# Arithmetic vs. Geometric Sequences

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## Objective

SWBAT identify the differences between arithmetic and geometric sequences from a table. SWBAT write an explicit or recursive formula for the function.

#### Big Idea

Relate geometric sequences to exponential functions and arithmetic sequences to linear functions.

## Warm up

10 minutes

I expect this Warm up to take about 10 minutes for my students to complete, including time to review the task with students.  The warmup enables a comparison of linear and exponential functions, which I use as a lead into the lesson.

I expect all of my students to recognize the pattern in each table, and to plot the points successfully. However, writing a formula that represents the function will be difficult for several students. When we review the warm up as a class, I will recommend that students take notes about writing explicit and recursive formulas for each function.  I review the content of the notes in the video below.

## Categorize Arithmetic and Geometric Sequences

15 minutes

After we review the warm up as a class, I have students add to their notes that a sequence is a list of numbers that typically has some pattern to it.  If we look at the output values in each table of the warm up, each one represents a sequence, and each of these sequences forms a pattern.

The linear function increases by a common difference d. When a sequence is formed by a common difference, we can call it an arithmetic sequence.

The exponential function increases by a common factor or common ratio. When a sequence is produced by a common factor, it is called a geometric sequence.

After defining these two types of sequences, I hand a stack of seven index cards to each pair or group. Each card is numbered and has a sequence of numbers on it (see Categorize each sequence for a list of the sequences). Each student works with their assigned elbow partner.  Groups of 3 students may be used if necessary.

The task: I ask my students to categorize each card as an arithmetic or a geometric sequence. On each card the student is to write the reason for their choice on their cards individually without discussing it with their partner(s). Each student works on one card at a time, sharing the work load until the cards are all categorized. When the cards are complete, the responses on the cards are shared with the partner(s).  Each student in the group critiques the category and reasoning of the other students (Math Practice 3). The cards are discussed, and a consensus must be met on each card before the categories of each card are shared with the class.

To bring this activity to a close, each group posts their selections on the board by writing the number of each sequence in the chosen category.  The information is discussed on any differences until the analysis of each of the 7 sequences are complete.

## Write Explicit and Recursive Formulas

15 minutes

After the class successfully classified each of the 7 sequences, the students continue to work with their elbow partner. I instruct students to write both a recursive and an explicit formula for each sequence.

This task is the most difficult part of the lesson for my students.  As students are working, I monitor their progress. I ask probing questions to help move the students forward, but I try to do so in a way that allows them to continue to engage in a productive struggle (MP1). Most students do well after persevering through the first couple of sequences.  They learn to better interpreting the structure of the arithmetic sequences and the geometric sequences (MP7). As the students make discoveries, I encourage them to add to the notes that they began during the warm-up section of the lesson.

If necessary, I create a workshop for students that are still struggling. Students rotate in-and-out of the workshop as they make progress with writing the formulas.  In the workshop, I work with a small group of students as they work on individual white boards.  I review each of the seven sequences, inviting students to join the workshop if they are encountering difficulty..

## Closure

10 minutes

After students complete their work, they are to complete the Exit Slip before leaving class. I use the Exit Slip as a formative assessment to check each student's understanding of the objective of this lesson. If need be, I may assign the Exit Slip as homework.