Continuing with the dot pattern from the Warm-up, the goal of this activity is for students to understand that, since sequences are functions, we can define sequences recursively and express them using function notation.

Students use a table to express regularity in repeated reasoning (MP8) as they write an expression for using . They also consider possible inputs for , reasoning that only integer values make sense. This leads to students expanding their understanding of sequences as functions whose inputs are restricted to the integers.

If students respond to the first question that any number or any positive number works for , without being more specific, consider asking:

• “Tell me more about how you decided what values make sense for .”
• “What types of values wouldn’t work for ?”

It is okay if students do not recall the term “integer” at this time, but they should understand that “any number” includes values such as 0.5 and , both of which could not be values of in this context.

The two important takeaways from this discussion are that sequences are a type of function whose domain is a subset of the integers and an understanding of what and mean.

Begin the discussion by inviting students to share values that do and do not make sense for , recording these for all to see. Make sure students understand that non-integer values do not make sense for the sequence represented by this dot pattern since there is no partial step between the steps that we can calculate. If students ask about , let them know that we could say that there is a Step 0 with 0 dots. , however, does not make sense here unless we attempt to define what a “negative dot” is. Tell students that an important aspect of working with functions is defining a domain that makes sense given the context and the model we choose to represent the context.

Next, invite students to share their equation for , focusing on how they reasoned about “in terms of the previous step” in order to get to .

After students share, highlight that:

• The equation for is known as a recursive definition because it describes a repeated, or recurring, process for getting the values of .
• The two other pieces needed for this type of definition are the starting value and what values can be.
• In this case, we can write the full definition as and for , where is an integer. Often times the “ is an integer” piece is left off when we know the function is a sequence, since the domain of a sequence is always a subset of the integers.

Using as the input to a function is likely an unfamiliar idea for students. They will continue to practice using function notation to define sequences recursively in the following activity, so they do not need to have mastery at this time.

Continuing the work of using function notation to write recursive definitions for sequences, students use this activity to practice writing such definitions. Students have previously seen several of the sequences in this activity since the focus of this activity is on the notation and not on making sense of the sequences. The sequence 1, 3, 7, 15, 31, . . . is from the Tower of Hanoi puzzle and is the only sequence in the activity that is neither arithmetic nor geometric.

Monitor for groups who wrote their definitions for the first two sequences in different, but correct, ways. For example, for the sequence 80, 40, 20, 10, 5:

• , 
• ,
• ,

In each of these approaches, the growth factor (or rate of change) is displayed in a different way, which corresponds to students’ previous work with writing non-recursive functions in more than one form. Showing multiple correct recursive definitions will help students build their understanding of recursively defined functions and discourage students from memorizing definitions.

This is the first time Math Language Routine 7: Compare and Connect is suggested in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations, and they record their method for all to see. They then compare and identify correspondences across strategies by means of a teacher-led gallery walk with commentary or teacher think-aloud (such as "I notice . I wonder "). A typical discussion question is “What is the same and what is different?” prompting students to compare their own strategy to the others. The purpose of this routine is to allow students to make sense of mathematical strategies and, through constructive conversations, develop awareness of the language used as they compare, contrast, and connect other ways of thinking to their own.

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Some students may not be sure where to start when defining a function recursively. Consider asking:

• “Tell me more about the recursive definition in the previous activity. What does each part mean to you?”
• “How could you write out in words how to use a term in the sequence to get the next term?”

The purpose of this discussion is for students to recognize that there can be multiple correct ways to write a recursive definition for a function, which matches how students have previously written non-recursive definitions for functions. Throughout the discussion, make sure that each recursive definition includes the starting value of the input and what input values are possible. Display the six sequences for all to see throughout the discussion to record the sequence type and student thinking next to each sequence.

Begin the discussion by displaying 2–3 approaches for sequences and from previously selected students for all to see. If time allows, invite students to share the sequence type and briefly describe their approach, then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the sequences have in common? How are they different?”
• “How does the growth factor (or rate of change) show up in each sequence?”
• “Are there any benefits or drawbacks to one representation compared to another?”

While the commutative property is not new to students, explicitly calling this out is a way to remind students that function notation like represents a specific value (the output of the function for an input of ) and should be treated just like a number. Repeat this process for the remaining sequences until the sequence type and at least one definition is written for each sequence.