Stay Focused

From Kirk Cousins, quarterback of the Washington Redskins:

Sometimes our guests ask why I have this hanging above my desk. It’s an old high school math quiz when I didn’t study at all and got a C+… just a subtle reminder to me of the importance of preparation. If I don’t prepare I get C’s!

Source: https://www.facebook.com/redskins/photos/a.118304319573.96677.102381354573/10155470824244574/?type=3&theater

Pizza Hut Pi Day Challenge: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the 2016 Pizza Hut Pi Day Challenge.

Part 1: Statement of the problem.

Part 2: Using the divisibility rules for 1, 5, 9, 10 to reduce the number of possibilities from 3,628,800 to 40,320.

Part 3: Using the divisibility rule for 2 to reduce the number of possibilities to 576.

Part 4: Using the divisibility rule for 3 to reduce the number of possibilities to 192.

Part 5: Using the divisibility rule for 4 to reduce the number of possibilities to 96.

Part 6: Using the divisibility rule for 8 to reduce the number of possibilities to 24.

Part 7: Reusing the divisibility rule for 3 to reduce the number of possibilities to 10.

Part 8: Dividing by 7 to find the answer.

 

Predicate Logic and Popular Culture: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on using examples from popular culture to illustrate principles of predicate logic. My experiences teaching these ideas to my discrete mathematics students led to my recent publication (John Quintanilla, “Name That Tune: Teaching Predicate Logic with Popular Culture,” MAA Focus, Vol. 36, No. 4, pp. 27-28, August/September 2016).

Unlike other series that I’ve made, this series didn’t have a natural chronological order. So I’ll list these by concept illustrated from popular logic.

green lineLogical and \land:

  • Part 1: “You Belong To Me,” by Taylor Swift
  • Part 21: “Do You Hear What I Hear,” covered by Whitney Houston
  • Part 31: The Godfather (1972)
  • Part 45: The Blues Brothers (1980)
  • Part 53: “What Does The Fox Say,” by Ylvis
  • Part 54: “Billie Jean,” by Michael Jackson
  • Part 98: “Call Me Maybe,” by Carly Rae Jepsen.

Logical or \lor:

  • Part 1: Shawshank Redemption (1994)

Logical negation \lnot:

  • Part 1: Richard Nixon
  • Part 32: “Satisfaction!”, by the Rolling Stones
  • Part 39: “We Are Never Ever Getting Back Together,” by Taylor Swift

Logical implication \Rightarrow:

  • Part 1: Field of Dreams (1989), and also “Roam,” by the B-52s
  • Part 2: “Word Crimes,” by Weird Al Yankovic
  • Part 7: “I’ll Be There For You,” by The Rembrandts (Theme Song from Friends)
  • Part 43: “Kiss,” by Prince
  • Part 50: “I’m Still A Guy,” by Brad Paisley
  • Part 76: “You’re Never Fully Dressed Without A Smile,” from Annie.
  • Part 109: “Dancing in the Dark,” by Bruce Springsteen.
  • Part 122: “Keep Yourself Alive,” by Queen.

For all \forall:

  • Part 3: Casablanca (1942)
  • Part 4: A Streetcar Named Desire (1951)
  • Part 34: “California Girls,” by The Beach Boys
  • Part 37: Fellowship of the Ring, by J. R. R. Tolkien
  • Part 49: “Buy Me A Boat,” by Chris Janson
  • Part 57: “Let It Go,” by Idina Menzel and from Frozen (2013)
  • Part 65: “Stars and Stripes Forever,” by John Philip Sousa.
  • Part 68: “Love Yourself,” by Justin Bieber.
  • Part 69: “I Will Always Love You,” by Dolly Parton (covered by Whitney Houston).
  • Part 74: “Faithfully,” by Journey.
  • Part 79: “We’re Not Gonna Take It Anymore,” by Twisted Sister.
  • Part 87: “Hungry Heart,” by Bruce Springsteen.
  • Part 99: “It’s the End of the World,” by R.E.M.
  • Part 100: “Hold the Line,” by Toto.
  • Part 101: “Break My Stride,” by Matthew Wilder.
  • Part 102: “Try Everything,” by Shakira.
  • Part 108: “BO$$,” by Fifth Harmony.
  • Part 113: “Sweet Caroline,” by Neil Diamond.
  • Part 114: “You Know Nothing, Jon Snow,” from Game of Thrones.
  • Part 118: “The Lazy Song,” by Bruno Mars.
  • Part 120: “Cold,” by Crossfade.
  • Part 123: “Always on My Mind,” by Willie Nelson.

