At the end of the popular musical Les Misérables, Inspector Javert falls from a bridge in the middle of Paris into the river below. As he falls, he sings...and sings...and sings.

In this lesson, students use quadratic functions to explore the mathematics of how objects fall. How high was Javert’s bridge, how fast was he traveling when he hit the water, and what’s the maximum height from which someone can safely jump?

Students will

Create a quadratic equation to model the relationship between the time an object falls and its distance

For a given time, evaluate the function to determine the height from which an object fell

Use the changes in quadratic distance to determine a (linear) model for speed vs. time

For a given time, evaluate the function to determine an object’s speed at impact

For a given speed, solve an equation to determine the object’s falling time and initial height

Discuss the maximum height from which someone can safely jump

Before you begin

Students should be able to distinguish between linear functions (e.g. y = 2x + 4) and non-linear functions (e.g. y = x^{2}). They should also have experience graphing linear functions.

What's an acceptable dating range? Students use linear equations and linear inequalities to examine the May-December romance, and ask whether the Half Plus Seven rule of thumb is a good one.

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

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)

How much should you bid 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’s the best strategy for creating a 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 Major League Baseball stadium 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)

How do noise-canceling headphones work? In this lesson, students use transformations of trigonometric functions to explore how sound waves can interfere with one another, and how noise-canceling headphones use incoming sounds to figure out how to produce that sweet, sweet silence.

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

How should pharmaceutical companies decide what to develop? In this lesson, students use linear and quadratic functions to explore how much pharmaceutical companies expect to make from different drugs, and discuss ways to incentivize companies to develop medications that are more valuable to society.

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

Do social networks like Facebook make us more connected? Students create a quadratic function to model the number of possible connections as a network grows, and consider the consequences of relying on Facebook for news and information.

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

Why are so many Americans dying from opiate overdoses? Students use exponential decay and rational functions to understand why addicted patients seek more and stronger opioids to alleviate their pain.

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

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), 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.

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