Lesson 5

Up until this lesson, students have defined sequences using mostly informal language such as “each new term is 3 more than the previous term” or “multiply the current term by 2 to get the next term.” The goal of this activity is for students to use that type of language to describe a pattern of dots and make sense of the general structure. In the following activity, students will continue working with the same pattern as they see how we can use function notation to recursively define sequences (MP1).

Tell students to close their books or devices for the warm-up. Display the image for all to see.

Select students to share their descriptions, recording these for all to see next to the image.

Next, say “The number of dots in a step is a function of the step number. Let's call this function \(D\). What is the value of \(D(4)\) and \(D(5)\)?” (10 and 15.) If needed, remind students that a function takes inputs from one set and assigns them to outputs from another set, assigning exactly one output to each input. In this case, the input is the step number and the output is the number of dots in the pattern. After some quiet think time, select students to share their values and explain their reasoning.

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 take the recursive definition they stated in the warm-up and express it using function notation. Students use a table to express regularity in repeated reasoning as they write an expression for \(D(n)\) using \(D(n-1)\) (MP8). They also consider possible inputs to \(D\), reasoning that only integer values make sense. This leads to expanding their understanding of sequences as functions whose inputs are restricted to the integers.

If students are unfamiliar with the term integer, remind them that the integers are all the counting numbers and their opposites, including zero. One way to write the integers is . . . ,-3, -2, -1, 0, 1, 2, 3, . . .

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 \(D(n)\) and \(D(n-1)\) mean.

Begin the discussion by inviting students to share values that do and do not make sense for \(n\), 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 \(D(0)\), let them know that we could say for the pattern that there is a Step 0 with 0 dots. \(D(\text-1)\), 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 what we are trying to do.

Next, invite students to share their equation for \(D(n)\), focusing on how they reasoned about “in terms of the previous step” in order to get to \(D(n)=D(n-1)+n\). Tell students that this is known as a recursive definition because it describes a repeated, or recurring, process for getting the values of \(D\). The two other pieces needed for this type of definition are what value to start at and what values \(n\) can be. In this case, we can write the full definition as \(D(1)=1\) and \(D(n)=D(n-1)+n\) for \(n\ge2\) where \(n\) is an integer. Often times the “\(n\) 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 \(n-1\) 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 writing recursive definitions for sequences using function notation started in the previous activity, the purpose of this activity is for students to practice writing such definitions. Several of the sequences selected are ones students have seen previously since the focus of this activity is on the notation and not on making sense of the sequence. 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 or geometric.

Monitor for groups who wrote their definitions for the first two sequences differently to share during the discussion. For example, for the sequence 80, 40, 20, 10, 5, . . .,  \(B(n)=B(n-1)\boldcdot0.5\), \(B(n)=\frac{ B(n-1)}{2}\), and \(B(n)=\frac12\boldcdot B(n-1)\) (all with \(n\ge2\)) are three possible definitions.

Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s for sequences \(A\) and \(B\) and decide if they are both correct, even if they are different. Follow with a whole-class discussion. 

Some students may not be sure where to start when defining a function recursively. Encourage them to look back at the previous activity. A good intermediate step is to write, in words, how to use a term in the sequence to get the next term.

The purpose of this discussion is for students to notice some common features of arithmetic and geometric sequences. Display the six sequences for all to see throughout the discussion to record the sequence type and student thinking next to each sequence.

For the sequences \(A\) and \(B\), invite previously identified groups to first share the sequence type and then the different ways they wrote their equations. While the commutative property is not new to students, explicitly calling this out is a way to remind students that things like \(A(n-1)\) represents a specific value (the output of the function \(A\) for an input of \(n-1\)) 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.

Conclude the discussion by inviting students to share things they notice about the definitions for the different sequence types. (The arithmetic sequences have adding or subtracting a term, the geometric sequences have multiplying by a term, and the sequence that is neither is a mix of adding and multiplying.) If not brought up by students, remind them that arithmetic sequences are a type of linear function and geometric sequences are a type of exponential function, so it makes sense to see only addition and subtraction for arithmetic sequences and only multiplication for geometric sequences.