For all and implication:

  • Part 8 and Part 9: “What Makes You Beautiful,” by One Direction
  • Part 13: “Safety Dance,” by Men Without Hats
  • Part 16: The Fellowship of the Ring, by J. R. R. Tolkien
  • Part 24 : “The Chipmunk Song,” by The Chipmunks
  • Part 55: The Quiet Man (1952)
  • Part 62: “All My Exes Live In Texas,” by George Strait.
  • Part 70: “Wannabe,” by the Spice Girls.
  • Part 72: “You Shook Me All Night Long,” by AC/DC.
  • Part 81: “Ascot Gavotte,” from My Fair Lady
  • Part 82: “Sharp Dressed Man,” by ZZ Top.
  • Part 86: “I Could Have Danced All Night,” from My Fair Lady.
  • Part 95: “Every Breath You Take,” by The Police.
  • Part 96: “Only the Lonely,” by Roy Orbison.
  • Part 97: “I Still Haven’t Found What I’m Looking For,” by U2.
  • Part 105: “Every Rose Has Its Thorn,” by Poison.
  • Part 107: “Party in the U.S.A.,” by Miley Cyrus.
  • Part 112: “Winners Aren’t Losers,” by Donald J. Trump and Jimmy Kimmel.
  • Part 115: “Every Time We Touch,” by Cascada.
  • Part 117: “Stronger,” by Kelly Clarkson.

There exists \exists:

  • Part 10: “Unanswered Prayers,” by Garth Brooks
  • Part 15: “Stand by Your Man,” by Tammy Wynette (also from The Blues Brothers)
  • Part 36: Hamlet, by William Shakespeare
  • Part 57: “Let It Go,” by Idina Menzel and from Frozen (2013)
  • Part 93: “There’s No Business Like Show Business,” from Annie Get Your Gun (1946).
  • Part 94: “Not While I’m Around,” from Sweeney Todd (1979).
  • Part 104: “Wild Blue Yonder” (US Air Force)
  • Part 106: “No One,” by Alicia Keys.
  • Part 116: “Ocean Front Property,” by George Strait.

Existence and uniqueness:

  • Part 14: “Girls Just Want To Have Fun,” by Cyndi Lauper
  • Part 20: “All I Want for Christmas Is You,” by Mariah Carey
  • Part 23: “All I Want for Christmas Is My Two Front Teeth,” covered by The Chipmunks
  • Part 29: “You’re The One That I Want,” from Grease
  • Part 30: “Only You,” by The Platters
  • Part 35: “Hound Dog,” by Elvis Presley
  • Part 73: “Dust In The Wind,” by Kansas.
  • Part 75: “Happy Together,” by The Turtles.
  • Part 77: “All She Wants To Do Is Dance,” by Don Henley.
  • Part 90: “All You Need Is Love,” by The Beatles.

DeMorgan’s Laws:

  • Part 5: “Never Gonna Give You Up,” by Rick Astley
  • Part 28: “We’re Breaking Free,” from High School Musical (2006)

Simple nested predicates:

  • Part 6: “Everybody Loves Somebody Sometime,” by Dean Martin
  • Part 25: “Every Valley Shall Be Exalted,” from Handel’s Messiah
  • Part 33: “Heartache Tonight,” by The Eagles
  • Part 38: “Everybody Needs Somebody To Love,” by Wilson Pickett and covered in The Blues Brothers (1980)
  • Part 46: “Mean,” by Taylor Swift
  • Part 56: “Turn! Turn! Turn!” by The Byrds
  • Part 63: P. T. Barnum.
  • Part 64: Abraham Lincoln.
  • Part 66: “Somewhere,” from West Side Story.
  • Part 71: “Hold On,” by Wilson Philips.
  • Part 80: Liverpool FC.
  • Part 84: “If You Leave,” by OMD.
  • Part 103: “The Caisson Song” (US Army).
  • Part 111: “Always Something There To Remind Me,” by Naked Eyes.
  • Part 121: “All the Right Moves,” by OneRepublic.

