Getting Started 
Misconception/Error The student shows no evidence of using the given equation to reason about the second equation. 
Examples of Student Work at this Level The student:
 Simply solves the second equation and makes no attempt to relate the solution to the first equation.
 Attempts to solve the given equation but does so incorrectly and makes a statement that does not address the question.

Questions Eliciting Thinking How can you use the equation you assume is true to determine if the other equation is true?
Suppose you solved the given equation? What would that tell you?
What if the equations had different solutions? What would that tell you? 
Instructional Implications Review the properties of equality and the properties of operations. Explain the reasoning process used in solving linear equations and that each step follows from the equality asserted in the previous step. Emphasize that appropriate application of the properties of equality enables one to rewrite an equation in an equivalent form. Provide the student with the steps of the solution of an equation and ask the student to justify each step using properties of equality and operations.
If needed, review the application of the properties of equality and the properties of operations to the process of solving an equation. Provide a variety of equations for the student to solve. Begin with simple onestep equations, then twostep equations, and finally, equations with rational numbers and expressions similar to . Ask the student to justify each step of the process of solving and provide feedback as needed.
Discuss various strategies for determining if the second equation in this task follows from the given equation (e.g., use properties of equality to rewrite the given equation as 3x = 12 and observe that 3x cannot equal both 12 and 12, so the second equation cannot follow from the given equation) or solve the given equation and see if its solution satisfies the second equation. If not, the second equation cannot follow from the given equation.
Consider using the MFAS tasks Justify the Process 1 (AREI.1.1) and Justify the Process 2 (AREI.1.1) if not used previously. 
Moving Forward 
Misconception/Error The student attempts to reason from the given equation but makes algebraic errors. 
Examples of Student Work at this Level The student attempts to solve Â but is unable to do so correctly. However, the student uses the result to correctly reason about the two equations.

Questions Eliciting Thinking Your reasoning is absolutely correct but you made a mistake in solving the equation . Can you find your mistake? 
Instructional Implications Review the application of the properties of equality and the properties of operations to the process of solving an equation. Provide a variety of equations for the student to solve. Begin with simple onestep equations, then twostep equations, and finally, equations with rational numbers and expressions similar to . Ask the student to justify each step of the process of solving and provide feedback as needed.
Consider using the MFAS tasks Justify the Process 1 (AREI.1.1) and Justify the Process 2 (AREI.1.1) if not used previously. 
Almost There 
Misconception/Error The student reasons from the given equation using an appropriate algebraic strategy but justification is either inadequate or incorrect. 
Examples of Student Work at this Level The student:
 Substitutes 4 for x in the given equation, Â but does not explain the significance of this value and its relationship to the second equation.
 Solves the second equation, 3x = 12, and then substitutes its solution, 4, for x in the given equation, but incorrectly assumes that the result suggests the given equation does not have a solution.

Questions Eliciting Thinking Why did you substitute 4 for x in ? Where did this value come from?
You used the word â€śitâ€ť in your explanation. What does â€śitâ€ť refer to?
Why did you say there is no solution to the first equation? What does that mean? Is it possible to solve for x in the first equation? What would you get? What does that tell you about the equation 3x = 12? 
Instructional Implications Provide feedback to the student concerning his or her justification. Assist the student in writing concise and complete justifications in which mathematical notation and terminology are used correctly. Remind the student that the justification should be correct and convincing. Provide additional opportunities to reason about the process of solving an equation and to provide justifications. 
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 reasons from the given equation using an appropriate algebraic strategy and provides adequate justification. The student:
 Solves the given equation, , and then substitutes its solution, 4, for x in the second equation, 3xÂ = 12, yielding the false statement, 12 = 12. The student explains that since 12 ? 12, then 3x = 12 cannot be true.
 Solves the second equation, 3x = 12, and then substitutes its solution, 4, for x in the given equation, , yielding the false statement, 8 = 4. The student explains that since 8 ? 4, then 3x = 12 cannot be true.
 Rewrites Â asÂ 3x = 12 and explains that since 3xÂ = 12 follows from the given equation it cannot also be true that 3xÂ = 12.
 Solves , and 3x = 12 and compares their solutions. The student reasons that since 4 ? 4, the second equation does not follow from the given equation.Â

Questions Eliciting Thinking How can you rewrite the second equation so that it follows from the first equation?
Can you find another equation that follows from the first equation? What must every equation that follows from the first equation have in common?
I see that you rewrote 3x = 12 as x = 4? How do you know that x = 4 follows from 3x = 12?
I see that you rewrote the given equation as 4x = x â€“ 12? How do you know that this follows from the given equation?
If you know that 3x ? 12, is it possible that ? 
Instructional Implications Provide the student with the steps of the solution of an equation and ask the student to justify each step using properties of equality and operations.
Challenge the student with equations containing mistakes in the solution process. Ask the student to identify the mistake and the property of equality associated with each mistake. Ask the student to construct a viable argument for an alternative solution strategy. 