##
* *Reflection: Grappling with Complexity
Comparing Linear and Exponential Functions - Section 1: Launch

In the interest of wanting students to learn and perform well, I sometimes unnecessarily reduce the complexity of a problem by spoon-feeding students too much information. In other words, I provide enough information to make a challenging problem easy. I must admit to this bad habit: telling students what approach or tool to use and guiding them down an obvious path.

In this lesson I try really hard to avoid doing this. I do not want to take away the opportunity for my students to learn by persevering and discovering.

**Where did I learn that students shouldn't struggle with math? I don't know.** Avoiding unnecessary scaffolding is something I find that I need to always have in mind. I know that my intentions are good: I do not want my students to experience frustration. I have learned that in order to do this, without giving away too much information requires planning. Here are some things that I consider:

- What do I expect my students to struggle with in this activity?
- What do I want my students to struggle with in this activity?
- What are the right questions to ask?
- What information can I share without giving away the problem?.

*Where did I learn that students shouldn't struggle with math?*

*Grappling with Complexity: Where did I learn that students shouldn't struggle with math?*

# Comparing Linear and Exponential Functions

Lesson 7 of 11

## Objective: SWBAT understand and demonstrate the differences between linear and exponential functions.

*60 minutes*

#### Launch

*5 min*

The opening activity to launch this lesson can be carried out in two ways:

- By handing out Entrance Tickets to students
- Projecting the Entrance Ticket on the board for all to see

Pairing can be done randomly. This task provides a good entry point because it sets an even ground for all students. It also involves students in discussion as they set up, complete, and interpret outcomes (**MP3**).

Once students begin the problem, some will start asking me what to do next. For this task, I am prepared to answer that I want them to discuss that task with their partner and figure it out themselves. I don't provide more help than this (**MP1**).

The primary strategy most of my students will begin with is using a table. Some students may start making calculations randomly on paper. After letting them think a little, I may hint that maybe they should organize their data better, but I will not demonstrate how to do this.

#### Resources

*expand content*

#### New Info / Exploration 1

*25 min*

As students continue to work on the Entrance Ticket, I walk around the room observing students writing their data and making sure their calculations are correct. I am planning to focus more on Option 2. Many of my students will not realize that all they have to do is multiply each amount by 1.5 to obtain the next amount. Most will find 50 percent of the amount and add it. To these students I ask plan to ask, "Instead of finding the percent of the amount of money and adding it, can you find a quicker way to obtain the next value?" The question gives away little, but hints strongly that their is something to discover.

My experience is that most of my students do figure out that each day's money can be multiplied by 1.5. This fact gives students a view of an important difference between the two options, and, the functions that can be used to model the option.

Once students are finished with their work I project the table CAM02020 on the whiteboard and ask students to compare their work to the table. I then ask these questions to the whole class:

**1. How do the amounts change each day in options 1 and 2? Do they increase by the same amount? **

**2. Which choice is the better choice, and was it always better throughout calculations? **

**3. In option 2, how much money would you have if your uncle's generosity went up to day 15? Explain how you got your answer. **

I then state that 1.5 is called the **growth factor** of this function. After giving students a chance to take in this new term, I ask the class (some student realize and say it before I ask) which option do they think represents a linear function and state their reason.

*expand content*

#### Exploration 2

*20 min*

Since the Entrance Ticket led most students to produce a table of values, I want my students to graph their data so they can visually interpret each function. There are two way to do this. One way is to have students plot the points in their graphing calculators and have the calculators draw the graph. The other way is by plotting points on graph paper by hand. I prefer using the graphing calculator because it much quicker.

I ask that students erase any data already in their calculators and give them the following instructions:

1. Press STAT ....highlight EDIT and press ENTER

2. In the L1 column, enter numbers from 0 to 14 (these are the days on my x axis)

3. In the L2 column, enter the amounts obtained in Option 1 for each day. (day 0 would be 10, and so on)

4. In the L3 column, enter the amounts obtained in Option 2 for each day.

5. Go to STAT PLOT, enter Plot 1 and turn it ON by highlighting ON and pressing enter.

6. Highlight the first or 2nd graph type and press enter (2nd is a line graph)

7. Go down to X list and enter L1 then in the Y list enter L2

8. Press Graph

Here's a video tutorial if these instructions don't suffice.

Every student should see the scatter plot which looks quite linear. The same can be done for STAT PLOT 2 but this time entering L1 and L3 for the X and Y lists respectively. Students should see that this graph is clearly a curve. To see how both graphs should appear click Graphs. Good questions to ask once students have both graphs on their calculator are:

- Where do these two graphs intersect?
- What information can we learn by thinking about the point where these two graphs intersect?

#### Resources

*expand content*

#### Closure

*10 min*

To end the lesson I project Closure Function Analogies on the board. Then, I will ask the class to take 5 minutes to complete the table. In the table, I provide the exponential equation for Option 2 in row one, they must find the linear equation for Option 1. After 5 minutes, I will call on volunteers and we will complete the chart as a class.

#### Resources

*expand content*

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- UNIT 1: Number Sense
- UNIT 2: Solving Linear Equations
- UNIT 3: Relationships between Quantities/Reasoning with Equations
- UNIT 4: Powers and Exponents
- UNIT 5: Congruence and Similarity
- UNIT 6: Systems of Linear Equations
- UNIT 7: Functions
- UNIT 8: Advanced Equations and Functions
- UNIT 9: The Pythagorean Theorem
- UNIT 10: Volumes of Cylinders, Cones, and Spheres
- UNIT 11: Bivariate Data

- LESSON 1: Relations that Function (Part 1)
- LESSON 2: Relations that Function (Part 2)
- LESSON 3: Functions Rule (Part 1 of 2)
- LESSON 4: Functions Rule (Part 2 of 2)
- LESSON 5: Crickets Tell Temperature
- LESSON 6: Linear? Yey or Nay
- LESSON 7: Comparing Linear and Exponential Functions
- LESSON 8: Time-Distance Graphs
- LESSON 9: Rates of Change
- LESSON 10: Sequences as Functions (Part 1 of 2)
- LESSON 11: Sequences as Functions (part 2 of 2)