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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Topic:
Building Functions (BF), Interpreting Functions (IF)

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.

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

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.

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

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.

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

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.

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 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.

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

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.

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)

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.

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

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.

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

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.

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

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.

Topic:
Building Functions (BF), 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)

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.

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

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.

Topic:
Building Functions (BF), Interpreting Functions (IF)

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.

Topic:
Functions (F), Interpreting Categorical and Quantitative Data (ID), Interpreting Functions (IF)

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.

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

Should Major League Baseball stadiums be standardized? Students use a quadratic function to model the trajectory of the average professional home run and debate whether Major League Baseball stadiums should all be designed the same.

Topic:
Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI), Seeing Structure in Expressions (SSE)

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.

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.

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