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.

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.

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.

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

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.

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

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.

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

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.

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.

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.

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

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.

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.

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.

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

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.

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.

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 log_1/_e(y/10⁷_).

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, 6^th 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 Euler, e 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 e^e.

e‘s abstract beauty casts a glow on Euler’s Identity:
eⁱ^ð + 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 Calculus, Differential Equations, Probability, Number Theory,
and other ancient realms:
y = e^x
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

Engaging students: Arithmetic sequences

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 Jessica Martinez. Her topic, from Precalculus: arithmetic sequences.

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

Perhaps some of us realized that we were pretty good at math at a young age, though I wonder if anyone was as good at math, or as fast a learner, as Carl Friedrich Gauss was. When Gauss was two he taught himself how to read; when he was three he checked and corrected his dad’s math whenever his father was calculating the payroll; and probably one of the most famous stories about Gauss was that when he was 9 or 10 he created a formula for an arithmetic progression just by glancing at a problem, which ultimately helped Gauss to start his lifelong education and career in mathematical theory. Gauss was sitting in an arithmetic class taught by a man named Buttner, who was said to dislike teaching peasant children, but he was so surprised and impressed that Gauss correctly calculated the solution to the sequence problem that Buttner started to take Gauss under his and wing and help him with his education. Turns out that the formula created by Gauss can be used to find any arithmetic progression. Later on, Gauss eventually earned a doctorate at the age of 22 with the financial help of the Duke of Brunswick; his dissertation was about the fundamental theory of algebra. Gauss had numerous and important contributions to the field of mathematics, but I won’t state them here for the fear of feeling highly insignificant to one of the greatest mathematicians of all time.

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

This is kind of a cool tidbit for any music lovers and any Led Zeppelin fans. So Led Zeppelin III had a crazy array of imagery on its album sleeve and volvelle, which was great way to further engage people who listened to their music. However, we know that records are a thing of past, replaced by CDs and now online streaming. So a musician named Bill Baird, who was inspired by the themes of Led Zeppelin’s third album, created a way to make his website and album just as aesthetically pleasing and interesting as LZ by using mathematical equations and formulas. The site where you can listen to his music has a sort of kaleidoscope design (inspired by LZ’s sundials and astronomical designs) that changes as the music plays, where the artwork “mimics the music” since music themes and sounds are constantly changing over time. However, even more interesting is that the music is different for every listener. The site uses an arithmetic sequence formula based on the listener’s location and the time they accessed the site along with the hand-mixed tracks by Bill to create a unique sequence (and thus a unique album) for every listener. If the database creates an already used track, it starts the process over again until it gets a new sequence.

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

One of my previous teachers suggested this as a great real world problem for arithmetic sequences: Halley’s Comet. The video itself is just a short clip in order to grab my students’ attention; the video describes the comet’s path, its velocity, and its appearances in history with fancy graphics and imaging which can appeal to the science and space lovers in my class along with history lovers. The path of the comet brings it around visible to earth about every 75 years. After we covered some basics on arithmetic sequences, I can present this video to my class and have them research the comet and how it can be represented with a sequence. Some of the questions I could ask them to answer could be: when did it last come? What are the next 3, 4, 5, etc. years it will visit? Have you or will you see it in your lifetime? Calculate its 50^th visit from its last visit, its 100^th visit? I could then challenge them to find some other natural phenomena that also follows an arithmetic sequence.

References

[Video file]. (2015, January 10). In The Legacy of Halley’s Comet. Retrieved November 18, 2016, from https://youtu.be/elsRH_utRdo

Carl Friedrich Gauss. (n.d.). Retrieved November 18, 2016, from http://www.sonoma.edu/math/faculty/falbo/gauss.html

Dial, C. (2016, October 27). Album Turns Into Something New Each Time It’s Played. Retrieved November 18, 2016, from http://www.psfk.com/2016/10/music-album-bill-baird-algorithm.html

Gauss: The Prince of Mathematics | Brilliant Math & Science Wiki. (n.d.). Retrieved November 18, 2016, from https://brilliant.org/wiki/gauss-the-prince-of-mathematics/

Howell, E. (2013, February 20). Halley’s Comet: Facts About the Most Famous Comet. Retrieved November 18, 2016, from http://www.space.com/19878-halleys-comet.html

Engaging students: Graphing the sine and cosine functions

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 Jessica Bonney. Her topic, from Precalculus: graphing the sine and cosine functions.

