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Unit 1, Lesson 6

Representing Sequences

Algebra 2

6.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

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Activity Narrative

The purpose of this Warm-up is for students to recall some of the ways functions can be represented, such as tables, graphs, equations, and descriptions. The focus here is on exponential and linear functions, and identifying the rates of change and growth factors.

Launch

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Activity

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For each sequence shown, find either the growth factor or rate of change. Be prepared to explain your reasoning.

5, 15, 25, 35, 45, . . .
Starting at 10, each new term is less than the previous term.
Graph, origin “O”. Horizontal axis,"term number," with scale from 0 to 6, by 1’s. Vertical axis, "value," with scale 0 to 16, by 2’s. Coordinates plotted (1 comma 16), (2 comma 8), (3 comma 4), (4 comma 2), and (5 comma 1).

for
1 0

2 0.1

3 0.2

4 0.3

5 0.4


1	0
2	0.1
3	0.2
4	0.3
5	0.4

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

The purpose of this discussion is to make sure students understand how to identify the growth factor or rate of change of sequences represented in different ways.

Display the questions for all to see throughout the discussion. For each problem, select students to share how they identified either the growth factor or rate of change. Highlight any comments linking these sequences to linear or exponential functions, such as that the rate of change would be the slope of the line that the terms of the sequence are on when graphed.

6.2

Activity

Instructional Routines

Extend It MLR8: Discussion Supports Take Turns

Materials

None

Activity Narrative

In this partner activity, students take turns matching a sequence to a definition. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

One sequence and one definition do not have matches. Students are tasked with writing the corresponding match.

Monitor for students using clear reasoning as they create a recursive definition for the sequence 18, 20, 22, 24 to share during the discussion.

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

Launch

Arrange students in groups of 2. Display the task for all to see. Tell students that they are going to match sequences to recursive definitions. If time allows, choose a student as a partner and demonstrate how to set up and do the activity. Otherwise, share these steps:

One partner picks a sequence.
They identify the matching definition and explain why they think it matches.
The other partner listens and makes sure they agree with the match and the reasoning.
If they don’t agree, the partners discuss until they come to an agreement.
The students swap roles.

Ensure that students notice that one sequence and one definition do not have matches, and that they are tasked with writing the corresponding match for each.

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frame for all to see: “I noticed , so I matched .” When students disagree, encourage them to challenge each other using these sentence frames:

“I agree because _____ .”
“I disagree because _____ .”

This will help students clarify their reasoning about each match.
Advances: Speaking, Listening, Conversing

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite groups of students to find matches for at least 4 of the sequences.
Supports accessibility for: Organization, Social-Emotional Functioning

6.3

Activity

Instructional Routines

Fit It Notice and Wonder

Materials

None

Activity Narrative

The purpose of this task is for students to write a recursive definition for a sequence that represents a mathematical context and to create other representations of the sequence.

Monitor for groups who create their recursive definitions in different ways to share during the whole-class discussion. For example, some students may first create a table showing step number and the associated values, while others may draw additional steps.

Allow students to use graph paper to sketch their graphs if needed. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Tell students to close their books or devices (or to keep them closed). Display the image for all to see. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing they notice and one thing they wonder. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the image.

If comparing the number of small squares between the steps is not brought up by students, ask “How would you describe the total number of small squares in Step 3 compared to the total number of small squares in Step 2?” (There are 9, or 3², more squares in Step 3 than in Step 2.)

Tell students to open their books or devices and arrange students in groups of 2. Encourage them to check in with their partner frequently as they work through the task.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with access to grid or graph paper to organize their work with representations of geometric and arithmetic sequences.
Supports accessibility for: Memory, Organization

Lesson Synthesis

Use this time to add more representations to the sequences poster. Display the poster for all to see. Arrange students in groups of 3–4, and assign each group one of the example sequences. Tell groups to create a graph showing the first five values of their sequence. If time allows, also invite groups to create a visual pattern for each sequence, such as one using dots or small squares. After work time, invite groups to share their representations. Add these to the poster for future reference.

to access Cool-down.

Student Lesson Summary

Here are some ways to represent a sequence. Each representation gives a different view of the same sequence.

List of terms: Here’s a list of terms for an arithmetic sequence : 4, 7, 10, 13, 16, . . . We can show this sequence is arithmetic by noting that the difference between consecutive terms is always 3, so we can say this sequence has a rate of change of 3.

Table A: A table lists the term number and value for each term . It can sometimes be easier to detect or analyze patterns when using a table.


1	4
2	7
3	10
4	13
5	16

Graph: The graph of a sequence is a set of points, because a sequence is a function whose domain is a part of the integers. For an arithmetic sequence, these points lie on a line since arithmetic sequences are a type of linear function.

<p>Graph of a sequence D on coordinate grid.</p>

Equation: We can define sequences recursively, using function notation to make an equation. For the sequence 4, 7, 10, 13, 16, . . . , the starting term is 4, and the rate of change is 3, so for . This type of definition tells us how to find any term if we know the previous term. It is not as helpful in calculating terms that are far away, like . Some sequences do not have recursive definitions, but geometric and arithmetic sequences always do.

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Lesson 6

Representing Sequences

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

HSF-BF.A.2

HSF-LE.A.2

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Launch

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSF-BF.A.2

HSF-LE.A.2

Building Toward

Standards Alignment

Building On

HSF-IF.A.3

Addressing

HSF-BF.A.2

HSF-IF.C

Building Toward