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Exponential Functions and The Tortoise and the Hare

Learning the connections between exponential functions and the old tale of the tortoise and the hare.

Learning Module


This learning module was designed to help students learn the mathematical concept of exponential functions. I have taught this unit in Algebra 1, typically around the beginning or middle of the second semester. Usually, exponential functions are taught in a way that focuses on a "drill and practice" approach. I think students lose a lot of the conceputal understanding that comes with exponential functions when it is taught this way. This learning module is a way to change the instruction to focus on both the procedural fluency and conceptual understanding of the content. Exponential functions is a topic covered in Algebra 1, typically at the high school level. However, some advanced students may experience this as early as 7th grade. Students in Algebra 1 are usually in 9th grade. This learning module will include videos from Khan Academy, the use of IXL as direct procedural fluency, and utilizing the graphing calculator feature on Desmos. The classroom would need to have access to one-on-one technology for students to full engage in this learning module. However, there are some aspects where students would be able to learn these concepts on traditional paper/pencilas well. The use of the technology would support an active learning approach where students are central to their learning, thus creating a community of learners that collaborate together; as opposed to the traditional teacherled/students follow classroom. This unit of learning should take approximately three weeks of 45 minute class periods or about one and a half weeks of 85 minute block class periods. 

The following are the learning objectives for this learning module. The students will be able to:

  • Explain the differences and similarities between exponential and linear functions. 
  • Relate input and output tables to their corresponding exponential graphs. 
  • Graph exponential functions.
  • Describe how the parameters of an exponential function affect its graph. 
  • Identify and apply ways in which exponential functions can model real-life situations.

The Common Core State Standards that align with this learning module are:

  • CCSS.MATH.CONTENT.HSA.CED.A.2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and
  • CCSS.MATH.CONTENT.HSF.IF.C.7 - Graph functions expressed symbollically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 
  • CCSS.MATH.CONTENT.HSF.IF.C.7.E - Graph exponential functions, showing intercepts and end behavior. 
  • CCSS.MATH.CONTENT.HSF.IF.C.9 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions.)
  • CCSS.MATH.CONTENT.HSF.LE.A.1 - Distinguish between situations that can be modeled with linear functions and with exponential functions. 
  • CCSS.MATH.CONTENT.HSF.LE.A.1.C - Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 
  • CCSS.MATH.CONTENT.HSF.LE.B.5 - Interpret the parameters in an exponential function in terms of a context. 


1. Introduction - Prior Knowledge

For the Student

Before studying exponential functions, students need to be able to: 

  1. Solve and graph linear equations/functions.
  2. Graph points on the coordinate plane. 
  3. Evaluate and simplify expressions involving exponents.

Assignment: Individually, follow the hyperlinks below to IXL and complete:

If you are struggling with these concepts, use the step-by-step instructions provided through IXL to help you understand and correct your mistakes. The image below shows an example of what this would look like if you miss a question on IXL:

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Discussion: Create a discussion post that summarizes each of the topics you just practiced on IXL: graphing points on the xy-plane, solving linear functions, graphing linear functions and properties of exponents. What were some things that you easily remembered? What were some topics you forgot and needed more practice on?

For the Teacher

Purpose: Algebra has many topics that build upon each other. Exponential functions is a combintation of applying knowledge of properties of exponents and applying prior knowledge of linear functions and graphing all together. Therefore, if students are not proficient in these areas, it will be very difficult for them to be successful throughout the exponential functions unit. This section should be used as a form of pre-assessment to gain a better understanding of where the students are at academically. 


Method: The students will complete the IXL's individually at first. I generally give them the first 10-15 minutes for independent practice to get an idea of where their level of understanding is at. Then, allow the students to collaborate in small groups to help each other work through these concepts. In this way, students who are already proficient in these areas can help other students in a peer tutoring method. Be walking amongst the students to monitor progress and provide assistance as needed. In addition, be aware of the teacher platform of IXL to help monitor student progress. With the IXL platform, teachers are able to monitor the students progress through the teacher dashboard with a feature called "real-time." This allows teachers to see instant feedback of which type of questions a student may be struggling with, if the student is "idle" and not participating, and how much time they are spending on each topic and question. An example of the teacher dashboard feature in IXL is further explained in the video below:

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Tips: Since some students may be proficient in these prerequestie skills already, partner these students with students that may be struggling to recall these concepts. In this way, every student is engaged in the activity. By doing so, the students struggling receive the explanations they need, and the students who are already proficient are deepening their understanding of the concepts by elaborating and teaching the concepts to other students. 


