# 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 Diana Calderon. Her topic, from Precalculus: introducing the number $e$.

How could you as a teacher create an activity or project that involves your topic?

– A project that I would want my students to work on that would introduce the number $e$ would be with having a weeklong project, assuming it is a block schedule, to allow the students to think about compound interest. the reason why we would use the compound interest formula to show $e$ is because, “It turns out that compounding weekly barely yields any more money than compounding monthly and at higher values of $n$, it gets closer and closer to what we recognize as the number $e$” The project would be about buying a car, the students would get to choose the car that they want, research multiple car dealerships, and they must figure out the calculations for compound interest in 24 months, 36 months, 48 months, 60 months and 72 months. For their final product they must have a picture/drawing of the car they chose to purchase, as well as choose the number of months they would like to finance for and the dealership they will purchase form. Finally, they must turn in a separate sheet with the calculations for the other months of finance they did not choose and why they chose not to choose them.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

– The number or better known as Euler’s number is a very important number in mathematics. Leonhard Euler was one of the greatest Swiss mathematicians from the 18th century. Although Euler was born in Switzerland, he spent much of his time in Russia and in Berlin. Euler’s father was great friends with Johan Bernoulli, who then became one of the most influential people in Euler’s life. Euler was also one who contributed to “ the mathematical notation in use today, such as the notation $f(x)$ to describe a function and the modern notation for the trigonometric functions”. Not only did Euler contribute to math, “He is also widely remembered for his contributions in mechanics, fluid dynamics, optics, astronomy, and music.” Euler was such an amazing mathematician that other mathematicians talked very highly of him such as Pierre-Simon Laplace who expressed how Euler is important in mathematics, “‘Read Euler, read Euler, he is master of us all’”

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?
– This video would be great to show to students to see how the number e is applied in different ways. The video starts off by talking about how we get $e$, “mathematically it is just what you get when you calculate 1 + (1/1000000)^1000000= 2.718 ≈ e and as the number gets bigger, you get Euler’s number, e=lim n→∞ f(n) (1+1/n)^n.” This is a really good video to show because the YouTuber talks about how when he was learning about the number e, he thought that it would never show up and then later realized that the better question was, when doesn’t it show up? He then proceeds to talk about how if you’re in high school then you start talking about it when it comes to compound interest, he then proceeds to give an example, “imagine you put \$1 in a bank that pays out 100% interest per year, that means after one year you’ll have two dollars but that’s only if th interest compounds once a year. If instead it compounds twice a year you get 50% after 6 months and another 50% after 6 more months.”, and so on, he explains up to daily and compounding every second, nanosecond and so on, the amount in that persons bank would become \$e (\$2.1718). He then gives a real-world examples of probability with the number e. I would stop this video at 3:09 because that would give enough insight to the students about other applications of the number e and why it useful for them to learn it and not just think about it as a button in their calculator.

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