Maximum or minimum of a function:

  • Part 12: “For the First Time in Forever,” by Kristen Bell and Idina Menzel and from Frozen (2013)
  • Part 19: “Tennessee Christmas,” by Amy Grant
  • Part 22: “The Most Wonderful Time of the Year,” by Andy Williams
  • Part 48: “I Got The Boy,” by Jana Kramer
  • Part 60: “I Loved Her First,” by Heartland
  • Part 92: “Anything You Can Do,” from Annie Get Your Gun.
  • Part 119: “Uptown Girl,” by Billy Joel.

Somewhat complicated examples:

  • Part 11 : “Friends in Low Places,” by Garth Brooks
  • Part 27 : “There is a Castle on a Cloud,” from Les Miserables
  • Part 41: Winston Churchill
  • Part 44: Casablanca (1942)
  • Part 51: “Everybody Wants to Rule the World,” by Tears For Fears
  • Part 58: “Fifteen,” by Taylor Swift
  • Part 59: “We Are Never Ever Getting Back Together,” by Taylor Swift
  • Part 61: “Style,” by Taylor Swift
  • Part 67: “When I Think Of You,” by Janet Jackson.
  • Part 78: “Nothing’s Gonna Stop Us Now,” by Starship.
  • Part 89: “No One Is Alone,” from Into The Woods.
  • Part 110: “Everybody Loves My Baby,” by Louis Armstrong.

Fairly complicated examples:

  • Part 17 : Richard Nixon
  • Part 47: “Homegrown,” by Zac Brown Band
  • Part 52: “If Ever You’re In My Arms Again,” by Peabo Bryson
  • Part 83: “Something Good,” from The Sound of Music.
  • Part 85: “Joy To The World,” by Three Dog Night.
  • Part 88: “Like A Rolling Stone,” by Bob Dylan.
  • Part 91: “Into the Fire,” from The Scarlet Pimpernel.

Really complicated examples:

  • Part 18: “Sleigh Ride,” covered by Pentatonix
  • Part 26: “All the Gold in California,” by the Gatlin Brothers
  • Part 40: “One of These Things Is Not Like the Others,” from Sesame Street
  • Part 42: “Take It Easy,” by The Eagles

log(an)

Source: https://www.facebook.com/MathAwesomeness/photos/a.342252885964433.1073741828.342251349297920/628770990645953/?type=3&theater

Trigonometry pun

Source: https://www.facebook.com/743378802479548/photos/a.743768985773863.1073741828.743378802479548/792810117536416/?type=3&theater

Engaging students: Geometric sequences

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 Zachery Hasegawa. His topic, from Precalculus: geometric sequences.

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How has this topic appeared in pop culture? (movies, TV, current music, video games, etc.)

Geometric sequences appear frequently in pop culture.  One example that immediately comes to mind is the movie The Happening starring Mark Wahlberg and Zoe Deschanel.  In the movie, there is a scene where a gentleman is trying to distract another woman from the chaos happening outside the jeep they’re traveling in.  He says to her “If I start out with a penny on the first day of a 31 day month and kept doubling it each day, so I’d have .01 on day 1, .02 on day 2, etc.  How much money will I have at the end of the month?” The woman franticly spouts out a wrong answer and the gentleman responds “You’d have over ten million dollars by the end of the month”.  The car goes on to crash just after that scene but as a matter of fact, you’d have exactly $10,737,418.20 at the end of the 31-day month.  This is an example of a geometric sequence because you start out with 0.01 and to get to the next term (day), you would multiply by a common ratio of 2.

