involving the use of the Distributive Property. 
Making Progress 
Misconception/Error The student does not apply the inverse operation correctly when considering the squared term. 
Examples of Student Work at this Level The student:
 Does not know that taking the square root is the inverse of squaring.
 Attempts to take the square root of each side of the equation but is unable to do so correctly.
 Does not simplify the equation completely, leaving the right side of the equation as .
 Does not extend the radical to the entire expression on the left side of the equal sign.

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 nine by two, will you get three?
What are you solving for in this equation? Is v the same thing as ? Would you consider Â simplified?
Is Â the same thing as ? Do you see anything that you need to change in your answer? What does your answer seem to indicate? 
Instructional Implications Explain that the inverse of squaring is taking the square root. 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. Be sure to emphasize that roots of quadratic equations occur in conjugate pairs and require the student to explicitly show both roots. Provide the student with additional opportunities to solve formulas for a variable in the quadratic term.
Express to the student the importance of accurately writing his or her answer. Provide feedback to the student regarding any of the following: Clearly writing the decimal point or comma in a numeral, extending the fraction bar completely across the numerator and/or denominator, and writing the exponent clearly as a superscript. Explain how not doing these could result in answers that are misinterpreted. 
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 . 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?
Can you find a solution of the equation ? Are there any other solutions? What is the square of 3? 
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 equation as .

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 v? 
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.
Challenge the student to find another way to solve for v. Provide assistance as needed. Ask the student to also try solving the equation for b and d. 