Lesson 12

The Number $e$

12.1: Matching Situations and Equations (5 minutes)

In this warm-up, students match equations with descriptions representing exponential functions in familiar contexts. When written in the form of $f(t) = a \boldcdot b^t$, all functions have the same value for the parameter $a$ and for the exponent. The goal is to prompt students to examine and interpret the different bases, setting the stage for thinking about $e$ as a base.

Match each equation to a situation it represents. Be prepared to explain how you know. Not all equations have a match.

$f(t) = 400 \boldcdot (0.5)^{0.1t}$

$g(t) = 400 \boldcdot (1.25)^{0.1t}$

$h(t) = 400 \boldcdot (0.75)^{0.1t}$

$j(t)=400 \boldcdot (2)^{10t}$

$k(t) = 400 \boldcdot (2)^{0.1t}$

A scientist begins an experiment with 400 bacteria in a petri dish. The population doubles every 10 hours. The function gives the number of bacteria $t$ hours since the experiment began.
A patient takes 400 mg of a medicine. The amount of medicine in her bloodstream decreases by 25% every 10 hours. The function gives the amount of medicine left in her bloodstream after $t$ hours of taking the medicine.
The half-life of a radioactive element is 10 years. There are 400 g of the element in a sample when it is first studied. The function gives the amount of the element remaining $t$ years later.
In a lake, the population of a species of fish is 400. The population is expected to grow by 25% in the next decade. The function gives the number of fish in the lake $t$ years after it was 400.

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Select students to share their responses and explanations for each description. If not brought up by a student, point out that all four examples of exponential equations show the same initial values and exponents but different bases and, as such, different growth factors. The numbers 0.5, 1.25, 0.75, and 2 were used. Tell students that, in this lesson, we will encounter another number that is used as a base in many contexts.

If time allows, invite students to write a situation to match $j$ and then select 2–3 students to share.

12.2: Notice and Wonder: Moldy Growth (10 minutes)

The purpose of this activity is to elicit the idea that $e$, while being a new symbol, is just a number and when that number is used as a base in an exponential expression, the function behaves just like those with other bases that represent exponential growth. While students may notice and wonder many things about these representations, the normal connections between them with regards to identifying the growth factor are the important discussion points.

Display the task statement, table, equation, and graph for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

A spot of mold is found on a basement wall. Its area is about 10 square centimeters. Here are three representations of a function that models how the mold is growing.

time (weeks)	area of mold (sq cm)
0	10
1	27
2	74
3	201
4	546

$a(t) = 10 \boldcdot e^t$

Coordinate plane, horizontal, time in weeks, 0 to 4 by 1, vertical area of mold in square centimeters, 0 to 300 by 50. A curve through points 0 comma 10, 1 comma 27, 2 comma 74 and 3 comma 201.

What do you notice? What do you wonder?

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Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

The function presented in the table is not, strictly speaking, exponential as the successive quotients have slightly different values. If students notice or wonder about this, tell them that the values have been rounded to the nearest whole number.

Then, tell students that $e$ is an important constant in mathematics. Its value is about 2.718. It is irrational, so it can’t be represented with a fraction, and its decimal representation never repeats or terminates. (Other examples of irrational numbers are $\pi$ and $\sqrt2$.) It is sometimes called Euler’s number, and is named $e$ after the 18th century mathematician Leonhard Euler.

Explain that $e$ is used by scientists, engineers, economists, and others. Tell students that they will learn much more about the many uses of $e$, its properties, and where it comes from in future courses. For now, it is enough to know that $e$ represents a number that is approximately 2.718.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. To help students share what they notice and wonder about the table and graph, provide sentence frames such as: “I notice that . . .”, “I wonder if . . . .”
Design Principle(s): Cultivate conversation

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of $e$. Place the symbol in the middle and surround by phrases and numbers that indicate the meaning, importance, and application of the symbol. For example, “2.718” and “used in science and engineering.”
Supports accessibility for: Conceptual processing; Language

12.3: $(1 + \text{tiny})^{\text{huge}}$ (20 minutes)

This activity serves as a brief and light introduction to how $e$ is visible in the behaviors of certain functions. The standards addressed in this course don’t require a deep understanding of the meaning of $e$, so the observations from this activity should not be assessed.

$e$ has a special connection to the expression $\left(1+\frac{1}{x}\right)^x$. Jumping to this expression would likely be an opaque process for most students, so they first investigate the range of behavior of its component pieces, for very small and very large positive values of $x$.

Entering increasingly larger or smaller values of $x$ in the functions and making observations about them is a chance to look for and express regularity through repeated reasoning (MP8).

Tell students that they are to analyze the behavior of some functions and see what observations could be made about $e$ along the way. Provide access to graphing technology and spreadsheet tools.

Arrange students in groups of 2. Tell students that they each should analyse the functions in different ways and then compare results. For example, one student could examine the functions numerically, using a table or a spreadsheet, while another student could examine them graphically.

Consider naming some tiny positive values of $x$ (for example, 0.1, 0.01, and 0.001) and large values of $x$ (for example, 100, 1,000, and 100,000) to students who may benefit from concrete examples.

Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer to capture their observations. Provide seven columns for students to organize their work. The first column label should be “function,” while the remaining six columns are divided into two sets; three for the three “tiny input” numbers and three for the “large input numbers.” Ask students to select three tiny input numbers and three large input numbers (either from the numbers given, or their own). Demonstrate sequencing the numbers for the column headers from left to right getting tinier, and from left to right getting larger. Explain to students that some may not need all six columns to draw conclusions. Students may especially want to capture observations and calculations for all six input values for $f(x)$ and $g(x)$.
Supports accessibility for: Language; Organization

Here are some functions. For each function, describe, in words, the outputs for very tiny, positive values of $x$ and for very large values of $x$.

