Getting Started 
Misconception/Error The student does not demonstrate an understanding of the definition of a function. 
Examples of Student Work at this Level The student is unable to correctly identify the graphs of functions and provides explanations that are unrelated to the definition of a function. For example, the explanation might be based upon:
 Whether the graph is linear or curved.
 The idea of â€śhaving slope.â€ť
 Whether the graph is â€śconstantâ€ť or not.
 Whether or not â€śthere are ordered pairs.â€ť
 Symmetry or continuity.
 Passing the vertical line test, but the student does not demonstrate an understanding of this test and how it relates to the definition of a function.

Questions Eliciting Thinking What is a function?
What is the vertical line test? How can you use the vertical line test to determine whether or not the graph represents a function? Can you show me how you are using the vertical line test?
Does a graph have to be a straight line to represent a function?
Does a graph have to be continuous to represent a function?
How did you determine that this graph did/did not represent a function? 
Instructional Implications 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, and in tables, or given 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 and nonexamples of graphed functions. Expose the student to a variety of graphs to include linear, quadratic, cubic, rational, trigonometric, absolute value, logarithmic, 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. 
Moving Forward 
Misconception/Error The student demonstrates a minor misconception by identifying one of the graphs incorrectly as being a function or not a function. 
Examples of Student Work at this Level The student:
 Is confused by the graph of the equation x = 3 and indicates that it represents a function.
 Does not understand the end behavior of the graph of the cubic function thinking that it becomes vertical and therefore, incorrectly indicates that it does not represent a function.
 Does not understand the difference between open and closed dots and indicates that the fourth graph is not a function.
 Does not realize that functions are not always continuous and indicates that the fourth graph is not a function.
Additionally, the student may not clearly explain or justify all answers.

Questions Eliciting Thinking Why did you indicate this graph was/was not a function?
What happens to this graph (pointing to the cubic function) as x gets progressively larger (or smaller)?
Could you tell that the point at (2, 4) is represented by an open dot (in the fourth option)? Do you know what an open dot signifies?
You said this graph (pointing to the fourth option) is not a function because it has multiple parts. Can you explain this in terms of the definition of a function? 
Instructional Implications Ask the student to list several points from the first graph on the worksheet in a table of values and to determine whether or not this table of values represents a function. Have the student justify his or her answer.
For the second graph, have the student draw a vertical line on the graph that intersects the graph in more than one point. Then have the student circle the points of intersection and list those points in a table of values. Ask the student to determine whether or not this table of values represents a function. Have the student justify his or her answer.
Discuss the graph of the cubic function with the student. Use a graphing calculator to graph a cubic function such as y = . Then ask the student to use the trace function to investigate the graph at its extremes. Have the student observe the coordinates of points at the extremes, so it becomes apparent that the graph is not vertical. Reinforce this by asking the student to consider what will happen when ycoordinates are calculated for two different values of xÂ (e.g., ask the student to determine ifÂ Â = Â whenÂ ).
Discuss the fourth graph with the student and explore why it represents a function. If needed, explain the difference between an open and a closed dot. Expose the student to additional examples of graphs that contain discontinuities, including some that are functions and some that are not functions. Ask the student to identify the ones that are functions and justify his or her choices. 
Almost There 
Misconception/Error The student accurately identifies all of the graphs that represent functions but is unable to explain or justify his or her choices. 
Examples of Student Work at this Level The student correctly distinguishes between the graphs that represent functions and those that do not. However, when justifying his or her decisions, the student provides an explanation that is unclear or incomplete.

Questions Eliciting Thinking How can the vertical line test be used to determine whether or not a graph represents a function?
Why must the vertical line only cross the graph at one point at a time? (Explanation should relate directly to the definition of a function.)
What are the implications of the vertical line test crossing the graph at more than one point? 
Instructional Implications Review the definition of a function emphasizing that for every input value there can be only one output value. 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 element in the domain is paired with more than one element in the range. Ask the student how the vertical line test enables one to identify such points. 
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 correctly determines that the graphs in a) and b) do not represent functions while the graphs in c) and d) represent functions. The student explains that a) and b) are not functions because either:
 The same xvalue is paired with more than one yvalue; or
 The graph fails the vertical line test.
The student explains that c) and d) are functions because either:
 Each xvalue is paired with exactly one yvalue; or
 The graph passes the vertical line test.
If the student justifies his or her choices by using the vertical line test, the student is able to explain why this test works. For example, the student says that if a vertical line intersects a graph in more than one point, then the points of intersection represent ordered pairs with the same xcoordinate but different ycoordinates. Consequently, the same xvalue is paired with more than one yvalue.

Questions Eliciting Thinking Does a horizontal line represent a function?
What do you think happens to the third graph as the values of x get progressively larger (or smaller)? 
Instructional Implications Introduce the student to the idea of onetoone functions. Encourage the student to go back through the task to identify the graphs of functions that are onetoone. Have the student sketch other examples of graphs that represent onetoone functions. Challenge the student to find a line test that can be used to identify onetoone functions. 