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Quantitative Skills and Reasoning Activity Workbook

Megan Patnott

Section 7.1 Building Exponential Models Prep Activities

Prep Activity 7.1.1.

Let’s use what we’ve learned about change factors and rates of change to find the population of three small towns. Resist the temptation to find the populations of the towns in Indiana and Ohio by first using the percent to find the number of people the town gains or loses, and then adding or subtracting! That’s a mathematically valid way to solve the problem, and would be fine if our real goal was to get the answers. But, our real goal is to practice using change factors and to have a couple of examples to help us see the connection between change factors and exponential models.
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(a)

The population of a small town in Indiana was 5,500 people in the year 2010 and census data showed that the population grew by approximately 2.5% every year until 2020.
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(ii)
Use the change factor to find the population in 2011.
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(iii)
Use the change factor to find the population in 2012 and 2013.
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(iv)
Use the change factor to find the population in 2020.
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(b)

The population of a small town in Ohio was 5,500 people in the year 2010 and census data showed that the population decreased by approximately 2.5% every year until 2020.
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(ii)
Use the change factor to find the population in 2011.
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(iii)
Use the change factor to find the population in 2012 and 2013.
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(iv)
Use the change factor to find the population in 2020.
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(c)

The population of a small town in Michigan was also 5,500 people in the year 2010, but it decreased by an average of 150 people a year until 2020.
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(i)
What is the rate of change in this town’s population?
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(ii)
Use the rate of change to find the population in 2011, 2012, and 2013.
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(iii)
Use the rate of change and the population in 2010 to build a linear model for the population of this town.
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(iv)
Use your linear model to find the population in 2020.
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(d)

What is different about the population change from year to year in the small town in Indiana and the population change from year to year in the small town in Ohio?
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(e)

What is different about the population change from year to year in the small town in Michigan and the population change from year to year in the small town in Ohio?
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Prep Activity 7.1.2.

In an expression of the form
\begin{equation*} c \times b^e\text{,} \end{equation*}
we call \(b\) the base, \(e\) the exponent or power, and \(c\) the coefficient. This vocabulary will be useful to us in the next lesson, so let’s practice it now.
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(a)

Identify the base, exponent, and coefficient in each of the expressions below.
\begin{equation*} 2 \times 1.5^4 \qquad 5500(1.025)^t \qquad \frac12 x^3 \qquad \left(\frac13\right)^y \end{equation*}
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(b)

Write an expression with base \(0.875\text{,}\) coefficient \(5\text{,}\) and exponent \(r\text{.}\)
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Prep Activity 7.1.3.

You’ll need to be able to do the following things for this lesson. Rate how confident you are on a scale of 1 - 5 (1 = not confident and 5 = very confident).
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Table 7.1.1.
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Skill or Concept: I can … Rating from 1 to 5
Apply the same percent change to a quantity repeatedly using the change factor.
Identify the base, exponent, and coefficient in an expression of the form \(c \times b^e\)
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