##
* *Reflection: Developing a Conceptual Understanding
Functions Applications: Money grows and the rainforest disappears - Section 3: Discussion

I like the One Million Dollar problem in this lesson to really drive home the idea of how powerful doubling can become and the overall concept of exponential growth. Sometimes if I am really looking to make a point with students, I take out a roll of credit card tape or paper and start by showing students a small amount. Then I have them help me double it again and again, and the tape stretches all the way across the classroom and soon it's down the hall. I think the idea students being able to *see* doubling is a powerful one and I really want students to understand how a quantity can get so big so fast.

*The Power of Doubling*

*Developing a Conceptual Understanding: The Power of Doubling*

# Functions Applications: Money grows and the rainforest disappears

Lesson 9 of 11

## Objective: SWBAT compare exponential growth and decay. SWBAT determine if an exponential function is a geometric sequence.

#### Opening

*5 min*

Today's lesson involves two of my favorite problems that allow students to puzzle with ideas of exponential growth and decay. Depending on how quickly the class works, I might give both problems to students or I might divide the class in two and give each half one of the problems. If I divide the class in two, the discussion period of the class can be used to have students share out their problem and work with the other class. If I want to differentiate this lesson for students, I might give the groups that are still developing their ideas about the difference between linear and exponential growth the money problem, and give the rainforest problem to students who are ready to try to quantify exponential decay.

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#### Investigation

*30 min*

We read through One Million Dollars or One Penny and the The Disappearing Rainforest together. Again, depending on how I've decided to divide up the class, I let students ask clarifying questions and then they get to work. As I circulate, I look out for:

One Million Dollars Problem

- Students may need to be encouraged to make three columns for their table. Students often forget that they will add the value of the days together, not just take the pennies given on one particular day. Although even with this misconception, in the end they should still take the penny option.
- Students may need some guidance about how to write an equation for the growth they are seeing. I'll ask them what they're multiplying over and over again to help them find the base for their exponent.
- Students may have trouble thinking about cents as related to dollars. I'll push them to think about how they can see the number of pennies they have as a dollar amount.

Rainforest Problem

- As in the Chew on This problem from the Sequences unit, students may have trouble thinking about how much rainforest remains, rather than how much disappears. I will probably let students go on for a while doing two steps (finding 7% and then subtracting) before I try to shift their thinking to keeping 93%.
- If they generate a three column table, we might be able to use the remaining amount of rainforest totals to find a common ratio.
- I'll have students look back at their work on Chew on This to help them write the explicit formula.

I may encourage students to use graphing software like desmos.com rather than having them do graphs by hand depending on their level of competency with graph making.

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#### Discussion

*20 min*

I have different groups share out their problems and findings here. In the money problem, we are looking to show again how quickly 1 cent can grow if it continues to double. Students are often shocked by the results of this problem. Again we can compare exponential and linear growth here. We can also talk about whether or not this function is growing discretely or continuously and if it fits the criteria to be an arithmetic or geometric sequence.

For the rainforest problem, we want to focus on writing the explicit equation for exponential decay (students have not learned the standard formula yet). I'll try to elicit from students that they are keeping 93% of the rainforest so .93 becomes our common ratio. We can talk about whether or not the rainforest would ever completely disappear and what this means mathematically versus in reality.

The focus of today's discussion should really be about students sharing their work and representations and defending their answers.

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#### Closing

*5 min*

To close today's class I'll ask students to respond to the following prompt on an exit ticket:

- What surprised you about your answers to today's problem? How do you understand how the math works to lead to such a surprising answer?

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- UNIT 1: Introduction to Algebra: Focus on Problem Solving
- UNIT 2: Multiple Representations: Situations, Tables, Graphs, and Equations
- UNIT 3: Systems of Equations and Inequalities
- UNIT 4: Quadratics!
- UNIT 5: Data and Statistics
- UNIT 6: Arithmetic & Geometric Sequences
- UNIT 7: Functions

- LESSON 1: A Re- Introduction to Functions
- LESSON 2: Fun with Functions
- LESSON 3: How Many Diagonals Does a Polygon Have?
- LESSON 4: Building Functions
- LESSON 5: Discrete and Continuous Functions
- LESSON 6: Equal Differences, Equal Factors
- LESSON 7: Linear and Exponential Functions Project
- LESSON 8: Comparing Rates of Growth
- LESSON 9: Functions Applications: Money grows and the rainforest disappears
- LESSON 10: Modeling Population Growth
- LESSON 11: Exponential Growth and Decay