2. Exponential Functions vs. Linear Functions

For the Student

How are exponential functions different from linear functions?

Learning Objective: Explain the differences between exponential and linear functions.

In your notes, make an input and output table for the exponential function y=3x for the x-values: -4, -3, -2, -, 1, 2, 3, 4, and 5.

Once you have made your input and output table in your notes, watch the Khan Academy video below on basic exponential functions. For the first video, stop at 5:17. Be sure to include all equations and graphs that are included in the videos into your notes as well. 

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Assignment - Create an update that discusses the following topics:

  1. In your own words, create a definition for exponential functions. 
  2. What is one specific difference between the graph of a linear function and the graph of an exponential function?
  3. Brainstorm: What is a real-life situation that you think could modeled by an exponential function? Explain why you think exponential functions could model it. 

For the Teacher

Purpose: So far in Algebra 1, students are drilled with the concept of linear functions. Because of this, it is essential for students to be able to differentiate between linear and exponential relationships. This lesson should be the foundation of the unit, and should focus on the key features of exponential functions.


Method: Having students use an input and output table will help them tap into their prior knowledge. They have seen and worked with input and output tables as early as 6th grade. By incorporating this, students can connect exponential functions to the concepts they have already mastered. As students work on their instructional videos, be walking amongst the students to ensure they are following the directions provided and to give clarification to any questions they may have. The goal is to ensure students are understanding the main differences between linear and exponential functions. If you notice students are struggling to identify these differences, check in with them and ask questions like "How would you describe the rate of change for a linear function? Would that be the same or different for an exponential function?" and "Describe how the exponential function looks in the graph? How is this the same as a linear function? How is it different?"


Tips: For students that finish this task quicker than others, encourage them to return to the first Khan Academy video at 5:18 to review over the word problem. This could be an extension or challenge problem as needed. For students that are struggling to identify the key differences among the two functions, create a small group for direct instruction. Replay the second Khan Academy video and pause it with them to add further description. This video focuses on the numbers behind the two functions and explains how the rates of change for each type of function are different. Creating a small study group to work with them will give them the more individualized instruction they may need. These two suggestions can occur simultaneously as well. Advanced students can either work independently or in small groups on the word problem while you work with the students that need extra support. 

3. Zombies and Exponential Functions

For the Student


Learning Objectives:

- Relate input and output tables to their corresponding exponential graphs.

- Graph exponential functions.

In your small groups, complete the sections named "Background" and "Make a Prediction" on the Zombie worksheet, provided below:


Zombie Simulation:

One student will be a zombie from Strain A, one will be a zombie from Strain B, and the rest of you are uninfected… for now.

  • Strain A Zombie: Place your yellow Post-It note on all the people you infect. Record how many people you infect on your worksheet. Remember to remind your new zombies that they cannot infect other people; they're just zombies.
  • Strain B Zombie: Place your green Post-It notes on all the people you infect. Give them their own green Post-It notes and remind your new zombies to also put the green Post-It on each person they infect.
  • Uninfected: If Strain A infects you, you're just a zombie roaming around - you cannot infect anyone else! If Strain B infects you, be sure to mark the new zombies with a green Post-It and give them some to spread around as well.

After each minute that passes, we will pause to collect data on how many of each strain of zombie there are. These data points will be recorded on your zombie worksheet in the tables provided.

After the simulation, return to your groups and graph your data points from both Strain A and Strain B. Complete the remaining questions on your zombie worksheet in your small group.



  1. Complete the exit slip for today's activity. 


  2. Create an update: Summarize your findings from the zombie simulation. Explain which functions are represented by Strain A and Strain B.
  3. Complete IXL - X.1: Evaluate an exponential function and X.2: Match Exponential functions and graphs.


For the Teacher

Purpose: This worksheet and activity solidifies the concepts from our introduction lesson on the differences between linear and exponential functions. The worksheet is beneficial for the "mathy" students, while the simulation activity is helpful for those students who need a visual representation or benefit from physical activity being tied to the concept being taught. The worksheet and activity in this lesson adds more detail on the numerical fluency behind exponential functions. However, at the same time, students are fully engaged in the conceptual understanding of how exponential functions grow as well. 

