Getting Started 
Misconception/Error The student is unable to identify a portion of the graph that could be removed so that the remainder represents aÂ function. Â 
Examples of Student Work at this Level Rather than removing a portion of the graph so that the remaining portion represents a function, the student says that the graph cannot represent a function because:
 It is not a straight line.
 It is a circle.
 It does not â€śshow change.â€ť
The student says to remove a portion of the graph that will not result in the remainder representing a function.

Questions Eliciting Thinking What is a function?
Does a graph have to be a straight line to represent a function?
Why doesnâ€™t the original graph represent a function?
How did you determine this remaining portion of the graph represents a function?
What is the vertical line test? How can you use the vertical line test to determine whether or not a graph represents a function?Â 
Instructional Implications Review the definition of a function with the student emphasizing that each input value can have only one output value. If the student was able to choose a portion of the graph that he or she thought represented a function, have the student identify several ordered pairs that have the same xcoordinate by using the vertical line test. Then ask the student to identify their ycoordinates. Relate this specific example to the definition of a function to explain why this portion of the graph does not represent a function.
Present the student with examples of graphs that represent functions and some that do not. Explain the rationale behind the vertical line test and assist the student in using the test to identify the graphs of functions. Expose the student to a variety of graphs to include linear, quadratic, cubic, rational, trigonometric, absolute value, logarithmic, exponential, cubic, 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. Of those that do not, ask the student to remove a portion of each graph so that the remaining portion represents a function.
Consider implementing MFAS task Identifying the Graphs of Functions (FIF.1.1). 
Moving Forward 
Misconception/Error The student illustrates removing a portion of the graph so that the remaining portion represents a function but is unable to explain his or her reasoning. 
Examples of Student Work at this Level The student illustrates the removal of a portion of the graph so that the remainder is a function. When explaining why the remaining portion of the graph is a function, the student:
 Says it passes the vertical line test but is unable to explain how the vertical line test relates to the definition of a function.
 References the definition of a relation (e.g., says, â€śEvery x has a yâ€ť) rather than the definition of a function.

Questions Eliciting Thinking What is the definition ofÂ a function?
How do you know the portion of the graph that you selected represents a function?
Why must the vertical line cross the graph at only one point at a time? (Explanation should relate directly to the definition of a function.)
What does it mean if a vertical line intersects 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.
If needed, assist the student in precisely describing the portion of the graph that was removed or the portion that remains. For example, suggest describing the portion that is removed as the points on the graph that lie in the first and second quadrants, orÂ points on the graph of the form {(x, y)0 <Â yÂ <Â 6}. 
Almost There 
Misconception/Error The student is imprecise in describing the portion of the graph that is to be removed. 
Examples of Student Work at this Level The student demonstrates an understanding of the graph of a function, but describes the portion of the graph imprecisely. The student describes the selected portion of the graph by saying:
 Remove the arc from 6 to 6.
 A semicircle would be a function.
 The bottom half or top half.

Questions Eliciting Thinking Can you show me the portion of the graph you are describing?
Specifically which points are remaining? What about (6, 0)? What about (0, 6)?
Is there another way that you could describe the portion of the graph that you chose? 
Instructional Implications Discuss with the student how one could misinterpret the description he or she gave of the portion of the graph that was chosen. Provide the student with some examples of a correct mathematical description of the graph that could represent a function. For example, suggest describing the portion that is removed as the points on the graph that lie in the first and second quadrants, orÂ points on the graph of the form {(x, y)0 <Â yÂ <Â 6}.
Present the student with several different portions of the graph of the circle. Ask the student to describe each portion and determine whether or not it could represent a function. Likewise, describe a portion of the graph and ask the student to identify that portion and determine if it represents 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 correctly identifies a portion of the graph that could be removed so that the remaining portion represents a function. For example, the student says to remove all points that lie in the first and second quadrant or remove points on the graph of the form {(x,Â y)0Â <Â yÂ < 6}. The student explains that the graph now representsÂ a function because each xvalue is paired with only one yvalue or it passes the vertical line test. If the student references the vertical line test, he or she is able to explain how the vertical line test detects a function by relating the test to the definition of a function. 
Questions Eliciting Thinking Are there other portions of the graph that would also represent a function?
Are there any points on the original graph that contain xvalues that are paired with only one yvalue? If you removed all but these points, would the new graph represent a function? 
Instructional Implications Introduce the student to the idea of onetoone functions. Challenge the student to find a line test that can be used to identify onetoone functions. Encourage the student to look at the portion of the graph that he or she chose to determine if it is onetoone. If the graph does not represent a onetoone function, have the student choose a portion of the graph that would. If the graph does represent a onetoone function, ask the student if there are other portions of the graph that could also represent a onetoone function. Have the student sketch examples of graphs that represent onetoone functions. 