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 student makes decisions for reasons that are unrelated to the definition of a function. For example, the student justifies decisions based on:
 The signs of the numbers in the tables or whether the numbers are whole or rational.
 Whether the relationship between inputs and outputs is linear.
 Whether there is an identifiable relationship between the values of x and y.

Questions Eliciting Thinking What is a function? What is the definition of a function?
How can you tell if a table of values represents a function?
Do you need to know the equation to decide whether or not the table could represent a function?
Does a functional relationship always involve proportionality? 
Instructional Implications Review the definition of a function emphasizing that each input can have only one output. Create with the student examples of relations represented by tables of values some of which are functions and some that are not. Model for the student how to determine, using the definition of a function, whether each one represents a function. Challenge the student to create tables of values that represent functions and tables of values that represent relations that are not functions.
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 develop explanations for why each does or does not represent a function based on the definition of a function. When justifying why a relation is not a function, ask the student to include with the explanation a specific example of an input with two different outputs. 
Moving Forward 
Misconception/Error The student can only provide a procedural explanation when justifying whether the tables of values could represent functions. 
Examples of Student Work at this Level The student:
 Explains in terms of the repetition of xvalues in the table.
 Graphs the ordered pairs and explains in terms of the vertical line test.

Questions Eliciting Thinking What is the definition of a function?
Why are you looking for repeating xvalues? How does this relate to the definition of a function?
How is the vertical line test related to the definition of a function? 
Instructional Implications Review the definition of a function with the student emphasizing that every input can only be paired with one output. Have the student review the values in the table to determine if any input is paired with more than one output. Model explaining, using the definiton of a function, why the first table of values represents a function while the second table does not. Provide a specific example of an input with two different outputs to show that the second table does not represent a function [e.g., (1, 1.8) and (1, 1.8)]. Encourage the student to do the same when explaining why a relation is not 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 therefore, is not a function.
Provide additional tables of values and ask the student to develop explanations for why each does or does not represent a function based on the definition of a function. When justifying why a relation is not a function, ask the student to include with the explanation a specific example of an input with two different outputs. 
Almost There 
Misconception/Error The student makes an error in reasoning using the definition of a function. 
Examples of Student Work at this Level The student concludes the first example does not represent a function because there is one input, four, that is paired with two outputs.

Questions Eliciting Thinking Can you explain why the first table does not represent a function?
What two different outputs are associated with the input, four? 
Instructional Implications Explain to the student why the first example actually represents a function. Consider using a mapping diagram to show that each input is paired with exactly one output. Provide additional tables of values and ask the student to develop explanations for why each could or could not represent a function based on the definition of a function. When justifying why a relation is not a function, ask the student to include with the explanation a specific example of an input with two different outputs. 
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 explains that:
 The first example represents a function because every xvalue is paired with exactly one yvalue. The student may explain that although the xvalue of four is listed twice in the table, it is paired with the same yvalue.
 The second example does not represent a function because the xvalue of one is paired with two different yvalues (1.8 and 1.8).

Questions Eliciting Thinking How could you determine if a relation described algebraically (e.g., y = 2x – 3) is a function? 
Instructional Implications Present the student with additional examples of relations, some of which are functions, presented in a variety of ways (using tables, graphs, algebraic equations, and verbal descriptions). Have the student indicate whether or not each represents a function and justify his or her answers. 