Getting Started 
Misconception/Error The student is unable to describe the relationship between a solution of the equation and the graph of the equation. 
Examples of Student Work at this Level The student writes that (4, 6):
 Has no relationship to the equationÂ y = 3x  6.
 Has the same slope as the equationÂ y = 3x  6.
 Is a point used to graph the line.
 Is a point on the line.
 And the equationÂ y = 3x  6Â are proportional/similar.
 And the equation y = 3x  6Â both contain a six.
 Is a function of the equation.
The student writes that (1, 2):
 Has no relationship to the equation y = 3x  6.
 Is not on the line.
 And the equation y = 3x  6 are not proportional/similar.
 Could be on a line that intersects or is parallel to y = 3x  6.
The student does not refer to a relationship at all and instead explains how to graph the equation using the slope and yintercept.

Questions Eliciting Thinking What do x and y represent in the equation y = 3x  6?
What do the points on this line represent?
Did you substitute four for x and six for y?
How did you graph a line before learning to use the slope and yintercept?
Could you make a table of values for this equation? What would the values in this table represent? 
Instructional Implications Explain to the student that if (4, 6) is a point on the graph of y = 3x  6, then (4, 6) is a solution of the equation. Stress to the student that any point on the graph of the equation is represented by an ordered pair that is a solution of the equation. Likewise, explain that every point that is a solution of the equation is on the graph of that equation, which in this case is a straight line. Describe the relationship between solutions of the equation and points on its graph as onetoone. Have the student use the equation to find another solution and then locate this point on the graph. Then have the student identify a point not on the graph of the equation and show that it is not a solution of the equation. Continue to emphasize the onetoone relationship between solutions of equations and points on their graphs as other equation types (e.g., absolute value, quadratic, and exponential) are introduced.
Consider implementing the MFAS task Finding Solutions (AREI.4.10) to further assess the student's understanding of the relationship between an equation and the graph of that equation. 
Moving Forward 
Misconception/Error The student understands the relationship between the two given points and the equation but is unable to generalize this relationship to other points on the line. 
Examples of Student Work at this Level The student writes that there could be a point on the line that would not be a solution to the equation.Â 
Questions Eliciting Thinking What must be true of a point if it represents a solution of the equation?
What must be true of a point if it does not represent a solution of the equation?
Could there be an ordered pair with an x value of 1000 that would be a solution to the equation? Why or why not?
How many solutions are there for the equation y = 3x  6?How could you describe or illustrate all of the solutions? 
Instructional Implications Provide the student with several solutions that are included on the part of the graph shown on the worksheet [e.g., (1, 9) and (5, 9)] and solutions that are beyond the part of the graph shown on the worksheet [e.g., (1000, 2994) and (500,1506)].Â Have the student verify that each point represents a solution of the equation. Explain to the student that every point on the line is a solution of the equation. Have the student select several points not on the line and confirm that these points are not solutions of the equation.
Describe the relationship between solutions of an equation and points on its graph as onetoone. Continue to emphasize the onetoone relationship between solutions of equations and points on their graphs as other equation types (e.g., absolute value, quadratic, and exponential) are introduced.
Consider implementing MFAS task Finding Solutions (AREI.4.10) to further assess the student's understanding of the relationship between an equation and the graph of that equation. 
Almost There 
Misconception/Error The student understands the relationship between a solution of the equation and its graph but does not provide a complete (or mathematical) explanation. 
Examples of Student Work at this Level The student writes that points on the graph of the equation:
 'Answer' the equation.
 'Connect with' the equation.
 Can be 'plugged into' the equation.Â

Questions Eliciting Thinking Can you explain further the relationship between a point on the line (or a point not on the line) and the equation of that line?
How does the answer you gave describe the relationship between the solution of the equation and its graph?
How do you know the relationship you described holds true for every point on the line? 
Instructional Implications Model for the student the use of mathematical reasoning and terminology in explaining the relationship between solutions of an equation and its graph. Describe the relationship between solutions of an equation and points on its graph as onetoone. Continue to emphasize the onetoone relationship between solutions of equations and points on their graphs as other equation types (e.g., absolute value, quadratic, and exponential) are introduced.
Provide the student with several solutions that are beyond the part of the graph shown on the worksheet [e.g., (1000, 2994) and (500, 1506)] and include solutions with noninteger coordinates [e.g., (, 5) and (, 4)]. Have the student verify that each point represents a solution of the equation. Explain to the student that every point on the line is a solution of the equation. Have the student select several points not on the line and confirm that these points are not solutions of the equation.
Consider implementing the MFAS task Finding Solutions (AREI.4.10) to further assess the student's understanding of the relationship between an equation and the graph of that equation. 
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 explains that since (4, 6) is a point on the line, it must be a solution of the equation y = 3x  6, and since (1, 2) is not on the line, it is not a solution of the equation. The student also understands that this relationship holds true for every point on or not on the line regardless of its location in the coordinate plane. 
Questions Eliciting Thinking How many solutions does the equationÂ yÂ = 3xÂ  6Â have?
Why does it make sense to represent the solutions of the equation y = 3x  6 as a line in the coordinate plane?
Can we represent the solution of every equation in two variables as a line in the coordinate plane?
Is there any value of x that could not be part of a solution ofÂ yÂ = 3xÂ  6? 
Instructional Implications Provide the student with an equation in one variable (e.g., 3x = 5xÂ + 8). Have the student compare the two equations, 3x = 5xÂ + 8 and y = 5xÂ + 8, by comparing: (1) the methods used to solve, (2) the solution set of each, and (3) the graph of each solution set. Have the student consider how x = 4 is graphed on a number line and how it is graphed in the coordinate plane (e.g., as a vertical line representing all points with an xcoordinate of 4). Ask the student if an equation in one variable could have infinitely many solutions and/or no solution. If needed, provide the student with examples of each.
Consider implementing the MFAS taskÂ Finding SolutionsÂ (AREI.4.10) to further assess the student's understanding of the relationship between an equation and the graph of that equation. 