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.

green lineHow 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.

 

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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.

 

 

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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.

Sources:
http://www.twc.edu/twcnow/blogs/student/10179/fun-fact-capitol-hill
http://thedistrict.com/sightseeing/other-washington-d-c-attractions/u-s-capitol/

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.

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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.

 

 

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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.

 

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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://illuminations.nctm.org/uploadedFiles/Content/Lessons/Resources/9-12/CuttingConics-AS.pdf

 

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

 

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

 

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

 

http://mste.illinois.edu/courses/ci399TSMsu03/folders/jmpeter1/Daily%20Assignments/Conic%20Sections/Ellipse%20Paper%20Folding(CI399).html

 

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,

from http://www.mathwords.com/f/foci_ellipse.htm

 

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

csep10.phys.utk.edu/astr161/lect/retrograde/aristotle.html

 

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

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

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.

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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.

ellipse3

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.

 

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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.

ellipse4

 

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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.

ellipse5 ellipse6

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.

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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.

ellipse1

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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.

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

https://www.youtube.com/watch?v=Yl8Qy79aLk0

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!

ellipse2

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.

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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.

http://www.quia.com/files/quia/users/porr06/PreAP-PreCal-Log-6.3.pdf

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

Engaging students: Graphing a hyperbola

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 Rebekah Bennett. Her topic, from Precalculus: graphing a hyperbola.

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Hyperbolas are one of the hardest things to find within the real world. Relating to students, the hyperbola is popularly known as the Hurley symbol; A widely known surf symbol that is now branded on clothes and surf boards. It is also used widely in designs to create patterns on large carpets or flooring. They can also be used when building houses to make sure that a curve on the exterior or interior of the house is mirrored exactly how the buyer wants. Hyperbolas can be found when building graphics for games such as the game roller coaster tycoon. This is a game where several different graphics must be formed so that any type of roller coaster can be created. Also, when playing the wii or xbox Kinect, hyperbolas are used within the design of the system. Since both game systems are based on movement and there are several different types of ways someone can move, the system must have these resources available so that it can read what the person in doing. Hyperbolas are commonly found everywhere with some type of design.

 

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To explore this topic, I would first show the students this video of the roller coaster “Fire and Ice” which is in Orlando, Florida at Universal Studios. This roller coaster was created so that when the two roller coasters go around a loop at the same time, they will never hit, making for a fun, adventurous time. This is what a hyperbola simply is; every point lies within the same ratio from focus to directrix. During the video point out the hyperbolic part of the roller coaster which is shown at the 49-51 second mark.

 

 

Now after watching the video, the students would be given about 8 minutes to explore by themselves or with a partner, how to create their own hyperbola. The student can use any resources he/she would like. Once the students have had enough time to explore, the teacher would then have the student watch an instructional video from Kahn Academy.

 

 

The video is very useful in teaching students how to graph a hyperbola because the instructor goes through step by step carefully explaining what each part means and why each part is placed where it is in the function. The video is engaging to the students since they don’t have to listen to their teacher say it a million times and then reinforce it. This is also helpful for the teacher because the student hears it from one source and then it is reinforced by the teacher, giving the teacher a second hand because it’s now coming from two sources not just one.

 

After the video, the students can now split up into groups of at least 3 and create their own “Fire and Ice” roller coaster from scratch. They will have the information from the video to help them know how to create the function and may also ask questions. The student may create their hyperbola roller coaster anyway they would like, using any directrix as well. But keep in mind that you would probably want to tell them it needs to be somewhat realistic or else you could get some crazy ideas. Once all the groups are finished, they will present their roller coaster to the class and be graded by their peers for one grade and then graded by the teacher for participation and correctness.

 

green lineFrom previous math courses, the student should already know the terms slope and vertex. The student should’ve already learned how to graph a parabola. Everything that a student uses to graph a parabola is used to graph a hyperbola but yet with more information. Starting from the bottom, a parabola is used because all a hyperbola technically is, is the graph show a parabola and its mirrored image at the same time. From here the student learns about the directrix, which is the axis of symmetry that the parabola follows. The student will now be able to learn about asymptotes which are basically what a directrix is in a hyperbola function. This opens the door to several graphs of limits that the student will learn throughout calculus and higher math classes.

 

Matrix transform

matrix_transform

Source: http://www.xkcd.com/184/

P.S. In case you don’t get the joke… and are wondering why the answer isn’t [a_2, -a_1]^T…  the matrix is an example of a rotation matrix. This concept appears quite frequently in linear algebra (not to mention video games and computer graphics). In the secondary mathematics curriculum, this device is often used to determine how to graph conic sections of the form

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,

where B \ne 0. I’ll refer to the MathWorld and Wikipedia pages for more information.

Engaging students: Parabolas

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 Claire McMahon. Her topic, from Precalculus: parabolas.

