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 DeForest Mitchell. His topic, from Precalculus: graphing sine and cosine functions.

C2: How has this topic appeared in high culture (art, classical music, theatre, etc.)?

There are multiple forms of art and music and theater using sine and cosine. I am going to focus on music. Tone and sound in itself is a reverb of sine functions with different wavelengths and amplitude. This is such a great importance not only in knowing sine functions but also being able to create music. Knowing what a tone is and why the sound is becoming higher/lower or quieter/louder. If the amplitude is shorter then the sound itself will be quieter and vise versa with the amplitude growing so will the over all sound. If the period is increasing or becoming longer, the sound will be deeper, while shorter periods will create sharper and higher pitch sounds. Once you combine the amplitude and the period in specific ways that’s how specific notes and tones are made. As shown below, there is a combination of amplitudes and periods to create a sound wave using sine functions. With this in mind mathematically you can predict what a tone or song will sound like just depending on the inflection of amplitude and frequency of the function.

B1: How can this topic be used in your students’ future courses in mathematics or science?

One big component to my personal understanding of graphing sine and cosine is seeing a physical model of it. Such as a spring in physics, or the light spectrum. There are websites such as https://www.mathsisfun.com/algebra/trig-sin-cos-tan-graphs.html to be able to represent a sine function being sketched as a weighted spring bounces up and down. There are different slides to adjust the stiffness and dampening of the spring to show the alternate forms of the same graph. Since and cosine are used in many different forms of sciences, such as wavelengths (as shown in the picture below). This is a great way to show the students that there is a reason to learn the subject and that there are practical uses for it in life outside of school. For example, with the wavelength spectrum a teacher can make the correlation that a sine function with a longer period would be like soundwaves that have a longer wavelength, also to show that there are only certain wavelengths that we can see and there can be a correlation to graph the sine period and that if you were to make it into a function that there is only a certain few period lengths for a sine function for humans to be able to see colors.

D1: 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.)

Circles have been around for millions of years and are to this day one of the simplest shapes. With Circles and by tracking the circumference of the circle you can make sine and cosine graphs. Joseph Fourier (1768 – 1830 A.D.) was a very influential person with the devilment of graphing sine and cosine functions. He was so due to his the foundning of the Fourier series. “Fourier series (thus the Fourier transform and finally the Discrete Fourier Transform) is our ancient desire to express everything in terms of circles or the most exceptionally simple and elegant abstract human construct. Most people prefer to say the same thing in a more ahistorical manner: to break a function into sines and cosines.” The summative way to say saying that Fourier came up with an equation to take any repeating series and be able to turn it into forms of sine and cosine to be able to graph. Which in turn creates a circle to better understand the said repeating series. This was a great mathematical find to show the correlation between any repeating series and the more well-known sine and cosine terms today. As shown below, with the Fourier series they were able to take a repeating series to “convert” them into a circle and then be able to graph the functions in terms of sine and cosine.

Resources:

https://www.gnu.org/software/gnuastro/manual/html_node/Fourier-series-historical-background.html

Engaging students: Fitting data to a quadratic 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 again comes from my former student Eduardo Torres Manzanarez. His topic, from Algebra: fitting data to a quadratic function.

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

One interesting project that could be done to invoke quadratic modeling is for students to develop a model that fits a business’ data of labor and output. The basic model of labor and output for a given company can be modeled by a quadratic function and it can be used to determine important figures such as the maximum output, minimum output, maximum labor, and minimum labor. The following image is an example of such a relationship.

In general, people would think that the more labor and resources used at the exact same time results in more product. If you have more product produced, then you accumulate more profit. These ideas are not wrong to be thought of but a key aspect that is missed in the thought process is that of land or otherwise known as workspace. The more employees you hire, the more space required so that these individuals can produce but space is limited just like any other resource. Lack of space inhibits production flow and therefore decreases product, decreases profits, and increases cost through increased wages. All of this does not occur until you pass the maximum of the model. So, both of these behaviors are shown and exhibited by a quadratic function. Students can realize these notions of labor and production by analyzing data of various companies. An activity that could show such a relationship in action is having one student create a small particular product such as a card with a particular design and produce as many as they can in a certain amount of time, with certain resources, and a workspace. Record the number of cards produced. Next, have two students create cards with the exact same time, resources, and workspace and record the amount produced. As more students are involved, the behavior of labor and production will be shown to be direct and then inverse to each other. The final piece for this activity would be for students to find realize what function seems to have the same shape as the data on a graph and for them to manipulate the function so that it fits on the data. Turns out the function will have to be a quadratic function.

