Happy Pythagoras Day!

Happy Pythagoras Day! Today is 8/15/17 (or 15/8/17 in other parts of the world), and 8^2+15^2=17^2.

We might as well celebrate today, because the next Pythagoras Day won’t happen for over 3 years. (Bonus points if you can figure out when it will be.)

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

 

 

Engaging students: Finding the 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 Lucy Grimmett. Her topic, from Precalculus: finding the equation of a circle.

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

I love doing activities to teach topics, or solidify students knowledge after direct teaches. The link below describes an activity/project a teacher did with her students. The students were asked to create an image using circles. They can use other images along with circles; however, circles are the main focus of the project. The picture had to include at least four circles. Once they had drawn their image containing the circles they were asked to find the equations of each of the circles in their picture. As an extra challenge students were asked to create a question to go with one of their circles that would aid another student in finding the equation.  This is where students have to truly put two-and-two together to create an in depth connection between the lesson, word problems, and furthermore the idea of the center of a circle and the length of the radius.

http://secondarymissrudolph.blogspot.com/2012/04/equations-of-circles-update.html

 

 

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

In Algebra II students learn about writing equations of graphs and how each piece of the equations manipulates the graph. This is a skill that continues into Pre-Calculus. Whether students are graphing circles, exponential functions, or trigonometric equations there is always variables that can be manipulated that manipulate the graph. Further, students will be required to find equations of hyperbolas, ellipses, and parabolas. These equations go hand and hand with one another when students are using a focus and directrix. As you know, mathematics constantly builds on itself. With students previous knowledge of quadratic equations specifically, they see how in the equation y = a(x-h)^2 + k, (h,k) is the “center” of the graph. The same goes for the equation of a circle. The point (h,k) is the literal center of the circle in the formula (x-h)^2+(y-k)^2=r^2.  With their previous knowledge of h affecting the x component and k affecting the y, students are able to grasp the concept more quickly, and more efficiently.

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

In Algebra II students learn about writing equations of graphs and how each piece of the equations manipulates the graph. This is a skill that continues into Pre-Calculus. Whether students are graphing circles, exponential functions, or trigonometric equations there is always variables that can be manipulated that manipulate the graph. Further, students will be required to find equations of hyperbolas, ellipses, and parabolas. These equations go hand and hand with one another when students are using a focus and directrix. As you know, mathematics constantly builds on itself. With students previous knowledge of quadratic equations specifically, they see how in the equation y = a(x-h)^2 +k, (h,k)  is the “center” of the graph. The same goes for the equation of a circle. The point (h,k) is the literal center of the circle in the formula  (x-h)^2+(y-k)^2 = r^2.  With their previous knowledge of h affecting the x component and k affecting the y, students are able to grasp the concept more quickly, and more efficiently.

 

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

There are so many online resources to use for mathematics teaching. Students have easy access to an online graphing calculator called Desmos. This allows students to play with different numbers in equations to see how they affect the graph. For examples, students can manipulate the radius and the center point. This will allow students to visually see how each variable contributes to the equation. Below are other links that are beneficial when teaching equations of circles. Khan Academy is a tool that is used by many educators, not only does Khan Academy include instructional videos, but it also has mini quizzes after that check for student knowledge. The second link below is an online resource which student can insert equations of circles and the program will give them the radius, center, a graph, and it will even give an explanation. There is a second mode on this resource which student put the radius and center and the site will return the equation, again, the website will give an explanation if needed. These resources are quick ways for student to see how the equation of a circle can change different pieces of circles.

https://www.khanacademy.org/math/geometry/hs-geo-circles/hs-geo-circle-standard-equation/e/equation_of_a_circle_1

http://www.mathportal.org/calculators/analytic-geometry/circle-equation-calculator.php

 

 

Engaging students: Using Pascal’s triangle

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 Lisa Sun. Her topic, from Precalculus: using Pascal’s triangle.

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

To introduce Pascal’s Triangle, I would create an activity where it involves coin tossing. I want to introduce them with coin tossing first before bringing in binomial expansions (or any other uses) because coin tossing are much more familiar to majority, if not all, students. Pascal’s Triangle can show you the probability of any combination of coin tossing (aka binomial distribution). Below are a few of the results and how they compare with Pascal’s Triangle:

Afterwards, I would ask the students guiding questions if they see anything interesting about the numbers that we gathered. I want them to notice that each number is the numbers directly above it added together (Ex: 1 + 2 = 3) and how those three numbers form a triangle hence, Pascal’s Triangle.

