Engaging students: Graphing 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 Christian Oropeza. His topic, from Precalculus: graphing sine and cosine functions.

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

An activity for students to understand how to graph sine and cosine could consist the use of website desmos (Reference 1). In this activity students will be in pairs and must complete a worksheet that list different forms of the equation sine and cosine that illustrate some of the transformations for sine and cosine. One student will enter the equation onto desmos and the other student will draw the graph on a separate worksheet. The pair will switch roles after each equation, so both students understand how to interpret and draw a given sine or cosine equation. After all the equations have been graphed and sketched, each pair will move on to the next part of the activity in which they must manipulate the equation asin(bx+c)+d and acos(bx+c)+d on desmos where a,b,c, and d are all numerical sliders that can be adjusted to help students visually interpret what transformation they represent. Finally, to prove that students understood the material, each pair will come up with a sketch of a transformation of sine or cosine and trade with another pair of students, in which they must figure out the corresponding equation that matches the given graph.

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

This topic comes up any subject that has sinusoidal waves, such as physics, calculus, and some engineering classes. For example, in calculus graphing the derivative of sine gives the graph of cosine. This shows students that the slope at any point on the sine curve is the cosine and the slope of any point on the cosine curve is the negative of the sine. The topic of sine and cosine is a crucial component in electrical engineering (EE). For EE, there’s a class called, circuit analysis that has a section named “Euler’s Sine Wave” and “Euler’s Cosine Wave”, which incorporates the use of Euler’s formula (Reference 2). Also, in electrical engineering, there’s a machine called a “signal generator”, which sends different types of signals as inputs to circuit. This machine can alter the frequency and amplitude of the signal, where amplitude represents the amount of voltage inputted into the circuit. In math, there’s a topic called “Fourier Series” that also incorporates sine and cosine (Reference 3).

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.

Desmos (Reference 1), can be used to show students the different transformations of both functions. This way students can visually understand what each component is and how each component affects the functions y=asin(bx+c)+d and y=acos(bx+c)+d, where a is the amplitude, b is the period, c is the phase shift, and d is the vertical shift. Vision learning (Reference 4), is also a great website for students when they are introduced to the topic of sine and cosine. This website goes over the history of sine by relating it to waves and circles. The website first goes over how Hipparchus calculated the trigonometric ratios and how that led to the sine function. This website gives students a background on how the functions sine and cosine came to be over time. Also, this website talks about how when the Unit Circle is placed on a Cartesian graph, this illustrates how sine and cosine take over a periodic trend, so students can see why the graphs of sine and cosine are infinite if the domain is all real numbers.

References:

Engaging students: Using right-triangle trigonometry

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 $a^2+b^2=c^2$ .

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.

References:

3Blue1Brown YouTube Video: https://youtu.be/IxNb1WG_Ido

Palen, Melanie. “What Is Trigonometry? Description & History of Trig.” Owlcation, Owlcation, 25 July 2018, owlcation.com/stem/What-is-Trigonometry.

Stapel, Elizabeth. “Right-Triangle Word Problems.” Purplemath, 2018, http://www.purplemath.com/modules/rghtprob.html

Engaging students: Ratios and rates of change

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Cameron Story. His topic, from Algebra: ratios and rates of change.

What interesting word problems using this topic can your students do now?

Since the most relatable example of a ratio is speed (meters per second, miles per hour, etc.), it’s easy to see how a teacher can make an interesting or engaging word problem out of this. First, however, let us take a look at an infamous word problem involving ratios/rates of change that is not inherently interesting on its own.

“Train A, traveling 70 miles per hour (mph), leaves Westford heading toward Eastford, 260 miles away. At the same time Train B, traveling 60 mph, leaves Eastford heading toward Westford. When do the two trains meet? How far from each city do they meet?” (“The Two Trains.” Mathforum.org, National Council of Teachers of Mathematics, mathforum.org/dr.math/faq/faq.two.trains.html.)

