##
* *Reflection: Discourse and Questioning
Special exponential "e" - Section 3: Euler's number and exponential functions

Students showed a lot of conviction when they were organizing the functions. Here are 2 examples of organizing functions.

The first class organized by looking at the term with exponents. The students separated the functions with **e** then put a group together that had x without an exponent and finally a group where x was raised to a power. Some of the students in this class disagreed and wanted to put the 2 rational expression together but were convinced by the class to organize the data this way.

Another class organized more like I expected. They stated that the A=P**e**^rt was different because it was the only function that had "multiple variables." At first the students did not know where they wanted the exponential y=**e**^x equation but a student said the exponent is x so it should go with the other exponential functions.

Some students did not want to put the linear equation with the other polynomial equations at first but a student explained to the class that the linear equation had an exponent of 1 on the x. This allowed the class to agree about the placement of the linear equation.

The students did struggle with this activity at first. They expected me to tell them how to organize the data. I reminded them that there are many ways to classify equations and as long as the class could defend the placement then the classification was correct.

In one class a students made the comment "we don't all think alike so we can see different ways to organize the equations."

*Discourse and Questioning: Organizing Functions*

# Special exponential "e"

Lesson 2 of 11

## Objective: SWBAT determine how Euler's number appears in different contexts.

## Big Idea: How can a number be divided so that raising the part to the power is was divided by gives you the largest product?

*45 minutes*

#### Bell work

*10 min*

Today we will focus on the number **e, **Euler's Number. I have directions for students to follow on the board. I expect that some students struggled with An Interesting Exploration of Numbers last night. Working in pairs today in class, students can help each other understand the questions. I give students about 5 minutes to work on the problems.

Now that students have discussed the worksheet in pairs I ask students to share their answers for question. The answers may be different since many students will not consider dividing the number by a decimal. If a pair puts up an answer with decimals the other students will start adjusting their results.

*expand content*

#### Discovering Euler's Number

*15 min*

As students start adjusting their results to include decimals I ask, "could I make a rule that would make our work more easier?"

Students discuss how to write a formula. Possible formulas are shared with the class along with justification for the formula. The class then decides which formula is the best. This formula is put into the calculator. Since we need to know the size of the piece, we also put 20/x into the calculator.

We look at the answer when we use whole numbers. The students notice that the answer is between 7 and 8. "How can we find decimals between 7 and 8?" We change the table settings to use a step of 0.1. Once we see a better answer we move to a step of 0.01. Students are seeing larger products.

**See if you can get a larger product**

Students work in pairs to find the largest product. I move around the room helping students that are confused or struggling with their calculators. I usually need to show students how to change the increment for the table. I may also need to remind students to change the start value of the table.

After about 2-3 minutes I ask students for the largest product and the value of 20/x.

Now that the class has seen how to work a problem, pairs are given different numbers to determine the largest product. Students find the largest product for their number and share their results in a table.

As students put results on the board students begin to see that the n/x column has answers that are close to or equal to each other. When students make comments about this I say, **I wonder if this is something special? **Of course students realize that it has to be or I would not make the comment. I sometimes have students that realize this number is ** e**. I ask:

**What do you mean e?****Do you know the mathematical name for e?**I explain that this number is called Euler's number because he discovered that this number kept coming up in his research.**Have you seen this number in formulas?**Some students remember the continuous interest formula which I put on the board.

*expand content*

I put several functions on the board. Groups are asked sort the functions by the type. Since I want to see how the groups separate the functions, I do not give students a list of functions to use. After a couple of minutes I ask groups what categories they used and how they place the functions. I will ask "**What is the difference between the polynomial functions and the exponential functions?"** Some groups may have struggled with y=e^x. **Is e a real number or a variable? So what type of function is y=e^x?**

To help see that the graph of y=**e**^x is an exponential function I have students graph y=**e**^x. For students that are visual learners seeing the graph helps the students understand the y=**e**^x is an exponential function. I now ask students to work in their groups and complete the following:

**Determine the key features of y=e^x (domain, range, intercepts, asymptotes, end behavior)****Compare the key features of y=e^x to the features of y=2^x****Which function is growing faster e^x or 2^x?****How can we determine which exponential function is growing faster without looking at the graphs?**

As students work I am informally assessing groups on understanding of the key features as well as how they explain the last question. I question students that seem to be confused. I am also identifying students who have good reasons or unusual explanations for the last question these students will share their responses with the class.

Once I feel that most groups have completed the task (about 4-5 minutes), I ask students to share their key features on the board. We discuss the second question. I work to have students realize that the key features are the same. "Why are the key features the same?" "Do all exponential functions have the same key features?"

We finally discuss how to determine which function is growing faster.

#### Resources

*expand content*

#### Resources

*expand content*

##### Similar Lessons

###### Comparing Growth Models, Day 1

*Favorites(2)*

*Resources(20)*

Environment: Suburban

###### The Skyscraper Problem

*Favorites(10)*

*Resources(14)*

Environment: Suburban

###### Graphing Linear Functions Using Given Information

*Favorites(30)*

*Resources(17)*

Environment: Urban

- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: The graphs of logarithmic and exponential functions
- LESSON 2: Special exponential "e"
- LESSON 3: Evaluating exponential and logarithms
- LESSON 4: Expanding and Condensing logarithms
- LESSON 5: Solving Exponential and Logarithmic Equations
- LESSON 6: Find intercept by solving equations
- LESSON 7: Change of Base Formula
- LESSON 8: Survey and Review Day
- LESSON 9: How do we use Logarithms and Exponentials
- LESSON 10: Review exponential and logarithmic functions
- LESSON 11: Exponential and Logarithm Assessment