How much should you bid in an auction? From Sotheby’s to eBay, auctions are a popular way to buy everything from paintings to baby clothes. Of course, the goal of an auction isn’t just to win the item; it’s to pay as little as possible for it.

In this lesson, students create polynomial functions to model the expected value of a given bid and determine the optimal amount someone should bid in any auction.

- Write and graph a quadratic equation to model a real-world situation
- Write and graph higher-order polynomial equations to describe a real-world situation
- Calculate and model expected value in a real-world context

Thumbnail image by Financial Times on flickr

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.

In basketball, which shot should you take? Students use probability and expected value to determine how much 3-point and 2-point shots are really "worth" to different NBA players.

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

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.

What does it mean for a playlist to be "random?" Students use probability to explore the idea of randomness, as well as the patterns that can emerge from random processes like shuffles.

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.

What is the likelihood of winning at craps? Students learn the rules of the popular casino game, and use probabilities to determine how likely players are to win big (or go broke).

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 have video game consoles changed over time? Students create exponential models to predict the speed of video game processors over time, compare their predictions to observed speeds, and consider the consequences as digital simulations become increasingly lifelike.

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.

How many people should you date before you settle down? Students use modeling with probability distributions to come up with a rule to try to maximize their relationship happiness.

When is it worth buying a Powerball ticket? Students count combinations and apply basic rules of probability and expected value to determine when the Powerball jackpot is large enough to justify the cost of playing the game.

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.

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.

How much should Nintendo charge for a video game console? Students use linear and quadratic models to analyze and discuss the relationship between the price of a Wii U console and profits for Nintendo.

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 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.

How should pharmaceutical companies decide which drugs to develop? Students create linear and quadratic functions to explore how much pharmaceutical companies profit from different drugs and consider ways to incentivize companies to prioritize medications that are valuable to society.

How has the human population changed over time? Students develop exponential models to analyze human population growth and explore the impact this growth will have in areas around the world.

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.

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. Students discuss the role that various parties played in creating the crisis and ways they can help to solve it.

What time should school start in the morning? Students use periodic functions to compare the alertness levels of adults vs. teenagers over the course of the day and debate the merits of starting school later.

Is higher education a good investment? Students write and solve systems of linear equations to determine how long it would take to pay off various degrees and discuss the pros and cons of different educational paths.

How accurate should government surveillance be? Students calculate conditional probabilities to determine the likelihood of false-positives and false-negatives, and discuss the tradeoffs between safety and accuracy.

How dangerous are heat and humidity? Students use polynomial functions to explore the heat index and discuss the life-and-death consequences that cities around the world will face in the coming years.

How have temperatures changed around the world? Students use periodic functions to compare long-term average monthly temperatures to recorded monthly temperatures, evaluate evidence of climate change, and discuss possible consequences.

How much would it cost to get all the toys in a Happy Meal? Students use trials, probabilities, and expected value to determine how many meals it takes to get a complete set of Happy Meal toys and debate whether McDonald’s should allow customers to pay a fee to choose their own figurine.

How do viruses spread through a population? Students use exponential growth and logarithms to model how a virus spreads through a population and evaluate how various factors influence the speed and scope of an outbreak.

How much should you trust your memory? Students use exponential decay to model memory fidelity and debate whether a bad memory is a good thing.

Should airlines overbook their flights? Students use compound probability and expected value to determine the optimal number of tickets an airline should sell and discuss whether airlines should be allowed to overbook their flights.

What’s the best way to play roulette? Students use probabilities and odds to analyze roulette payouts and debate the optimal strategy for winning the game (including not playing at all).

How much does age matter in a relationship? Students use a system of linear inequalities to explore the popular dating rule-of-thumb, ‘half plus seven’, and debate how important age -- and other factors -- are in healthy relationships.

What’s the ideal size of a soda can? Students create rational functions to explore the relationship between volume, surface area, and cost to determine the optimal size of a soda can.

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

How have video game consoles changed over time? Students create exponential models to predict the speed of video game processors over time, compare their predictions to observed speeds, and consider the consequences as digital simulations become increasingly lifelike.

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