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

A fun activity for students to learn how to graph the sine and cosine function would be having them build the graph using spaghetti and yarn. Students would start out with a simple warm-up to help them recall the different values of sine and cosine on the unit circle depending on the given angle. After the warm-up, I would then pair students off into groups of two and have them create the graphs, one creating the sine graph and the other creating the cosine graph. The first step in this activity would be for students to take their yarn and wrap it around the unit circle, marking each significant angle on the yarn with a marker. Next, students will create the x and y-axis on their paper, making the x-axis along the center of the paper (labeling it Θ) and the y-axis about 1/3 of the way from the left-end of the paper (labeling it either cosΘ or sinΘ). They then lay the yarn on the x-axis, with the end on the origin, which represents 0 radians, and using the marks they made on the yarn they will mark and label each point on the x-axis. Going back to the unit circle, students will then measure the major angles of either sine or cosine with spaghetti. This part is used to help solidify their understanding that the values of x and y correspond to cosine and sine. After measuring and cutting the spaghetti, students will then glue the spaghetti down to the matching angle on the coordinate plane. Once they have finished gluing their pasta down, students will take a marker and draw the curve. To end the lesson, I would have the students do a think-pair-share, answering the following question: Why is the function curve wider that the unit circle? After, I would have students compare their graphs and demonstrate how they found their graph.

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

Graphing the sine and cosine functions is a topic that students will carry on with them throughout the rest of their future science and mathematics courses. For starters, they will need to know how to do this for all advanced calculus or trigonometry classes they will take in high school or even in college. An example of this would be, when the students learn how to derive the tangent, cotangent, secant, and cosecant functions and graphs. Next, students will use this more in depth in their future physics courses. They will be able to relate waveforms and vibrations to that of specific sine and cosine graphs. Vibrations are graphs with the equations y=sin(t) or y=cos(t), and the time needed for one oscillation across the x-axis is referred to as a period. Waveforms are graphs with the equations y=sin(x) or y=cos(x), and the distance needed for one oscillation across the x-axis is referred to as a wavelength. As you can see, this particular topic in pre-calculus is an important piece in laying the foundation in their future academics and beyond.

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

For starters, the word trigonometry comes from the Greek word trignon, meaning “triangle”, and metron, meaning “to measure.” Before the 16^th century, trigonometry was mainly used for computing the unaccounted for parts of a triangle when the other parts were given. When it comes to ancient civilizations, Egyptians had a collection of 84 algebra, arithmetic, and geometry problems called the Rhind Papyrus. This showed that the Egyptians had some knowledge about the triangle, almost like a “pre-trigonometry”. It wasn’t until the Greeks, that trigonometry began to make sense. Hipparchus was the first to construct a table of the values of trigonometric functions. The next key contribution to trigonometry as we know it came from India. The author of the Aryabhatiya used words for “chord” and “half-chord” which was later shortend to jya or jiva. Following this, Muslim scholars translated the words into Arabic, which was then translated into Latin. An English minister, Edmund Gunter, first used the shortened term that we know, sin, in 1624. In 1614, John Napier invented logarithms, the final major contribution of classical trigonometry.

References:

https://www.britannica.com/topic/trigonometry

http://betterlesson.com/lesson/437440/graphs-of-sine-and-cosine

http://www.algebralab.org/lessons/lesson.aspx?file=trigonometry_trigperiodfreq.xml

Engaging students: Solving Equations with Rational Functions

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 Heidee Nicoll. Her topic, from Precalculus: solving equations with rational functions.

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

To jog the students’ memory of what rational functions look like and what some of their properties are, I would do a relay race with them. The class would be divided into two groups, and each group would have a different rational function, not anything too difficult, but something for which they could easily compute values, something like f(x)=-2/x and g(x)=3/x. On the board would be two large papers, each with a table of values and a blank graph. The x-values would be filled in, but the y-values would be blank. The students would line up, and the first student in each line has to compute the y-value for the first given x-value, then grab the one marker for his/her team, go up to the board and write that value in the table. The next student will compute the next value, and so on. The students would be able to use the calculators on their phones if necessary, but they would not be able to use graphing calculators since they would be able to just plug the function in and look at the table. Once the teams had all the y-values written down, the next student would have to come up to the board and plot the first point on the graph, and so on, until all the points were plotted. The very last student would connect the dots to make a curve. Then we could have a class discussion about vertical asymptotes, and how they show up in the table as an error or undefined value. We could talk about what they remember of end behavior, horizontal asymptotes, x- and y-intercepts, and that could lead into the rest of the lesson.

