"The One with the Exponential Growth": A Friends-Themed Math Problem

Problem Overview

In this problem-based learning scenario, students will explore exponential functions through the lens of the popular TV show "Friends." They'll analyze how viral social media content spreads, specifically looking at how clips, memes, and quotes from "Friends" can grow exponentially in viewership across different platforms. Students will model this growth mathematically, make predictions, and analyze the implications of exponential growth in digital media.

Learning Objectives

• Model real-world situations using exponential functions

• Write exponential functions in the form f(x) = a·bˣ

• Interpret the meaning of a and b in context

• Compare linear and exponential growth

• Use technology to graph exponential functions

• Make predictions using exponential models

• Solve exponential equations algebraically and graphically

Real-World Context

Since arriving on streaming platforms, "Friends" has experienced a renaissance with a new generation of viewers. Clips, quotes, and memes from the show regularly go viral on platforms like TikTok, Instagram, and Twitter. When content goes viral, it often follows an exponential growth pattern as people share it with their networks, who then share it with their networks, and so on.

Social media managers and content creators need to understand these growth patterns to predict viewership, plan marketing strategies, and capitalize on trending content. Understanding exponential functions helps explain why some content reaches millions of viewers in just days while other content grows more slowly.

Problem Statement

Central Question: You work for Warner Bros. as a social media analyst tracking the popularity of "Friends" content across different platforms. Several clips from the show have recently been posted, and you need to model their growth, predict their future viewership, and recommend which clips the company should promote further.

Scenario: Warner Bros. has simultaneously released three classic "Friends" clips on different social media platforms:

1. "Pivot!" (Ross moving the couch) on TikTok

2. "Joey Doesn't Share Food!" on Instagram

3. "Smelly Cat" performance on Twitter

Each clip is being shared at different rates. Your job is to:

• Collect and analyze the viewership data for each clip

• Create mathematical models using exponential functions

• Predict future viewership

• Recommend which clip(s) to promote with additional marketing

• Explain how the exponential growth of these clips compares to other growth patterns

Guided Questions

Part 1: Data Analysis

The table below shows the number of views (in thousands) for each clip over the first 5 days:

a. For each clip, calculate the ratio of views from one day to the next. What do you notice?
b. What type of growth model would best fit each clip's data? Explain your reasoning.

Part 2: Creating Exponential Models

For each clip:
a. Identify the initial value (a) and the growth factor (b) for an exponential function in the form f(t) = a·b^t, where t is the number of days since release.
b. Write the exponential function that models the number of views after t days.
c. Use technology to graph your functions. How well do they fit the data points?

Part 3: Making Predictions

Using your models:
a. Predict the number of views each clip will have after 10 days.
b. After how many days will each clip reach 5 million views?
c. The "Pivot!" clip is growing faster than the others. If this trend continues, how many more views will it have than "Joey Doesn't Share Food!" after 14 days?

Part 4: Analysis and Recommendations

Warner Bros. has a budget to promote one of these clips further:
a. Which clip would you recommend they promote? Justify your answer mathematically.
b. If the promoted clip's growth rate increases by 20%, how would this affect its viewership after 14 days?
c. Compare the exponential growth of these clips to a hypothetical linear growth model. Why is understanding the difference important for social media marketing?

Part 5: Student Choice

Choose ONE of the following extensions:
a. Research and analyze the actual social media performance of a "Friends" clip or meme that went viral. How does its growth compare to your models?
b. Create your own "Friends"-themed content idea and design an exponential growth model that would represent your ideal viral spread.
c. Investigate how the addition of a decay factor would affect your model (as viral content eventually loses popularity). Modify your function to include this factor.

Expected Solution Path

Sample Approach 1: Analytical Student

This student might take a systematic approach:

1. Data Analysis:

   • Calculate growth ratios: "Pivot!" ≈ 2.5, "Joey" = 2.0, "Smelly Cat" ≈ 2.4

   • Recognize consistent ratios indicate exponential growth

2. Model Creation:

   • "Pivot!": f(t) = 10 · 2.5ᵗ

   • "Joey": f(t) = 15 · 2ᵗ

   • "Smelly Cat": f(t) = 5 · 2.4ᵗ

3. Predictions:

   • Use the models to calculate precise values for day 10

   • Solve equations like 10 · 2.5ᵗ = 5,000,000 using logarithms

   • Compare long-term projections using algebraic methods

4. Recommendation:

   • Recommend "Pivot!" based on highest growth rate

   • Calculate exact differences between projections

   • Provide quantitative justification for marketing decisions

Sample Approach 2: Visual/Technological Student

This student might rely more on graphical representations:

1. Data Analysis:

   • Plot the data points for each clip

   • Observe the curve shapes visually

   • Use technology to confirm exponential patterns

2. Model Creation:

   • Use graphing calculator or spreadsheet to find best-fit exponential functions

   • Verify models by overlaying them on the original data points

   • Adjust parameters as needed for better fits

3. Predictions:

   • Use graphing to visualize future growth

   • Find intersections between horizontal lines (target views) and functions

   • Create visual comparisons of the three growth trajectories

4. Recommendation:

   • Use visual projections to demonstrate differences

   • Create compelling graphs showing the impact of increased promotion

   • Present visual evidence for marketing decisions

Sample Approach 3: Context-Focused Student

This student might emphasize real-world applications:

1. Data Analysis:

   • Connect growth patterns to actual social media sharing behaviors

   • Research typical viral content growth rates for comparison

   • Consider platform differences (TikTok vs. Instagram vs. Twitter)

2. Model Creation:

   • Incorporate platform-specific factors into models

   • Consider audience demographics for "Friends" content

   • Adjust models based on typical engagement patterns

3. Predictions:

   • Consider practical limitations to exponential growth

   • Factor in potential saturation points

   • Make realistic projections based on platform constraints

4. Recommendation:

   • Consider ROI for different platforms

   • Analyze which clip resonates best with target demographics

   • Provide context-rich justification for marketing decisions

Extension Opportunities

1. Logarithmic Transformation:

   • Convert exponential equations to linear form using logarithms

   • Analyze how logarithmic scales help visualize exponential data

   • Solve exponential equations using logarithms

2. Compound Growth Models:

   • Explore how continuous compounding relates to the number e

   • Compare discrete vs. continuous exponential models

   • Investigate the formula P(t) = P₀e⁽⁻ʳᵗ⁾

3. Logistic Growth:

   • Research how viral content eventually reaches saturation

   • Modify exponential models to include carrying capacity

   • Create logistic growth models: P(t) = K/(1+Ae⁽⁻ʳᵗ⁾)

4. Cross-Platform Analysis:

   • Investigate how content spreads across multiple platforms

   • Model the combined growth across all platforms

   • Analyze how sharing between platforms affects overall growth

5. Financial Applications:

   • Connect to compound interest and investment growth

   • Calculate how Warner Bros. could monetize viral content

   • Analyze the financial impact of successful social media campaigns

Success Criteria: Students will successfully complete this project when they:

1. Create accurate exponential models for all three clips

2. Make reasonable predictions using their models

3. Justify their marketing recommendations with mathematical evidence

4. Compare exponential growth to other growth patterns

5. Connect their mathematical work to the real-world context of social media and viral content

6. Present their findings clearly using appropriate mathematical notation and terminology