Getting Started 
Misconception/Error The student has little or no understanding of the concept of a function. 
Examples of Student Work at this Level The studentâ€™s response does not address the relationship between the variables. 
Questions Eliciting Thinking Do you know what a relation is?
Do you know what makes a relation a function?
How many outputs can be associated with one input?
Explain why you thought this relation was/was not a function. 
Instructional Implications Review with the student the definition of a function emphasizing that for every input value there can be only one output. Discuss domain and range, using a variety of terms including input/output values and independent/dependent variables.
If the student completed the tables, have the student write each row of values as ordered pairs, emphasizing again that every xvalue can be paired with only one yvalue.Â Next, ask the student to plot each ordered pair in the coordinate plane. Assist the student in understanding why one table or diagram results in a function while the other does not. Relate the vertical line test to the definition of a function. Make sure the student understands that when an xvalue is paired with more than one yvalue, the points on the graph will align vertically.
Provide the student with several relations represented by either a table of values or a mapping diagram. Be sure to include several examples and nonexamples of functions. Have the student determine whether the relation is a function and justify his/her answer. 
Moving Forward 
Misconception/Error The student understands that a function is a relation but does not understand its distinguishing characteristics. 
Examples of Student Work at this Level The student justifies his or her answer with a statement such as:
 Every yvalue can only be paired with one xvalue.
 The x and yvalues cannot be the same number in a function.
 It is not a function because the points do not make a straight line.
 The xvalue andÂ yvalue should always be the same in a function.
 Every input has an output, so it is a function.
The student:
 Pairs the same xvalue with the same yvalue twice in the table and writes that it is not a function because the xvalues cannot repeat.
 Pairs the same yvalue with different xvalues and writes that it is not a function because the yvalues cannot repeat.
 Pairs the same xvalue with two different yvalues and writes that it is a function because the relation has two of the same xvalues.
 Justifies the relation is not a function because the relationship between the x and yvalues is not the same for every ordered pair (e.g., â€śThe range must always be twice the domainâ€ť).

Questions Eliciting Thinking What is a function? What makes a relation a function?
Can you explain why you thought this relation was/was not a function?
What did you mean by the xvalues cannot repeat?
Are all functions linear? 
Instructional Implications Review with the student the definition of a function emphasizing that for every input value there can be only one output. Have the student list the values he or she wrote for the domain and range as ordered pairs, emphasizing again that eachÂ xvalue can only be paired with exactly one yvalue.
Provide the student with several relations represented by either a table of values or a mapping diagram. Be sure to include several examples and nonexamples of functions that illustrate the most common misconceptions listed in the examples of student work. Have the student determine whether the relation is a function and justify his or her answer.
Provide the student with an equation that represents a linear function. Have the student create a table of values for this function by substituting several values into the function. Select an xvalue the student has already used and substitute that value again and record the result in the table. Ask the student if this table of values represents a function.
Consider implementing MFAS task Identifying Functions (FIF.1.1). 
Making Progress 
Misconception/Error The student demonstrates an understanding of the concept of a function in his or her explanation but does not understand how to interpret the mapping diagram. 
Examples of Student Work at this Level The student identifies Diagram 4 as not representing a function because two different values in the domain are paired with the same value in the range.
The student identifies Diagram 3 as not being a function because the domain and range do not have the same number of elements. 
Questions Eliciting Thinking What is a function?
Can you rewrite the x and yvalues from your mapping diagram as a list of ordered pairs?
Explain why you thought this relation was/was not a function. 
Instructional Implications Provide the student with several mapping diagrams that represent examples and nonexamples of functions. Explain how the mapping diagram shows the pairing of elements from the domain with elements from the range. Have the student translate the mapping diagram into a list of ordered pairs and determine whether or not the relation is a function.
Consider implementing MFAS task Identifying Functions (FIF.1.1). 
Almost There 
Misconception/Error The student is able to determine which examples represent y as a function of x and which do not but provides an incomplete or nonmathematical explanation. 
Examples of Student Work at this Level The student writes a statement such as:
 If you put a value in you should always get out the same number.
 If you plug inÂ x, you get the same y every time.
 All of the different numbers have different outcomes.
 If x = y, then it has to equal that every time.

Questions Eliciting Thinking Can you explain further why you thought this relation was/was not a function? 
Instructional Implications Review with the student the definition of a function emphasizing that for every input value there can be only one output. In a function, the same input value will always be paired with the same output value. Model the use of mathematical reasoning and mathematical terminology in explaining why a relation is or is 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 writes a statement such as:
 Yes, because each x is only paired with one y.
 No, because the input has more than one output.

Questions Eliciting Thinking How could the domain in Table 2 be changed so that it represents a function? How about Diagram 4? 
Instructional Implications Introduce the student to the concept of onetoone functions. Ask the student if there are any examples on the worksheet of onetoone functions. Have the student rewrite any nonexamples, so they represent onetoone functions. 