Getting Started 
Misconception/Error The student does not understand the concept of domain. 
Examples of Student Work at this Level Instead of describing the domain, the student:
 Elaborates on the description of the relationship between the variables.
Â
 States that the domain of each is â€śa set of numbers.â€ť
 Confuses the domain with the range or identifies the domain as containing a third variable.
Â

Questions Eliciting Thinking What do you know about the term domain?
What do you know about function notation? Which part of a function denotes the domain? The range?
How do you determine the independent and dependent variables from function notation? How do the independent and dependent variables relate to the domain and range of a function? 
Instructional Implications Describe the domain of a function as the set of all input values (or values of x) for which the function is defined. Assist the student in understanding that the domain of the first function is the set of possible numbers of cupcakes sold and the domain of the second function is an interval of time. Guide the student to identify a number system, (e.g., integers or real numbers) that describes the kinds of numbers in the domain as well as the specific domain elements. Ask the student to consider the lower and upper limits on the domain and how to describe them (e.g., for the first function, the lower limit is zero since it is possible that no cupcakes will be sold and the upper limit depends on the number of cupcakes prepared for sale). Provide the student with additional opportunities to describe domains of functions from verbal descriptions.
Review function notation and explain that E(c) represents the earnings when c cupcakes are sold. Be sure the student understands that c represents values of the independent variable, and these values comprise the domain of the function.
Provide instruction on ways to describe sets of numbers (e.g., using inequalities, setbuilder notation, and interval notation). Emphasize that when describing the domain of a function, one must consider the number system from which the domain is drawn as well as the specific elements in the domain. Provide a verbal description of a number set [e.g., the set of real numbers from 4 to 17 (inclusive)] and ask the student to use both set builder notation and interval notation to represent the set. 
Making Progress 
Misconception/Error The student does not completely and accurately describe the domain. 
Examples of Student Work at this Level The student:
 Describes the domain as â€śthe set of all xvaluesâ€ť without specifically describing the values.
 Verbally describes the domain (e.g., as the 'set of all cupcakes soldâ€ť) but gives no indication of the kinds of numbers in the domain or any limitations on the numbers.Â
 Characterizes numbers that are or are not reasonable but does not describe any limits on the domain.
Â

Questions Eliciting Thinking What kinds of numbers are in the domain?
What specific values are in the domain?
Are there any upper or lower limits on the values in the domain? 
Instructional Implications Guide the student to identify a number system (e.g., integers or real numbers) that describes the kinds of numbers in the domain as well as the specific domain elements. Ask the student to consider the context and identify any lower or upper limits on the domain or any specific values that must be excluded.
If needed, review ways to describe sets of numbers (e.g., using inequalities, setbuilder notation, and interval notation). Provide a verbal description of a number set [e.g., the set of real numbers from 4 to 17 (inclusive)] and ask the student to use both set builder notation and interval notation to represent the set.
Provide the student with additional opportunities to describe domains of functions from verbal descriptions. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level For the first function, the student describes the domain as the set integers 0, 1, 2, 3, 4, â€¦, n. The student explains that zero is the lower limit since no cupcakes might be sold and, n, the upper limit, depends on the number of cupcakes available for sale.
For the second function, the student describes the domain as an interval of real numbers with a lower limit of zero and an upper limit that depends on how long it takes the runner to complete a 20mile run. 
Questions Eliciting Thinking How would you describe the range of each of these functions? 
Instructional Implications If needed, review ways to describe sets of numbers (e.g., using inequalities, setbuilder notation, and interval notation). Provide a verbal description of a number set [e.g., the set of real numbers from 4 to 17 (inclusive)] and ask the student to use both set builder notation and interval notation to represent the set. 