 

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

Geometric sequences are popularly found in Book IX of Elements by Euclid, dating back to 300 B.C.  Euclid of Alexandria, a famous Greek mathematician also considered the “Father of Geometry” was the main contributor of this theory.  Geometric sequences and series are one of the easiest examples of infinite series with finite sums.  Geometric sequences and series have played an important role in the early development of calculus, and have continued to be a main case of study in the convergence of series.  Geometric sequences and series are used a lot in mathematics, and they are very important in physics, engineering, biology, economics, computer science, queuing theory, and even finance.

 

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How can technology (specifically Khan Academy/YouTube) be used to effectively engage students with this topic:

 

I really like the video that Khan Academy does on YouTube about Geometric Sequences.  This particular video is a very good introduction to Geometric Sequences because he explains the difference between Geometric Sequences and Series, which I thought to be helpful because I always got the two confused with each other.  Mr. Khan starts out by explaining what exactly a Geometric Sequence is. He describes a Geometric sequence as “A progression of numbers where each successive number is a fixed multiple of the one before it.” He goes on to give numerical examples to specifically show you what he means.  He explains that a1 is typically our first term; a2 is the second term, etc.  He then explains that to get from a1 to a2, you will multiply a1 by the “common ratio” usually represented by “r. For example, “3, 12, 48, 192” is a finite geometric sequence where the common ratio, r, is 4 because to go from 3 to 12 or from 12 to 48, you multiply by 4. He goes on to explain that a Geometric Sequence is a list (sequence) of numbers (terms) that are being multiplied by a common ratio and that a Geometric Series is the sum of the terms (numbers) in the Geometric Sequence.  Using the same numbers as from the Geometric Sequence above, the geometric series is “3+12+48+192”.

 

 

References:

 

 

 

 

Engaging students: Graphing parametric equations

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 Perla Perez. Her topic, from Precalculus: graphing parametric equations.

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

 

Graphing calculators and online sketch pads can allow students to be able to visually see how parametric equations look like. Let’s say the teachers is just introducing parametric equations for the first time to his/her students. Since technology is advancing at a fast rate using an online sketchpad such as, http://www.sineofthetimes.org/the-art-of-parametric-equations-2/, allows student/anyone to explore without much thought. Using the sketch pad above, have students create different figures by moving the blue, green, and red slides. The figures represent can range from:

 

Have a couple student present a figure they like to the class and begin asking simple question like: Why did you chose this picture? Which slides did you use to get here? What did you observe of this sketch? What happens to the equations when you move one of the slides? After the last students shows their figure, bring the class together. Explain that the goal for the next few classes is to be able to graph similar graphs. Although the parametric equations on this sketchpad may be advanced for a high school pre-calculus class, this allows students to get a broader sense of where their high school education can grow.

 

Resources:

http://www.sineofthetimes.org/the-art-of-parametric-equations-2/

 

 

 

 

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

 

Jules Lissajous (1822–1880) was a professor of mathematics in Lycée Saint-Louis from 1847 till 1874. During this time, though, he mainly focused on the study of vibrations and sound. It is because of him (mainly) that we can visually see sound. The way this was done was by parametric equations specially x=asin(w1t+z2), y=bsin(w2+z2), where a, b are amplitudes w1, w2 angular frequencies, z1, z2 phases, and t is time. Using this he was able to create, “patterns formed when two vibrations along perpendicular lines are superimposed”. To a high school student essentially resembles a coordinate plane. Even though he won the Lacaze Prize in 1867 for what many call, “beautiful experiments”, a man named Nathaniel Bowditch produced them using a compound pendulum in 1815. Even though the work isn’t new, Lissajous did his work independently. Jules has helped advanced our studies of not only math but in physics.

 

Resources:

http://www.hit.bme.hu/~papay/edu/Lab/Lissajous.pdf

http://www-groups.dcs.st-and.ac.uk/history/Biographies/Lissajous.html

http://www.s9.com/Biography/lissajous-jules-antoine/

 

 

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What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

 

Students in algebra learn how to graph a function in a coordinate plane as well as be able to solve for an equation from a graph. Before, students begin to graph parametric equations for the most have some understanding of what a parametric equation is, at least in most situations. To take the students to the next level and build on their understanding, interesting world problems such as the one below can help the process go more smoothly. This worksheet begins with something students should be able to complete and expand to what graphs of a parametric equations look like.

