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:
 Says a function is â€śan equation like y = mx + b.â€ť Upon questioning, the student is unable to elaborate or describe any important properties of a function.
 Describes a function in any of a number of incorrect ways such as:
 Functions are two points that depend on each other to form a line.
 A function is an equation that you solve and graph.
 A function is a shape or line that never intersects.

Questions Eliciting Thinking What are the components of functions? Can you describe what a function does with these components?
Can you give an example of a function? What is domain? What is range?
Must all functions be linear? Is it possible to have a nonlinear function? If so, how would it look? Can you sketch a nonlinear function for me?
How does a function look when presented in a table or a mapping diagram? What does the graph of a function look like? 
Instructional Implications Provide instruction on the definition of a function. Explain that a function is a relation in which each element of the domain is paired with exactly one element of the range or a rule that assigns to each element in the domain a single element of the range. Use the â€śfunction machineâ€ť analogy to help the student visualize that a function is like a machine that takes a value as an input and transforms it into an output value. Then explain that the input and output can be described as an ordered pair. Implicit in the function machine analogy is the defining property of functions (i.e., each element of the domain is paired with exactly one element of the range since each input is transformed into a single output). Emphasize that the domain is the set of all inputs and the range is the set of all outputs. Eventually, introduce function notation and guide the student to use correct notation when working with functions.
Provide examples of functions described in a variety of formats: verbal descriptions, twocolumn tables, equations, and graphs. Emphasize that each format is a way to show the pairings of elements from the domain with elements in the range (or, returning to the function machine analogy, inputs with outputs). Be sure to include examples of both linear and nonlinear functions as well as examples of relations that are not functions.
Model defining functions using appropriate mathematical terminology. 
Making Progress 
Misconception/Error The student can describe an important property of functions but offers an incomplete or incorrect definition. 
Examples of Student Work at this Level The student explains that:
 The graph of a function will pass the vertical line test which means that â€śxvalues do not repeat.â€ť
 A function includes â€śinput and output values, and the inputs never repeat.â€ť
 A function â€śrepresents a relationship.â€ť
 A function pairs â€śincomesâ€ť and â€śoutcomes.â€ť

Questions Eliciting Thinking What you said about the graph of a function is true, but what exactly is a function?
How does a function look? How is a function described?
What does the domain of a function represent? What does the function do to the elements in its domain?
How are the domain and range of a function related?
What does f(x) mean? 
Instructional Implications Guide the student in a discussion of a more precise definition of a function. Explain that a good definition places the object of the definition in a larger class of objects as well as precisely describes what distinguishes it from other objects in that larger class. For example, a function is a relation (the larger class of objects to which it belongs) that pairs each element of the domain with exactly one element of the range (the property that distinguishes it from other relations). Be sure the student understands terminology associated with functions such as domain and range, and the student uses notation appropriately. 
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 defines a function in an appropriate way. For example, a function is a relation in which each element in the domain (input value) is paired with exactly one element in the range (output value). The student might further explain that if f is a function, and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x.

Questions Eliciting Thinking What do you know about function notation? What does f(x) mean?
Can you describe a specific example of function? How would you describe its domain and its range? 
Instructional Implications Challenge the student to describe an example of a function using each of the following formats: a verbal description, a twocolumn table, an equation, and a graph. Require that not all examples be linear. Then ask the student to describe the domain and range of each function. 