# 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 Peter Buhler. His topic, from Precalculus: graphing an ellipse.

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

One project that could be assigned to students during the unit on conic sections could be to challenge students to either find or make an ellipse. This could be with a household object, a computer simulated object, or it could be something such as the movement of the planets around the sun. Students would be expected to visually display their object(s) of choice, as well as provide an equation for the ellipse. For example, if the student chose to use a deflated basketball or football, students would use the actual units found when measuring the object and then create an equation for that ellipse. Of course, students would also be expected to graph the ellipse using the appropriate equation, and then check the graph with the actual object (if possible). This project would allow students to be creative in choosing something of ellipse form, and would allow them to further explore the graphing and equation-building of an ellipse.

How can this topic be used in your students’ future courses in mathematics or science?

While graphing an ellipse is a topic within the Pre-Calculus curriculum, it also has applications within other topics as well. One of these is the unit circle, which is also taught in most Pre-Calculus courses. The unit circle is simply an ellipse where both major and minor axis are of length 1, as well as the center at (0,0). Students can be encouraged to draw comparisons between the two topics. Not only can they rewrite the equation of an ellipse to fit the unit circle, but students can also use the distance formula to calculate sine and cosine values on the unit circle. They can then use the distance formula on various forms of an ellipse, and compare and contrast between the two.

Later on in a students’ mathematical career, some students may encounter ellipse used in three dimensions in Calculus III, in an engineering course, or even in an astronomy course. Ellipses have many applications, and students may benefit from you (as the teacher) perhaps mentioning some of these applications when going over the unit on conic sections.

How has this topic appeared in high culture?

One particularly intriguing application of an ellipse (among many applications) is in the design of a whispering gallery. This is essentially a piece of architecture that is designed in the shape of an ellipse so that when someone is standing at one focus of the ellipse, they can clearly hear someone whispering from the exact location of the other focus. Some of examples of these “whispering rooms include St. Paul’s Cathedral, the Echo Wall in Beijing, and in the U.S. Capitol building. It has been commonly noted that President John Quincy Adams would eavesdrop on others while standing in the Capitol, simply due to the physics of sound waves traveling inside an ellipse shaped building.

On a more personal business, I can remember multiple visits to the Science Museum in Fair Park, where various forms of sciences were displayed in formats that children (and adults!) could interact with. There was one exhibit that was set up for several years that also incorporated this ellipse-shaped architecture. I remember it clearly, due to the fact that I was so fascinated with how I could stand 30 yards from someone and be able to hear their whisper clearly. This could also be a class project or even a class trip that would allow students to hypothesize why this works the way it does. It can be noted that this would work for both Physics and Math classes, as it has applications to both.

# 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 Marissa Arevalo. Her topic, from Precalculus: graphing an ellipse.

How have different cultures throughout time used this topic in their society?

In a philosophy paper, I wrote a about the usage of ellipses that applied not only to the field of mathematics but theology and science (more specifically astronomy), and the implications it had throughout time. Throughout centuries, mankind has argued over the ways of the universe and whether or not we are the center of that universe or if something else is. From the times of Ancient Greece, Aristotle believed that the center of our universe revolved around a form of unchanging matter that did not obey the laws of the planet earth. Ptolemy rejected this idea and created a model of a universe centered around Earth itself where the other planets revolved around us, but he could not answer as to why the planetary orbits did not follow a circular path. Later on in the 14th and 15th centuries Copernicus and Galileo respectively argued for a system that orbited the sun rather than the Earth. This idea went against the beliefs of the church and their research caused Galileo to be held into persecution for his radical ideas (Copernicus died before any due harm came to him). It was not until Johannes Kepler, under the tutelage of his teacher Tycho Brahe, observed the motion of the planet Mars and noted that the path did not actually follow a circular path but an elliptical one. His findings disproved his teacher, who was a firm advocate of the church and believed in a geocentric model, showing that the planets were centered around the sun. Sir Isaac Newton’s Laws of Gravity later proved Kepler’s theories, and to this day are known as Kepler’s Laws of Planetary Motion.