This word problem is:

Begin by having students fill the table out. Take a moment to check student’s results. Next, challenge the students to attempt to plot and describe the graph as asked in part b and c. From there on, the instructor can go over what these parts mean. This is a great way to start having student connect their knowledge of equations to a graph to even more topics in-depth.

 

Resources:

http://www.austincc.edu/lochbaum/11-3%20Parametric%20Equations.pdf

Engaging students: Computing trigonometric functions using a unit 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 Sarah Asmar. Her topic, from Precalculus: computing trigonometric functions using a unit circle.

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

Learning the unit circle can be very challenging for many students. One must know all the elements of the circle and need to know how to apply it. Therefore, I have come up with a few activities to make learning the unit circle more fun and engaging. One activity that would be great when teaching students how to memorize the unit circle and all the elements of it is the game of “I Have Who Has.”  I would create a stack of note cards that would have one element of the unit circle on it. For example, one card will have 90° while another will have π. I will do that for all the elements on the unit circle. Then, I would pass out one note card to each student. One student will begin by saying “I have 2π, who has 0° or 360°?” Then, the student that has the card with those two elements on it will say what they have and ask who has the next element. This will go on until all of the elements have been said and it returns to the student that started the game.  Another activity I found that would help students see the unit circle in a more colorful way is if they created it on a paper plate using colored yarn or colored markers. The x and y axis would be in one color and the rest would be in different colors. They would label each line/angle with the correct degree and radian, and the correct (x, y). Here is the link to a picture of what I would want the students to do: https://s-media-cache-ak0.pinimg.com/736x/74/e9/23/74e9232e7389804ce4df2ea6890e0ff9.jpg

 

 

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

Students first see trigonometry in Geometry class as sophomores in high school, but they typically go into more depth during pre-calculus. One way to compute trigonometric functions using the unit circle is by using right triangles. You can find the angle measurement by drawing a right triangle on the unit circle and connect two points. The two special right triangles (30-60-90 and 45-45-90) can be used to form the unit circle. Students would need to recall the rules from geometry to figure out the side lengths of the triangles. With this, students are forced to remember what they were taught in geometry class in order to compute trigonometric functions. If students can see how using the two special triangles creates the unit circle, then it might make more sense to them as where all the measurements/elements came from.

 

 

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

Technology plays a big role in education these days. Students and teachers are encouraged to use technology in the classroom. Khan Academy is one of my favorite websites. He creates very detailed videos about every mathematical topic. I found a few videos on his website to show my students that would help them understand how to use sine, cosine, and tangent with the unit circle. He even has a video that shows a way to remember the unit circle. Another way to implement technology use with tis topic would be with the graphing calculator. Students tend to believe the calculator more than their own teacher. If they saw that the calculator gave them the same exact values as the found using the unit circle, I think they would be amazed and understand how the calculator finds them as well. They might see themselves as smart as the calculator if they can figure out the values by hand and then using the calculator to check their work. I also, might try to find a funny YouTube video that would help the students remember the parts of the unit circle. Once they have the unit circle memorized, it is much easier using it to compute trigonometric functions. Students tend to be more engaged and willing to do something when technology is involved.

 

References:

 

https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/unit-circle-definition-of-trig-functions/v/a-way-to-remember-the-entire-unit-circle-for-trigonometry

 

 

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: Exponential Growth and Decay

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 Madison duPont. Her topic, from Precalculus: exponential growth and decay.

