# Lesson 12

The Number $e$

### Lesson Narrative

This lesson marks students’ first encounter with the number $$e$$. In the coming lessons, students will work with exponential functions that model real-life situations. Many such functions are expressed using $$e$$ as the base. Students are not expected to construct exponential functions with base $$e$$, and understanding $$e$$ in depth is beyond the scope of this course. Students are, however, expected to be able to interpret and graph such functions using technology. They will also return to logarithms as a way to solve exponential equations with base $$e$$, relating back to earlier work.

In the first activity, students notice the letter $$e$$ being used as the base of an exponential function with a growth factor of about 2.7 and learn some basic facts about the mathematical constant. Next, they investigate the behavior of some functions for certain values of input and observe some patterns emerging (MP8). They notice that a particular expression, $$\left(1 + \frac{1}{x}\right)^x$$, is closely related to the value $$e$$.

### Learning Goals

Teacher Facing

• Comprehend that $e$ represents a constant whose value is approximately 2.718.

### Student Facing

• Let’s learn about the number $$e$$.

### Required Preparation

Acquire devices that can run Desmos and GeoGebra (recommended) or other graphing and spreadsheet technology. It is ideal if each student has their own device. (Desmos and GeoGebra are available under Math Tools.)

### Student Facing

• I know that $e$ is an irrational constant, like $\pi$, that has a value of about 2.718.

### CCSS Standards

Building On

• $e$ (mathematical constant)
The number $$e$$ is an irrational number with an infinite decimal expansion that starts $$2.71828182845\ .\ .\ .$$, which is used in finance and science as the base for an exponential function.