Method: Allow students to work in their small groups to complete the "Background" and "Make a Prediction!" sections of the worksheet. This can be introduced as a warm up or bell ringer activity to start the class. Monitor student answers as they work to ensure they are headed in the correct thought process. For the simulation, assign one student the role of "Strain A Zombie," one student as "Strain B zombie," and the remaining students as the "uninfected." Be clear on verbal instructions explaining each person's role in the simulation. Allow at least 15 minutes for the simulation (7 minutes of timed "Zombie attacks" and extra time between each interval to record the number of each strain of zombie). Regroup students into their small groups after the simulation so they can collaborate to complete their work. During small group work time, walk amongst the student groups to provide assistance as needed. Also make sure all group members are participating in the discourse. 

Tips: Your role during the simulation is to ensure all students are engaged solely in the activity. I suggest having a visual timer for each minute that students are being "attacked" by the zombies. This often helps students remember they are still in the classroom setting and not at recess. There are many free online timers, and usually have many themes you can chose from. 

4. Exploring the Graphs of Exponential Functions

For the Student

Learning Objectives:

- Describe how the parameters of an exponential function affect its graph.

- Graph exponential functions.

The generic formula for an exponential function is: \(f(x) = ab^x+c\)

In this formula a, b, c are all variables that change the way the graph of the exponential function looks. 

Use the Geogebra Activity to explore how each variable affects the graph of an exponential function. In the activity, the purple graph represents the original function that has a=1 and c=0. The blue graph represents the graph when a, b, and c are changed. As seen in the image below, you can differentiate between the two graphs to make conclusions about how the parameters change the graph. 

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  1. Create an update that summarizes your findings. How does 'a' change the graph? How does 'b' change the graph? How does 'c' change the graph? Be specific in your descriptions and include media that support your findings. 
  2. Comment on at least 2 of your peer's updates. Were their findings similar to yours? How were they different?
  3. Complete the following worksheet to practice graphing exponential functions.



For the Teacher

Purpose: Many students struggle to understand how a graph's appearance is affected by the parameters of its corresponding equation. By exploring the different parameters themselves, students are able to draw their own conclusions about how different numbers affect a graph. When students are given the opportunity to create their own connections to concepts based on prior knolwedge or previous lived experience, their retention of the content is much higher. 


Method: Students should work collaboratively in their small groups on this exploration activity. The instructor's role is to monitor student progress as they work. The goal is for the students to create their own conclusions, not be influenced by the knowledge of the teacher. The students have the prior knowledge required to complete this exercise without being simply told what the "answers" are. Thus, the instructor should use questioning strategies to guide the students' thought processes, but not completely dictate them. Many students struggle to manipulate the Geogebra activity properly. Because of this, the teacher should look for students that are struggling to operate the Geogebra activity and operate the sliders. 


Tips: If you are not familiar with Geogebra, I strongly suggest working through the activity yourself prior to this lesson. In this way, you will be able to better predict the misconceptions students may have as they work through the exercise. Push students to make connections based on their prior knowledge. Giving students direct answers can seem helpful, but it is more beneficial for them to make their own conclusions. Because of this, try answering their questions with follow up, leading questions. In this way, students are lead into the correct thought process and able to do so by creating their own summaries. If you struggle to think of some good leading questions, the following link provides beneficial information that could get you started. 

5. Applications of Exponential Functions

For the Student

Exponential Growth and Decay

Learning Objective: Identify and apply ways in which exponential functions can model real-life situations.

Watch the video below to learn about exponential growth and decay. Take notes on the examples completed in the videos. 

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The following SlideShow explains the mathematical formulas of exponential growth and decay. The PDF below is blank. Work through this PDF in your small groups and attempt to complete the examples on your own. 


Once you have had the chance to attempt the word problems in the notes, check your group's work in the PDF below:



  1. Create an update describing 3 different types of real-world examples that model exponential functions. They can be growth or decay. Include media into your update that further explains and supports your examples provided. 
  2. Comment on at least 2 of your peers' updates. 
  3. Complete the worksheet below as homework.



For the Teacher

Purpose: This lesson focuses on how exponential functions are modeled in a real-world context. It is important for students to recognize how the mathematical concepts learned connect to the world outside of the classroom. 


Method: Allow the students to work through the examples in the Khan Academy video on their own/in their small groups. This will give them the opportunity to work through these examples collaboratively and engage in discourse to solve them. In addition, as the students work through the slide-show PDF, they will be able to productively struggle through the examples posed. As the students work through these activities, the instructor should be walking amongst the small groups to ensure all students are engaged and productively struggling, not only struggling with the content. The goal is for students to problem solve through these examples, but receive the support they need from their instructor as needed. This is a more individualized approach that can fit the learning needs of all students. 


