# Equal Differences, Equal Factors

## Objective

SWBAT define linear and exponential patterns of growth.

#### Big Idea

Is that function a sequence? Students learn how to differentiate between arithmetic and geometric sequences and linear and exponential functions.

## Opening

10 minutes

The purpose of this lesson is for students to use what they have already learned about arithmetic and geometric sequences to understand linear and exponential functions.

I begin class with a Warm Up question where students have 3 minutes or so to recall what they know about sequences. I ask them to respond to the following prompt:

• What are the key characteristics of an arithmetic sequence?
• What are the key characteristics of a geometric sequence?

Students may need a reminder that in order to be a sequence, the In column of values have to increase consecutively. This is a key piece for them to understand the difference between sequences and functions that are not considered sequences.

I let them know that in today's lesson, we are going to zoom out a little bit to look at a broader category of functions. In today's class we will be looking at constant differences and constant ratios over equal intervals.  I ask students to come up with an example of table that might show this (starting with equal differences over equal intervals).

## Investigation

30 minutes

Next, we read through Sorting Out the Change together. I let students get to work and circulate around the room. Issues I watch for:

• In many of the problems, students will see a change but it may not be linear or exponential. I ask them to see if they can describe the pattern anyway and possibly represent it algebraically.
• In Question #5, students may need encouragement to experiment with different values of a and see what happens to the resulting function.
• Question #6 poses an interesting situation to students.  There are equal intervals in the x column, but no change in the f(x) column. I might encourage students to look at a graph and then see what they think. There are great opportunities here for students to present arguments about what kind of change this table represents.
• Students often struggle with Question #7. I might use a paper plate to represent the Ferris wheel and ask them to look at various heights as the wheel turns.
• The table in Question #9 is out of order and students may need a hint about organizing the x values.
• Question #10 also presents an interesting scenario for students as the algae population increase exponentially and then levels off.  We might examine two different scenarios to use one as an example of exponential growth.
• Students can compare Question #13 to Question #6 and again come up with arguments about what kind of change this might represent.

## Discussion

15 minutes

We start today's discussion by having students share out a question they think represents equal differences over equal intervals.  I have a small group present their work and other students can challenge or agree with the group's ideas. I explain to students that this type of pattern represents a linear function.

Next, we do the same with a question that shows equal factors over equal intervals.  I explain to students that this type of pattern represents an exponential function.

It is worth reminding students here that arithmetic and geometric sequences are kinds of linear and exponential functions, but that not all functions are sequences.  I remind students that in order to be a sequence, the x values must increase by consecutive terms.

The remainder of the discussion we spend looking at the other questions and getting explanations and/or ideas from students about whether or not they show equal differences or equal factors or neither.  Many of the questions can lead to rich discussions among students.

## Closing

5 minutes

Because the key idea of today's lesson is for students to zoom out and take a look at functions and compare them to sequences, I want to end today's class with a reflection that will help students remember this idea.  I ask students to complete an exit ticket in response to the following prompt:

• What is the difference between an arithmetic sequence and a linear function?  How can you identify each one?

## Citations

Sorting Out the Change is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.

http://www.mathematicsvisionproject.org/secondary-1-mathematics.html