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 writes a statement such as:
 Neither variable is a function of the other because 'the domain and range need to be numbers' or 'you cannot graph a food item.'
 Food item is not a function of price because the food items must always be the domain.

Questions Eliciting Thinking What is a function?
What does it mean for one value to be a function of another value?
What is the domain/range in the first problem?
What is the domain/range in the second problem?
Do theÂ independent andÂ dependent variables have to be numerical? 
Instructional Implications Introduce the definition of a function as a relation in which every input is paired with exactly one output. Have the student list associated values of the independent and dependent variables as ordered pairs, first with price as the independent variable, [e.g., (1.79, pizza slice)] and then with price as the dependent variable [e.g., (pizza slice, 1.79)]. Explain that since each food item has exactly one price, then price is a function of food item. However, since each price is paired with more than one food item, then food item is not a function of price.
Be sure the student understands what it means for one variable to be a function of another variable. Explain the phrase â€śis a function ofâ€ť in mathematical terms, that is, by applying the mathematical definition of a function. Provide the student with mathematical examples (e.g., since P = 4s, the perimeter of a square is a function of the length of a side of the square) and realworld examples (e.g., the amount of tax applied to a purchase is a function of the total cost of the purchase) and model the use of the phrase â€śis a function of.â€ť
Elicit realworld examples of functional relationships from the student encouraging him or her to also consider nonnumerical variables as the domain or range. Discuss whether or not the converse relationship in each example is also a function. Then provide the student with several examples of relations. For each example, have the student determine ifÂ a is a function of b and if b is a function of a. 
Moving Forward 
Misconception/Error The student understands that a function is a relation but does not clearly understand its distinguishing qualities. 
Examples of Student Work at this Level The student writes a statement such as:
 The food determines price, but price does not determine food.
 The xvalues cannot repeat, therefore it is not a function.
 The xvalues do not repeat, therefore it is a function.
 The food item has to be the xvalue and the price has to be the yvalue.

Questions Eliciting Thinking What is a function?
Explain why you thought this relation was/was not a function?
If price is the domain, what element(s) in the range would be paired with $2.99? With $3.89?
If the food items on the menu compose the domain, what element(s) in the range would be paired with Cheeseburger? With Bacon Burger? 
Instructional Implications Review the feature of a relation that makes it a function, that is, that every input is paired with exactly one output. Have the student list associated values of the independent and dependent variables as ordered pairs, first with price as the independent variable, [e.g., (1.79, pizza slice)] and then with price as the dependent variable [e.g., (pizza slice, 1.79)]. Explain that since each food item has exactly one price, then price is a function of food item. However, since each price is paired with more than one food item, then food item is not a function of price.
Provide additional examples of relations represented by a table of values in a realworld context. Be sure to include both examples and nonexamples of functions. Have the student determine whether each relation is a function and justify his or her answer. 
Almost There 
Misconception/Error The student demonstrates an understanding of the concept of a function but misinterprets the phrase â€śa is a function of bâ€ť and confuses the independent and dependent variables. 
Examples of Student Work at this Level When asked if the food item is a function of price, the student says, â€śYes, because each food item has only one price.â€ť

Questions Eliciting Thinking What does it mean for one value to be a function of another value?
Can you rewrite the menu as a set of ordered pairs? How would the ordered pairs differ in the two problems? 
Instructional Implications Explain what it means for one variable to be a function of another variable. Explain the phrase â€śis a function ofâ€ť in mathematical terms, that is, by applying the mathematical definition of a function. Provide the student with mathematical examples (e.g., since P = 4s, the perimeter of a square is a function of the length of a side of the square) and realworld examples (e.g., the amount of tax applied to a purchase is a function of the total cost of the purchase) and model the use of the phrase â€śis a function of.â€ť
Provide the student with additional examples of relations. For each example, have the student determine ifÂ a is a function of b and if b is a function of a.
Introduce and use function notation to describe functional relationships and model explaining that y = f(x) means that y is a function of x. 
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 food item is not a function of price since there are several food items paired with the same price (e.g., both the cheeseburger and bacon burger are $2.99). However, price is a function of food item since each food item has only one price.

Questions Eliciting Thinking How could the menu be changed so that food item is a function of price?
If every item on the menu had the same price, would price still be a function of the food item? 
Instructional Implications Introduce the student to the concept of onetoone functions. Encourage the student to generate realworld and mathematical examples of onetoone functions. 