# Understanding exponential growth and decay

Create and analyze a simple exponential function arising from a real-world or mathematical context (F-lE.A.2$^\star$).

Students have been introduced to exponential functions in Exponential Functions 1. For some students this introduction could have occurred a year or two previously. Therefore this unit starts with an activity that re-introduces exponential growth and decay in an engaging real-world or mathematical context. The context can be modeled by an exponential function with domain contained in the integers, thus providing a review of previous experience as needed for students.

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WHAT: Students are shown a video of someone photocopying a dollar bill with the copier set to reduce size, repeatedly photocopying the reduced image with the same setting. They engage in several components of the modeling cycle (MP.4) as they guess how big the image is after 9 iterations and draw their guess on a diagram. They then calculate the answer to that question and various follow-up questions (F-LE.A.2$^\star$).

WHY: The purpose of this activity is re-activate students’ understanding of situations where a quantity changes by constant factor at each stage in a sequence. This activity provides a concrete context where the quantities, the dimensions of the image, can be easily measured and where students are asked a question designed to stimulate their curiosity.

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WHAT: From the Mathalicious website: “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 (F-LE.A.1$^\star$, F-LE.A.2$^\star$) to explore urbanization in the U.S., as well as what different levels of urbanization might mean for future life in the country.”

WHY: This is a low bar/high ceiling lesson in an engaging modeling context (MP.4) that can serve to re-orient students with prior learning about the nature of linear and exponential growth, approximating growth rate based on data, and building and interpreting functions. Many opportunities are presented to engage in constructing arguments MP.3. The last two questions invoke a recursive rule that offer a nice challenge if time permits.

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