Acetaminophen is one of the most popular over-the-counter pain relievers in the country, but it’s also one of the most common causes of liver failure. There isn’t a big difference between helpful and harmful dosages, and sometimes even following manufacturer recommendations isn’t enough to keep people out of harm’s way.

In this lesson 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.

Students will

Given acetaminophen’s half-life, calculate its hourly rate of elimination from the body

Write an exponential rule for determining how much of the drug remains in a person’s body after a given time

Use logarithms to determine the required time until a person’s body contains a given amount of the drug

Describe how the amount of medication changes over time during a period of continued use

Discuss some of the dangers of exceeding the recommended dosage guidelines

Discuss what drug manufacturers should do to help prevent accidental acetaminophen overdoses

Before you begin

Students should be familiar with writing rules describing exponential decay. They will need to use logarithms to solve for variable exponents, and they should be able to apply basic rules of exponents and logarithms, for instance that (a)^{bc} = (a^{b})^{c} and log(a^{b}) = b log(a).

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.

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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.

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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.

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

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.

Topic:
Interpreting Functions (IF), 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.

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

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.

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

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 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.

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Linear, Quadratic, and Exponential Models (LE), Reasoning with Equations and Inequalities (REI)

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)

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

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)

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), Interpreting Functions (IF)