# Geometric Sequences and Exponential Functions

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## Objective

SWBAT determine an exponential function that can be used to model a geometric sequence.

#### Big Idea

This lesson uses pictures as a way to build a concrete understanding of exponential growth.

## Launch

5 minutes

Have students make a list on their own to answer this warmup question.  Explain to students that they are only trying to generate ideas. There are no right or wrong answers to this prompt.

Once students seem to have written down everything they can notice, have them turn and talk to compare their list with a partner. Encourage them to add new items to each list (MP3).

Based on the previous lesson, students should notice that each stack is three times as large.  I think that that the color-coded diagram is helpful in this instance because each part of the stack (white, red, and pink) gets three times as large with each subsequent stack (MP2)

## Direct Instruction

25 minutes

During this direct instruction we are trying to develop a connection between the geometric sequence and the function that models it.  There are questions on each of the slides.  Let the students work to answer the questions with their partner and try to justify their solutions (MP3).  Guide the class to correct answers through student questioning and by showing student work using a document camera or other means.  Try to avoid the urge to tell students how to answer each of the questions.

Slide 1

From the pre-algebra lesson on exponents, students should be able to determine that 3^0 is equal to 1.  Make sure that all students have access to graph paper or a white board with a graph on it to show their work.  Students will notice once they start graphing that the function is not linear.  Explain to them that they should be drawing a "smooth" curve that passes through each of the points.

Slide 2 and 3

This concept can be approached in two ways.  To graph the points to the left of the y-axis students need to understand negative exponents (see pre-algebra lesson on exponents).  They can also see the pattern of working backwards by dividing.  If each larger term is derived by multiplying by 3, we can work backwards by dividing by 3.  So working backwards from 1 you would go to 1/3, 1/9, 1/27, etc.  They can connect these values to the values of the output from the function( 3^-1, 3^-2, 3^-3, etc) (MP7).

Slide 4 and 5

This series of slides introduce the idea of exponential decay.  Some students may have difficulty seeing that each subsequent term in this series is being multiplied by 1/2.  The question on slide 5 refers to "asymptotic behavior" (Not that you would ever call it that).  This is a good time to do a think-pair-share and see what students can come up with on their own.  They can then try to justify their idea with a partner and ultimately the whole class (MP3).  This concept is one that will be built both throughout this course and in higher level mathematics (MP2).

Slide 6

This slide contains a lot.  First, students need to graph the data they have so far to the right of the y-axis 1, 1/2, 1/4, 1/8, 1/16.  Next, they need to think about the value of the function at the negative integer values (1/2)^-1.  This could be a difficult concept for students to grasp so also have them work backwards by multiplying by 2 (inverse of multiplying by 1/2).  This will give them the values of 2, 4, 8, and 16 to the left of the y-axis.  Students can make note of the difference between the graph of the first function and the graph of the second function noting the similarities and differences (MP7).

## Closure

10 minutes

This ticket out the door will give you a good sense of where all students in your class are with understanding this concept.  The three questions are at three different levels of understanding.  The first should be completed by most students who can identify the common ratio.  The second will show if students can generalize the pattern (although you will have to examine the student work because they may figure out the tenth term by just continuing to count on.  Either way this is good information to gather).  The third question will be tough to answer for some students because of the coefficient to the common ratio (f(x)=2(3)^x).  This assumes that the first number plugged in is zero.  Since we are writing this in function notation that is a good assumption to make.  If we are using "1" as the first input the function would be f(x)=2(3)^(x-1).