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

In previous courses, students should have already learned about binomial expansions. (Ex: (a+b)2 = a2+ 2ab + b2). This topic extends their prior knowledge even further because Pascal’s Triangle displays the coefficients in binomial expansions. Below are a few examples in comparison with Pascal’s Triangle:

If any of the students are having difficulties expanding any of the binomials or remembering the formula, they can remember Pascal’s Triangle. Using the Pascal’s Triangle for solving binomial expansions can aid the students when it comes to being in a stressful environment (ex: taking a test). Making a connection between their prior knowledge on binomial expansion and Pascal’s Triangle, I believe it would give the students a deeper understanding as to how Pascal’s Triangle was formed.

 

 

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C2: How has this topic appeared in high culture?

There’s a computer scientist, John Biles, at Rochester University in New York State who used the series of Fibonacci numbers to make a piece of music. How do the Fibonacci numbers relate to Pascal’s Triangle you ask? Well, observe the following:

As you can see, the sum of the numbers diagonally gives you the Fibonacci numbers (a series of numbers in which each number is the sum of the two preceding numbers).

John Biles composed a piece called PGA -1 which is based on a Fibonacci sequence. Note that on a piano, from middle C to a one octave C, there are a total of eight white keys (a Fibonacci number). Also, when you do a chromatic C scale which includes all the black keys, there are a total of five black keys (another Fibonacci number) which are also separated in a group of two and three black keys (see the pattern?). When you’re creating chords, let’s take the C chord for example, it consists of the notes C, E, and G. Notice that harmonizing notes are coming from the third note and the fifth note of the whole C scale. So following similar ideas on the use of these numbers/sequences, John Biles was able to compose music.

Here is John Biles full article: http://igm.rit.edu/~jabics//Fibo98/

Here is his composed song: http://igm.rit.edu/~jabics//Fibo98/PGA-1.mp3

The following may be a bit extra, but I also want to include this youtube link of this blogger who was very precise and compared the sequences to current pop music:

[I found this to be super interesting!]

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

Hundreds of years before Blaise Pascal (mathematician whom Pascal’s Triangle was named after), many mathematicians in different societies applied their knowledge of the Triangle.

Indian mathematicians used the array of numbers to represent short and long sounds in poetic meters in their chants and conversations. A Chinese mathematician, Chu Shih Chieh, used the triangle for binomial expansions. Music composers, like Mozart and Debussy, used the sequence to compose their music to guide them what notes to play that would be pleasing to the audience. In the past, arithmetic composing was frowned upon however contemporary music to this day is now filled with them. When Pascal’s work on the triangle was published, society began to apply the knowledge of the Triangle towards gambling with dice. In the end, all cultures began to use Pascal’s Triangle similarly in their daily lives.

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

The Youtube video above is a great tool for students who are visual learners. This video is to the point and clear with the message as to what Pascal’s Triangle is, the uses of it, and who aided in the discovery of it. I also believe the characters that were being used in this video would be appealing to students. This video was filled with facts that I want my students to know therefore, I would like them to follow along and write down important facts about Pascal’s Triangle. I would like to conclude that technology can be a “force multiplier” for all teachers in their classroom. Instead of having the teacher being the only source of help in a classroom, students can access web site, online tutorials, and more to assist them. What’s great is that students can access this at any time. Therefore, they can re-watch this video again once they’re home when they need a refresher or didn’t understand something the first time.

 

References:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html#othermusic

http://www.mathsisfun.com/pascals-triangle.html

http://ualr.edu/lasmoller/pascalstriangle.html

 

Engaging students: Computing logarithms with base 10

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 Katelyn Kutch. Her topic, from Precalculus: computing logarithms with base 10.

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How has this topic appeared in the news?

http://www.seeitmarket.com/the-log-blog-trading-with-music-and-logarithmic-scale-investing-14879/ . This website gives an insight into logarithms that many students would not know and I think that what is has to say is quite interesting. While this may not be a news article, it includes many methods in which logarithms can and are being used in the world. It also gives some insight into the history of logarithms. I feel like showing the students this website would get them interested in logarithms because they can see what logarithms can do, like tell us the magnitude of an earthquake on the Richter Scale. Students may not find logarithms interesting, but I feel like most would find this interesting.

