### Summary

Students will investigate logistic functions as mathematical models of real-world situations, such as zombie outbreaks, and make comparisons with exponential growth models. Students will learn about the properties of the logistic curves and how to graph logistic functions by hand. This lesson can be taught in Algebra 2 (similarly sequenced in a Precalculus course) after students have mastered exponential and logarithmic functions. This lesson can also be used as a review in a Calculus course before students learn about differential equations.

### Essential Question(s)

How can logistics functions be used to model real-world phenomena?

### Snapshot

**Engage**

Students watch a video and read a scenario about the spread of zombie infection and then make a prediction.

**Explore**

Students simulate a zombie outbreak and gather data.

**Explain**

Students use an online calculator to generate a logistic function to model their data, then formalize their understanding of logistic functions, and learn how to sketch a logistic curve by hand.

**Extend**

Students apply what they learned by graphing logistic functions and identifying key characteristics of their graphs.

**Evaluate**

Students determine if exponential or logistic functions would best model given growth scenarios.

### Materials

Lesson Slides (attached)

Exploring a Zombie Outbreak handout (attached; one per pair; printed front/back)

Using Key Characteristics handout (attached; one per student; printed front only)

Exit Ticket handout (attached; one per student; printed front only)

Pencils

Paper

Student devices with Internet access

Graphing calculator (optional; one per pair)

### Engage

15 Minute(s)

Introduce the lesson using the attached **Lesson Slides**. Display **slide 3 **to share the lesson's essential question with students. Go to **slide 4 **to share the lesson's learning objectives. Review each of these with students to the extent you feel necessary.

Go to **slide 5** and inform students that they are going to watch a silent video and need to be ready to share what they think the video represents. Show **slide 6** and click the image to view the "Simulation of a Zombie Outbreak!" video. (Clicking the image opens a browser tab to view the video.) After students watch this silent video, ask them to share what they think the video represents.

If students struggle to make the connection to zombies, ask what fictional events or pop culture TV shows that could be represented by the map.

Have students find a partner or assign partners, then read the following prompt on **slide 7**:

*When the first human on Earth got infected, no one believed it. We all thought zombies could never be real. As time went by, it was obvious that we had underestimated the swiftness with which human populations could decrease. Our town was one of the last to get hit, but it was brutal. By the time the first infection came, so many residents had left or died of starvation that only 2,000 of us remained to see it.*

*If there is one zombie among us, how long will it take for all of us to become infected?*

Have students work with their partner to write everything they know about zombies and to predict how long it would take to turn a population of 2000 community members into zombies starting with only 1 zombie.

Have students share their responses and explain how they found their solution. Be sure to encourage the variety of approaches. At this time, do not correct or hint to students that they are incorrect. Use student responses to determine if students need a quick refresh on exponential growth.

### Explore

20 Minute(s)

Show **slide 8** and read the following prompt:

*Our town quickly turned from a safe hiding space into the worst place to try to survive. With zombie populations increasing and resources depleting, several of us decided to run. We spent days hiding out in empty grocery stores and vacated homes to plan our escape, and we were able to make it out of town to a remote cabin that a family had abandoned. By the time our plan was executed, there were only 20 of us left.*

Show **slide 9** and continue reading the prompt below:

*We’ve been safe here for a while. I’m a little worried about Joe, though—he’s been extremely tired ever since we arrived. He said he wasn’t feeling well. What if he has the zombie infection and doesn’t know it yet?*

*How long will it take for all of us to become infected? Let’s run a simulation and find out!*

Show **slide 10** and preview the activity with the students. Imagine a bag of red and blue tiles where the red tiles represent the zombies and the blue tiles represent the humans. The total population is 20. Draw two tiles from the bag, representing a random interaction between two community members. It could be 2 humans (2 blue tiles), 2 zombies (2 red tiles), or 1 human and 1 zombie (1 blue and 1 red tile) interacting.

Display **slide 11** and explain the procedure for what to do with the interactions:

If you draw 2 humans (2 blue tiles) or 2 zombies (2 red tiles)—if two of the same types of community members interact—then record your result, (the digital simulation will automatically place the tiles back in the bag as long as "with replacement" is still selected), and repeat the drawing.

If you draw 1 human (1 blue tile) and 1 zombie (1 red tile)—if a human interacts with a zombie—then record your result, remove the human (blue tile) and replace it with a zombie (red tile), and repeat the drawing.

Now that students have a sense of how the simulation works, pass out one copy of the attached **Exploring a Zombie Outbreak** handout to each pair of students and review the directions on the handout about how to record their results. Show **slide 12**, and direct students to go to https://bit.ly/3LEgtbh. Here, students are using the CPM Probability Generators, not to calculate probability, but to run their zombie outbreak simulation. On the left, students have a quick-reference to the directions for how the simulation tool works. Guide students to make sure that the number of red tiles is 1 and the number of blue tiles is 19 for their first interaction (drawing of tiles) by right-clicking on the bag.

Display **slide 13** and continue explaining the process of the simulation. Direct students to work with their partner and start their simulations, recording their results on their handout. As students begin working, remind them to check that the number of humans plus the number of zombies always equals 20.

Once students complete their simulation, show **slide 14**. Direct students to go to desmos.com, click "Graphing Calculator," add a table, and enter their data into the table.

### Explain

15 Minute(s)

Ask students to compare this trend of data to what they know about exponential growth: *How are they the same? How are they different?*

Now ask students what they think caused the data to not continuously grow.

Display **slide 15** and use this slide to direct students on how to generate a curve that models their data.

Display **slide 16** and explain that the graph on their screen is a logistic curve with two asymptotes. Clarify to students that the asymptotes on the slide are not visible in Desmos because they represent where the function is approaching but not what the function equals. Facilitate a discussion about where the asymptotes are and why they are at *y* = 0 and *y* = 20. Direct students’ attention to the maximum growth point and explain this vocabulary term. In Calculus, this point is known as the point of inflection, where the graph changes concavity. Discuss the characteristics of the graph and parameters of the function to the extent you see necessary.

Show **slide 17** and review the procedure for graphing a logistic function by hand.

### Extend

20 Minute(s)

Display **slide 18** and pass out a copy of the attached **Using Key Characteristics** handout to each pair of students. Instruct students to work with their partner to graph the first two equations and label the key characteristics of the graph. For questions 3–4, students are provided a graph and asked to write what they know about the graph. Encourage students to use academic vocabulary.

As time allows, ask for volunteers to explain the process and reasoning.

### Evaluate

10 Minute(s)

Use the Exit Ticket strategy to individually assess what students have learned from the lesson. Show **slide 23** and pass out a copy of the attached **Exit Ticket** handout to each student. Students are asked to read a scenario and decide if an exponential or logistic function would best model the growth described in the scenario.

After students have submitted their work, unhide and show **slides 24–27**. Give students time to reflect on their thinking. Use student responses to see what misconceptions still exist.

### Resources

K20 Center. (n.d.). Bell Ringers and Exit Tickets. Strategies. https://learn.k20center.ou.edu/strategy/125

K20 Center. (n.d.). Desmos Studio. Tech tools. https://learn.k20center.ou.edu/tech-tool/2356

K20 Center. (n.d.). CPM Probability Generators. Tech Tools. https://learn.k20center.ou.edu/tech-tool/2317

Vallinder, M. (2010, April 23). Simulation of a Zombie Outbreak! [Video]. YouTube. https://youtu.be/t_AlsNx5UhA