Getting Started 
Misconception/Error The student does not understand graphical representations of relations and functions. 
Examples of Student Work at this Level The student makes a decision about the graphed relations but his or her explanation does not indicate an understanding that the graphs convey sets of ordered pairs consisting of inputs and their corresponding outputs. The student’s decision is based on:
 How many times the graph intersects the yaxis.
 Whether or not the graph is symmetric across the yaxis.

Questions Eliciting Thinking What is a relation? How does the graph of a relation convey inputs and outputs?
When is a relation a function?
How can you tell whether or not a graph can represent a function?
What is the line test? How can you use it? 
Instructional Implications Review the definition of a relation (e.g., a set of ordered pairs consisting of an input and a corresponding output). Emphasize that each point on the graph of a relation is represented by an ordered pair and it is the set of these ordered pairs that comprise the relation. Within each ordered pair, the xcoordinate can be thought of as the input and the ycoordinate is the output.
Review the definition of a function emphasizing that each input value can have only one output value. Expose the student to a variety of functions both linear and nonlinear presented algebraically, graphically, in a table, and by verbal descriptions. Ask the student to explain why each represents a function. Provide feedback as needed.
Explain the rationale behind the vertical line test and assist the student in using the test to identify the graphs of functions. Present the student with additional examples of graphed relations, some of which are functions. Expose the student to a variety of graphs to include linear, quadratic, cubic, rational, absolute value, exponential, step, and piecewise. Include both horizontal and vertical lines. Have the student indicate whether or not each graph represents a function and to use the definition of a function to justify his or her answers. 
Making Progress 
Misconception/Error The student can determine whether a relation is a function from its graph but can only provide a procedural explanation. 
Examples of Student Work at this Level The student correctly determines whether the graphs represent functions but explains in terms of the “vertical line test.” The student is unable to relate the vertical line test to the definition of a function or explains in terms of the “repetition of x values” as if he or she were inspecting a table of ordered pairs and simply looking for xvalues that occur more than once. 
Questions Eliciting Thinking What is the definition of a function?
How is the vertical line test related to the definition of a function?
How is the repetition of an xvalue related to whether or not a relation is a function? 
Instructional Implications Review the definition of a function. Have the student draw a vertical line on the second graph that intersects the graph in more than one point. Then have the student circle the points of intersection and write those points as ordered pairs. Model explaining why the second graph does not represent a function in terms of the definition of a function.
Help the student relate the definition of a function to the vertical line test. Model explaining how the vertical line test can be used to determine whether or not a graph represents a function. Have the student identify points on the graph that indicate an input is paired with more than one output. Ask the student how the vertical line test enables one to identify such points. Be sure the student understands that when a vertical line intersects a graph in more than one place, there is a value of x that has more than one associated yvalue and is, therefore, not a function. 
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 first graph represents a function because every xvalue is paired with exactly one yvalue or every input has only one output. The student states that the second graph does not represent a function because there are xvalues that are paired with more than one yvalue or inputs that have more than one output. The student may reference specific examples on the graph. 
Questions Eliciting Thinking What portion of the second graph could you remove so that it would represent a function? If part of the graph is removed, does it still represent the original relation? 
Instructional Implications Present the student with additional examples of graphed relations, some of which are functions. Expose the student to a variety of graphs to include linear, quadratic, cubic, rational, absolute value, exponential, step, and piecewise. Include both horizontal and vertical lines. Have the student indicate whether or not each graph represents a function and justify his or her answers. 