This is a distance-over-time that most students or past students are familiar with, but why is this problem still being used? There are a few issues I have with this example. Firstly, I cannot think of very many students who could honestly get excited about trains, especially now in the modern era of vehicular travel. I am willing to bet that most of your high school math students have never even been on a train; and if they have, it was most likely an underwhelming experience. This example also lacks creativity. Giving the trains actual names or having them traveling between real world places would have been a step in the right direction.

So how can we change this example to become engaging to students? Firstly, let’s replace the trains with modern cars, and crank up the speed. Every student is familiar with cars, and fast-moving cars (in my opinion) is much more exciting. One could easily imagine using modern rockets as the vehicle as well, and replacing the towns with interplanetary destinations. Next, instead of naming the cars Car A and Car B, we can use actual modern electric cars such as the Model 3 from Tesla Motors. Take a look of the following word problem I came up with instead (you may notice the stakes of the situation described is objectively more engaging then a problem about train travel):

“Tesla is hoping to feature one of its new cars in a commercial, in which a car attempts to race underneath a falling refrigerator in dramatic fashion. In the commercial, the car must travel at top speed, traveling over 25 meters of track from start to finish. As soon as the car passes the starting line, the fridge is dropped from 10 meters up in the air above the finish line, at a rate of 20 meters per second. The top speeds (in meters per second) of the Tesla Model 3 and the Tesla Roadster are shown below. Which car should Tesla pick to safely beat the falling fridge?”

The reason a creative approach works better is that it increases the student engagement; students do not want to do word problems, so it is our job as teachers to make them interesting.

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

Creating an activity around rates of change allows for a lot of creativity. For example, one could take a physical approach, in which students record how fast they can run (only requires a stop watch and a set distance) and using that to plot their data on a distance vs. time graph.

It is important to remember that ratios can represent far more than just speed. Some relatable examples include rate of hair growth, number of hours studied per week, or even how many gallons of water drank in a day. For my Tesla commercial word problem, I used a website (desmos.com) to flesh out this one problem into an engaging classroom activity. Having your classroom activities on interactive platforms that evoke teamwork and cooperation in your students is key to student engagement.

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

Desmos Classroom Activities (at teacher.desmos.com) is an incredibly useful tool that teachers can use to quickly create any activity for their students. These activities can even be done on smart phones, which removes some of the hassle of getting computers in the classroom. When creating an activity, teachers also have access to a wide range of tools including (but not limited to) animation, student inputs, information slides (for presentation), and even interactive functions that allow students to modify given equations.

The main benefit of using Desmos for classroom activities is that the teacher has full and complete access to viewing student progress. Instead of walking around the room trying to hunt down students who need help, the teacher can view which students are stuck on which problems. The teacher can then approach the issue fully prepared, and know exactly which students are having problems before their hands even hit the air.

I created a Desmos activity available for use in a lesson about ratios or rates of change (link: https://teacher.desmos.com/activitybuilder/custom/5b887ad92c2ff330af6b87c0) which uses the same Tesla commercial word problem I gave before. Using this website, I was able to build this world problem into a somewhat-realistic and animated simulation, asking critical questions in order to build upon the underlying mathematical concepts. Feel free to adapt my lesson (Desmos has a copy/edit feature for activities) for any vehicle, scenario, or speed.

References:

“Desmos Classroom Activities.” Desmos Classroom Activities, 2010, teacher.desmos.com/.

Story, Cameron. “Ratios and Rates of Change Activity.” Desmos Classroom Activities, 30 Aug. 2018, teacher.desmos.com/activitybuilder/custom/5b887ad92c2ff330af6b87c0.

“The Two Trains.” Mathforum.org, National Council of Teachers of Mathematics, mathforum.org/dr.math/faq/faq.two.trains.html.

Vertical line test

Source: https://scontent-dfw5-1.xx.fbcdn.net/v/t1.0-9/44665774_2388027201214092_5864075778743861248_n.jpg?_nc_cat=100&_nc_ht=scontent-dfw5-1.xx&oh=ee5e18af1ccda3f69c03ed9bdc35a280&oe=5CA63AEC

Fibonacci joke

Source: https://www.facebook.com/Tysonism/photos/a.313776598684093/1921522124576191/?type=3&theater

Cow-culus

Source: https://www.facebook.com/PunsOnly/photos/a.441764155928376/1469094213195360/?type=3&theater

Matrix Jokes

A lot more Matrix jokes can be found at https://mathwithbaddrawings.com/2018/03/07/matrix-jokes/

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 Danielle Pope. Her topic, from Precalculus: arithmetic sequences.

