Getting Started 
Misconception/Error The student is unable to describe the relationship between a solution of the equation y = 2xÂ + 4 and the graph of the equation. 
Examples of Student Work at this Level The student:
 Identifies the relationship as being positive or negative depending upon the location of the point in the coordinate plane.
 Identifies the relationship as being proportional.
 Explains how to graph the equation using the slope and yintercept.
The student writes:
 There is no relationship.
 a = 2 and b = 4.
 (a, b) is the same as the slope of the line.
 (a, b) is a function of the graph.
 (a, b) is on a line parallel to the graph and (c, d) is on a line that intersects the graph.
 (a, b) solves/answers the equation.Â

Questions Eliciting Thinking What do x and y represent in the equation y = 2x + 4?
In the ordered pair (a, b), what does a represent? What does b represent in the equation? How is the ordered pair related to the equation?
What does this line represent?
What is the solution set of the equation y = 2xÂ + 4? 
Instructional Implications Explain to the student that if (a, b) is a solution of the equation y = 2x+4, then (a, b) lies on the graph of that equation. Stress to the student 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. Likewise, explain that any point on the graph of the equation is represented by an ordered pair that is a solution of the equation. Describe the relationship between solutions of the equation and points on its graph as onetoone. Have the student use the equation to find one of its solutions 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.
If necessary, explain to the student that in the ordered pair (a, b), b represents the ycoordinate associated with an xcoordinate of a. In this case, b does not necessarily refer to the yintercept as it does when referring to the slopeintercept form of the equation of a line (i.e., y = mx + b). Challenge the student to identify the value of a for which b is the yintercept.
Consider implementing the MFAS task What Is the Point? (AREI.4.10) to further assess the student's understanding of the relationship between points on a line and solutions of the equation of that line. 
Moving Forward 
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 the solution of the equation:
 Will fit on the line.
 Determines where the line is.
 Will make the line.
The student says there is no relationship between an ordered pair that is not a solution and the graph of the equation.

Questions Eliciting Thinking What is the relationship between a solution of the equation and its graph? Can you provide more information about this relationship?
What do you mean by the point will fit on/will make the line?
What do you mean by the point determines where the line is?
Could the point (c, d) be on the line? Why or why not? 
Instructional Implications Model for the student the use of mathematical reasoning and mathematical 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.
Consider implementing the MFAS taskÂ What Is the Point?Â (AREI.4.10) to further assess the student's understanding of the relationship between points on a line and solutions of the equation of that line. 
Almost There 
Misconception/Error The student understands the relationship between solutions of the equation and its graph but does not graph and /or label the point correctly. 
Examples of Student Work at this Level The student:
 Does not graph a point.
 Graphs the point but does not label it.
 Graphs the point but labels it incorrectly.

Questions Eliciting Thinking Can you graph and label an ordered pair that is a solution of the equation and one that is not a solution of the equation? Explain why the point you graphed is, or is not, a solution of the equation.
Can you identify the point that you graphed? Why is it important to label points that are graphed in a coordinate plane?
You made a mistake labeling this point. Can you identify and correct your mistake? How could this mistake be avoided in the future?
Does the point you graphed correspond with the ordered pair that you wrote? Why or why not? 
Instructional Implications Provide feedback to the student and allow the student to revise his or her work. Review with the student how to graph and label a point. Remind the student that in an ordered pair the first number represents the xcoordinate and the second number represents the ycoordinate [e.g., (x, y)]. Remind the student of the importance of attending to precision when identifying the coordinates of a point in the coordinate plane.
Consider implementing the MFAS taskÂ What Is the Point?Â (AREI.4.10) to assess the student's understanding of a line representing the set of all solutions of the equation of that line. 
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 recognizes that since the point (a, b) is a solution of the equation, it lies on the graph of the line. The student also understands that since the point (c, d) is not a solution of the equation, it does not lie on the graph of the line. The student is able to identify a solution of the equation and graphs and labels the point correctly. The student also graphs and correctly labels a point not on the line and understands that this point is not a solution of the equation. 
Questions Eliciting Thinking How many solutions does the equation y = 2x + 4 have?
Why does it make sense to represent the solutions of the equation y = 2x + 4 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 of y = 2xÂ + 4?
How do you know the relationship you described holds true for every point on or off the line? 
Instructional Implications Provide the student with an equation in one variable (e.g., 2x = 2xÂ + 4). Have the student compare the two equations, 2x = 2x + 4 and y = 2x + 4, 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 = 1 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 one). 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 What Is the Point?Â (AREI.4.10) to assess the student's understanding of a line representing the set of all solutions of the equation of that line. 