Getting Started 
Misconception/Error The student does not demonstrate an understanding of the concept of a function. 
Examples of Student Work at this Level The student does not understand that a function is a relation that pairs each input with exactly one output. For example, the student:
 Guesses and provides no justification.
 Substitutes numbers for the variables and tries to solve.
 Assumes all functions are linear or first degree so does not recognize the second example as that of a function.
 Provides an explanation that does not indicate an understanding of the concept of a function.

Questions Eliciting Thinking What is a relation? What is a function? What makes a relation a function?
How are functions described?
Can you construct a table that represents a function? 
Instructional Implications Review the definition of a function emphasizing that each input value is paired with only one output value. Review linear functions and their equations. Be sure the student understands that every linear equation, except those of vertical lines, represent functions since each input will always have only one output. Guide the student to recognize the form of a linear function presented algebraically and the form of the equation of a vertical line.
Guide the student to explore unfamiliar equations by finding ordered pair solutions and reasoning about the possibility of the same input having more than one output. Model doing soÂ by exploring the equation . Ask the student to consider any operation(s) involving the inputs (i.e., squaring them). Then ask the student if squaring a particular input could ever result in two different outputs. Allow the student to square a variety of rational and irrational numbers in order to explore this possibility. Then have the student consider the equation . Show the student how the same input (e.g., 9), can result in two different outputs (i.e., 3 and 3). Write this input and its two outputs as ordered pairs. Emphasize that once an example of this type has been identified, then one knows that the relation cannot be a function and there is no need to continue.
Encourage the student to also graph equations in order to determine whether or not they represent functions. 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. 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. Emphasize that even when using the vertical line test, the student should use the definition of a function to justify his or her answers.
Expose the student to a variety of functions both linear and nonlinear presented algebraically, graphically, in tables, and by verbal descriptions. 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. 
Making Progress 
Misconception/Error The student demonstrates an understanding of the concept of a function but does not have an effective strategy for determining whether a relation described algebraically is a function. 
Examples of Student Work at this Level The student correctly identifies only the first two examples as functions. However, the studentâ€™s justification is incomplete or contains errors. Upon questioning, the student explains that in order for a relation to be a function, each input can only be paired with one output but struggles to provide a coherent justification for each example.

Questions Eliciting Thinking How can you use an equation to explore whether an input is paired with more than one output?
How does the vertical line test work? What does it mean when a vertical line intersects the graph in more than one place?
Are all functions linear? Can some functions be nonlinear? 
Instructional Implications Guide the student to explore unfamiliar equations by finding ordered pair solutions and reasoning about the possibility of the same input having more than one output. Model doing so by exploring the equation . Ask the student to consider any operation(s) involving the inputs (i.e., squaring them). Then ask the student if squaring a particular input could ever result in two different outputs. Allow the student to square a variety of rational and irrational numbers in order to explore this possibility. Then have the student consider the equation . Show the student how the same input (e.g., 9), can result in two different outputs (i.e., 3 and 3). Write this input and its two outputs as ordered pairs. Emphasize that once an example of this type has been identified, then one knows that the relation cannot be a function and there is no need to continue.
Encourage the student to also graph equations in order to determine whether or not they represent functions. 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. 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. Emphasize that even when using the vertical line test, the student should use the definition of a function to justify his or her answers.
Expose the student to a variety of functions both linear and nonlinear presented algebraically, graphically, in tables, and by verbal descriptions. 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 identifies the first two relations as functions and explains that in each case, for each input there will always be only one output. The student says the third relation is not a function since an input can have two different outputs. When asked, the student can provide an example such as (4, 2) and (4, 2). 
Questions Eliciting Thinking You said the graph of the first equation is a line. Why does that make it a function? Are all lines examples of functions?
What if a relation pairs two different inputs with the same output? Could it still be a function?
What if a relation pairs every input with the same output? Could that be a function? What would its graph look like? 
Instructional Implications Challenge the student to write equations that can represent functions and equations that cannot represent functions. 