Over the last two centuries, more and more people in the U.S. have been moving out of the country and into cities. The urban population, as a percent, has grown from about 6% in 1800 to over 80% by the end of the last decade. But the rate of growth hasn’t been constant. So how have cities been growing and changing over the past 200 years?

In this lesson students use recursive rules and linear and exponential functions to explore urbanization in the U.S., as well as what different levels of urbanization might mean for future life in the country.

Students will

Given a graph of real-world data, calculate potential linear and exponential rates of change

Develop linear and exponential models for urban population growth and evaluate them for different years

Informally compare the goodness-of-fit for the two models and use them to make predictions

Given a recursive rule to model urbanization, compare and contrast its behavior to the previous models

Vary the parameters of a recursive rule to achieve different long-term behavior

Discuss how different urbanization trends might affect the future of life in the U.S.

Before you begin

Students should be able to describe the difference between linear and exponential functions in terms of rates of change, as well as write explicit formulas based on those rates. Students will also be exposed to recursive rules, so some familiarity with that concept would be helpful, though this lesson could serve as an introduction.

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Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF)

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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.

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Building Functions (BF), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)

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.

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Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI), Similarity, Right Triangles, and Trigonometry (SRT)

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Building Functions (BF), Linear, Quadratic, and Exponential Models (LE), Seeing Structure in Expressions (SSE)

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

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

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

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 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)

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

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Topic:
Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

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.

Topic:
Building Functions (BF), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE), Seeing Structure in Expressions (SSE)

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

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

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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 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.

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

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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)