We utilize these laws and other properties in order to define what it means to be a planet, therefore a planet:

1. Must be round in physical shape
2. Must have an elliptical path around the sun
3. Must be able to clear anything that comes into its orbital path

These properties defined all of our planets, except Pluto, who it was discovered to be smaller than other things that existed in its orbit in the Kuiper Belt and therefore cannot have the third property. While the orbital pattern of Pluto followed the guidelines of the other planets (though with a greater eccentricity), Pluto was too small, therefore removing it off the list of planets in the solar system in 2006 and was defined as a “dwarf planet”. While the students of this time may not relate to the surprise of the reclassification of Pluto in our solar system, it is still relatable to today’s society as this long debate of the way planets move and how our universe was created greatly impacts science even today as we make new discoveries over other celestial bodies in our universe.

How can technology (YouTube, Khan Academy [khanacademy.org]. Vi Hart, Geometers, graphing calculators, etc.) be used to effectively engage students with this topic?

A website that can be utilized for students to get more involved in their own learning would be Gizmos where the students can be given a small exploration sheet in which they can compare the graph of ellipse to its equation and what exactly affects the shape of the ellipse as different aspects are altered. The students can also manipulate the graph and watch the standard form of the equation change over time. The site allows the  student to also see the pythagorean and geometric relationships and definitions of an ellipse as the equation is altered. One very important key feature on the exploration of the geometric definition is that the student is able to plot the purple point that moves along the edge of the figure in different locations to show the relationship between the lengths of foci from the edge. The only downside may be that while the teacher can use the site for a short free trial, they may have to make payments in order to continue using it. Desmos is another website that can graph ellipse equations, but it does not provide the ability to see the geometric definition applied to the graph of the function.

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

Another idea that would have the students thinking about the geometric concepts surrounding the properties of an ellipse would be for the teacher to have worksheets in which the students would show the representation of where an ellipse could be formed in the cutting of double-napped cone with a plane. The students could lead discussions in their own ideas and how an ellipse, hyperbola, parabola, and circle are created if you literally sliced the cones into pieces. The teacher could have either a physical model of the cones or have the students create the physical model of the cones with play-doh and cut the cones with cardboard/plastic-wear/dental floss (preferred) and describe the shape that was created in by the cuts made. (Another idea is to make the cones with Rice Krispies or scones and jam/chocolate) The good thing about this play-doh is approximately 50 cents at Wal-Mart and provide a nice way for students to make mistakes and restart without having to clean that big of a mess up. The students will be more involved in the material if they are able to create physical models and form their own ideas on things that many teachers do not address in their lessons. This is coming from personal experience of not knowing certain geometric properties of conic sections until taking college courses.

References:

https://www.desmos.com/calculator

https://www.pinterest.com/pin/480759328950528032/

https://www.pinterest.com/pin/343540277799331864/

https://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=132

Cain, F. (2012). Why Pluto is No Longer a Planet – Universe Today. Retrieved March 22, 2016, from http://www.universetoday.com/13573/why-pluto-is-no-longer-a-planet/

Helden, A. V. (2016, February 17). Galileo. Retrieved March 22, 2016, from http://

http://www.britannica.com/biography/Galileo-Galilei

Jones, A. R. (n.d.). Ptolemaic system. Retrieved March 22, 2016, from http://

http://www.britannica.com/topic/Ptolemaic-system

Leveillee, N. P. (2011). Copernicus, Galileo, and the Church: Science in a Religious World.

