# Model with exponential functions

## • Model situations of growth and decay with exponential functions expressed in various different forms given a graph, a description of the situation, or two input-output pairs (including reading these from a table) (F-LE.A.2$^\star$). • Recognize that exponential functions have a constant percent growth or decay rate per unit interval (F-LE.A.1a$^\star$). • Interpret the parameters of an exponential function in a context (F-LE.B.5$^\star$).

Now that they are familiar with the basic form $f(x)=ab^x$ of an exponential function, students start to work with exponential functions expressed in other ways. They learn the relationship between the growth (or decay) factor and the growth (or decay) rate; if $r$ is the growth rate then $1 + r$ is the growth factor. They model more complex situations where they must derive the growth factor in various ways given data about the context.

1 Predicting the Past

WHAT: Students read a description of a plant tripling in size each week and are led to construct a function to model the height of the plant in terms of weeks. Their model must account for weeks before observations started which provide a context for negative domain values. They are asked to interpret values of the function.

WHY: The purpose of this instructional task is to provide an opportunity for students to use and interpret the meaning of a negative exponent in an exponential function. In Grade 8, students understand expressions with negative exponents in terms of how they interact operationally, for example, the fact that $(2^5)(2^{-4}) = 2^1$ must mean that $2^{−4}$ is equal to $1/2^4$. In this unit, they can express regularity in repeated reasoning (MP8) and create an exponential function to describe a relationship. They can then interpret and evaluate the function in order to understand the meaning of a negative exponent in a context.

2 Moore's Law and Computers

WHAT: Students are given instructions for an investigation where they roll 30 dice, set all the 1’s aside, write down how many dice are left, and repeat. They make predictions about the outcome, conduct the dice-rolling ten times, use technology to view a scatterplot of the number of dice remaining versus the roll number, and write a function $d(x) = ab^x$ to model the relationship. They graph their model on the same set of axes as the scatterplot and make observations.

WHY: This task provides concrete experience with exponential decay and leads students to write a function to model exponential decay. A key insight is that since approximately $1/6$ of the dice are removed after each roll, the value of b should be $(1 – 1/6)$, or $5/6$. Although this task does not require students to make decisions about choosing a model or a technological tool it does provide experiences valuable to building their skills in doing so (MP4, MP5).

3 DDT-cay

WHAT: In DDT-cay, students are given a function $a(t) = 9(0.5)^t$ modeling the decay of the chemical DDT, which has a half-life of approximately 15 years. Students evaluate the function for $t = 0$, $-1$, and $1$, interpret the meaning of those values in the context, and interpret more precisely the meaning of $t$ in the definition of the function. In All Your Base are Belong to Us, students find the base of a function $p(r) = 64(b)^r$ modeling the number of players in each round of a tournament and find and interpret values.

WHY: Both of these tasks incorporate ideas from the previous two activities: exponential decay and negative exponents. They are fairly simple contexts with straightforward questions, and could serve as formative assessment tasks or performance assessments.

4 All Your Base Are Belong to Us

WHAT: In DDT-cay, students are given a function $a(t) = 9(0.5)^t$ modeling the decay of the chemical DDT, which has a half-life of approximately 15 years. Students evaluate the function for $t = 0$, $-1$, and $1$, interpret the meaning of those values in the context, and interpret more precisely the meaning of $t$ in the definition of the function. In All Your Base are Belong to Us, students find the base of a function $p(r) = 64(b)^r$ modeling the number of players in each round of a tournament and find and interpret values.

WHY: Both of these tasks incorporate ideas from the previous two activities: exponential decay and negative exponents. They are fairly simple contexts with straightforward questions, and could serve as formative assessment tasks or performance assessments.

## External Resources

1 Billions and Billions

#### Description

WHAT: Students find out the total population and the number of births and deaths in 2011. They consider whether the population is likely to continue growing by a constant amount or a constant percentage (F-LE.A.1a$^\star$). They then build an exponential model for human population growth in the form $P=P_0(1+r)^t$ (F-LE.A.1$^\star$). They use their model to predict what the world population will be in the future, and consider questions about what life will be like with more and more people (MP4).

WHY: In this activity students learn the relationship between the percent growth rate $r$ and the growth factor $1 + r$ in an exponential model. They learn to construct an exponential model for a real world situation where the data available naturally leads to the percent growth rate (MP4).

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