Lesson 5

Sequences are Functions

5.1: Bowling for Triangles (Part 1) (5 minutes)

Warm-up

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).

Launch

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

Student Facing

Describe how to produce one step of the pattern from the previous step.

A dot pattern.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

5.2: Bowling for Triangles (Part 2) (15 minutes)

Activity

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.

Launch

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem.
Supports accessibility for: Visual-spatial processing

Student Facing

Here is a visual pattern of dots. The number of dots \(D(n)\) is a function of the step number \(n\).

A dot pattern.
  1. What values make sense for \(n\) in this situation? What values don't make sense for \(n\)?
  2. Complete the table for Steps 1 to 5.
    \(n\) \(D(n)\)
    1 1
    2 \(D(1)+2=3\)
    3 \(D(2)+3=6\)
    4  
    5  
  3. Following the pattern in the table, write an equation for \(D(n)\) in terms of the previous step. Be prepared to explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Consider the same triangular pattern.

  1. Is the sequence defined by the number of dots in each step arithmetic, geometric, or neither? Explain how you know.

  2. Can you write an expression for the number of dots in Step \(n\) without using the value of \(D\) from a previous step?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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, . . .

Activity Synthesis

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.

5.3: Let's Define Some Sequences (15 minutes)

Activity

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.

Launch

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. 

Student Facing

Use the first 5 terms of each sequence to state if the sequence is arithmetic, geometric, or neither. Next, define the sequence recursively using function notation.

  1. \(A\): 30, 40, 50, 60, 70, . . .
  2. \(B\): 80, 40, 20, 10, 5, 2.5, . . .
  3. \(C\): 1, 2, 4, 8, 16, 32, . . .
  4. \(D\): \(1, \frac12, \frac14, \frac18, \frac{1}{16},\) . . .
  5. \(E\): 20, 13, 6, -1, -8, . . .
  6. \(F\): 1, 3, 7, 15, 31, . . .

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. 
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

Make one class display for geometric and arithmetic sequences. This display should be posted in the classroom for the remaining lessons within this unit. It should look something like what is shown here. Since linear and exponential functions are review, two definitions are given here for each example sequence, one recursive and one for the \(n^{\text{th}}\) term. If students ask about the second, let them know that later lessons will review writing definitions of that form. Lastly, leave enough room on the poster to add on other representations, such as tables, graphs, or a visual pattern at the end of the next lesson.

Sequence: A List of Numbers

Geometric (exponential function)

Multiply each term by the growth factor.

Example sequence: 2, 6, 18, 54, . . .

Growth factor: 3

Recursive: \(f(1)=2, f(n)=3 \boldcdot f(n-1)\), \(n\ge2\)

\(n^{\text{th}}\) term: \(f(n)=2\boldcdot3^{n-1}\), \(n\ge1\)

Example sequence: 160, 40, 10, 2.5, . . .

Growth factor: \(\frac14\)

Recursive: \(h(1)=160, h(n)=\frac14 \boldcdot h(n-1)\), \(n\ge2\)

\(n^{\text{th}}\) term: \(h(n)=160\boldcdot\frac14^{n-1}\), \(n\ge1\)

Arithmetic (linear function)

Add to each term the rate of change.

Example sequence: 2, 7, 12, 17, . . .

Rate of change: 5

Recursive: \(g(1)=2, g(n)=5+g(n-1)\), \(n\ge2\)

\(n^{\text{th}}\) term: \(g(n)=2+5(n-1)\), \(n\ge1\)

Example sequence: 9, 5, 1, -3, . . .

Rate of change: -4

Recursive: \(k(1)=9,k(n)=\text-4 + k(n-1)\), \(n\ge2\)

\(n^{\text{th}}\) term: \(k(n)=9-4(n-1)\), \(n\ge1\)

5.4: Cool-down - Define This Sequence (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Sometimes we can define a sequence recursively. That is, we can describe how to calculate the next term in a sequence if we know the previous term.

Here’s a sequence: 6, 10, 14, 18, 22, . . . This is an arithmetic sequence, where each term is 4 more than the previous term. Since sequences are functions, let's call this sequence \(f\) and then we can use function notation to write \(f(n) = f(n-1) + 4\). Here, \(f(n)\) is the term, \(f(n-1)\) is the previous term, and + 4 represents the rate of change since \(f\) is an arithmetic sequence.

When we define a function recursively, we also must say what the first term is. Without that, there would be no way of knowing if the sequence defined by \(f(n) = f(n-1) + 4\) started with 6 or 81 or any other number. Here, one possible initial condition is \(f(1) = 6\). (It could also make sense to number the terms starting with 0, using \(f(0) = 6,\) and we'll talk more about this later.)

Combining this information gives the recursive definition: \(f(1) = 6\) and \(f(n) = f(n-1) + 4\) for \(n \ge2\), where \(n\) is an integer. We include the \(n\ge2\) at the end since the value of \(f\) at 1 is already given and the other terms in the sequence are generated by inputting integers larger than 1 into the definition.