Lesson Plan

# Exploring Exponential Equations #WithMathICan

Students will further explore the concepts of exponential equations through the use of this lesson and activity. This will be accomplished by making predictions, simulating a fish population decay through the use of technology, and graphing data.
Jamilee K.
Classroom teacher
Cabell Midland High School
Ona, United States
My Subjects Math
Objectives

Students will be able to...

graph exponential functions expressed symbolically and show key features of the graph find patters in data that represent geometric sequences and exponential functions

recognize patterns as sequences and that sequences are functions

Subjects
Math

#### 1 Introduction/Hook

Activity: Conversing

Project the following question on the board and have students write down their choice: "Would you rather be given one million dollars right now, or be given one penny today and each day be given double what you were given the day before for thirty days?". Give students a couple minutes to think it through, make a decision, then write down what they choose as well as why they chose it. Ask for a volunteer from each choice to explain why they chose the one they did. After letting the students discuss their reasoning, project a calculator on the board and show mathematically how much they would have received with the penny option after 30 days (approximately 5 million on the 30th day, not to mention all they received the days before). Ask the students what type of function this represents (assuming they have some previous knowledge of exponential functions). Give the students another minute to write down their new thoughts on the situation and discuss how it models an exponential function.

#### 2 Activity

Hand out the Fish Pond Simulation Activity sheet to all students, ask students what a simulation means, and describe how it will be used today. Also ask students to describe what fish population decay is and explain that we will be simulating fish population decay with the use of pennies. Give students a minute or two to look over activity sheet and ask questions if needed. Questions 1 - 6 are the steps to follow to complete the activity and questions 7-13 are to be answered afterwards.

Ask students to predict how many years they think it will take them to get to only one fish and have them right that down somewhere on the worksheet. We will revisit this later.

Students can be paired into partners or groups depending on what works best for you.

Instruct students to go to www.random.org/coins using their iPad. Have them pick 100 coins to flip and the US penny (or other coin) then click flip coins. Start the table at year 0 with 100 fish. Students will need to count how many tails they have facing upward because these represent the live fish left after the first year. Students will record this number for year one in the table. At the bottom of the screen students will click "go back" and change the number of coins to flip to reflect the number of tails/live fish they have now. Click flip coins and continue this pattern until 1 coin remains.

Once the table is filled with as many years as it took to get to one fish at the end, students will continue answers questions 7-13 on the activity sheet based of their findings. Students will graph their data both by hand on the back of the activity sheet (need graph paper), and using the Desmos graphing calculator app in the next step.

Student Instructions

Attached is the Student Activity Sheet

Fish Pond Simulation Activity – Student Guide

Plants, mammals, birds, fish, and insects all contribute to the existence of life on Earth. Using mathematics and mathematical models, we can build and refine models to help us predict how the size of a population will change over time. Mathematical models don’t produce an exact answer, but they can help us understand tither the patterns or the trends that exist.

You are going to use pennies (digital or real) to simulate and model population decay in a fish pond.

1. Count the number of “fish” (pennies) in your pond, and record the number under year 0 in the Population Decay table.
2. Gently pour the pennies onto your desk and spread them out.
3. The heads represent the dead fish so discard them to the side. Count the number of live fish after year 1 (tails) and record this number under year 1 in the Population Decay table.
4. Place the live fish (tails) back into the cup and start again.
5. Gently pour the fish (pennies) on your desk spreading out as needed, and discard the dead fish and count the live fish recording this number under year 2 of the Population Decay Table.
6. Continue this pattern until there are no fish remaining.
 Population Decay Year Number of Fish

1. Write a sentence or two describing any patterns you see in this fish population.

1. If you started with more fish, say 200, do you think you would see a different pattern? Why or why not?

1. Based off the patterns you see, what type of function is modeled by this simulation? How do you know?

1. Graph your data on the back of this sheet. Be sure to include an appropriate scale and labels before graphing your data.

1. What is the y-intercept of the function?

1. What is the asymptote?

1. Describe the end behavior.

#### 3 Graphing

Once finished with the Fish Pond Simulation Activity Sheet, instruct students to open the Desmos app on the iPad and click on the "+" symbol to add a table. Students are to add each ordered pair from their table on their activity sheet to the table on the app. Once all data is entered into the table, have the students readjust the graph to see all of their points. Have students compare their graph by hand with the one on the Desmos app. Discuss differences and similarities to graphing using technology versus by hand.

#### 4 Wrap Up

Activity: Conversing

Once finished with the activity, teacher will bring attention back to the center of the classroom and go over the project asking students to their answers.

Start by asking who predicted the number of years it would take to reach 1 fish correctly. For ones who did not predict correctly, ask students to volunteer why they picked their predicted number and what they learned in the process. Then proceed to go through answers to numbers 7-13 from students.

Ask for a couple students to project their graphs made by hand on the white board/Smart board using a document camera.

Answer any lingering questions students may have before taking up activity.

#### 5 Ticket out the Door

Kahoot!
Free, Paid

Possible survey questions to include are:

- Overall, what were your thoughts on the activity?

I really liked it / I somewhat liked it / I did not like it / I very much did not like it

- What part of the activity did you enjoy the most?

Bell Work / Coin Flipper / Graphing by hand / Graphing using Desmos

​- What type of function did this simulation represent?

Linear / Exponential