Student Pulse, 3(5), 1-2. Retrieved March 15, 2016, from http://www.studentpulse.com/

articles/533/copernicus-galileo-and-the-church-science-in-a-religious-world

Rosenburg, M. (2015, April 22). Scientiflix. Retrieved March 22, 2016, from http://

scientiflix.com/post/117082918519/keplers-first-law-of-motion-elliptical-orbits

Simmons, B. (2016, February 21). Mathwords: Foci of an Ellipse. Retrieved March 22, 2016,

The Universe of Aristotle and Ptolemy. (n.d.). Retrieved March 22, 2016, from http://

Westman, R. S. (2016, February 21). Johannes Kepler. Retrieved March 22, 2016, from http://

http://www.britannica.com/biography/Johannes-Kepler

# Math Maps The Island of Utopia

Under the category of “Somebody Had To Figure It Out,” Dr. Andrew Simoson of King University (Bristol, Tennessee) used calculus to determine the shape of the island of Utopia in the 500-year-old book by Sir Thomas More based on the description of island given in the book’s introduction.

Paper by Dr. Simoson: http://archive.bridgesmathart.org/2016/bridges2016-65.html

# 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 Kristin Ambrose. Her topic, from Precalculus: finding the foci of an ellipse.

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

In order to help my students visualize what foci are and the role they play in ellipses, I could do an activity that involves my students constructing ellipses given the foci. This can be done with two thumbtacks, string (tied into a loop), paper, and a pencil. What you do is place a piece of paper on top of a cork board, then stick two thumbtacks into the board and put a loop of string around them. Then take a pencil and pull the string tight, so that it makes a triangle. Then draw an ellipse by moving the pencil around the two thumbtacks, while keeping the string pulled tight to make a triangle shape. The picture below depicts how the activity should work.

I would give my students time to change the distance between the thumbtacks and create other ellipses, so that they could see how the distance between the two thumbtacks affects the shape of the ellipse. In keeping with the style of ‘discovery’ based learning, only after my students had created a few different ellipses would I explain that the thumbtacks are actually the ‘foci’ of the ellipse. I think this activity would help my students have a better visual of what foci actually are and how they affect the shape of ellipses. It would also help my students to understand why the sum of the distance between each foci and any point on the ellipse is always constant. I believe this would be a good segue into discussing how to find the foci of an ellipse.

How can this topic be used in your students’ future courses in mathematics and science?

Ellipses tend to come up in topics like Physics and Astronomy. Specifically in Astronomy, ellipses become important when learning about orbits. An orbiting satellite follows an elliptical shape around an object called the primary. The primary simply means the body being orbited and is typically located at one of the two foci of an ellipse. A good website that describes this phenomenon is http://www.braeunig.us/space/orbmech.htm. This website explains different types of orbits and how they relate to different conic sections, ellipses being one of them. In our solar system, the Earth orbits the sun, with the sun lying at one of the foci on the ellipse. In elliptical orbits, the center of mass is located at a focus of the ellipse, but since the sun contains most of the mass in our solar system, the center of mass is located almost at the sun; therefore the planets orbit the sun. Below is an illustration of this concept.

How can technology be used to effectively engage students with this topic?

A really neat tool to use in mathematics is a computer application called GeoGebra. It is free to download and useful for a lot of mathematical topics. For the topic of foci and ellipses, I would have students create an ellipse in GeoGebra using the ellipse tool. Once the ellipse is created, students can grab the foci points and pull them around to change the shape of the ellipse. Students can also grab the point ‘C’ and move it around to change the shape of the ellipse. The nice thing about GeoGebra is that not only does it show the shapes and points on the graph it also states the coordinates of the points in the ‘Algebra’ section. As students are exploring the different ways they can change the shape of the ellipse, they can also see how the coordinates change. On my GeoGebra ellipse, I also added a point ‘D’ which is the center of the ellipse. I created this point by typing D = (A+B)/2 in the ‘input’ section. Once ‘D’ is created, as students move the foci around, the location of ‘D’ will change as well, so students can see how the center of the ellipse and the location of the foci are interconnected. I think this tool would be a great way to get started on the topic of how to find foci, and it helps students to visualize how the shape of the ellipse, the foci, and the center of the ellipse are all interconnected. Below are some pictures of different ellipses I created in GeoGebra.

