Getting Started 
Misconception/Error The student does not demonstrate an understanding of a recursively defined sequence. 
Examples of Student Work at this Level The student is unable to correctly find the values of the terms of the recursively defined sequence. For example, the student:
 Calculates terms using another recursive rule such as S(n) = (n â€“ 1)S(n â€“ 1) â€“ 5.
 Calculates terms using an explicit rule such as S(n) = 33Â â€“ 6n.

Questions Eliciting Thinking What does it mean for a sequence to be defined recursively?
What is the value of the sequence when nÂ = 1?
What does S(n  1) mean? 
Instructional Implications Review the difference between defining sequences explicitly and recursively. Provide examples of each and guide the student to calculate the first few terms. Review notation used to define a recursive sequence, such as S(n), S(n  1), or S(n + 1). Guide the student to understand that S(n  1) refers to the term preceding S(n) and that S(n + 1) refers to the term after S(n). Explain that the definition of a particular recursive sequence contains two components: a statement of the first term and a formula for calculating successive terms from preceding terms. Remind the student that S(1) is given as 27. Then explain that the given formula indicates that S(2) = S(1)  5 = 27  5 = 22. Have the student find the next three terms of the sequence and complete the table.
Provide additional examples of recursively defined sequences and ask the student to find the first five terms of each sequence. 
Moving Forward 
Misconception/Error The student does not demonstrate an understanding of the concept of a function. 
Examples of Student Work at this Level The student is able to use the definition of the sequence to find the first five terms. However, when asked to explain why the sequence is a function, the student says:
 There is a constant rate of change.
 Inputs and outputs are different.
 There is an xvalue and a yvalue.
 Each value of y depends on a value of x.

Questions Eliciting Thinking What are the defining qualities of a function? What can you do to determine if a relation is a function?
I agree that there is a constant rate of change in the S(n) values. Can you explain how thisÂ makes the sequence a function?
What did you mean by, â€śThe inputs and outputs are differentâ€ť?
Is any relation that relates xvalues to yvalues a function? 
Instructional Implications Review the definitions of relation and function emphasizing that a function is a relation in which every input value is paired with only one output value. Provide examples of relations that are functions and relations that are not functions described in a variety of ways (e.g., as 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 concept of a function and a test used 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 by 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 contains the same xcoordinate but a different ycoordinate. 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 including 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 justify his or her answers.
Consider implementing the MFAS tasksÂ Identifying Functions (FIF.1.1) andÂ Which Sequences Are Functions? (FIF.1.3). 
Almost There 
Misconception/Error The student is unable to correctly describe the domain of the function. 
Examples of Student Work at this Level The student is able to use the definition of the sequence to find the first five terms and can explain why the sequence is a function. However, when stating the domain of the function, the student:
 Limits the domain to only the values of n shown in the table.
 Describes the domain as n = 1 without including the set of numbers to which n belongs.
 Confuses domain with range.

Questions Eliciting Thinking Can S(6) be calculated? Is six in the domain? How many terms of this sequence can be calculated?
What kinds of numbers are in the domain? Can 1.5 be an element of the domain?
What is the difference between the domain and the range of a function? 
Instructional Implications Explain that the terms of a sequence comprise the range of a function while their term numbers comprise the domain. Illustrate this using the table of values for the sequence. Emphasize that the domain of a sequence is always a subset of the integers. Model a verbal description of each domain (e.g., â€śThe domain is the set of integers greater than or equal to oneâ€ť). Also, show the student how to use notation to write each domain [e.g., {1, 2, 3, 4, â€¦} or {xÂ :Â xÂ Integers and ].
Consider implementing the MFAS task Which Sequences Are Functions? (FIF.1.3). 
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 completes the table of values as follows:
The student states the sequence is a function because for every value of n there is one value of S(n)and the domain of the function is the Counting numbers, the Natural numbers (N), or the positive Integers.

Questions Eliciting Thinking How did you determine the sequence is a function?
Can the domain of the function contain negative values? Why or why not? 
Instructional Implications Challenge the student to write an explicit function for the recursivelydefined sequence given in the problem. Remind the student to include a statement defining the constraints on the domain of the sequence. Provide additional opportunities for the student to work with sequences.
Consider implementing the MFAS taskÂ Which Sequences Are Functions? (FIF.1.3). 