In 2004, the social network Facebook launched to little fanfare. Today, over 10% of the world’s population has a Facebook account, and the company is worth billions of dollars. But where, exactly, does the value of a social network come from?

In this lesson, students will come up with a quadratic function to model how a network’s value (measured in terms of the number of connections between users) grows with the number of people using the network. They’ll also discuss who gets the most value out of a social network: the users themselves, or the advertisers vying for likes, comments, and greenbacks.

Students will

Calculate the number of connections that can be formed in a small group of people

Derive a general formula for the number of connections in a network with p people in it, and use this as a proxy for a network’s value

Estimate Facebook’s value over time based on the growth of its user base

Discuss whether a large social network is better for users or advertisers

Explore an alternative to Facebook’s social network model, and discuss which approach is more valuable

Before you begin

Students should be able to write and evaluate quadratic functions. The beginning of the lesson can be used to introduce some basic concepts from graph theory (such as edges and vertices), though the lesson can also be used without making any explicit mention of these ideas.

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Topic:
Building Functions (BF), Creating Equations (CED), Expressions and Equations (EE), Functions (F), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

How have video game console speeds changed over time? Students write an exponential function based on the Atari 2600 and Moore's Law, and see whether the model was correct for subsequent video game consoles.

Topic:
Building Functions (BF), Interpreting Categorical and Quantitative Data (ID), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)

How has the human population changed over time? Students build an exponential model for population growth, and use it to make predictions about the future of our planet.

Topic:
Building Functions (BF), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)

How much should people pay for donuts? Students use linear, rational, and piecewise functions to describe the total and average costs of an order at Carpe Donut.

Topic:
Building Functions (BF), Interpreting Functions (IF)

What's the ideal size for a soda can? Students use the formulas for surface area and volume of a cylinder to design different cans, calculate their cost of production, and find the can that uses the least material to contain a standard 12 ounces of liquid.

How far away from the TV should you sit? Students use right triangle trigonometry and a rational function to explore the percent of your visual field that is occupied by the area of a television.

Topic:
Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI), Similarity, Right Triangles, and Trigonometry (SRT)

How can you make money in a pyramid scheme? Students learn about how pyramid schemes work (and how they fail), and use geometric sequences to model the exponential growth of a pyramid scheme over time.

Topic:
Building Functions (BF), Linear, Quadratic, and Exponential Models (LE), Seeing Structure in Expressions (SSE)

Why hasn't everyone already died of a contagion? And, if vampires exist, shouldn't we all be sucking blood by now? Students model the exponential growth of a contagion and use logarithms and finite geometric series to determine the time needed for a disease to infect the entire population. They'll also informally prove that vampires can't be real.

Topic:
Creating Equations (CED), Linear, Quadratic, and Exponential Models (LE), Seeing Structure in Expressions (SSE)

How much do you really pay when you use a credit card? Students develop an exponential growth model to determine how much an item really ends up costing when purchased on credit.

Topic:
Building Functions (BF), Creating Equations (CED), Linear, Quadratic, and Exponential Models (LE)

Could Inspector Javert have survived the fall? Students use quadratic models to determine how high the bridge was in Les Misérables, and how fast Javert was traveling when he hit the water.

Topic:
Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF)

What's the optimal bidding strategy in an auction? Students use probability, expected value, and polynomial functions to develop a profit-maximizing bidding strategy.

Topic:
Building Functions (BF), Interpreting Functions (IF)

How has the iPod depreciated over time? Students compare linear and exponential decay, as well as explore how various products have depreciated and what might account for those differences.

How can we improve our calendar? Students examine some other ways to keep track of dates, and use number sense and function concepts to convert between different calendars.

Topic:
Building Functions (BF), Functions (F), Interpreting Functions (IF)

What are the odds of creating the best March Madness bracket? Students use probability to discover that it’s basically impossible to correctly predict every game in the tournament. Nevertheless, that doesn’t stop people from trying.

Topic:
Conditional Probability and the Rules of Probability (CP), Creating Equations (CED), Linear, Quadratic, and Exponential Models (LE)

In which MLB ballpark is it hardest to hit a home run? Students find the roots and maxima of quadratic functions to model the trajectory of a potential home-run ball.

Topic:
Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

How has the urban population changed over time, and will we all eventually live in cities? Students use recursive rules along with linear and exponential models to explore how America's urban areas have been growing over the last 200 years.

Topic:
Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE), Reasoning with Equations and Inequalities (REI)

When should NFL teams go for it on fourth down? Students use quadratic functions to develop a model of expected points. They then apply this model to determine when teams should punt the ball, and more importantly, when they shouldn’t.

Topic:
Building Functions (BF), Interpreting Functions (IF), Using Probability to Make Decisions (MD)

How much should Nintendo charge for the Wii U? Students use linear functions to explore demand for the Wii U console and Nintendo's per-unit profit from each sale. They use those functions to create a quadratic model for Nintendo's total profit and determine the profit-maximizing price for the console.

Topic:
Creating Equations (CED), Building Functions (BF), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

Which size pizza should you order? Students apply the area of a circle formula to write linear and quadratic formulas that measure how much of a pizza is actually pizza, and how much is crust.

Topic:
Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

How have temperatures changed around the world? Students use trigonometric functions to model annual temperature changes at different locations around the globe and explore how the climate has changed in various cities over time.

Topic:
Building Functions (BF), Interpreting Functions (IF)

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Mathalicious lessons provide teachers with an opportunity to teach standards-based math through real-world topics that students care about.

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