# 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 Donna House. Her topic, from Precalculus: graphing an ellipse.

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

A great hands-on activity for learning about an ellipse is created with some cardboard, a string, some tape, a yarn needle (or something to make a hole in the cardboard), and a marker.

To create the “ellipse boards,” take a piece of cardboard about one foot by one foot. You do not have to use a square piece and it can be larger or smaller. Just make certain the cardboard is large enough for the graph to be clearly seen. (I prefer white cardboard because it is easier to see the marks, but regular cardboard will also work if you use dark markers.)

Next, using the marker, make two marks for the foci. Thread the string (or yarn) through the yarn needle and poke a hole through one of the foci, pulling the string to the back side of the cardboard. Tie a knot in the string and tape it to the back of the board.

Now, thread the other end of the string through the yarn needle and poke a hole through the other focus. Decide how long the string needs to be to create a nice ellipse. (Remember the string must be 2a long – whatever length that is. Unless you really want the ellipse to be a certain size, the length of the string can vary. The farther apart the foci are, the more elongated the ellipse will be. This can also lead to a discussion about what happens to the shape of the ellipse as the foci get very close to each other!) Make certain the drawing will not fall off the edge of the board. Then tie a knot in this end of the string and tape it down. Each ellipse board will have a different sized ellipse unless you VERY carefully measure the foci and the string. I think having different sizes is better (and much easier to do) and shows the students that the formula for an ellipse works. Now the boards are ready for the students! (The students can put together their ellipse boards in class or you can have them pre-made to save time.)

The fun part is the actual drawing of the ellipse. This, however, is not as easy as it looks! To draw the ellipse, use the marker to stretch the string taut and let the string guide your drawing. Be sure to draw one before class so you will be able to give the students suggestions as they draw their own ellipses.

On their boards, the students can find the center, draw the major and minor axes, can find the vertices, and can easily see that the foci are on the major axis. Using the string, you can prove that the sum of the distances from any point on the ellipse to each of the foci is always 2a, and, using the Pythagorean Theorem, the students can see how to find the foci.

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

Since an ellipse is created when a cylinder is cut at an angle, ellipses are commonly encountered in construction. An example is creating a right angle while joining two pipes to build the corner of a fence. One joining method is to cut each pipe at a 45° angle then weld them together. Students could be asked to determine the length of the major and minor axes of the resulting ellipse when a 2” diameter pipe is cut at a 45° angle.

This same idea is used to make holes in walls or tile for some light fixtures, plumbing fixtures (like shower heads), vent pipes, etc.

I also found the following class project. This could be done in small groups by giving each group the main problem and letting them brainstorm to come up with the solution. I think this would be wonderful to stimulate creativity in the classroom.

How can technology be used to effectively engage students with this topic?

To engage the students, begin by showing the first 3 1/2 minutes or so of this video from YouTube:

Note that the doctor actually touches the peppermint while the sound waves are on!

But what does this have to do with an ellipse?

A unique characteristic of the ellipse is that shockwaves emitted from one focus will
reflect off the ellipse and go through the other focus. Using this characteristic, medical engineers have created a device called a lithotripter (as shown in the video) which can break up kidney and gall stones with minimal damage to the surrounding tissue. This eliminates the need for traditional surgery. Mathematics continues to make life easier!

As illustrated in the diagram above, when an energy ray reflects off a surface, the angle of incidence is equal to the angle of reflection.

Here is a short article explaining how the medical device works. (The above illustration comes from this article.) Using the computer, project the article onto the screen to show to the class.

http://mathcentral.uregina.ca/beyond/articles/Lithotripsy/lithotripsy1.html

This not only shows how technology can be used to engage students, it also shows how this topic is used in technology!

# 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.

# 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.

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

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.

What is Halley’s Comet?

How has this topic appeared in high culture?

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