I’ll use this one-liner when I ask my students to do something that’s a little conventional but nevertheless within their grasp. For example, consider the following calculation using a half-angle trigonometric identity:

That’s certainly a very complicated calculation, with plenty of predictable places where a student might make an inadvertent mistake.

In my experience, one somewhat surprising place that can trip up students seeing such a calculation for the first time is the very first step: changing into . Upon reflection, perhaps this isn’t so surprising: students are very accustomed to taking a complicated expression like and making it simpler. However, they aren’t often asked to take a simple expression like and make it more complicated.

So I try to make this explicitly clear to my students. A lot of times, we want to make a complicated expression simple. Sometimes, we have to go the other direction and make a simple expression more complicated. Students should be able to do both. And, to try to make this memorable for my students, I use my one-liner:

“In the words of the great philosopher, you gotta know when to hold ’em and know when to fold ’em.”

Yes, that’s an old song reference. My experience is that most students have heard the line before but unfortunately can’t identify the singer: the late, great Kenny Rogers.

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 Cody Luttrell. His topic, from Precalculus: using right-triangle trigonometry.

A.1 Now that students are able to use right triangle trigonometry, there is many things that they can do. For example, they know how to take the height of buildings if needed. If they are standing 45 feet away from a building and they have to look up approximately 60 degrees to see the top of the building, they can approximate the height of the building by using what they know about right triangle trigonometry. Ideally, they would say that the tan(60 degree)= (Height of building)/(distance from building = 45). They can now solve for the height of the building. The students could also use right triangle trigonometry to solve for the elevation it takes to look at the top of a building if they know the distance they are from the building and the height of the building. It would be set up as the previous example, but the students would be using inverse cosine to solve for the elevation.

A.2 An engaging activity and/or project I could do would be to find the height of a pump launch rocket. Let’s say I can find a rocket that states that it can travel up to 50 feet into the air. I could pose this problem to my students and ask how we can test to see if that is true. Some students may guess and say by using a measuring tape, ladder, etc. to measure the height of the rocket. I would then introduce right triangle trigonometry to the students. After a couple of days of practice, we can come back to the question of the height of the rocket. I could ask how the students could find the height of the rocket by using what we have just learned. Ideally, I would want to here that we can use tangent to find the height of the rocket. By using altimeters, I would then have the students stand at different distances from the rocket and measure the altitude. They would then compute the height of the rocket.

D.1 In the late 6th century BC, the Greek mathematician Pythagoras gave us the Pythagorean Theorem. This states that in a right triangle, the distance of the two legs of a right triangle squared added together is equal to the distance of the hypotenuse squared (). This actually was a special case for the law of cosines (). By also just knowing 2 side lengths of a right triangle, one may use the Pythagorean Theorem to solve for the third side which will then in return be able to give you the six trigonometric values for a right triangle. The Pythagorean Theorem also contributes to one of the most know trigonometric identities, . This can be seen in the unit circle where the legs of the right triangle are and and the hypotenuse is 1 unit long. Because Pythagoras gave us the Pythagorean Theorem, we were then able to solve more complex problems by using right triangle trigonometry.

I used these shirts as props when teaching Precalculus this week, and they worked like a charm.

After deriving the three Pythagorean identities from trigonometry, I told my class that I got these hand-made his-and-hers T-shirts for my wife’s birthday a couple of years ago. If you can’t see from the picture, one says and the other .

After holding up the shirts, I then asked the class what mathematical message was being communicated.

After a few seconds, someone ventured a guess: “We add up to 1?”

I answered, “That’s right. Together, we’re one.”

Whereupon the class spontaneously reacted with a loud “Awwwwwwwwww.”

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 Cameron Story. His topic, from Precalculus: using right-triangle trigonometry.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Most right-angle trigonometry word problems involve giving two measurements of a triangle (angle, sides or both) and asking the students to solve for the missing piece. I argue that these problems are fine for practice, but one has to admit these problems encourage “plugging and chugging” along with their formula sheets.

To make things interesting, I would use something along the lines of this word problem from purplemath.com:

“You use a transit to measure the angle of the sun in the sky; the sun fills 34′ of arc. Assuming the sun is 92,919,800 miles away, find the diameter of the sun. Round your answer to the nearest mile,” (Stapel, 2018).

This is incredible! Using trigonometry, students can find out the diameter of the entire sun just by knowing how far away it is and how much of the sky the sun takes up. If you were to use this word problem in a experimental type of project, I strongly recommend using the moon for measurement instead; you can probably guess why measuring the sun in the sky is a BAD idea.

What are the contributions of various cultures to this topic?

One amazing culture to contribute to the study of triangles and trigonometry were the Ancient Babylonians, who lived in what is now Iraq about 4,000 years ago. Archaeologists have found clay tablets from 1800 BC where the Babylonians carved and recorded various formulas and geometric properties. There were several such tablets found to have been lists of Pythagorean triples, which are integer solutions to the famous equation .

The Greeks, while going through their own philosophical and mathematical renaissance, gave the namesake for trigonometry. Melanie Palen, writer for the blog Owlcation, makes is very clear why trigonometry “… sounds triangle-y.” The word trigonometry is derived from two Greek words – ‘trigonon’ which means ‘triangle’ and ‘metron’ meaning ‘measure.’ “Put together, the words mean “triangle measuring”” (Palen, 2018).

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

In the YouTube video “Tattoos on Math” by the YouTube channel 3Blue1Brown (link: https://youtu.be/IxNb1WG_Ido), Grant Sanderson offers a unique perspective on the six main trigonometric functions. In the video. Grant explains how his friend Cam has the initials CSC, which is how we notationaly represent the cosecant function. Not only is this engaging because most students wouldn’t even think of seeing tattoos in math class, but also because Grant always backs up the mathematical content in his videos with beautiful animations.

Students know how sine and cosine functions are represented geometrically; these are just the “legs” of a right-angled triangle. Most students, however, only see the other four trigonometric functions as formulas to be solved. However, as Grant cleverly explains and visualizes in this video, all of these functions have geometric representations as well when paired with the unit circle. This video (moreover, this entire YouTube channel) can be helpful to those visual-learning students who need more than a formula to be convinced of something like the cosecant function.

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.

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

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.

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.

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.

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:

Sin^{2} (a) + Cos^{2}(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. Cos^{2}(a) = 1 – Sin^{2}(a), Sin^{2}(a) = 1 – Cos^{2}(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.

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:

,

,

.

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.

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

for all vectors and . Consequently,

,

which means that the angle

is defined. This is the measure of the angle between the two vectors and .

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

.

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 implies that is a constant almost surely.

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

.

Furthermore, if , then for some constants and , where . On the other hand, if , if , then for some constants and , where .

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.