Lesson 3

Different Types of Sequences

3.1: Remembering Function Notation (5 minutes)

Warm-up

The purpose of this warm-up is to informally assess strategies and understandings students currently have for interpreting function notation which they learned about in an earlier course. Students will use function notation when they define sequences with equations in later lessons, so this warm-up is an opportunity for practice.

Launch

Display the first two sentences and ask students to read them quietly. Then, display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer before displaying the next problem. Keep all problems displayed throughout the warm-up. Follow with a whole-class discussion.

Student Facing

Consider the function \(f\) given by \(f(n) = 3n - 7\). This function takes an input, multiplies it by 3, then subtracts 7.

Evaluate mentally.

  • \(f(10)\)
  • \(f(10) - 1\)
  • \(f(10 - 1)\)
  • \(f(5) - f(4)\)

Student Response

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

Anticipated Misconceptions

Some students may believe that function notation "distributes," that is, that expressions like \(f(5) - f(4)\) are equivalent to \(f(5-4)=f(1)\). Ask these students to work out the value of the expression both ways in order to make clear that \(f(5) - f(4)\) and \(f(5-4)\) are not equivalent expressions with a reminder that function notation is not multiplication, but rather a way to write the output of a function for a specific input.

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see.

3.2: Three Sequences (15 minutes)

Activity

The purpose of this activity is for students to contrast three different types of sequences and to introduce the term arithmetic sequence.

Monitor for students using precise language, either orally or in writing, during work time to invite to share during the whole-class discussion.

Launch

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.

Conversing, Speaking, Writing: MLR 2 Collect and Display. As students discuss the sequences, circulate and listen to student talk about the patterns they notice. Write down common or important phrases you hear students say about each type onto a visual display, for example “add 10 each time” or “multiply each term by 2.” Collect the responses into a visual display. Throughout the remainder of the lesson, continue to update collected student language and remind students to borrow language from the display as needed. In the lesson synthesis, after the term “arithmetic sequence” is introduced, ask students to sort the collected language into three groups, one for language used to describe arithmetic sequences, a second for geometric sequences, and the third for neither.
Design Principle(s): Support sense-making; Maximize meta-awareness
Representation: Internalize Comprehension. Activate or supply background knowledge about sequences. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

Student Facing

Here are the values of the first 5 terms of 3 sequences:

  • \(A\): 30, 40, 50, 60, 70, . . .
  • \(B\): 0, 5, 15, 30, 50, . . .
  • \(C\): 1, 2, 4, 8, 16, . . .
  1. For each sequence, describe a way to produce a new term from the previous term.
  2. If the patterns you described continue, which sequence has the second greatest value for the 10th term?
  3. Which of these could be geometric sequences? Explain how you know.

Student Response

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

Student Facing

Are you ready for more?

Elena says that it’s not possible to have a sequence of numbers that is both arithmetic and geometric. Do you agree with Elena? Explain your reasoning.

Student Response

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

Anticipated Misconceptions

Students may need help identifying a pattern for the two non-geometric sequences. Ask, “What can you say about the change between consecutive terms in the sequence?” Encourage students to use any language they wish to describe the pattern—they do not need to use an equation at this time. Some students may benefit from creating a table where they can see the term number and the value of the term side by side, particularly for Sequence B.

Activity Synthesis

The purpose of this discussion is to compare different types of sequences and introduce students to the term arithmetic sequence. Begin the discussion by asking students how \(A\) and \(C\) are alike and different. They might offer things like:

  • \(C\) is geometric but \(A\) is not.
  • In \(A\), you always add 10 to get from term to term, but in \(C\), you always multiply by 2.
  • In \(C\), the growth factor is 2. In \(A\), you get the next term by adding 10 to the previous term.

Tell students that sequence \(A\) is an example of an arithmetic sequence. Here are two ways to know a sequence is arithmetic:

  • You always add the same number to get from one term to the next.
  • If you subtract any term from the next term, you always get the same number.

Share that the constant in an arithmetic sequence is called the rate of change or common difference. In sequence \(A\), the rate of change is 10, because \(40=30+10\), \(50=40+10\), \(60=50+10\), and \(70=60+10\).

Some students may notice the similarity between an arithmetic sequence and a linear function. Invite these students to share their observations, such as how both are defined by a constant rate of change. Tell students that arithmetic sequences are a type of linear function and that their knowledge of linear functions will help them describe arithmetic sequences during this unit. If students do not bring up the connection to linear functions, ask "What do you remember about linear functions?" Record student responses for all to see and invite comparisons between linear functions and arithmetic sequences.

3.3: Representing a Sequence (15 minutes)

Activity

The purpose of this activity is for students to create another representation of a given sequence and to give them an opportunity to use the vocabulary they have learned for geometric and arithmetic sequences.

Monitor for students who create Mai's graph in order to understand her reasoning and for students who reason about whether the sequence is defined by a rate of change or a growth factor to invite to share during the discussion.

Student Facing

Jada and Mai are trying to decide what type of sequence this could be:

term number value
1 2
2 6
5 18

Jada says: “I think this sequence is geometric because in the value column each row is 3 times the previous row.”

Mai says: “I don’t think it is geometric. I graphed it and it doesn’t look geometric.”

Do you agree with Jada or Mai? Explain or show your reasoning.

Student Response

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

Activity Synthesis

Invite previously selected students to share their reasoning, displaying any graphs created for all to see. If the idea of slope or average rate of change does not come up, ask students how they could have found the missing points for an arithmetic sequence just given \((2,6)\) and \((5,18)\). Draw in the slope triangle and show the computation for the average rate of change.

Lesson Synthesis

Lesson Synthesis

Ask students to discuss with a partner: “How are arithmetic and geometric sequences alike and different?” After they have had a few minutes to discuss, ask several students to share the things that came up. Some things to highlight:

  • They are both types of sequences. That is, they are both lists of numbers.
  • To get from one term to the next for both arithmetic and geometric sequences, you “do the same thing each time.”
  • For geometric sequences, you always multiply by the same number to get the next term. For arithmetic sequences, you always add the same number to get the next term.
  • Geometric sequences have growth factors (the quotient of any term and the previous term), but arithmetic sequences have a constant rate of change (the difference of any term and the previous term).

3.4: Cool-down - Do What’s Next (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Consider the sequence 2, 5, 8, . . . How would you describe how to calculate the next term from the previous?

In this case, each term in this sequence is 3 more than the term before it.

A sequence. The number 2, followed by 5, 8, and two blanks. Starting underneath 2, and repeating to the right 4 times, an arrow going to the right labeled "plus 3."

A way to describe this sequence is: the starting term is 2 and the \(\text{current term} = \text{previous term}+3\).

This is an example of an arithmetic sequence. An arithmetic sequence is one where the value of each term is the value of the previous term with a constant added. If you know the constant to add, you can use it to find other terms.

For example, each term in this sequence is 3 more than the term before it. To find this constant, sometimes called the rate of change or common difference, you can subtract consecutive terms. This can also help you decide whether a sequence is arithmetic.

For example, the sequence 3, 6, 12, 24 is not an arithmetic sequence because \(6-3 \neq 12-6 \neq 24-12\). But the sequence 100, 80, 60, 40 is because if the differences of consecutive terms are all the same: \(80-100 = 60-80 = 40-60 = \text-20\). This means that the rate of change is -20 for the sequence 100, 80, 60, 40.

It is important to remember that while the last two lessons have introduced geometric and arithmetic sequences, there are many other sequences that are neither geometric nor arithmetic.