# Engaging students: Introducing the number e

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Nada Al-Ghussain. Her topic, from Precalculus: introducing the number e. How can this topic be used in your students’ future courses in mathematics or science?

Not every student loves math, but almost all students use math in his or her advanced courses. Students in microbiology will use the number e, to calculate the number of bacteria that will grow on a plate during a specific time. Biology or pharmacology students hoping to go into the health field will be able to find the time it takes a drug to lose one-half of its pharmacologic activity. By knowing this they will be able to know when a drug expires. Students going into business and finance will take math classes that rely greatly on the number e. It will help them understand and be able to calculate continuous compound interest when needed. Students who do love the math will get to explore the relation of logarithms and exponentials and how they interrelate. As students move into calculus, they are introduced to derivatives and integrals. The number e is unique, since when the area of a region bounded by a hyperbola y= 1/x, the x-axis, and the vertical lines x=1 and x= e is 1. So a quick introduction to e in any level of studies, reminds the students that it is there to simplify our life! What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

In the late 16th century, a Scottish mathematician named John Napier was a great mind that introduced to the world decimal point and Napier’s bones, which simplified calculating large numbers. Napier by the early 17th century was finishing 20 years of developing logarithm theory and tables with base 1/e and constant 10^7. In doing this, multiplication computational time was cut tremendously in astronomy and navigation. Other mathematicians built on this to make lives easier (at least mathematically speaking!) and help develop the logarithmic system we use today.

Henry Briggs, an English mathematician saw the benefit of using base 10 instead of Napier’s base 1/e. Together Briggs and Napier revised the system to base 10, were Briggs published 30,000 natural numbers to 14 places [those from 1 to 20,000 and from 90,000 to 100,000]! Napier’s became known as the “natural logarithm” and Briggs as the “common logarithm”. This convinced Johann Kepler of the advantages of logarithms, which led him to discovery of the laws of planetary motions. Kepler’s reputation was instrumental in spreading the use of logarithms throughout Europe. Then no other than Isaac Newton used Kepler’s laws in discovering the law of gravity.

In the 18th century Swiss mathematician, Leonhard Euler, figured he would have less distraction after becoming blind. Euler’s interest in e stemmed from the need to calculate compounded interest on a sum of money. The limit for compounding interest is expressed by the constant e. So if you invest \$1 at a rate of interest of 100% a year and in interest is compounded continually, then you will have \$2.71828… at the end of the year. Euler helped show us many ways e can be used and in return published the constant e. It didn’t stop there but other mathematical symbols we use today like i, f(x), Σ, and the generalized acceptance of π are thanks to Euler. How can technology be used to effectively engage students with this topic?

Statistics and math used in the same sentence will make most students back hairs stand up! I would engage the students and ask them if they started a new job for one month only, would they rather get 1 million dollars or 1 penny doubled every day for a month? I would give the students a few minutes to contemplate the question, without using any calculators. Then I would take a toll of the number of the students’ choices for each one. I would show them a video regarding the question and idea of compound interest. Students will see how quickly a penny gets transformed into millions of dollars in a short time. Money and short time used in the same sentence will make students fully alert! I would then ask them another question, how many times do you need to fold a newspaper to get to the moon? As a class we would decide that the thickness is 0.001cm and the distance from the Earth to the moon would be given. I would give them some time to formulate a number and then take votes around the class, which should be correct. The video is then played which shows how high folding paper can go! This one helps them see the growth and compare it to the world around them. After the engaged, students are introduced to the number e and its roll in mathematics.

Money: watch until 2:35:

Paper:

References:

http://mathworld.wolfram.com/e.html

http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/

http://www.math.wichita.edu/history/men/euler.html

http://www.maa.org/publications/periodicals/convergence/john-napier-his-life-his-logs-and-his-bones-introduction

http://www.purplemath.com/modules/expofcns5.htm

http://ualr.edu/lasmoller/efacts.html