Getting Started 
Misconception/Error The student does not understand whenÂ a linear equation in one variable has one, no, or infinitely many solutions. 
Examples of Student Work at this Level The student attempts to solve each equation but is unable to determine the correct number of solutions for some or all of the equations and provides either no rationale or an incorrect one. 
Questions Eliciting Thinking What does it mean for a number to be a solution of an equation?
Can you show me how you solved the first equation? If the equation is now 3x â€“ 6 = 3x â€“ 6, what does that mean about the possible values of x?
Why does the second equation have no solutions (or infinitely many solutions)? Can you think of a number that can be multiplied by either 2 or 2 and result in the same product?
Letâ€™s look at the third equation. Is it possible for seven more than a given amount to be the same as the given amount? 
Instructional Implications Review the concept of a solution of an equation. Explain that a solution of an equation is any quantity that can be substituted for the variable that results in a true statement. Model for the student how the first equation has many possible (i.e., infinitely many) solutions by demonstrating how different values of x result in a true statement. Solve the second equation with the student and explain that 2(0) + 7 = 7 and 2(0) + 7 = 7 so x = 0 is a solution. Challenge the student to find another solution and to explain why no other value of x can make the original equation true. Attempt to substitute a value for x in the third equation to show the student how, regardless of the value, multiplying it by two and adding seven will never result in the same quantity as simply multiplying the value by two. Guide the student to use reasoning to determine the outcomes for each of these equations. Assist the student in summarizing when a linear equation has a single solution, infinite solutions, or no solutions based on the structure of the equation.
Provide additional opportunities to solve equations with one, no, or infinitely many solutions.
Consider implementing the MFAS tasks Linear Equations  1, Linear Equations  2, and Linear Equations  3.
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Moving Forward 
Misconception/Error The student correctly identifies the number of solutions for each equation but provides an incomplete or incorrect rationale. 
Examples of Student Work at this Level The student correctly identifies the number of solutions ofÂ each equation but provides:
 No rationale.
 An incorrect rationale.
 An incomplete rationale.

Questions Eliciting Thinking How did you determine how many solutions each equation has?
What did you mean by â€śthere is no variable for the solutionâ€ť (or any other unclear statement)?
If two linear functions have the same slope and same yintercepts (or the same yintercepts and different slopes or the same slopes and different yintercepts), what must be true of their graphs? How does this relate to the solutions of these equations? 
Instructional Implications Assist the student in relating the algebraic result of solving each equation to the number of solutions. Guide the student to use reasoning to determine the outcomes for each of these equations. Model explaining that:
 The first equation is an identity which means that the statement is true for all values of x and therefore has infinitely many solutions.
 The only solution of the second equation is x = 0.
 The third equation has no solutions because it produces a false statement or because there are no values of x for which the statement can be true.
Provide additional opportunities to solve equations with one, no, or infinitely many solutions. 
Almost There 
Misconception/Error The student is unable to distinguish an equation with a solution of zero from an equation with no solutions. 
Examples of Student Work at this Level The student correctly identifies equations with infinitely many and no solutions but states that the equation in problem #2 has no solutions. The student:
 Transforms the equation to 2x = 2x or x = x and declares that there is no solution because the two sides of the equation are not the same.
 Correctly solves the equation getting x = 0 but states that there is no solution.

Questions Eliciting Thinking What does it mean for an equation to have no solutions?
Your equation still has variables on both sides; can you solve the equation for x? 
Instructional Implications Model solving the equation in problem #2. Remind the student that zero is a number, not the absence of a number; a solution of zero is just as valid as any other numerical solution. If the student explains that an equation has no solutions because the values on each side of the equal sign appear to be opposites, remind the student that zero is its own opposite. Explain that equations that have variables on both sides of the equal sign can still be transformed to a simpler form. The student should withhold judgment on the equation until this has been completed. Challenge the student to write another equation for which zero is a solution.
Provide additional opportunities to solve equations with aÂ solution of zero. 
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:
 Correctly transforms the equation in #1 to an identity (e.g., 3x â€“ 6 = 3x â€“ 6, 3x = 3x, 6 = 6, x = x, 3 = 3, or 0 = 0) and interprets this to mean that the statement is true for all values of x and therefore has infinitely many solutions.
 Identifies that x = 0 in question #2 and states that this equation has one solution because the only value of x that makes the equation true is zero.
 Transforms the equation in question #3 to 7 0 or reasons from its given form to conclude that the equation has no solutions because it produces a false statement or because there are no values of x for which the statement can be true.Â

Questions Eliciting Thinking Can you write an example of an equation that has no solutions?
What would happen if you graphed y = 2x + 7 and y = 2x + 7? How would the graphs be related? 
Instructional Implications Have the student explore the relationship between the expressions on each side of each equation in the coordinate plane. For example, guide the student to rewrite 2x + 7 = 2x + 7 as y = 2x + 7 and y = 2x + 7. Then have the student explore the relationship between the two functions by graphing them on the same set of axes. Guide the student to observe that the graphs have the same yintercept but different slopes. Consequently, these lines have one point of intersection which occurs at their yintercepts. Because of this, the equation 2x + 7 = 2x + 7, has one solution given by the xcoordinate of the point of intersection (i.e., x = 0). Challenge the student to investigate and analyze the first and third equations in terms of their graphs.
Consider implementing the MFAS task Equation Prototypes. 