Engaging students: Graphing an ellipse

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 Samantha Smith. Her topic, from Precalculus: graphing an ellipse.

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How has this topic appeared in pop culture?

Football is America’s favorite sport. There is practically a holiday for it: Super Bowl Sunday. I do not think students realize how much math is actually involved in the game of football, from statistics, to yards, the stadium and even the football itself. The video link below explores the shape of the football and of what importance the shape is. As you can see in the picture below, a 2D look of the football shows us that it is in the shape of an ellipse.

 

The video further explains how the 3D shape (Prolate Spheroid) spins in the air and is aerodynamic. Also, since it is not spherical, it is very unpredictable when it hits the ground. The football can easily change directions at a moments notice. This video is a really cool introduction to graphing an ellipse; it shows what the shape does in the real world. Students could even figure out a graph to represent a football. Overall, this is just a way to engage students in something that they are interested in.

football

https://www.nbclearn.com/nfl/cuecard/50824 (Geometric Shapes –Spheres, Ellipses, & Prolate Speroids)

 

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

Halley’s Comet has been observed since at least 240 B.C. It could be labeled as the most well-known comet. The comet is named after one of Isaac Newton’ friends, Edmond Halley. Halley worked closely with Newton and used Newton’s laws to calculate how gravitational fields effected comets. Up until this point in history, it was believed that comets traveled in a straight path, passing the Earth only once. Halley discovered that a comet observed in 1682 followed the same path as a comet observed in 1607 and 1531. He predicted the comet would return in 76 years, and it did. Halley’s Comet was last seen in 1986 so, according to Halley’s calculations, it will reappear in 2061.
Halley’s Comet has an elliptical orbit around the sun. It gets as close to the sun as the Earth and as far away from the sun as Pluto. This is an example of how ellipses appear in nature. We could also look at the elliptical orbits of the different planets around the sun. Students have grown up hearing about Newton’s Laws, but this is an actual event that supported and developed those laws in relation to ellipses.

http://www.universetoday.com/48991/halleys-comet/

halley

 

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How has this topic appeared in high culture?

whisperinggallery

Through my research on ellipses, the coolest application I found is Statuary Hall (the Whispering Gallery) in our nation’s capital. The Hall was constructed in the shape of an ellipse. It is said that if you stand at one focal point of the ellipse, you can hear someone whispering across the room at the other focal point because of the acoustical properties of the elliptical shape. The YouTube video below illustrates this phenomena. The gallery used to be a meeting place of the House of Representatives. According to legend, it was John Quincy Adams that discovered the room’s sound properties. He placed his desk at a focus so he could easily hear conversations across the room.

The first link below is a problem students can work out after transitioning from the story of the hall. Given the dimensions of the room, students find the equation of the ellipse that models the room, the foci of the ellipse, and the area of the ellipse. This one topic can cover multiple applications of the elliptical form of Statuary Hall.

Click to access PreAP-PreCal-Log-6.3.pdf

http://www.pleacher.com/mp/mlessons/calculus/appellip.html

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