"Demystifying e" Day #1
Lesson 15 of 21
Objective: This lesson serves as the foundational lesson for the next several days. It provides the connection between the important mathematical constant 2.718… and exponential functions. e is both derived and defined through investigation in the lesson where students seemly “stumble upon” its overarching idea. More than many other lesson, it is extremely important that this lesson is teacher DRIVEN and student MOTIVATED. Due to the nature of the content, and the students relative inexperience with high level mathematical conversations – the teacher must be careful to steer the lesson in the intended direction. The PowerPoint and student notes guide are powerful tools for making sure this happens! Despite the lesson being teacher driven, it must also be STUDENT motivated. Through the scaffolding activities and analogies provided, the students will be engaged in deep conversations throughout much of the period.
The basis for this lesson is that the mathematical constant “e” is possibly the second most misunderstood and underappreciated constant in mathematics… second only to . When I have taught Geometry in the past, I do a similar lesson to this in an effort to expose students to the REAL definition and significance of pi – the ever present ratio between the circumference and the diameter of a circle. The students always leave fascinated at what they should already know! Many people believe that this understanding is harder to achieve with e, because they are fearful of finding the words and examples to make sense of the behind the scenes mathematics. For the sake of the students, however, this lesson works to achieve EXACTLY THAT before throwing them into the calculations or defining a natural logarithm! Without careful explanation, the same mistakes that are sometimes made when teaching can also be made when teaching e. Don’t miss an opportunity to turn students ON to mathematics, not turn them OFF!
When I first taught this lesson early in my career (last year), I felt inadequately prepared to help lead an in-depth investigation and analysis of e. Although I knew the mathematics, my concern was whether or not I was well versed and well-practiced enough to keep the students from being confused and frustrated! I hope that you find the rest of this narrative to give you the confidence you need to run a “college style” lesson. Although the primary tasks of the day are lecture and discussion, I believe that it comes at a perfect time in the framework of the content. I have found that it really is a refreshing day for students and teachers!
The lesson begins by having the students list a textbook (or internet) definition of e on a www.polleverwhere.com poll that I create. As the students respond, poll results are streamed live (anonymously) to the projector screen. All students can feel comfortable responded to the question, regardless of their ability level.
After 4-5 minutes, we take time to read over the responses with the class. When I am doing this, I make sure to read the definition in a “boring” tone, and in a way that shares minimal insight into the real mathematical significance at hand. Typical responses look like the following:
The mathematical constant e is the base of the natural logarithm.
The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718281828459.
Quickly, I tell the students to snap out of it – and of course I am not going to let them off the hook without a clear and proper mathematical understanding of the SIGNIFICANCE of e! I remind them of the time that we put into pi in our Geometry class, and that they had all better be able to give a MEANINGFUL definition of pi, not just 3.14. There is nothing significant about 3.14, and the same is true about 2.718. What is significant, however, are the ideas presented on the way to the number’s derivation.
Hence, the lesson’s title: “Demystifying e”
Additional Notes: PLEASE check out the slide show! It really provides a good illustration for the flow of the lesson:
1) Slide #4’s “student ID” analogy really hit home with my kids. I ask them if they could picture an environment where I called them “#134563” instead of their name. Their school ID number is insignificant in terms of WHO they really are. The same can be said about pi and e.
2) Perhaps you can think of your own way to kick off the lesson, but using the analogy to pi really helped to provide the students with the direction and goal of the lesson. It also allowed us to further define the lesson’s title, demystifying.
Summing it all Up
Please, PLEASE take time to look through the PowerPoint if you have not already. It is a critical piece to this lesson. The lesson summary is located on slides #21-24. It engages the students in summary conversations at the following levels:
1) Mathematical: the definition is revisited yet again, and a connection is made to a future concept that the students will study in later math classes – a limit.
2) Conceptual: like a “speed limit” for continuously compounded interest, and a frame of reference for us to utilize
3) Practical: So, if we start with $1.00 and compound continuously at 100% return we get 1e. If we start with $2.00, we get 2e. If we start with $11.79, we get 11.79e.