Geometric Sequences

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SWBAT translate among verbal, algebraic, numeric and graphical representations of geometric sequences.

Big Idea

There are easier ways to generate the 100th term of a geometric sequence than listing all 99 terms before it. In this lesson, students learn to work flexibly with explicit and recursive expressions of a geometric sequence.

Warm-Up and Review

20 minutes

The warm-up asks students to calculate the quantities that have increased or decreased by a certain percent.  Students benefit from reviewing how to express a growth factor as the rate of increase added to one and a decay factor as the rate of decrease subtracted from one.  Understanding this method of calculating percent increase and decrease is essential for solving applied problems with geometric sequences and series formulas.

Because some students will finish this work more quickly than others, I assign the first 5 problems in this warm-up to all students and offer the final two (which are substantially more difficult) as challenge exercises for students who finish early.  Students will need about 10 minutes to work through the warm-up.

While students work, I display homework answers on the overhead and circulate around the room to assign a score to each student's work according to my homework rubric.  As I do this, I note what parts of the assignment were difficult and make sure we review those as a group.  I ask for volunteers to put some problems on the board or show students the solution myself if one part of the assignment was difficult for many students.

After the homework discussion, I ask students to pair up to compare answers to the warm-up. I ask about strategy and highlight the efficiency of multiplying by 1 plus the rate when calculating a percent increase and 1 minus the rate when calculating a percent decrease.  I send students a quick poll with one additional tip calculation problem after this discussion and ask students to respond with the one step expression for calculating the tip (NOT the final answer).

Explore and Extend

40 minutes

This lesson parallels that of the previous day.  Here, we focus our attention on geometric patterns and learn the formal methods of defining geometric number patterns explicitly and recursively.  In a prior lesson, we sorted sequence strips into piles according to the math used to move from one term to the next.  In this lesson, we focus on the geometric sequences from this activity.  Using the sequence strips labeled (b), (g), (h) and (j), I ask students to work together to come up with a rule that expresses the term in terms of n, the term's position in the sequence [MP7].  As students work together, I offer support and hints as necessary.

When students have had time to develop these rules, I ask them to share their ideas using a quick poll on the TI NSpire Navigator System. When we have a collection to look at, we check each idea together to see if substituting 1 for n yields the correct initial term, 2 for n yields the correct second term, etc.  This leads to a discussion of sequence notation and how it is similar to and different from exponential function notation. [MP3]  Students need to connect this new notation to what they have learned in Algebra 1, so it is important to help students understand that the geometric sequence is an exponential function whose only input values are the positive counting numbers.

After the group discussion of student findings, I write the formulas on the board and we work a few examples of using the formulas to find the nth term, the term number, or the common ratio.  We also examine the graph of a geometric sequence and compare it to the graph of an exponential function.  Students take note of the formulas, illustrations and examples that I write on the board and actively participants in the discussion.

Practice and Discuss

20 minutes
Students work in table groups to complete Geometric sequence WS A, which provides practice with the geometric sequence formulas.  During this time I circulate with a clipboard to note problems that students are having both with the math content and with remaining on task.  I stop to ask questions and give hints as necessary.  After students have worked for 10 or 15 minutes, I share answers to the exercises and answer any questions that came up for students.


Closure and Assignment

10 minutes

To summarize the work with geometric sequences, I ask students to help me generate a list of all the problem types that they encountered in the group work.  We briefly discuss the idea that each exercise has different given information and that this makes some exercises more challenging than others.  In general, students are most challenged by problems in which the nth term is given and the goal is to find the number of terms.  This type of problem sometimes requires students to "guess and check" because they have not yet learned about logarithms.  In the video reflection, Foreshadowing Logarithms, I outline my method of working around this.  

For homework, I choose problems like those assigned for classwork and ask students to check their answers on Edmodo before they arrive in class the next day.