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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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)