Engaging students: Defining sine, cosine and tangent in a right 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 Jessica Williams. Her topic, from Geometry: defining sine, cosine and tangent in a right triangle.

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

I’ve actually had the opportunity to teach this lesson to my 10th graders last semester. It is a difficult concept for the students to understand, however if you teach it in a way the students are actively engaged, it helps extremely. Prior to this lesson, the students knew about the hypotenuse and knew the other sides lengths as “legs.” We started by calling 3 students up to the front to hold up our three triangle posters. (triangle cut outs with the 90 degree angle showing and then there was an agle missing). We asked the students how we could find a missing angle given only one side length. For starters, I demonstrated on one triangle by placing a spray water bottle at the missing angle given, and spray the water across.
I will then ask one student to come up to help me demonstrate on the other two triangles. We asked the student where the water is spraying. All of them said words along the lines of “across, away from the angle.” We eventually got to the word opposite. Then we called two students up to demonstrate with the water bottle to determine which side is opposite. If we always know the hypotenuse is the leg across from 90 degree angle, and the opposite side is the one across from the missing angle, then we discovered the last leg must be the adjacent side. Which adjacent means, “next to” or “beside”. Next, we teacher-lead the students through a SOH-CAH-TOA foldable under the doc cam. This was important because they used to later to answer multiple questions using smart pals. Smart pal questions on the board, allowed for EACH student to have to answer and show their work on their smart pal in order to hold it up once we asked for answers. This allowed for formative assessment for the teachers and for the students to see if they were correctly answering the questions. Next we incorporated a “find someone who” Kagan structure tool, which allowed the students to all be actively engaged and answering questions regarding the task. Then we explained and went over misconceptions as a class. It was a very successful lesson overall, and the students were all actively engaged the entire time!

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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?

Trigonometry was originally developed for the use of sailing as a navigation method. The origins can be traced back to ancient Egypt, the Indus Valley, and Mesopotamia. This was over 4000 years ago. Measuring angles in degrees, minutes, and seconds comes from the Babylonian’s base 60 system of numbers. In 150 B.C.E, Hipparchus made a trigonometric table using sine to solve triangles. Later on, Ptolemy extended the trig calculations in 100 C.E. Also, in interesting fact is the ancient Sinhalese used trig to calculate for water flow. Persian mathematician Abul Wafa introduced the angle addition identities. As you can see, there are MANY different mathematicians who distributed to the topic of trigonometry. A lot of them built upon previous work and discovered new formulas, identities, etc. It’s amazing to see how even trigonometry is used to every day life. You always hear people say, “When will I ever use this is life?” and it bugs me to hear this. However, I always have examples of how math is used in our everyday world and from a past long ago that advanced us to where we are today.

 

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

I would show this video that I found on youtube. I would exclude the first movie example that involves shooting, however the rest are great examples.

Showing movie clips to students is always a great way to grab their attention. Visually showing them that math is a part of movies, and every day life shows them that it is important. This video would also be great to use as practice problem, but blur out one of the side lengths or angles missing. You could play the movie scene then pause it on the part with the triangle and have the students solve for missing angle or side length. It would be a fun activity for the students and involve great practice. You could even make this a homework assignment. It’s engaging to watch and keeps the student’s attention while doing homework. The video shows that math is involved in dancing, buildings, etc. This activity also can excite students to try to find math in the movies or tv shows that the watch. You could assign the students to pay attention to to the next couple of shows or movies the watch and to bring back to class an example or two of how math was incorporated in it. Mathematics goes unnoticed because it is honestly part of our everyday norm/lifestyle.

References: https://www.youtube.com/watch?v=LYNN0OYDUB4

http://www.newworldencyclopedia.org/entry/Trigonometry

 

I Have a Tan

Source: https://www.facebook.com/WowSoPunny/photos/a.985057168177019.1073741829.984774068205329/1634769573205772/?type=3&theater

Trigonometry pun

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

Engaging students: Graphing the sine and cosine functions

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

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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.

 

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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.

 

 

 

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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?

 

            For starters, the word trigonometry comes from the Greek word trignon, meaning “triangle”, and metron, meaning “to measure.” Before the 16th 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: Deriving the double angle formulas for sine, cosine, and tangent

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

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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.

 

 

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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.

 

 

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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

My Favorite One-Liners: Part 106

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Years ago, when I first taught Precalculus at the college level, I was starting a section on trigonometry by reminding my students of the acronym SOHCAHTOA for keeping the trig functions straight:

\sin \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Hypotenuse}},

\cos \theta = \displaystyle \frac{\hbox{Adjacent}}{\hbox{Hypotenuse}},

\tan \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Adjacent}}.

At this point, one of my students volunteered that a previous math teacher had taught her an acrostic to keep these straight: Some Old Hippie Caught Another Hippie Tripping On Acid.