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

In the future, the topic of arithmetic sequences will be built upon by introducing another sequence, the geometric sequence. A geometric sequence is just a sequence of multiples instead of increasing by a constant. The next topic introduced will be finding the sum of a sequence of numbers. This will be introduced as a series. The summation symbol will also be introduced to kids and they will learn that new notation. Summations will bring along many formulas for finding the leading coefficient and will show up later in Calculus 2 classes when talking about convergence and divergence of series. Another one of the things that kids will always be doing with sequences and series is finding the general form of a given sequence or series. Through school, this idea will never change the sequence and series will just get harder to identify.

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

An arithmetic sequence is a set of numbers that have a constant difference between each term. One of the main people that come up when researching these sequences is Carl Friedrich Gauss. Many math-loving people know him as the “Prince of mathematicians”. He is famous for coming up with the equations to solve the sum of an arithmetic sequence. This comes as no surprise that he came up with this formula. The surprising thing about this realization is that he made it at an age young enough to still be in grade school. Stories say that Gauss was asked to solve for the sum on the board in grade school and used the formula of M ( M + 1 ) / 2 to solve for the correct answer. This just goes to show that anyone can, in fact, contribute to the greater good of mathematics at any age.

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

One of the first civilizations that utilized sequences was the Egyptians. They used the sequence of multiples of 2 to do their multiplication. The basic sequence is 1, 2, 4, 8, 16, 32, … and we are trying to solve 24 x 13 with the process pictured below.

The process behind this is to write the multiple of 2 sequences down the left side of the paper until you reach the largest multiple of 2 without going over the second number being multiplied, in this case, 13. Once that is done set the first term on the right side equal to the first number being multiplied, in this case, 24. Next, multiply the right side by 4 until you get the same amount of terms on the left side. Lastly find the sum of numbers on the left that add to 13, which are 1, 4, and 8. Add the corresponding multiples from the side, 24 + 96 + 192 = 312. The right side sum of the corresponding numbers checked on the left gives the product of the original problem, i.e. 312. This trick is cool to show just on its own but it’s also cool because it uses something as simple as a specific list of numbers aka a sequence of numbers.

References

http://www.softschools.com/facts/scientists/carl_friedrich_gauss_facts/827/

https://rabungapalgebraiii.wikispaces.com/Arithmetic+Sequences+and+Series

Click to access egyptian_arithmetic.pdf

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 Deanna Cravens. Her topic, from Precalculus: introducing the number $e$ .

How was this topic adopted by the mathematical community?

The number e is a relatively newer irrational number if compared to pi. However, it first made its appearance very subtly in 1618. Napier was working on a table of natural logarithms, however it was not noted that the base was e. There were a few other appearances of e but mathematicians had not truly made a connection to it. Eventually in 1683, Jacob Bernoulli was looking at a business application dealing with continuously compounded interest and recognized that the log function and the exponential function were inverses. In 1690, a letter was written by Leibniz and e officially had a name, except it was called ‘b’ at the time. As it comes to no surprise, Euler had his hand in discovering e. He published Introductio in Analysin infinitorum in 1748 where he showed that e is the limit of $(1 + 1/n)^n$ . Now Euler did not explicitly prove that e is irrational, however most people accepted it at that point, but it was indeed later proven.

How could you as a teacher create an activity or project that involves your topic?
Where does the number e come from? Well, the answer is a business application dealing with continuously compounded interest. However, students in a pre-calculus class can easily discover the number e without having to use the calculus behind it. Simply give students this short activity at the beginning of class.

