Getting Started 
Misconception/Error The student is unable to identify the xcoordinate of the point where the graphs intersect. 
Examples of Student Work at this Level The student identifies:
 Both coordinates of the point of intersection.
 The ycoordinate of the point where the graphs intersect.
Additionally, the student is unable to show why x = 1 is a solution of the equation .

Questions Eliciting Thinking What does it mean for graphs to intersect?
Can you show me the point where the graphs intersect? What are the coordinates of this point? What is the xcoordinate of this point?
Can you show that x = 1 is a solution of f(x) = 2x  1? What does it mean for this value to be a solution of this equation?
What does it mean to be a solution of a system of equations? 
Instructional Implications Review the concept of a solution of an equation in two variables. Be sure the student can distinguish between the x and ycoordinates of points. Assist the student in developing an understanding of the onetoone relationship between solutions of an equation in two variables and points on its graph (AREI.4.10). Provide the student with an equation in two variables and its graph. Ask the student to identify the coordinates of a point on the graph and demonstrate how the coordinates satisfy the equation. Then ask the student to identify the coordinates of a point not on the graph and ask the student to demonstrate that the coordinates do not satisfy the equation. Emphasize that the coordinates of every point on the graph satisfy the equation and every solution of the equation is represented by a point on the graph.
Review the graphical representation of the solution of a system of linear equations in two variables. Explain that since the point of intersection of the graphs is a point on each equationâ€™s graph, it represents a solution of each equation in the system. Since it represents a solution of each equation in the system, it is, by definition, a solution of the system. Provide the student with an example of a system of equations and its graph. Ask the student to identify the coordinates of the point of intersection of the graphs and demonstrate how the coordinates satisfy each equation in the system.
If necessary, review function notation. Provide frequent opportunities for the student to use function notation to develop an understanding of its meaning and the conventions surrounding its use.
Consider implementing the MFAS tasks Finding Solutions (AREI.4.10) and What Is the Point? (AREI.4.10). 
Moving Forward 
Misconception/Error The student is unable to explain, in general, why the xcoordinate of the point of intersection is a solution of the equation f(x) = g(x). 
Examples of Student Work at this Level The student finds the xcoordinate of the point of intersection and shows that it is a solution of the equation . However, the student is unable to explain, in general, why the xcoordinate of the point of intersection is a solution of the equation f(x) = g(x).
The student:
 Explains in terms of â€śtrial and error.â€ť
 Does not attempt an explanation.
 Describes the outcome when showing x = 1 satisfies the equationÂ without explaining why.

Questions Eliciting Thinking If you substitute 1 for x in function f, what will you get [i.e., what is f(1)]? How is this related to the graph of function f?
If you substitute 1 for x in function g what will you get [i.e., what is g(1)]? How is this related to the graph of function g?
Why should x = 1 be a solution of the equation ? 
Instructional Implications Review what it means for an ordered pair of numbers to be a solution of a system of equations and relate the solution to the graph of the system. Provide additional opportunities for the student to demonstrate that the coordinates of the point of intersection of the graphs of the equations in a system satisfy each equation in the system.
Be sure the student understands function notation. Model explaining why, in regard to specific instances, the xcoordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). For example, with regard to the functions given in this task, explain that since (1, 3) is a point on the graph of function f, then f(1) = 3; since (1, 3) is a point on the graph of function g, then g(1) = 3. Consequently, f(1) = g(1).
Consider implementing the MFAS task Graphs and Solutions  2 (AREI.11). 
Almost There 
Misconception/Error The student provides an imprecise explanation for why, in general terms, the xcoordinate of the point where the graphs of the equations y = f(x) and y = g(x) intersect is the solution of the equation f(x) = g(x). 
Examples of Student Work at this Level The studentâ€™s response indicates an understanding of the concept, but the student does not make use of appropriate mathematical terminology and notation to provide a precise and clear explanation.

Questions Eliciting Thinking What is the significance of the point of intersection in terms of the solutions of the equations?
What is f(1)? What is g(1)? How are these values related to the graphs of the functions?
Suppose the point of intersection is given by (a, b). What is f(a)? What is g(a)? 
Instructional Implications Be sure the student understands function notation. Model explaining why, in regard to specific instances, the xcoordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). For example, with regard to the functions given in this task, explain that since (1, 3) is a point on the graph of function f, then f(1) = 3; since (1, 3) is a point on the graph of function g, then g(1) = 3. Consequently, f(1) = g(1). Then extend this explanation to the general case by supposing that (a, b) describes the coordinates of the point of intersection of the graphs of functions f and g. Explain that since (a, b) is a point on the graph of function f, then f(a) = b and since (a, b) is a point on the graph of function g, then g(a) = b. Therefore, f(a) = g(a). Assist the student in understanding the relationship between the explanation for specific cases and the explanation for the general case.
Consider implementing the MFAS task Graphs and Solutions  2 (AREI.11). 
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 states that the xcoordinate of the point of intersection of the two graphs is 1 and shows that x = 1 satisfies the equation by substituting 1 for x and demonstrating that each side of the equation results in the same value, 3. The student explains that, in general, the point of intersection of the graphs of f and g, (a, b), is a solution of each function so that f(a) = b and g(a) = b. Therefore, f(a) = g(a). 
Questions Eliciting Thinking Do you think the graphs of these two functions could intersect in another point?
Is it possible for two graphs to intersect in more than one point?
Is it possible for two linear functions to intersect in more than one point? 
Instructional Implications Allow the student to use a graphing calculator or other graphing technology to further explore systems of equations and their graphs. Have the student use graphing technology to find approximate solutions of systems of equations.
Consider implementing the MFAS task Graphs and Solutions  2 (AREI.4.11). 