##
* *Reflection: Problem-based Approaches
Zombies: Exploring Exponential Growth - Section 2: Investigation

While students were acting out the activity in this lesson I wanted them to be making predictions so they could start to see the patterns that emerged. Rather than using the idea with the red dot, I organized the activity in a slightly different way. I stood on one side of the room (1 infected) and had my students all stand on the other side. Then I told them to walk across the room. When they did, I would walk across in the opposite direction and tag one student who would then be infected with the virus. That infected student would stay on the other side of the room with me (2 infected). We would then repeat the activity, but now, both me and the student would walk in the opposite direction and each tag one other student who would now be infected (4 infected). We continue the activity until all students are infected.

After the first two rounds, I asked students to think about how many students would be infected in the next round. I had students do a quick Think-Pair-Share where they are standing and I listen in to see if they are starting to see the pattern. We did this after the fifth round as well to allow students to process the situation they were acting out. This really helped students to see how the number of infected people grew more slowly at first but then more rapidly as the activity went on. As an additional aid, you could show students the Youtube video below. It helps to show how the growth rate of an exponential situation increases over time.

**Source Url**: https://www.youtube.com/watch?v=e-3Li7iMqMM (accessed June 19, 2014)

# Zombies: Exploring Exponential Growth

Lesson 4 of 13

## Objective: SWBAT generate an exponential function based on a set of data. SWBAT determine if growth is linear or exponential.

## Big Idea: By acting out exponential growth students will realize how quickly the dependent variable can increase in an exponential model.

*40 minutes*

#### Launch

*5 min*

At the start of class, students will read through the scenario regarding the zombie attacks: exponential_zombies_launch

Once students have read through both slides, I will ask them to predict the way the zombie attacks will work. I plan to have students do a **think-pair-share** to determine how many days it will take to infect the whole class. One or two students will share out their ideas and justifications. But, I do not indicate if students are correct or incorrect in their thinking (MP3).

Then, I organize the class by having all students go to one side of the room while I, the one infected with Solanum, will go to the other side of the room.

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#### Investigation

*27 min*

**Playing the Game**: To begin with, I (the teacher) is the only one infected. This will be indicated by a red dot on my hand. I will shut the lights off and turn them back on to indicate the beginning of day 1. At this point I will choose one other student to infect. I put a red dot on their hand and give them a red marker. They then go to one side of the room with you, other students remain where they are. Repeat the process by turning the lights out and back on again to indicate the beginning of day 2. Both you and the student find one other student each to infect. Put a red dot on their hand and give them a red marker. Repeat this process until there are no other students to infect. The process should lead to an exponential sequence 1, 2, 4, 8, 16...

**Analyzing the Game**: Once all students have been infected, I will have them fill out the exponential_zombies_worksheet table using the values from the class. Each output should be a multiple of 2 with the exception of the last day when all students become infected (unless there are 16, 32, 64, etc. students in the class). Students will be generalizing their work to determine a function that models this particular scenario. Students will also be extending their work to solve for the number of days it would take to infect a larger group.

Next, students need to think about how the values and function would change if there were 7 infected people to begin with. Have them fill out the table and determine the common ratio (which is still 2). If students are having difficulty with the numbers have them make a diagram that would show the spread of the disease. This could be as simple as 7 dots with two branches coming off of each one. Then two branches coming out from each of those, etc.

When discussing the work, pay particular attention to question #3 where students discuss why the attacks seem to be exponential. There could be a wide variety of answers here but guide students towards the understanding that there is a multiplicative connection between the various outputs as opposed to an additive connection (MP2). Students may also need some guidance with writing the function for question #5. Have students check their answer to #5 by actually plugging the values from the table into their function and determining if the function holds true for all x-values (MP6).

*expand content*

#### Close

*8 min*

I will ask my students to work on this **ticket out the door** with a partner. The first three questions provide a window on whether students are grasping the concept of writing and evaluating exponential functions. The final question is interesting because if part c is evaluated correctly the total number of people is greater than the number of people on the planet (**MP2**, **MP3**). This could lead to an interesting discussion about how exponential growth works. The fact that even when a quantity doubles it does not continue to do so forever (as in the paper folding problem). The growth eventually slows down and levels off. The exponential model is good for showing how a quantity can grow over a certain domain. This concept will be explored more in future lessons.

#### Resources

*expand content*

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- LESSON 1: Getting Started: Investigating Exponents
- LESSON 2: Geometric Sequences
- LESSON 3: Geometric Sequences and Exponential Functions
- LESSON 4: Zombies: Exploring Exponential Growth
- LESSON 5: More With Exponential Growth
- LESSON 6: Graphing Exponential Decay Functions
- LESSON 7: Effect of Changing b in f(x) = (b)^x
- LESSON 8: Transforming Exponential Functions
- LESSON 9: Comparing Geometric and Arithmetic Sequences
- LESSON 10: Solving Equations Involving Exponents
- LESSON 11: Comparing Linear and Exponential Functions Day 1
- LESSON 12: Comparing Exponential and Linear Functions Day 2
- LESSON 13: Modeling with Exponential Functions