B1) How can this topic be used in your students’ future courses in mathematics or science?

Fitting data onto a quadratic function is useful in analyzing behavior between variables. In various mathematical courses, data is provided but in science usually one must come up with data through an experiment. Particularly there are many situations in physics where this is the case and relationships have to be modeled by fitting data onto various functions. Doing quadratic modeling and even linear modeling early on is a good introduction into other models that are used in the many fields of science. Not every experiment is recorded perfectly and hence there can never be a perfect model. Through analytical skills presented in this topic, it scaffolds students to find a model for bacteria growth, a model for velocity, a model for the position of an object, and a model for nuclear decay in the future and what to expect the behavior of these models to be. This topic in combination with limits from calculus builds onto piece-wise models for probability and statistics.

E1) How can technology be used to effectively engage students with this topic?

Technology such as graphing calculators, Excel, Desmos, and TI-Nspires can be used to create the best model possible based on least-squares regression. This technology is engaging in developing models, not because of the lack of convoluted math that deals with squaring differences but rather the focus on analyzing particular models such as a quadratic model. They could be engaging for students when students can input particular sets of data they find interesting and need a way to model it. Furthermore, students can use technology to develop beautiful graphs that can be easily interpreted than rough sketches of these models. TI-Nspire software can be used by a teacher to send a particular data set to students and their own TI-Nspires. Students can then insert a quadratic function on the graphing application and manipulate the function by changing its overall shape by the mouse cursor. This allows students to dictate their own particular models and allows for comparison between models as to which is more accurate for particular data.

References

Engaging students: Adding, subtracting, and multiplying matrices

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 again comes from my former student Diana Calderon. Her topic, from Algebra: adding, subtracting, and multiplying matrices.

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

As a teacher I would do a foldable activity in which the students will have to fill in the blank in the front of the foldable that would allow them to discover how addition, multiplication and subtraction work for matrices. Once they open the foldable, they would have to do different examples and get to also create one. Out of the problems that the students create, as a teacher, I would choose one of each and allow them to go up to the board and explain how they did it and address any misconceptions that may have happened when they were discovering how the concepts work. I plan on doing my foldable with color coding so that the students can see where the numbers in the columns and rows changed when the matrices were added, multiplied, or subtracted, I will most likely limit the matrices to vary from 2×1,2×2,2×3,3×2,and 3×3.

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

– The topic of adding, subtracting and multiplying matrices allows students to extend their knowledge when it comes to adding, multiplying or subtracting polynomials. I can show the students how a polynomial is similar to a 1×1 matrix. Another subject that they may see something similar to matrices would be in Biology with the punnett squares, it can be as basic as doing it for one generation or two and then go from there on. As said in the article “Use of Matrices to Determine Genetic Probability” by Andrew Almendarez, “Through prescribed manipulations and interpretations matrices can be used to represent and solve physical problems. One such problem is finding the probability of a certain genotype within a population over multiple generations.”, this also ties into probability which they most likely learned the previous year. It would be good to tell them that if they are interested in the medical lab field for example, “trying to breed cows that produce the most milk. If cows of a certain genotype were known to produce more milk than others it is useful to know how many cows of that genotype there will be after a number of generations, and what will maximize the proportion of that genotype in the future. This is where the Punnett is used in conjunction with matrices”.

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

The topic of matrix multiplication came up when I looked in the news. Recently, apple has been one of the most popular brands when it comes to computers, cellular devices, TV, ear phones, etc. With that being said, every year or so they release a new “it” item. This month they are releasing the new iPhone 11, which overall in my opinion is the best cellular device one can get, it has a sleek professional design, great camera, a huge amount of storage embedded within itself and many other useful resources that one utilizes in their everyday life. In the article “iPhone 11: Apple’s A13 Bionic Chip Enables More Than Just Faster Speed” it mentions matrices and how “The A13 Bionic has a whopping 8.5 billion second-generation seven-nanometer transistors, up from 6.9 billion in the previous generation. It can perform one trillion operations per second, thanks in part to new machine learning accelerators that can run matrix multiplication six times faster.”. For me it is amazing to know just how fast these devices can calculate anything and everything one wants to find out instead of doing it by hand.