$a(x) = 1^x$

$b(x) = \text-x$

$d(x) = \frac{1}{x}$

$f(x) = \left(\frac{1}{x}\right)^x$

$g(x) = \left(1 + \frac{1}{x}\right)^x$

$h(x) = e^{x}$

$k(x) = 1 + x$
Remember that $e \approx 2.718$. What does the function $g$ have to do with the number $e$?
What do you notice about the relationship between $h$ and $k$ for very small, positive values of $x$?

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Are you ready for more?

Complete the table to show the value of each expression to the nearest hundred-thousandth. Two entries have already been completed as an example.

$x$	$2^x$	$e^x$	$3^x$
0.1	1.07177	1.10517	$\phantom{1.11612}$
0.01
0.001
0.0001

What do you notice about the values in the table?

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Students may substitute $0$ for $x$ in order to see what happens for small positive values of $x$. Focus their attention on functions $d$, $f$, and $g$ where the $0$ does not belong to the domain. For the other functions, encourage students to look at a graph (using technology for example). Does the value when $x = 0$ help figure out what values each function takes for small positive values of $x$?

Invite students to share their observations for the first question and the supporting work (a table, a spreadsheet, or graphs) that led to these observations, if available.

Focus the conversation on the responses to the last two questions. If not already observed by students, highlight that:

The value of function $g$ gets closer and closer to $e$ as $x$ gets larger and larger. Display a graph representing $g(x)= \left(1 + \frac{1}{x}\right)^x$ and trace it out for large values of $x$. The curve essentially flattens out as $x$ increases.

Here is a graph of $g$ along with a graph of the horizontal line given by $y = e$.

As the input grows, the two graphs are difficult to distinguish. The function $g$ has a horizontal asymptote of $e$. (The exact value of the horizontal asymptote for $g$ is one way to define the number $e$.)
Point out that, when $x$ is a very large value, $\frac 1x$ is very small (as we saw in function $d$). So in the expression $\left(1 + \frac{1}{x}\right)^x$, we are expressing $e$ as $e \approx(1 + \text{tiny})^{\text{huge}}$. (It is critical that the "huge" number is the multiplicative inverse of the "small" number.)
The values of $h$ and $k$ are very close for very small, positive values of $x$. In other words, raising $e$ to an increasingly small positive exponent $x$ gives essentially the same values as adding 1 to $x$. Display a graph such as the following, or use graphing technology and zoom in toward smaller and smaller values of $x$.

Tell students that these behaviors turn out to be useful in mathematical and scientific applications, and they will learn a lot more about them in future math courses.

Conversing, Representing: MLR 7 Compare and Connect. Invite students to make connections between the various representations and approaches. Ask students compare how the number $e$ is represented in the table or graph of the function $g$. Listen for and amplify the language students use to describe how the value of $g$ approaches the number $e$ in the table or graph.
Design Principle(s): Maximize meta-awareness; Optimize output (for comparison)

Lesson Synthesis

To help students build a better sense for the constant $e$, consider discussing questions such as:

“How is $e$ like $\pi$?” (They are both are irrational numbers and cannot be represented as a fraction. If written as a decimal, the decimal numbers don’t repeat or terminate. They are both important in math.)
“How is $e$ different than $\pi$?” ($e$ is about 2.718 and $\pi$ is about 3.14. $\pi$ is the ratio of the circumference and the radius of a circle. It shows up a lot in geometry. We’re not quite sure yet how $e$ came about, but we saw it used in an exponential function.)
“In the moldy wall problem, we saw the mold growth represented with $a(t)=10 \boldcdot e^t$, where $t$ is time in weeks and $a(t)$ is the mold-covered area in square centimeters. How would you describe how fast the mold was growing to someone who doesn’t know about $e$?” (Each week, the area it covers is growing by a factor of about 2.7. It more than doubles, but doesn’t quite triple.)

12.4: Cool-down - What Did You Learn about $e$? (5 minutes)

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Student Lesson Summary

Scientists, economists, engineers, and others often use the number $e$ in their mathematical models. What is $e$?

$e$ is an important constant in mathematics, just like the constant $\pi$, which is important in geometry. The value of $e$ is approximately 2.718. Just like $\pi$, the number $e$ is irrational, so it can’t be represented as a fraction, and its decimal representation never repeats or terminates. The number is named after the 18th-century mathematician Leonhard Euler and is sometimes called Euler’s number.

$e$ has many useful properties and it arises in situations involving exponential growth or decay, so $e$ often appears in exponential functions. In upcoming lessons, we will work with functions that are expressed using $e$.

Lesson 12

12.1: Matching Situations and Equations (5 minutes)

Warm-up

Student Facing

Student Response

Activity Synthesis

12.2: Notice and Wonder: Moldy Growth (10 minutes)

Activity

Launch

Student Facing

Student Response

Activity Synthesis

12.3: $(1 + \text{tiny})^{\text{huge}}$ (20 minutes)

Activity

Launch

Student Facing

Student Response

Student Facing

Are you ready for more?

Student Response

Anticipated Misconceptions

Activity Synthesis

Lesson Synthesis

Lesson Synthesis

12.4: Cool-down - What Did You Learn about $e$? (5 minutes)

Cool-Down

Student Lesson Summary

Student Facing