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

Desmos online graphing calculator is quite nifty. The functions show up in different colors, and you can graph points as well as lines and curves. I found a sort of online worksheet on Desmos talking about rational functions, and modified it. This is the link to the modified version: https://www.desmos.com/calculator/zi62lrxnim It leads the student step by step, as they click on each function to see it on the graph, through looking at the vertical asymptotes, x- and y-intercepts, any holes or slant asymptotes, and at the very end gets them thinking about intersections and solving equations. The purpose would be to remind the students of all the properties of rational functions that we should think about when solving, and how graphing the functions to get a solution is a viable option. In the activity, the students are also asked to move a few slides to graph the correct asymptotes. In this way they are not just taking in information, but are required to provide some answers of their own. All of this information should be already learned, so it would just be a review for the students as they take what they already know and learn how to apply it to solving equations with rational functions.

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

This is a paragraph from Encyclopedia Britannica about Apollonius of Perga and his contributions to geometry.

Greek geometry entered its golden age in the 3rd century. This was a period rich with geometric discoveries, particularly in the solution of problems by analysis and other methods, and was dominated by the achievements of two figures: Archimedes of Syracuse(early 3rd century bc) and Apollonius of Perga (late 3rd century bc). Apollonius is best known for his Conics, a treatise in eight books (Books I–IV survive in Greek, V–VII in a medieval Arabic translation; Book VIII is lost). Apollonius presented a comprehensive survey of the properties of these [parabolas, hyperbolas, and ellipses]. A sample of the topics he covered includes the following: the relations satisfied by the diameters and tangents of conics (Book I); how hyperbolas are related to their “asymptotes,” the lines they approach without ever meeting (Book II); how to draw tangents to given conics (Book II); relations of chords intersecting in conics (Book III); the determination of the number of ways in which conics may intersect (Book IV); how to draw “normal” lines to conics (that is, lines meeting them at right angles; Book V); and the congruence and similarity of conics (Book VI). (Knorr).

We would read it as a class and I would point out that a hyperbola is the parent function for rational functions, y=1/x, and that when we are talking about asymptotes, we are using information that Apollonius worked on and studied.

Works Cited

Biographical Dictionary. n.d. Image. 18 November 2016.

Knorr, Wilbur R. Encyclopedia Britannica: Greek Mathematics. n.d. Website. 18 November 2016.

Original Desmos Activity: https://www.desmos.com/calculator/3azkdx4llk

Modified Desmos Activity: https://www.desmos.com/calculator/zi62lrxnim

Engaging students: Deriving the double angle formulas for sine, cosine, and tangent

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 Daniel Adkins. His topic, from Precalculus: deriving the double angle formulas for sine, cosine, and tangent.

How does this topic extend what your students should have already learned?

A major factor that simplifies deriving the double angle formulas is recalling the trigonometric identities that help students “skip steps.” This is true especially for the Sum formulas, so a brief review of these formulas in any fashion would help students possibly derive the equations on their own in some cases. Listed below are the formulas that can lead directly to the double angle formulas.

A list of the formulas that students can benefit from recalling:

Sum Formulas:
- sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
- cos(a+b) = cos(a)cos(b) – sin(a)sin(b)
- tan(a+b) = [tan(a) +tan(b)] / [1-tan(a)tan(b)]

Pythagorean Identity:
- Sin² (a) + Cos²(a) = 1

This leads to the next topic, an activity for students to attempt the equation on their own.

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

I’m a firm believer that the more often a student can learn something of their own accord, the better off they are. Providing the skeletal structure of the proofs for the double angle formulas of sine, cosine, and tangent might be enough to help students reach the formulas themselves. The major benefit of this is that, even though these are simple proofs, they have a lot of variance on how they may be presented to students and how “hands on” the activity can be.