Citations:
• Use of Matrices to Determine Genetic Probability

• iPhone 11: Apple’s A13 Bionic Chip Enables More Than Just Faster Speed
https://www.inverse.com/article/59239-a13-chip-faster-more-efficient

Visualizing Vectors

From the Math Values blog of the Mathematical Association of America:

Anyone who has taught linear algebra knows how easy it is for students to get absorbed in performing matrix computations and memorizing theorems, losing the beauty of the structures in this foundational subject. James Factor and Susan Pustejovsky of Alverno College in Milwaukee, WI, bring back the visual beauty of linear algebra through their NSF-funded project Transforming Linear Algebra Education with GeoGebra Applets.

The applets are freely available in the GeoGebra book Transforming Linear Algebra Education https://www.geogebra.org/m/XnfUWvvp. Each topic is packaged with a video to show how the applets work, the applet, and learning activities.

Left-Hand Rule?

Misleading pictures in math textbooks always send 10,000 volts of electricity down my spine. Thanks to the right-hand rule, the cross product should be pointing down, not up. This comes from the 2007 edition of Glencoe’s “Advanced Mathematical Topics,” a high-school Precalculus book.

For what it’s worth, this is the same line of textbooks that, in a supplementary publication, said that the rational numbers are not countable.

Once upon a time in algebra class…

Side note: Yes, there’s only one true exponential curve on the graph. Yes, the spread of COVID-19 is best modeled with a logistic growth curve or an SEIR model. Nevertheless, this comic absolutely rings true.

Slightly Incorrect Ugly Mathematical Christmas T-Shirts: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on slightly incorrect ugly mathematical Christmas T-shirts.

Part 1: Missing digits in the expansion of $\pi$.

Part 2: Incorrect computation of Pascal’s triangle.

Part 3: Incorrect name of Pascal’s triangle.

The IRS Uses Geometric Series?

I recently read the delightful article “The IRS Uses Geometric Series?” by Michelle Ghrist in the August/September 2019 issue of MAA FOCUS. The article concerns a church raffle for a \$4000 ATV in which the church would pay for the tax bill of the winner. This turned out to be an unexpected real-world application of an infinite geometric series. A few key quotes: According to the IRS rules at the time,

…winnings below a certain level [were] subject to a 25% regular gambling withholding tax…

My initial thought was that the church would need to pay $0.25 \times \4000 = \1000$ to the IRS. However, I then wondered if this extra $\1000$ payment would then be considered part of the prize and therefore also subject to 25% withholding, requiring the church to give $0.25 \times \1000 = \250$ more to the IRS. But then this $\250$ would also be part of the prize and subject to withholding, with this process continuing forever.

I got quite excited about the possibility of an infinite geometric series being necessary to implement IRS tax code. By my calculations… [gave] an effective tax rate of 33-1/3%.

I then read more of the instructions, which clarified if the payer pays the withholding tax rate for the payee, “the withholding is 33.33% of the FMV [Fair Market Value] of the noncash payment minus the amount of the wager.” It was satisfying to discover the behind-the-scenes math leading to that number…

In any event, I am glad to know that the IRS can properly apply geometric series.”

Here’s a link to the whole article: http://digitaleditions.walsworthprintgroup.com/publication/?m=7656&l=1#{%22issue_id%22:606088,%22page%22:%2214%22}

Note: The authors notes that, in January 2018, the IRS dropped the two above rates to 24% and 31.58%.

Decimal Approximations of Logarithms: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the decimal expansions of logarithms.

Part 1: Pedagogical motivation: how can students develop a better understanding for the apparently random jumble of digits in irrational logarithms?

Part 2: Idea: use large powers.

Part 3: Further idea: use very large powers.

Part 4: Connect to continued fractions and convergents.

Part 5: Tips for students to find these very large powers.