The Number e
Lesson 14 of 14
Objective: SWBAT approximate the value of the number e using a limiting process. SWBAT use natural logarithms to solve exponential equations.
These have the same format as the two from the previous lesson, so the students should be looking for improvement. Be sure to provide a couple of minutes in between for reflection and informal discussion of strategies. Also, be sure to emphasize the importance of progress over perfection.
What is Interest?
I like to start with a conversation about savings accounts and interest payments. I'll ask these questions with the aim of making sure that everyone has an intuitive feel for what interest payments are and how compound interest differs from simple interest. Students will volunteer answers and I will keep the discussion going until I'm satisfied that the class understands the answers to my questions.
- If I borrow $100, and you want me to pay it back with 10% interest, what does that mean?
- I owe $110.
- Simple Interest: I borrow $100 from you and you will change me 10% interest on the original amount each week until I pay you back. What does this mean? What type of equation would model this situation (if I make no payments)? (MP 4)
- I owe $110, $120, $130, ... depending on how many weeks it takes. Linear equation.
- Compound Interest: I borrow $100 from you and you charge me 10% interest compounded weekly until I pay you back. What does this mean? Can we use a linear equation in this case?
- I owe $110, $121, $133.10, ... depending on how many weeks it takes. Not linear, but exponential.
- Investment: If I invest $100 in a bank that pays 2% interest compounded monthly, what does this mean for me?
- I earn a little bit each month. The value is $102, $104.04, $106.12, .... as the months go by. Exponential again.
Now the class is ready for a really strange scenario!
Hand out The Number e and ask the students to take a minute to read the directions on their own. They may begin solving the problems as soon as they're ready. We'll talk about the history of this problem in the next section of the lesson.
After a couple of minutes of individual time (during which I'll be checking for understanding), I'll let the class begin working in small groups. The most important thing at this stage is the relationship between the interest rate and the number of times interest is compounded. The pattern should be apparent but make sure everyone sees that the full 100% is divided by the number of times interest is compounded. (MP 8) (I've found in practice that most students see the pattern on their own, but I do have to point it every once in a while.)
Once this is understood, students should be able to complete the table fairly quickly and most will happily assume that the value approaches a limit.
To help everyone keep pace, I will check solutions with the class periodically. I will also project a copy of the chart on the board and ask students to begin filling it in after about 10 minutes. We'll use this table to reference during the discussion later.
I have found that many students will write their equations as sums like this: 1 + 0.25 + 0.3125 + ... This will lead to big problems when they try to write a general equation, or when the number of times interest is compounded grows very large. They may need some help seeing that adding 25% is the same as multiplying by 125%. The earlier you can catch this, the better. Check out this video for another common difficulty.
Defining the Number e
I'll begin the summary discussion/lecture by sharing a graph like this one with the class (the GeoGebra file is available as a resource). Along with the table that has already been completed, this graph will help everyone to see that it is reasonable to say that the expression (1 + 1/n)^n approaches a limiting value as n grows infinitely large.
With this, I'll write the definition of the number e on the board and explain how it arises from what the students have just done. See my Teacher's Notes on the second page for an outline of the mini-lecture I'll deliver. The points included are as follows:
- the definition of e as a limit
- the history of the discovery of the number e
- the use of e for continuous exponential growth/decay
- the general form of the exponential function
- the natural logarithm
The number e is so important - as is the natural logarithm - that our students need these definitions. Unfortunately, much of its important cannot be seen clearly until calculus, so my students simply have to take it on faith. That's the reason for a lecture on this topic rather than a guided discovery.
If time permits, I'll start diving into some of the more wonderful and surprising properties of the number e and the natural logarithm (e.g. the logarithmic spiral in nature, e as the limit of sequences, e as a continued fraction, etc.). This is also a great place to bring up some of the applications of natural exponential growth, such as radioactive decay, Newton's Law of Cooling, and ideal population growth.
Class will end with this conversation, and I don't usually assign homework except to complete any unfinished parts of today's worksheet.
A great online resource for the teacher or students is this post at Better Explained.