Needless to say, I’ve been passing this pearl of wisdom on to my students ever since.

My Favorite One-Liners: Part 104

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

I use today’s quip when discussing the Taylor series expansions for sine and/or cosine:

\sin x = x - \displaystyle \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \dots

\cos x = 1 - \displaystyle \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \dots

To try to convince students that these intimidating formulas are indeed correct, I’ll ask them to pull out their calculators and compute the first three terms of the above expansion for $x=0.2$, and then compute \sin 0.2. The results:

This generates a pretty predictable reaction, “Whoa; it actually works!” Of course, this shouldn’t be a surprise; calculators actually use the Taylor series expansion (and a few trig identity tricks) when calculating sines and cosines. So, I’ll tell my class,

It’s not like your calculator draws a right triangle, takes out a ruler to measure the lengths of the opposite side and the hypotenuse, and divides to find the sine of an angle.

 

My Favorite One-Liners: Part 100

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Today’s quip is one that I’ll use surprisingly often:

If you ever meet a mathematician at a bar, ask him or her, “What is your favorite application of the Cauchy-Schwartz inequality?”

The point is that the Cauchy-Schwartz inequality arises surprisingly often in the undergraduate mathematics curriculum, and so I make a point to highlight it when I use it. For example, off the top of my head:

1. In trigonometry, the Cauchy-Schwartz inequality states that

|{\bf u} \cdot {\bf v}| \le \; \parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel

for all vectors {\bf u} and {\bf v}. Consequently,

-1 \le \displaystyle \frac{ {\bf u} \cdot {\bf v} } {\parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel} \le 1,

which means that the angle

\theta = \cos^{-1} \left( \displaystyle \frac{ {\bf u} \cdot {\bf v} } {\parallel \!\! {\bf u} \!\! \parallel \cdot \parallel \!\! {\bf v} \!\! \parallel} \right)

is defined. This is the measure of the angle between the two vectors {\bf u} and {\bf v}.

2. In probability and statistics, the standard deviation of a random variable X is defined as

\hbox{SD}(X) = \sqrt{E(X^2) - [E(X)]^2}.

The Cauchy-Schwartz inequality assures that the quantity under the square root is nonnegative, so that the standard deviation is actually defined. Also, the Cauchy-Schwartz inequality can be used to show that \hbox{SD}(X) = 0 implies that X is a constant almost surely.

3. Also in probability and statistics, the correlation between two random variables X and Y must satisfy

-1 \le \hbox{Corr}(X,Y) \le 1.

Furthermore, if \hbox{Corr}(X,Y)=1, then Y= aX +b for some constants a and b, where a > 0. On the other hand, if \hbox{Corr}(X,Y)=-1, if \hbox{Corr}(X,Y)=1, then Y= aX +b for some constants a and b, where a < 0.

Since I’m a mathematician, I guess my favorite application of the Cauchy-Schwartz inequality appears in my first professional article, where the inequality was used to confirm some new bounds that I derived with my graduate adviser.

My Favorite One-Liners: Part 76

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here’s a problem that might arise in trigonometry:

Compute \cos \displaystyle \frac{2017\pi}{6}.

To begin, we observe that \displaystyle \frac{2017}{6} = 336 + \displaystyle \frac{1}{6}, so that

\cos \displaystyle \frac{2017\pi}{6} = \cos \left( \displaystyle 336\pi + \frac{\pi}{6} \right).

We then remember that \cos \theta is a periodic function with period 2\pi. This means that we can add or subtract any multiple of 2\pi to the angle, and the result of the function doesn’t change. In particular, -336\pi is a multiple of 2 \pi, so that

\cos \displaystyle \frac{2017\pi}{6} = \cos \left( \displaystyle 336\pi + \frac{\pi}{6} \right)

= \cos \left( \displaystyle 336\pi + \frac{\pi}{6} - 336\pi \right)

= \cos \displaystyle \frac{\pi}{6}

= \displaystyle \frac{\sqrt{3}}{2}.

Said another way, 336\pi corresponds to 336/2 = 168 complete rotations, and the value of cosine doesn’t change with a complete rotation. So it’s OK to just throw away any even multiple of \pi when computing the sine or cosine of a very large angle. I then tell my class:

In mathematics, there’s a technical term for this idea; it’s called \pi throwing.

My Favorite One-Liners: Part 40

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

In some classes, the Greek letter \phi or \Phi naturally appears. Sometimes, it’s an angle in a triangle or a displacement when graphing a sinusoidal function. Other times, it represents the cumulative distribution function of a standard normal distribution.

Which begs the question, how should a student pronounce this symbol?

I tell my students that this is the Greek letter “phi,” pronounced “fee”. However, other mathematicians may pronounce it as “fie,” rhyming with “high”. Continuing,

Other mathematicians pronounce it as “foe.” Others, as “fum.”