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

http://mathequalslove.blogspot.com/2014/01/introducing-logarithms-with-foldables.html . This website gives multiples games that teachers can do with logarithms, not just base 10, but for all logarithms. The teacher had foldables that the students put their notes in for logarithms and personally, as a kinesthetic learner, that is something that I loved when teachers did it. It helped me visually put down the notes and it was something that I could keep to refer to. The teacher also had Log War, Log Bingo, and Log Speed Dating. Students always respond better when a sense of fun is involved in the lesson and this teacher proved that when one of her students asked about another game involving the subject. The games are ones that students interact with the teacher, with each other, and it enhances their own thinking because they are having to do calculations, correctly, in order to win the game. This seems like a wonderful website to pull from when wanting to do something fun with a lesson.

 

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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?

In 1614 a Scottish mathematician by the name of John Napier published his discovery for logarithms. Napier worked with an English mathematician by the name of Henry Briggs. The two of them adjusted Napier’s original logarithm to the form that we use today. After Napier passed away, Briggs continued their work alone and published, in 1624, a table of logarithms that calculated 14 decimal places for numbers between 1 and 20,000, and numbers between 90,000 and 100,000. In 1628 Adriaan Vlacq, a Dutch publisher, published a 10 decimal place table for values between 1 and 100,000, which included the values for 70,000 that were not previously published. Both men worked on setting up log trigonometric tables. Later, the notation Log(y) was adopted in 1675, by Leibniz, and soon after he connected Log(y) to the integral of dy/y.

 

 

Engaging students: Introducing the number e

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 Jillian Greene. Her topic, from Precalculus: introducing the number e.

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

 

By this point in their mathematics career, the students have had plenty of experience with simple and compound interest formulas. Whether or not they discovered it them themselves through exploration in a class or their teacher just gave it to them, they’ve used it before. Now we can do an exploration activity that will connect that formula to the number e, and then to the limit. The activity will say: what if you invested $1 for 1 year at 100% compound interest? It’s a pretty good deal! But how much does the number of compounding periods affect the final value? Using the formula they have, A=P(1+r/n)^nt, they will calculate how much money they will make if it’s compounded:

  • Yearly
  • Biannually
  • Quarterly
  • Weekly
  • Daily
  • Hourly
  • Every minute
  • And every second

The first time it’s compounded, the final value will be $2. However, the more compounding periods you add, the closer to e you’ll get. For instance, weekly would be A=1(1+1/52)^52=2.69259695. Every second will get you A=1(1+1/31536000)^31536000=2.71828162, which is pretty to 2.718. The last three calculations will actually begin with 2.718. We can have some discussion with this as a class, bringing in the concept of limits. Then we can assess and see if anyone has seen this number before. If not, they can pop out their calculators and you can have them type “e” and then hit enter, and blow their minds.

 

 

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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?

Though Euler does not receive credit for the first discovery of the number e, he does receive credit for naming it and first publishing it. Some say the e means exponential, some say he’d already published uses for a-d, and some say he named it after himself. He is quoted directly for saying “For the number whose logarithm is unity, let e be written, which is 2,7182817… [sic] whose logarithm according to Vlacq is 0,4342944… “ regarding the number e. He also has a couple of other choice quotes that illustrate his humor, ie “[upon losing the use of his right eye] ‘Now I will have less distraction.’” And “”Sir,

hence God exists; reply!” In response to the French philosophe Diderot, who was trying to convert the court of Catherine the Great of Russia to atheism. Diderot had no idea what Euler was talking about and left the court to a chorus of laughter.”  Back to e, however. If Euler did not first discover it, who did? A man name John Napier did the best he could to discover e. Napier was alive from 1550-1617, so he did not have access to a rich history of advanced algebra. Logarithm tables existed, some close to natural log, but none to identify this mystical number. Napier was merely trying to find an easier way to approach multiplication (and consequently exponentiation). His work, Construction of the Marvelous Rule of Logarithms, he states that X=Nap log y, where Nap log (107)=0. In today’s terms, with today’s math, we can translate that to Nap log y = 107 log1/e(y/107).