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The parabola took a long time to get to us and took a few thinkers to really get the idea down.  This website that I found really nailed the dates and also simplified the rational that led up to the parabola as we know it today.  The history of the parabola is as follows:

The parabola was explored by Menaechmus (380 BC to 320 BC), who was a pupil of Plato and Eudoxus. He was trying to duplicate the cube by finding the side of the cube that has an area double the cube. Instead, Menaechmus solved it by finding the intersection of the two parabolas x2=y and y2=2x. Euclid (325 BC to 265 BC) wrote about the parabola. Apollonius (262 BC to 190 BC) named the parabola. Pappus (290 to 350) considered the focus and directrix of the parabola. Pascal (1623 to 1662) considered the parabola as a projection of a circle. Galileo (1564 to 1642) showed that projectiles falling under uniform gravity follow parabolic paths. Gregory (1638 to 1675) and Newton (1643 to 1727) considered the properties of a parabola.

This really got me to thinking what it really took to figure out the derivation of the formula and even for the graph of the parabola.  I find it interesting that the idea had to travel through seven genius minds to come to all of the properties that the parabola holds to this day.

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This same website led me to another use of the parabola, other than to describe a projectile’s path.  The use of suspension bridges relies heavily on a parabolic model.  Other parabolic models would include the satellite dishes and even all types of lights.  Have you ever thought that every single place that light bulb reflects is a reflection off a point from the focus to the parabola to create your beam of light!!  Pretty cool!!  So you might ask why do I need to know anything about parabolas?  There is your answer; it’s used in everyday life.  Here are a couple of examples from the website that I found interesting:

One of the “real world” applications of parabolas involves the concept of a 3D parabolic reflector in which a parabola is revolved about its axis (the line segment joining the vertex and focus). The shape of car headlights, mirrors in reflecting telescopes, and television and radio antennae (such as the one below) all utilize this property.

Antenna of a Radio Telescope

All incoming rays parallel to the axis of the parabola are reflected through the focus.

Flashlights & Headlights

In terms of a car headlight, this property is used to reflect the light rays emanating from the focus of the parabola (where the actual light bulb is located) in parallel rays.

Here are the specs on the suspension bridge:

Hold up a chain by both ends and you’ll get a curve. What kind of curve is it? You might say it is a parabola – Galileo Galili believed it was a parabola. Yet, Galileo was wrong!!!! That curve is NOT a parabola. It is a catenary.It makes sense that you would think that the curved chain is a parabola. Both the catenary and the parabola have similar properties. Both curves have a single low point. They both have a vertical line of symmetry, they at least appear to be continuous and differentiable throughout, and the slope is steeper as we move away from the low point, but it never becomes vertical.So, how is the curve of the cable in a suspension bridge a parabola? When the structure is being built and the main cables are attached to the towers, the curve is a catenary. But when the cables are attached to the deck with hangers, it is no longer a catenary. The curve of the cables become the curve of a parabola. Unlike the catenary, which is curving under its own weight, the parabola is curving not just under its own weight, but also curving from holding up the weight of the deck. The cable of a suspension bridge is under tension from holding up the bridge.Therefore, the cables of a suspension bridge is a parabola, because the weight of the deck is equally distributed on the curve.

I never really knew that there was a difference between the two and now I know that there are certain properties that made it down through the ages that hold true today.  This was a very enlightening subject matter.

Website used: http://www.carondelet.pvt.k12.ca.us/Family/Math/03210/page2.htm

Engaging students: Equation of a circle

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 Alyssa Dalling. Her topic, from Precalculus: the equation of a circle.

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

A fun way to engage students and also introduce the standard form of an equation of a circle is the following:

  • Start by separating the class into groups of 2 or 3
  • Pass each group a specific amount of flashcards. (Each group will have the same flashcards)
  • Each flashcard has a picture of a graphed circle and the equation of that circle in standard form
  • The students will work together to figure out how the pictures of the circle relate to the equation

This will help students understand how different aspects of a circle relate to its standard form equation. The following is an example of a flashcard that could be passed out.

equationofcircle

Source: http://www.mathwarehouse.com/geometry/circle/equation-of-a-circle.php

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C. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Circles have been used through history in many different works of art. One such type is called a tessellation. The word Tessellate means to cover a plane with a pattern in such a way as to leave no region uncovered. So, a tessellation is created when a shape or shapes are repeated over and over again. The pictures above show just a few examples of how circles are used in different types of art. A good way to engage students would be to show them a few examples of tessellations using circles.

Source: http://mathforum.org/sum95/suzanne/whattess.html

equationofcircle4equationofcircle2equationofcircle3

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

Khan Academy has a really fun resource for using equations to graph circles. At the beginning of class, the teacher could allow students to play around with this program. It allows students to see an equation of a circle in standard form then they would graph the circle. It gives hints as well as the answer when students are ready. The good thing about this is that even if a student goes straight to the answer, they are still trying to identify the connection between the equation of the circle and the answer Khan Academy shows.

http://www.khanacademy.org/math/trigonometry/conics_precalc/circles-tutorial-precalc/e/graphing_circles