I have an example worksheet demonstrating this with the first two double angle formulas attached below. This is in extremely hands on format that can be given to students with the formulas needed in the top right corner and the general position where these should be inserted. If needed the instructor could take this a step further and have the different Pythagorean Identities already listed out (I.e. Cos²(a) = 1 – Sin²(a), Sin²(a) = 1 – Cos²(a)) to emphasize that different formats could be needed. This is an extreme that wouldn’t take students any time to reach the conclusions desired. Of course a lot of this information could be dropped to increase the effort needed to reach the conclusion.

A major benefit with this also is that even though they’re simple, students will still feel extremely rewarded from succeeding on this paper on their own, and thus would be more intrinsically motivated towards learning trig identities.

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

When it comes to technology in the classroom, I tend to lean more on the careful side. I know me as a person/instructor, and I know I can get carried away and make a mess of things because there was so much excitement over a new toy to play with. I also know that the technology can often detract from the actual math itself, but when it comes to trigonometry, and basically any form of geometric mathematics, it’s absolutely necessary to have a visual aid, and this is where technology excels.

The Wolfram Company has provided hundreds of widgets for this exact purpose, and below, you’ll find one attached that demonstrates that sin(2a) appears to be equal to its identity 2cos(a)sin(a). This is clearly not a rigorous proof, but it will help students visualize how these formulas interact with each other and how they may be similar. The fact that it isn’t rigorous may even convince students to try to debunk it. If you can make a student just irritated enough that they spend a few minutes trying to find a way to show you that you’re wrong, then you’ve done your job in that you’ve convinced them to try mathematics for a purpose.

After all, at the end of the day, it doesn’t matter how you begin your classroom, or how you engage your students, what matters is that they are engaged, and are willing to learn.

Wolfram does have a free cdf reader for its demonstrations on this website: http://demonstrations.wolfram.com/AVisualProofOfTheDoubleAngleFormulaForSine/

References

Engaging students: Using Pascal’s triangle

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 Daniel Herfeldt. His topic, from Precalculus: using Pascal’s triangle.

A great activity for Pascal’s triangle would be to first have the students find a pattern of odds and evens. The first thing that you would do is to print out blank Pascal’s triangle. You would give each student a paper for them to fill out. They would have to first fill out the triangle themselves. This would give them practice on which numbers to add as well as further see a pattern of what the next one would potentially look like. After they finish, they would have to color in all of the odd numbers a certain color, and followed by coloring all of the even ones a different color. From here, they will see that once you color it is, the even numbers will make an upside down triangle. Next to the biggest triangles, you will see smaller triangles. An example is shown below. When the students have finished, you will show them why it is like that. Then explain what the name of the colored triangle is, which is called the Sierpinski Triangle.

Pascal’s Triangle is used all over mathematics. It is mainly recognized as how to find the coefficients of binomials, as well as a lot of other uses for binomials. What students and many other people do not know, is that this triangle can be used for much more. For example, you are able to use Pascal’s triangle to find the Fibonacci sequence. Although it may be a little harder to find than the coefficients of binomials, it is still possible. If you add up the numbers in a diagonal pattern from right to left, you will be able to find the Fibonacci sequence. Below will be a picture of how this is implemented. Another way that this will help in future courses is that it allows you to find squares of a number easily. If you look at the 3^rd diagonal row, adding two consecutive numbers from left to right will give the square of a number. A picture of this will also be posted below. Another way that this is implemented in future courses is statistics and probability. This triangle can be used to find the probability of many different things. This is only a few ways that the triangle can be used in future courses, considering that there are plenty of other ways it can be used. In all, this is a very important topic for someone that is pursuing mathematics.

Fibonacci sequence:

Triangular numbers:

This video would be a great way to either start a lesson on Pascal’s Triangle or to review the lesson before a test. The video shows different ways that you can implement the triangle to solve different things in mathematics. If this was the video to start the lesson, I would have each student take out a notebook and writing utensil while watching the video. Throughout the video the students would have to find at least three different ways a person may use Pascal’s triangle that they found particularly interesting. This should lead to most of the ways to be picked by at least one student. After they share their answers, explain further why these work. This could make students more intrigued with the subject. If the video was for a review of the topic, I would also have the students have out a writing utensil and a notebook. For this instance, I would have each individual write down what they had forgotten about Pascal’s triangle. From here the teacher will review the points that were most forgotten, serving as a review.