Getting Started 
Misconception/Error The student does not understand the meaning of fÂ (22) or fÂ (a). 
Examples of Student Work at this Level The student does not interpret fÂ (22) and fÂ (a) as representing values of the dependent variable. For example, the student:
 Says fÂ (22) is the number of minutes the cell phone has been in use.
 Interprets function notation in terms of multiplication.
The student incorrectly interprets fÂ (22) in terms of the dependent variable. For example, the student says fÂ (22) means that there is 22% of the battery life left.

Questions Eliciting Thinking What are the two variables being related in this problem?
What is the difference between fÂ (22) and 22 in the context of this problem?
How do you read the symbols f (m)? What do you think the m stands for? 
Instructional Implications Provide direct instruction on function notation and interpreting function notation in the context of a problem. Emphasize that a function associates elements of the domain with elements of the range and that the notation reflects this association. Clarify where elements from the domain and elements from the range appear in function notation. Emphasize that fÂ (x) represents a value of the dependent variable associated with an independent variable value of x. Expose the student to problems that involve two different functions, f and g, so there is an opportunity to observe the usefulness of naming functions. Provide additional opportunities for the student to encounter, use, and interpret function notation in problem contexts, so it becomes familiar and comfortable. 
Moving Forward 
Misconception/Error The student does not understand the meaning of fÂ (a) = 50. 
Examples of Student Work at this Level The student interprets fÂ (22) as representing a value of the dependent variable, that is, as a percent of battery life remaining. However, the student is unable to correctlyÂ interpret the equation fÂ (a) = 50.

Questions Eliciting Thinking What is the significance of the number 22 in the notation fÂ (22)?
If fÂ (22) represents a percent of battery life, what should fÂ (a) represent?
What does the 50 refer to in this problem? What is the relationship between a and 50? 
Instructional Implications If needed, clarify the meaning of the number 22 in the notation fÂ (22). Then relate the meaning of fÂ (22) to the meaning of fÂ (a). Guide the student to observe how function notation relates elements from the domain with elements from the range. Clarify where elements from the domain and elements from the range appear in function notation. Provide additional opportunities for the student to encounter, use, and interpret function notation in problem contexts. 
Almost There 
Misconception/Error The studentâ€™s explanation does not make clear the association between values of the independent and dependent variables. 
Examples of Student Work at this Level The student does not make explicit the relationship between specific values of the domain and range implicit in function notation. For example, the student describes fÂ (22) as a percent of battery life remaining but does not make clear that it is the percent after 22 minutes of use. Or, the student explains that fÂ (a) = 50 means that there is 50% of the battery life remaining but does not make clear that this occurs after a minutes of use.

Questions Eliciting Thinking What is the purpose of the 22Â in the expression fÂ (22)?
On your paper you stated that 50 represented the percentage of battery life remaining; what does the a represent? Do the a and the 50 go together in any way? 
Instructional Implications Emphasize the relationship between the independent and dependent variables and how to interpret them in the context of a problem. Model interpreting function notation in the context of problems by making the association between values of the independent and dependent variables explicit. Expose the student to other students' explanations and interpretations that are complete and correct. Provide additional examples of functions in which the student is given a specific input and asked to identify the associated output (and vice versa) using function notation. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level With regard to the first question, the student says f (22) represents the percent of battery life remaining after the cell phone has been in use for 22 minutes. With regard to the second question, the student says thatÂ fÂ (a) = 50 means that after a minutes of use, the phone willÂ have 50% battery life remaining. 
Questions Eliciting Thinking Does it matter which letters or symbols are chosen when writing function notation?
How could you rewrite the ordered pair (25, 90) using function notation? What would this ordered pair mean in the context of this problem? 
Instructional Implications Provide the student with examples of the misuse of function notation [e.g., f(x) = y + 3; for some function g, g = x â€“ 8; or (d)h = d2]. Ask the student to explain the notational error and correct it.
Give the student a variety of representations of functions (e.g., verbal descriptions, graphs, algebraic rules). Then ask the student to use function notation to find and describe output values given corresponding input values, and vice versa. 