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 Biviana Esparza. Her topic, from Precalculus: graphing a hyperbola.

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B2. How does this topic extend what your students should have learned in previous courses?

Prior to learning about conics and hyperbolas in precalculus, students should be able to identify different shapes and figures and learn to identify cross sections of prisms, pyramids, cylinders, cones, and spheres, from geometry class. In algebra 2, students learn to write quadratic equations and learn vocabulary such as vertex, foci, directrix, axis of symmetry, and direction of opening, all which are used when dealing with hyperbolas as well.

 

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How has this topic appeared in pop culture?

The sport of baseball originates back before the Civil War and has come to be known as America’s pastime. On average, 110 balls are used in a major league baseball game, because the balls are usually tossed out if they’ve touched the dirt. Baseballs have a rubber or cork center, wrapped in yearn, and covered with leather sown together tightly by 108 stitches of red string. The stitches are in a hyperbola shape if looked at from a certain angle and depending on how the pitcher has held the stitches, different pitches are thrown.

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E1. How can technology be used to effectively engage students with this topic?

Desmos is a great, interactive website that has many activities that can be used in the classroom. One of the activities it has is called Polygraph: Conics. The Desmos activity is similar to the board game Guess Who? in which students are in pairs and will ask yes or no questions to guess the graph of a hyperbola or ellipse of their choosing. This activity encourages students to make good questions and use precise vocabulary and academic language when describing conics, specifically over ellipses and hyperbolas, so that they can win the game.
https://teacher.desmos.com/polygraph/custom/560ad28c9e65da5615091ec7

References:
https://teacher.desmos.com/polygraph/custom/560ad28c9e65da5615091ec7
https://en.wikipedia.org/wiki/Baseball_(ball)

https://www.teksresourcesystem.net/module/content/search/item/681138/viewdetail.ashx

 

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.