One of the good things about this activity is that it gives a brief snippet of the history of e before students begin to calculate it. Then, students can easily use a calculator and plug in the listed values in the table into the equation $(1+1/n)^n$ . As the numbers get increasingly large, students will notice that they will all appear to be getting closer to 2.718… which is now known as the number e. As a teacher it is important to note that e is like pi, it is an irrational number that goes on forever and doesn’t have any sort of repeating pattern, yet it is extremely important in mathematics.

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

This video would be excellent to show students who are asking, “why is e so important or where does it come from?” The video starts out by stating what e is approximately equal to. Then it gives a brief history about e and talks about compounded interest. It does a great job at explaining compounded interest. It is executed in a way where pre-calculus students can easily understand the concept. It also uses good visual cues to show how it would work. Next it lists several applications of e. These applications include: statistics through the normal curve, biology by modeling population growth, and physics by the exponential decay of a radioactive material. Overall, it does a great job showing the importance of e in real world applications. Thus, showing the importance of e to a pre-calculus students.

References:
1. http://www.classzone.com/eservices/home/pdf/student/LA208CAD.pdf
2. http://www-history.mcs.st-and.ac.uk/HistTopics/e.html
3. https://www.youtube.com/watch?v=R0oUeLQIbIk

Engaging students: Synthetic Division

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 Deetria Bowser. Her topic, from Precalculus: synthetic division.

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.

It is tough to find websites or technology that help with synthetic division, due to the fact that most websites consists of a long list of instruction, which is not engaging. One website that does seem helpful to a student learning synthetic division is: http://emathlab.com. Under the Algebra tab one can select the option for polynomial functions, then synthetic division. Once selected, a synthetic division activity will pop up. In this activity the student is given a polynomial (of third degree most of the time) divided by a degree one polynomial. The student is then expected to correctly fill the cells with the correct numbers for synthetic division. If they do not get the correct number, the cell turns red and they have to keep trying until they get the answer correct. This activity will be beneficial to students because they will be able to get a feel on the correct placement of numbers when using synthetic division. Additionally, this tool will help them realize what to do when they get polynomials, such as $x^3-1$ . Finally this online tool will allow the students to evaluate themselves.

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

The idea of synthetic division is used to find the zeros of a function. One may need to find zeros of a function in a variety of mathematics and science courses. For example, in physics, one may need to find the roots of a trajectory equation. To find said roots, one could use synthetic division. Also an example of finding roots could be used to help in computer programming. On math.stackexchange.com a programming student presents the following problem: “I am currently programming a simulation for a pinball game and want to calculate the time when the ball hits a circle (if they collide at some point). For the calculation part, I’m adding the radius of the ball to the radius of the circle, so that i only have to check if the midpoint of the ball collides with the circle. Of course, the circle is displayed with it’s original radius.

Now for the ball’s (midpoint) trajectory i’ve got these two equations who define the movement of the ball on the x- and y-axis (depending on the gravitational acceleration):

$x(t)=s_x+v_x t, \quad y(t)=s_y+v_y t− \frac{1}{2} g t^2$ ,

with $(s_x,s_y)$ = starting point of the ball, $(v_x,v_y)$ = initial velocity, $g$ = gravitational acceleration and $t$ = time.

To check for collision, I took these two equations and put them into the equation of a circle. Once multiplied out the student got something of the form: $a t^4+bt^3+ct^2+dt+e=0$ . If the coefficients $a,b,c,d,e$ are rational numbers, then he will be able to use synthetic division to find all of the roots, and successfully create his game.

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

In previous courses, students are taught to find zeros by either graphing, guessing and plugging in a number for x and hoping that the result is zero, or using long division. Synthetic division provides a more systematic way of finding zero’s than just guessing, and can prove to be quicker than graphing and using long division. Additionally, synthetic division can expand on the idea of showing something is not a factor. For instance, when one tries to synthetically divide the polynomial $x^4-3x^2+5x-7$ by $x-2$ one will get a remainder of 7. This is another way of proving that $x-2$ is not a factor of $x^4-3x^2+5x-7$ . Also, one now knows what the polynomial $x^4-3x2+5x-7$ is when $x = 2$ . Synthetic division, extends the idea of finding factors and non-factors of polynomials, as well as solutions to polynomials at a specific $x$ .