Getting Started 
Misconception/Error The student does not understand when a linear equation in one variable has one, no, or infinitely many solutions. 
Examples of Student Work at this Level Some or all of the equations the student writes do not have the indicated number of solutions. Additionally, the student is unable to provide adequate explanations for responses. 
Questions Eliciting Thinking What makes a number a solution of an equation?
How can you tell if an equation has just one solution? Can you determine if the equation 3x + 1 = 16 has a solution?
What does the equal sign mean? Can 2x + 4 ever have the same value as 2x?
What happens when you substitute a number for yÂ in the equation 5y + 8 = 5y + 8? 
Instructional Implications Review the concept of a solution of an equation and what it means for a linear equation to have exactly one solution, infinitely many solutions, or no solutions. Assist the student in identifying the algebraic result associated with each case. Provide equations such as x + 2 = 5, x + 3 = x + 3, and x + 3 = x + 4. Guide the student to solve each equation and relate the outcome to the number of solutions. Model the use of 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 write and solve equations with one, no, or infinitely many solutions.
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Making Progress 
Misconception/Error The student correctly writes equations that have the given numbers of solutions but provides incomplete or incorrect rationales. 
Examples of Student Work at this Level The student correctly writes an equation that has each of the given numbers of solutions but provides no rationale, an incorrect rationale, or an incomplete rationale. 
Questions Eliciting Thinking How did you determine how many solutions each equation has?
What features of this equation indicate the number of solutions it will have? How do you know? 
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. For the first equation written, explain that there is only one possible value of x that makes the equation true and this value can be found by solving the equation. Ask the student to solve his or her equation and then check the solution to determine if it makes the equation true. For the second equation written, explain that there are no values of x that can make this equation true. Use the studentâ€™s equation to demonstrate this. For the third equation, introduce the term identity and explain that any value of x will make the equation true. Again, use the studentâ€™s equation to demonstrate this.
Provide additional opportunities to write and solve equations with one, no, or infinitely many solutions. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level For question #1, the student writes an equation such as xÂ  2Â = 8 (an equation with exactly one solution) and explains:
 There is only one possible value of xÂ that makes the equation true.
 x can only be 10 (or another value) to make this a true statement.
For question #2, the student provides an equation such as x + 4 = x and explains:
 There are no values of x that would make this true.
 If I add 4 to x it can no longer be equal to x.
For question #3, the student provides an equation such as x + 3 = x + 3 and explains:
 The equation is true no matter what value is substituted for x.
 If I add three to a number, I will get the same result when I add three to it again. It doesn"t matter what value I start with.

Questions Eliciting Thinking If an equation can be transformed to a false statement like 7 = 0, why does that mean that it has no solutions?
How can you tell from the structure of the equation how many solutions it will have? 
Instructional Implications Ask the student to solve multistep equations with rational coefficients and constants with one solution, no solution, or infinitely many solutions. 