# Modeling with exponential functions

Build exponential functions to model real world contexts (F-LE.A.1$^\star$, F-LE.A.2$^\star$, F-BF.A.1$^\star$).

Having seen the purpose of various different expressions for exponential functions, students now start to build functions in those forms in order model real-world contexts. Contexts may include Moore’s law for computer processor speeds, population growth, and temperature change. Students should also consider whether an exponential or a linear model is appropriate in various contexts.

## Tasks

WHAT: Students construct various exponential models for human population growth by assuming a model of the form $P=P_0 b^t$ and solving for $b$ using data from different periods within the period 1804–2012. Comparing the different models they decide if an exponential model is suitable for the entire period (MP.4).

WHY: The purpose of this task is to illustrate an important aspect of the modeling process, choosing different models, evaluating them, and possibly rejecting them if they do not fit the situation.

WHAT: In this task are given data about the boiling point of water at various elevations. They use a graphing utility with the capability to vary parameters to construct linear functions that model the data well over small ranges of elevation. The models do not work well over the full range of the data. In this latter case students use an exponential function expressed in the form $f(t)=ae^{ct}$ to model the data.

WHY: The purpose of this task is to give students practice in choosing models and evaluating whether a model works well or not for a given situation (an important aspect of MP.4). They discover the important fact that whether a model fits well or not depends on the range of data use; data that looks linear on a small scale might not be so on a larger scale. The task also gives students practice in building exponential functions with base e and seeing the effect on the graph of varying parameters.

## External Resources

#### Description

WHAT: A series of questions where students are given a pair of points and asked to write an exponential function in the form $y = ab^x$ that passes through them (F-LE.A.1$^\star$). They are also asked to respond to interpretation questions like, “How can you tell from the two points whether b is greater than or less than 1?”

WHY: The purpose of this task is to illustrate another way of finding exponential models: using two data points to solve for the base, $b$. The materials include a warm-up and support for differentiation. By working on this skill out of context, students can focus on fluency with useful algebraic moves.

#### Description

WHAT: Computer scientist Gordon Moore made a prediction in the 1960’s that computer processor speeds would double every two years. When it comes to the processors in video game consoles, how good was that prediction? In this activity, students construct and graph an exponential function that models Moore’s Law; analyze what it predicts for game consoles; research and create a scatterplot of how console speeds have actually changed over time; graph and interpret a function that best fits the scatterplot; and compare Moore’s prediction with reality (F-LE.A.2$^\star$, F-LE.B.5$^\star$, N-Q.A.2$^\star$, MP.4).

WHY: Exponential functions are used to analyze a claim in the context of game consoles, about which many students are knowledgeable and interested. This lesson brings together many important new concepts and skills learned in this unit.

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#### Description

WHAT: From the website: “What mathematical principles describe the spread of disease? How can we mathematically model an epidemic? In this lesson, students explore the fundamental mathematical concepts underlying the spread of contagious diseases. Using a simple exponential model (F-BF.A.1a$^\star$), students compare and contrast the effects of different transmission rates on a population and refine their understanding of the nature and characteristics of exponential growth. Students can then compare their projections with actual Ebola data from West Africa, to create context for analyzing the strengths and limitations of this simplified model.”

WHY: The spread of disease provides an authentic and engaging context for students to apply their understanding of exponential functions using appropriate tools (MP.5) and hone their modeling skills (MP.4).

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