# 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 Kenna Kilbride. Her topic, from Precalculus: introducing the number e. How can this topic be used in your students’ future courses in mathematics or science?

Students will add on to this constant from calculus up to differential equations and even further. In Calculus I students use the number e to solve exponential functions and logarithm function. Calculus II uses the number e when computing integrals. In Complex Numbers you see the number e written as the Taylor series

$latex e^x = \displaystyle \sum_{n=0}^\infty \frac{x^n}{n!} Differential equations utilizes the number $e$ in $y(x) = Ce^x$. The number $e$ can be utilized in many other areas since it is considered to be a base of the natural logarithm. The number $e$ is also defined as: $e = \displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^x$ Also the number $e$ can be seen in the infinite series$latex e = \sum_{k=0}^\infty \frac{1}{k!}

The number e can be seen in many different areas of mathematics and with many different series and equations. Stirling’s approximation, Pippenger product, and Euler formula are just a few more examples of where you can see the number e.

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

http://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegralsCompTest.aspx What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

Introducing this constant can be a very hard thing for a teacher to do and using a word problem that involves a satellite that students can comprehend what they do in the sky will help.

A satellite has a radioisotope power supply. The power output in watts is given by the equation

P = 50e^(-t/250)

where t is the time in days and e is the base of natural logarithms.

Then when introducing, e, you can give them problems that they can easily solve without fully understanding what e is. Give them problems such as, how much power will be available in a year. The solution is:

P = 50e^(-365/250)

= 5Oe^(-1.46)

= 50 x 0.232

= 11 .6

Once e has been more formally introduced and the students can then become more familiar (this should only be added on when the students fully understand e) you can add onto this problem by giving them questions such as, what is the half-life of the power supply? Students must use natural log to solve this equation:

25 = 50e^(-t/250)

for t and obtain

– t/250 = ln O.5

= -0.693

t = 250 x 0.693

= 173 days

http://er.jsc.nasa.gov/seh/math49.html What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

John Napier was born in Scotland around 1550. Napier started attending St. Andrews University at the age of 13. After leaving St. Andrews without a degree he attended Cambridge University. Later he studied abroad, presumably in Paris. In 1614 Napier invented logarithms and later exponential expressions. Along with mathematics, Napier was interested in peace keeping and religion. Napier died on April 4, 1617 of gout.

Euler contributed to e, a mathematical constant. He was born 1707 in the town Basel of Switzerland. By the age of 16 he had earned a Master’s degree and in 1727 he applied for a position as a Physics professor at the University of Basel and was turned down. Due to extreme health problems by 1771 he had lost almost all of his vision. By the time of his death in 1783, the Academy of Sciences in Petersburg had received 500 of his works.

http://www.macs.hw.ac.uk/~greg/calculators/napier/great.html

http://www.pdmi.ras.ru/EIMI/EulerBio.html

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