Special exponential "e"

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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?

Bell work

10 minutes

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.



Discovering Euler's Number

15 minutes

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.


Euler's number and exponential functions

15 minutes

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.



5 minutes

I want to end the lesson by connecting what we did with logarithms to this lesson.  Students are given a prompt to find the inverse of y=e^x. I want students to have the inverse written as a logarithm with base e. The natural logarithm notation will be introduced tomorrow.