Tips: If you notice some students and/or student groups are more advanced with these exercises compared to others, assign them to new groups to collaborate and teach each other. This creates a sense of a learning community that depends on each other, and not only the teacher as the sole source of knowledge. 

6. The Tortoise and The Hare

For the Student

The Fable of The Tortoise and The Hare

Learning Objective: Identify and apply ways in which exponential functions can model real-life situations.

In your small groups, you are given the following task: to solve the tale of The Tortoise and The Hare! However, this time, there is a third contestant - some unknown animal. The only materials you have (to start with) are the flipbook provided and your extensive knowledge of mathematics. Use both the flipbook and directions provided to help you solve the tale.


Be prepared to present your group's findings to the class!



Once you have finished your exploration of the flipbook, create an update that discusses the following:

  • Your group's method used to solve the problem posed.
  • The data collected from your exploration (whether it was a graphical representation, set of data points, input and output tables, etc.). Present this in a visually organized way.
  • Your group's conclusion about the tale: Who won the race? How do you know? What kind of animal is the mystery third participant? Why do you think so? Add data to support your responses. 

For the Teacher

Purpose: This activity focuses on building problem solving skills for the students. The students are not given many discrete instructions or numbers for their problem. They will be able to apply their knowledge of both linear and exponential functions to problem solve their way to a solution.


Method: Introduce the problem posed with the PowerPoint presentation provided below (or one similar to it). As mentioned previously, this is a problem solving activity. As the instructor, you do not want to influence the students' thought processes at all. Utilize guiding questions that are related to linear and exponential functions to get the students thinking mathematically. In addition, use the actual tale of The Tortoise and The Hare to help students realize the connections between the rate of change a hare would have compared to the rate of change a tortoise would have, but do so without directly telling them. 


Tips: Student groups will have numerous different ways of solving this problem; so do not stop them if their approach differs from how you would solve it. Provide different tools for the students to use like rulers, graph paper, calculators, etc. I provided 3 different examples of student work from previous times I have implemented this activity to reference as needed. 




7. Project - The Tortoise and the Hare

For the Student

The Tortoise and The Hare Presentations

Use this time in class to prepare for your group presentations on The Tortoise and The Hare Project. Watch the video below to help your group master key features of a presentation. 

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Peer Feedback

Each small group will partner with another group and present to each other. Students will provide their peers with feedback based on the rubric provided below.


Once your group receives their feedback, adjust your presentation to improve it before your final presentation. 

For the Teacher

Purpose: The students will need ample time to practice their presentations of their findings from their Tortoise and the Hare problem posed. The more time students have to practice before they present, the better their presentations will be.


Method: Have small groups watch the video provided for tips on their group presentations. Then, have the groups practice their presentations together. As students are working, walk amongst them to ensure each group member has a role in the presentation. After about 20-30 minutes of practice, engage students in a peer editing process. Pair up groups to watch each other's presentations and provide feedback on the rubrics provided. In this way, students are able to receive feedback on what features of their work and presentation they need to improve on. 


Tips: As students are engaged in their peer editing groups, try to watch small sections of each group's presentation and provide feedback to them as well. When students are able to receive feedback and improve on their work, they are engaged in a self-reflecting activity as well. 

8. Survey

For the Student

Complete the student survey in the link here.

For the Teacher

Instruct students to complete the student survey using the link provided.


"Exponential growth & decay word problems" Khan Academy. Retrieved from:

Meade, Lynn. 2017 "Tips on Group Presentations." Retrieved from:

Moore, Janet. 2017. "NASA Supernova and The Tortoise and The Hare Presentation" Illinois State University Department of Mathematics. 

"Questionign Strategies." The University of Illinois Urbana Champaign Center of Innovation for Teaching and Learning, 2019. Retrieved from:

"Intro to exponential functions." Khan Academy. Retrieved from:

"Exponential vs. linear growth." Khan Academy. Retrieved from:

Doering, Suzanne. "Exponential Functions - Explore Parameters." Geogebra. Retrieved from:

Sawalha, Yamamah. (2018) "The Effects of Teaching Exponential Functions Using Authentic Problem Solving On Students' Achievement And Attitude." Wayne State University.