Getting Started 
Misconception/Error The student does not demonstrate an understanding of the concept of a function or holds multiple misconceptions about functions. 
Examples of Student Work at this Level The student:
 Says that functions are lines or must be linear.
 Says that passing the vertical line test is the definition of a function.
 Makes a statement with some numerical values with no reference to functions.
 Uses a vocabulary word such as domain, input, or vertical line test without any explanation.
 Uses incorrect terminology to characterize a function (e.g., says that a function is “a data that is continuous”).

Questions Eliciting Thinking What can you tell me about a function? Do you know any vocabulary words that would apply?
Can you draw a picture that would represent a function? Can you draw one that is not a function?
Can you construct a table that would represent a function?
Can you tell me any other way to describe functions?
You mentioned the vertical line test. How does that test work? 
Instructional Implications Review the definitions of relation and function emphasizing that a function is a relation in which every input value can only be paired with one output. Provide examples of both functions and relations that are not functions described in a variety of ways (e.g., tables of values, mapping diagrams, algebraic rules, graphs, and verbal descriptions). Be sure to include many nonlinear examples of functions. Guide the student to carefully consider each example to determine whether or not it represents a function. Model explaining and justifying the reasoning behind the determination.
Explain the difference between defining the term function and using a test to detect functions such as identifying repeated values of x in a table or using the vertical line test. Explain the rationale behind the vertical line test directly relating it to the definition of a function. Be sure the student understands that if a vertical line intersects a graph in more than one point, each of the points of intersection contain the same xcoordinate but different ycoordinates. Consequently, the same value of x has been paired with more than one value of y, so the graph cannot represent a function. 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, absolute value, exponential, step, and piecewise. Include both horizontal and vertical lines. Have the student indicate whether or not each graph represents a function and to justify his or her answers.
Caution the student to be careful when examining relations given as tables of values. Point out the difference between an input having more than one output and the input occurring more than once with the same output (e.g., the relation {(1, 3), (2, 6), (3, 9), (1, 3), (5, 15)} is a function). Although an input, one, occurs twice, it is paired with just one unique output value. 
Making Progress 
Misconception/Error The student demonstrates an understanding of the concept of a function but is unable to provide a complete and precise definition. 
Examples of Student Work at this Level The student defines a function in terms of the vertical line test but gives no clear understanding that each input has only one output.
The student uses drawings to explain a function but is unable to explain using correct mathematical terminology.
The student indicates that a domain value does not repeat and that makes the relation a function.

Questions Eliciting Thinking You mentioned the vertical line test. How does that test work?
What must be true of the inputs and their outputs in order for a relation to be a function?
Does the graph of a function have to be a straight line? 
Instructional Implications Provide the student with the definition of the term function emphasizing that a function is a relation in which every input value can only be paired with one output. Explain the difference between defining the term function and using a test to detect functions such as identifying repeated values of x in a table or using the vertical line test. Explain the rationale behind the vertical line test directly relating it to the definition of a function. Caution the student to be careful when examining relations given as tables of values. Point out the difference between an input having more than one output and the input occurring more than once with the same output (e.g., the relation {(1, 3), (2, 6), (3, 9), (1, 3), (5, 15)} is a function). Although an input, one, occurs twice, it is paired with just one unique output value.
Provide additional examples of both functions and relations that are not functions described in a variety of ways (e.g., tables of values, mapping diagrams, and graphs). Ask the student to determine which examples are functions and to justify his or her decisions by referencing the definition rather than a test for detecting relations that are not functions. 
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 says that a function is:
 A rule that assigns exactly one output to each input,
 A relation in which each input has exactly one output, or
 A mapping from one set (called the domain) to another set (called the range) that assigns each element of the domain exactly one element of the range.
The student may elaborate with examples and nonexamples.

Questions Eliciting Thinking Does it matter if there is an output that is paired with more than one input? Is {(0, 5), (1, 5), (2, 5), (3, 5), (4, 5)} a function?
Is a vertical line a function? 
Instructional Implications Ask the student to explore relations described by algebraic rules to determine whether or not they are functions. Be sure to specify which variable represents the input and which the output (e.g., give the student the rule q + 2 = p). Ask the student if this rule represents a function when the inputs are given by p and the outputs are given by q. Then ask the student if it is still a function if the inputs are given by q and the outputs are given by p. 