How has the value of the original iPod changed since it was announced in 2001? In this lesson students will compare **linear vs. exponential decay**, and will use eBay to explore how various products have depreciated over time. In the end, students will see which Apple products have depreciated the most and the least over time, and discuss what might account for the differences.

- Write, evaluate, and graph linear models for iPod depreciation
- Write, evaluate, and graph exponential models for iPod depreciation
- Compare and contrast linear versus exponential depreciation models
- Compare and contrast exponential depreciation models across different Apple products

- Colored pencils (optional)

Apple, eBay

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.

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.

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.

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.

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.

Could Inspector Javert have survived the fall? Students use quadratic models to determine how high the bridge was in *Les Misérables*, and explore the maximum height from which someone can safely jump.

How much can you trust your memory? Students construct and compare linear and exponential models to explore how much a memory degrades each time it's remembered.

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.

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.

How much Tylenol can you safely take? Students use exponential functions and logarithms to explore the risks of acetaminophen toxicity, and discuss what they think drug manufacturers should do to make sure people use their products safely.

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.

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.

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.

How much more do graduates earn, and is college worth the cost? Students use systems of linear equations to compare different educational options.

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.

How much should companies pay their employees? Students graph and solve systems of linear equations in order to examine the effects of wage levels on labor and consumer markets, and they discuss the possible pros and cons of increasing the minimum wage.

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.

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.

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.

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.

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

How do the rules of an election affect who wins? Students calculate (as a percent) how much of the electoral and popular vote different presidential candidates have received, and add with integers to explore elections under possible alternative voting systems.

In basketball, should you ever foul at the buzzer? Students use probabilities to determine when the defense should foul...and when they should *not*.