Lesson 13

In this lesson, students are expected to use calculators with an \(e\) button. This warm-up is meant to smooth out any technical difficulties with using their calculator to perform desired calculations.

Students may not have had much experience with entering an expression (rather than a single number) for the exponent and may not realize that parentheses may be needed to group the parts of the expression together. Expect calculation errors that stem from this issue.

Depending on the number of digits their calculator allows and the settings for displaying decimals, students may also notice that their calculation results have a different number of decimal places than those on the task statements.

Arrange students in groups of 1. Provide access to calculators that have buttons for \(e\) and exponent.

Invite students to share some issues they came across when evaluating the expressions with a calculator and how to resolve them. Make sure that students recognize that grouping symbols may be needed when entering an expression as an exponent, and that calculation results may vary because of the level of precision of the calculator.

This activity exposes students to growth factors written as \(e^r\). An important takeaway for students is understanding that when we assume \(r\) is a continuous growth rate, that is, the rate \(r\) is happening at every moment and not just each unit of time \(t\), then \(e^r\) is how we can write the growth factor. If, however, we assume that the rate \(r\) is happening for each unit of time \(t\), then using a growth factor of \(1+r\) is appropriate.

This work prepares students to work with equations of the form \(f(t)= P \boldcdot e^{rt}\), which they will encounter in future activities throughout the rest of the unit. In the next activity, students will focus on interpreting the parameters of exponential equations written in this form. At this stage, students are not expected to understand \(e\) in depth. That work is reserved for future, more-advanced courses. The goal here is to connect this new form of expressing an exponential function to students’ prior work by centering student thinking around the idea of how a growth rate is being applied.

Read the introduction to the activity and display the equations for \(P\) and \(Q\). Ask students to think of at least one thing they notice and at least one thing they wonder about the functions. Give students 1 minute of quiet think time, and then invite students to share, recording and displaying their responses for all to see. Tell students that during this activity they will investigate pairs of functions that model the exponential growth of the same population, but make different assumptions about how the population is growing. 

Arrange students in groups of 2–4 and provide access to graphing technology and a spreadsheet tool, in case requested. Ask groups to complete the first question and then pause for a brief whole-class discussion.

Invite students to share their tables and observations. Highlight the observations that the predictions produced by the two models are fairly close, but not identical, and that the difference seems to increase for larger values of \(t\).

Next, tell students that they will now use graphs to compare the predictions of the two models for colonies that are growing at slower and faster rates. Suggest that they identify the two graphs for each colony with labels and different colors (if possible and simple to do). Consider asking students to split up the graphing work to optimize time.

Select groups of students to share their observations and graphs (or consider displaying the graphs in the Student Response for all to see). Here are some questions for discussion:

• “Which pair of graphs seemed most alike?” (Colony 4's graphs were almost the same until you got to large values of \(t\). At the end of 2 year2, \(v(24)\) and \(w(24)\) were both about 11.3 thousand.)
• “Which pair of graphs seemed least alike?” (Colony 1's graphs looked different after just a few months. After 2 years, \(f(24) \approx32.3\) while \(g(24)\approx33.2\).)
• “For Colony 2, what is the growth factor for 1 month predicted by each model?”(\(k\) predicts a growth factor of 1.03 while \(l\) predicts a growth factor of \(e^{0.03}\), which is a bit more than 1.03.)

Tell students that exponential functions that involve small but ongoing growth (such as population growth or inflation) can be modeled in different ways. A factor written in the form of \(e^{0.03}\) highlights that the \(r\) percent growth rate (for example, 3%) is applied continuously. The \((1+r)^t\) model predicts a change by a factor of \(1 + r\) for each unit of time. While these sound similar, they are not equivalent.

Students will learn more about \(e\) in future courses. For now, it is sufficient simply to know that scientists often find it helpful to use \(e^r\) to model exponential growth and decay in cases where the growth or decay is assumed to happen continuously.

This activity further familiarizes students with the way continuously compounded models are typically written. It makes explicit the format of an equation used to represent such a model, which students will encounter more of in future lessons. Students also interpret the parameters of such equations in context and use the given structure to complete partially built equations. In this course, though, students are not assessed on generating a model to represent a situation characterized by continuous growth.

Only examples of growth situations are included in the activity. Decay situations are intentionally excluded so students don’t have to make sense of a negative factor in the exponent of an expression while also deciphering a new form for writing exponential functions. In synthesizing the activity, however, the teacher could choose to mention that in decay situations the \(r\) takes on a negative value to represent a percent decrease.

Arrange students in groups of 2. Give students a few minutes of quiet work time, and then ask them to discuss their responses with their partner. Follow with a whole-class discussion.

Some students may struggle relating the given exponential expression to the given situation. Draw their attention to what each part of the expression means. For example, 13 is the value of the expression when \(t = 0\). For the second question, the value when \(t = 0\) is the United States population, in millions, in 1964. What is this? (192). The coefficient of \(t\) is the continuous growth rate. What is the continuous growth rate predicted by the model? (It's 1.4% which corresponds to 0.014.)

Select students to share how they completed and interpreted the missing information, where possible drawing attention to the structure of the exponential model given. Invite students to discuss how this form is like and unlike the equations students have seen prior to this point (\(f(x) = a \boldcdot b^x\)). In particular, highlight that the growth factor, \(e^r\), is expressed in terms of both \(e\) and \(r\). This is different than growth at a rate of \(r\) per unit time which is expressed only in terms of \(r\) (as \(1+r\)).

This activity is optional. Use this activity to allow students to practice analyzing graphs of exponential functions involving \(e\) and setting an appropriate graphing window on their graphing technology in order to produce useful graphs.

Students have graphed exponential functions and analyzed the graphs to solve problems prior to this point. Though the functions here use base \(e\), the analytical work is fundamentally the same as that when the base is another number. Students will also have opportunities to graph exponential functions with base \(e\) later in the unit.

Provide access to devices that can run Desmos or other graphing technology.

Focus the discussion on how students set the graphing window and how they used the graphs to answer the questions.

Depending on the graphing technology used, students may use the following strategies:

• Visually estimate the value of the output given a certain input, or the input given an output.
• Use a tracing tool to trace the graph and identify the coordinates when the input or output has a certain value.
• To find an unknown input: graph a horizontal line at a given \(y\)-value (for example, \(y=100\), a doubled bacteria population), find the intersection of that line and the graph of the exponential function, and identify the \(x\)-value at that intersection.
• To find an unknown output: use a table or type something like \(b(10)\) after \(b\) was defined.

Some students may also notice that they could calculate the output value given an exponential equation and an input value, but had to rely on the graph to find an unknown input value because they were unsure how to solve for \(x\) when the equation has \(e\) as a base. Tell students that in a future lesson they will learn how to use logarithms to solve this type of equation algebraically similar to how they have used logarithms previously.