Getting Started 
Misconception/Error The student does not understand what it means for f(x) to equal g(x). 
Examples of Student Work at this Level The student completes an algebraic manipulation unrelated to finding solutions of the equation f(x) = g(x).

Questions Eliciting Thinking What does it mean for f(x) to equal g(x)? Can you write out the equation f(x) = g(x) using the functions in this problem?
Using the graph, can you find f(1)? g(2)? 
Instructional Implications 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.
Be sure the student understands the meaning of function notation such as f(2). Use the functions given in this task to show the student that f(x) = g(x) becomes the equation . Then model how to use the graph to find the solutions of the equation. Explain that since (2, 0) is a point on each equationâ€™s graph, then both f(2) = 0 and g(2) = 0 so that f(x) = g(x) when x = 2. Explain that any value of x for which f(x) = g(x) is a solution of the equation f(x) = g(x) which is in this instance. Make clear that the solutions of the equation f(x) = g(x) are the xcoordinates of the points of intersection of the graphs.
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 attempts to solve the equation f(x) = g(x) but is unable to do so correctly. 
Examples of Student Work at this Level The student attempts to solve the equation but makes algebraic errors.

Questions Eliciting Thinking Can you use the graph to find the solutions of the equation ?
What kind of an equation is ? Can you explain how you tried to solve it? 
Instructional Implications Be sure the student understands the meaning of function notation such as f(2). Then model how to use the graph to find the solutions of the equation. Explain that since (2, 0) is a point on each equationâ€™s graph, then both f(2) = 0 and g(2) = 0 so that f(x) = g(x) when x = 2. Explain that any value of x for which f(x) = g(x) is a solution of the equation f(x) = g(x) which is Â in this instance. Make clear that the solutions of the equation f(x) = g(x) are the xcoordinates of the points of intersection of the graphs.
Review the methods of solving a quadratic including factoring and using the quadratic formula. Provide additional examples of quadratic functions for the student to solve and provide feedback.
Consider implementing the MFAS task Graphs and Solutions  1 (AREI.4.11) if not used previously. 
Almost There 
Misconception/Error The student cannot use the graph to solve the equation f(x) = g(x). 
Examples of Student Work at this Level The student solves the equation Â algebraically but is unable to relate the solutions to the graphs of the functions or use the graphs to locate the solutions.

Questions Eliciting Thinking Can you use the graph to find the solutions of the equation ?
How do the solutions you found algebraically relate to the graph? 
Instructional Implications Be sure the student understands the meaning of function notation such as f(2). Then model how to use the graph to find the solutions of the equation. Explain that since (2, 0) is a point on each equationâ€™s graph, then both f(2) = 0 and g(2) = 0 so that f(x) = g(x) when x = 2. Explain that any value of x for which f(x) = g(x) is a solution of the equation f(x) = g(x) which is Â in this instance. Make clear that the solutions of the equation f(x) = g(x) are the xcoordinates of the points of intersection of the graphs.
Consider implementing the MFAS task Graphs and SolutionsÂ  1 (AREI.4.11) if not used previously. 
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 identifies the solutions as x = 2, and x = 3. The student explains that these are the xcoordinates of the points of intersection of the graphs. Upon questioning, the student further explains that the points of intersection, (2, 0) and (3, 5), represent solutions of both equations so that f(2) = 0 = g(2) and f(3) = 5 = g(3).
The student may initially describe the solutions as ordered pairs of numbers [e.g., as (2, 0) and (3, 5)] but upon questioning clearly states that solutions of the equation are x = 2, and x = 3.

Questions Eliciting Thinking You described the solutions of the equation f(x) = g(x) as ordered pairs. What, specifically, does this equation look like for these functions? Are the solutions of this equation ordered pairs of numbers?
Why should the xcoordinates of the points of intersection satisfy the equation f(x) = g(x)?
I see you found the solutions by looking at the graph. Could you find the solutions another way?
How could you verify that the coordinates of the points of intersection satisfy each function?
What are the solutions of the system of these two equations? 
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  1 (AREI.4.11) if not used previously. 