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Carpe Donut in Charlottesville, Virginia, has an interesting pricing scheme. You can buy one donut for $2, two donuts for $3, three for…well, you get the idea. This means that two people could pay less by purchasing their donuts together. Three people could do even better. So how does the average cost per donut change, and how much should we be paying for each?

In this lesson, students use linear, rational, and piecewise functions to describe the total and average cost of an order at Carpe Donut. Through this, they’ll see the benefits of buying in bulk, and will find a least-expensive way to purchase these delicious donuts.

- Use linear and piecewise-linear functions to describe a pricing scheme
- Model average cost using rational functions
- Make formal or informal limit arguments to explain end behavior of total and average cost functions

Carpe Donut in Charlottesville, VA

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.

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 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 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.

How much should you bid in an auction? Students create polynomial functions to model the expected value of a given bid and determine the optimal amount someone should bid in any auction.

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 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.

Should the government increase the minimum wage? Students use systems of linear equations to explore the relationship between wage and labor, analyze the economics of fast-food restaurants, and debate whether the federal government should increase the minimum wage.

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 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.

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*.