 

 

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

After some discussion on this topic, if my class is a pre-AP or particularly curious class, I will have them go around and read this poem about e out loud. Then from this poem, I can have the students split up into groups. Each group will be responsible for dissecting this poem for certain things and then presenting their most interesting/exciting/relatable findings. One group will tackle the names; what history lesson is given to us here? Another group will handle applications; what did the various figures say we can do with e? The final group will report back on different representations of e; what all is e equal to? My expectations here would be for the students to see the insanely vast history and application of this number and gain some appreciation. I would expect to see Napier, Euler, and Leibniz for sure from the first group. From the second group, I would expect continuous compound interest, 1/e in probability and statistics, and calculus. The third group would be expected to present the numerical value of e, the limit that e is equal to, its infinite sum representation, and Euler’s identity. A number worthy of a 500 word poem and a slew of historical mathematicians must be important.

 

The Enigmatic Number e

by Sarah Glaz

It ambushed Napier at Gartness,
like a swashbuckling pirate
leaping from the base.
He felt its power, but never realized its nature.
e‘s first appearance in disguise—a tabular array
of values of ln, was logged in an appendix
to Napier‘s posthumous publication.
Oughtred, inventor of the circular slide rule,
still ignorant of e‘s true role,
performed the calculations.

A hundred thirteen years the hit and run goes on.
There and not there—elusive e,
escape artist and trickster,
weaves in and out of minds and computations:
Saint-Vincent caught a glimpse of it under rectangular hyperbolas;
Huygens mistook its rising trace for logarithmic curve;
Nicolaus Mercator described its log as natural
without accounting for its base;
Jacob Bernoulli, compounding interest continuously,
came close, yet failed to recognize its face;
and Leibniz grasped it hiding in the maze of calculus,
natural basis for comprehending change—but
misidentified as b.

The name was first recorded in a letter
Euler sent Goldbach in November 1731:
“e denontat hic numerum, cujus logarithmus hyperbolicus est=1.”
Since a was taken, and Euler
was partial to vowels,
e rushed to make a claim—the next in line.

We sometimes call e Euler‘s Number: he knew
e in its infancy as 2.718281828459045235.

On Wednesday, 6th of May, 2009,
e revealed itself to Kondo and Pagliarulo,
digit by digit, to 200,000,000,000 decimal places.
It found a new digital game to play.

In retrospect, following Euler‘s naming,
e lifted its black mask and showed its limit:
e=limn→∞(1+1n)ne=limn→∞(1+1n)n
Bernoulli‘s compounded interest for an investment of one.

Its reciprocal gave Bernoulli many trials,
from gambling at the slot machines to deranged parties
where nameless gentlemen check hats with butlers at the door,
and when they leave, e‘s reciprocal hands each a stranger’s hat.

In gratitude to Eulere showed a serious side,
infinite sum representation:
e=∑n=0∞1n!=10!+11!+12!+13!+⋯e=∑n=0∞1n!=10!+11!+12!+13!+⋯

For Euler‘s eyes alone, e fanned the peacock tail of
e−12e−12’s continued fraction expansion,
displaying patterns that confirmed
its own irrationality.

A century passed till e—through Hermite‘s pen,
was proved to be a transcendental number.
But to this day it teases us with
speculations about ee.

e‘s abstract beauty casts a glow on Euler’s Identity:
eið + 1 = 0,
the elegant, mysterious equation,
where waltzing arm in arm with i and π,
e flirts with complex numbers and roots of unity.

We meet e nowadays in functional high places
of CalculusDifferential EquationsProbabilityNumber Theory,
and other ancient realms:
y = ex
e
 is the base of the unique exponential function
whose derivative is equal to itself.
The more things change the more they stay the same. 
e
 gathers gravitas as solid under integration,
∫exdx=ex+c∫exdx=ex+c
a constant c is the mere difference;
and often e makes guest appearances in Taylor series expansions.
And now and then e stars in published poetry—
honors and administrative duties multiply with age.

 

 

References:

http://www.maa.org/press/periodicals/convergence/the-enigmatic-number-iei-a-history-in-verse-and-its-uses-in-the-mathematics-classroom-the-annotated

 

http://www.maa.org/publications/periodicals/convergence/napiers-e-napier

 

http://www-history.mcs.st-and.ac.uk/HistTopics/e.html