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

Through an EDSE 4000 assignment (for which we were to find a Higher Level Task,) I found a fantastic activity that demonstrates exponential growth and decay in an exploratory, hands-on manner. The link to the website with the lesson plan as well as the activity can be found below. This activity is beneficial to the students for several reasons. The first is that they use a variety of materials and methods: hands-on manipulatives (M&Ms), technology (graphing calculators), and written work. This provides students with varied learning styles a chance to participate in and understand the concept of exponential growth and decay. Consequently, the students are able to experience how quickly exponential growth and decay occurs as the number of M&Ms they are having to count, collect, shake, and dump on their desk grows or shrinks rapidly. They then are able to see how this real-life phenomenon can be measured mathematically through an equation and represented mathematically in a graph. Another reason why I enjoyed this activity was because the worksheet had them make conjectures, analyze data, and find relationships between factual and actual information. This activity was conducted in my EDSE 4000 class and proved to even interest colleagues because the likelihood of getting an exponential relationship from probability of M&Ms facing a certain way seemed unlikely and intriguing. There were a few tips I took away from conducting the activity in my class that may be helpful to remember when conducting this activity again. First, be sure to instruct students not to eat any of the M&Ms until after they complete both the growth and decay portion. Second, inform students of how to count morphed or faded M&Ms prior to the activity. Third, the students will need to be slightly informed about exponential functions in order to make conjectures or determine theoretical functions as required in the worksheet. Fourth, going over how to use the calculator as directed prior to or during the activity may help the activity run more smoothly. Lastly, skittles do not work as well with this activity because they make a significantly sticky mess as they melt in hands. Overall, the hands-on exploration and intellectual reasoning utilized in this activity makes exponential growth and decay interesting, entertaining, and relatable.

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How has this topic appeared in the news?

Exponential growth and decay is largely recognized in the news media regarding the Exponential Growth in Technology. The links below provide intriguing information about the study of how quickly and steadily technology is growing. Morris’ Law is referenced often to provide some explanation for the startlingly rapid growth of technology and decay of previous forms of technology. Also, provided on these sites are videos of Ray Kurzweil discussing his theories of technology being able to duplicate patterns and behaviors of the human brain even more powerfully than that of a human in the near future due to the exponential pattern of technology’s growth. This would likely be interesting to students as technology is a growing part of their lives, lives that may become even more dependent on technology in this coming generation’s lifetime. All of this plausible reality being convincingly calculated from a simple exponential pattern that can be introduced in a high school classroom is pretty amazing, and possibly even powerful, to the minds of future students that can apply this knowledge to the technology phenomenon (or maybe even in other topics of our society) in their future careers. Another video found on the thatsreallypossible.com site has Dr. Albert Bartlett discussing the relevance and impact of “simple” exponential relationships applied to our global community’s resources and economy that are not just hypothetical, but that have happened, and are likely to happen. Using these sites you not only show students the power and importance of exponential growth and decay, you also inform them as global citizens and expose them to realistic problems and ideas that will need to be solved or explored in their lifetime or near future, which is arguably the essence of teaching.

 

 

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

The graphing applet found at mathwarehouse.com (referenced below) is extremely useful in extending student knowledge of the principles of exponential growth and decay. Using this for an activity can help students compare and contrast changing elements of the function without working separate (seemingly unrelated) examples on their own or in groups. Not only is the applet beneficial because you can graph several factors at a time, but you also have clear, graphical representation of the algebraic manipulations along side the algebra. This can be useful for students that learn visually or are ELLs. Activities can be easily carried out by projecting the applet onto a SMART board for full-class evaluation and discussion, having students perform exercises in groups and recording findings for notes, or even just helping students understand differences in homework problems, and hard to understand textbooks notation that are not making sense to students with verbal or written explanations. This being a free website students can access at home on their computer, smart phone, tablet, etc. can be resourceful to students that do not have a graphing calculator and can also be helpful to students as they work through problems independently and try to understand the behaviors of exponential growth and decay outside of the classroom. Because of the applet’s accessibility, aesthetic set up, and ease in manipulation, I recommend this as a useful technology resource both for the teacher and the student as they explore exponential growth and decay.

 

Pleather, D. (n.d.). Precalculus Lesson Plans and Work Sheets. Retrieved November 17, 2016, from http://www.pleacher.com/mp/mlessons/algebra/mm.html.

Document: M&M_GrowthDecayActivity

http://bigthink.com/think-tank/big-idea-technology-grows-exponentially

http://www.thatsreallypossible.com/exponential-growth/

http://www.mathwarehouse.com/exponential-growth-and-decay/interactive-exponential-graph-applet.php