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: Finding the area of 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 again comes from my former student Andrew Cory. His topic, from Geometry: finding the area of a right triangle.

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

Finding the area of a right triangle opens up the door to all sorts of applications in the future. The next step is the Pythagorean theorem which is used constantly throughout many math courses. The study of right triangles also opens up the world of trigonometry with students will be using in nearly every math course they go on to take. Once knowledge is learned of right triangles, other triangles can be manipulated to look like right triangles, or to create right triangles within normal triangles. Triangles are even utilized when determining things about other shapes as well, such as dividing rectangles into 2 triangles and other manipulations. If they go on to pursue geometry further, the Pythagorean theorem is one of the first couple of theorems proved and used in book 1 of Euclid.

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

Pythagoras was a Greek philosopher that contributed to right triangles. He is credited with discovering possibly one of the most important right triangle properties. A legend says that after he discovered the Pythagorean theorem, he sacrificed an ox, or possibly an entire hecatomb, or 100 cattle, to the gods. The legitimacy of this legend is questioned because there is a widely held belief that he was against blood sacrifices. The Pythagorean theorem was known and used by Babylonians and Indians centuries before Pythagoras, but it is believed he was the first to introduce it to the Greeks. Some suggest that he was also the first to introduce a mathematical proof, however, some say this is implausible since he was never credited with proving any theorem in antiquity.

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

Applications such as Geogebra can be used for any type of geometry activity. It is a great way to help kids visualize what is happening with shapes in geometry, something that is usually a struggle for students. For helping students understand how to find the area of a right triangle, it can easily be shown that if you take a rectangle, or a square, and cut it in half diagonally, you get two right triangles. And since the area of a right triangle is half of the area of a rectangle or square. The various ways that shapes can be manipulated virtually can be a big help for students that learn in different ways. Being able to view shapes in different ways opens doors for students who traditionally struggle seeing a shape in their head, and using it to solve their problems.

Sources

https://en.wikipedia.org/wiki/Pythagoras#In_mathematics

Engaging students: Defining the terms perpendicular and parallel

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 Diana Calderon. Her topic, from Geometry: defining the terms perpendicular and parallel.

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

– This topic of parallel and perpendicular appears in art in the early 1900’s, late 1910’-1930’s. The movement was widely known as De Stijl, which in Dutch means “the style”. This movement had characteristics of “abstract, pared-down aesthetic centered in basic visual elements such as geometric forms and primary colors.” , the two main artists of this artistic movement were Theo can Doesburg and Piet Mondrian. The artistic movement started because of a reaction to the end of World War I, “Partly a reaction against the decorative excesses of Art Deco, the reduced quality of De Stijl art was envisioned by its creators as a universal visual language appropriate to the modern era, a time of a new, spiritualized world order”. As seen below, there are multiple lines, all of which are either perpendicular to each other or parallel, “De Stijl artists espoused a visual language consisting of precisely rendered geometric forms – usually straight lines, squares, and rectangles–and primary colors.”.

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.)

– When we think of geometry a lot of people instantly think of triangles, SOHCAHTOA, and other 2D or 3D shapes. But when I think of geometry I think of the Greeks and Euclid, the literal father of geometry, only because I learned about him in Dr. Cherry’s class. With that being said, Euclid was one of the first mathematicians to define the term parallel, in specific, parallel lines. In 300 BCE Euclid stated some definitions for the basics of geometry, then give five postulates, “The postulates (or axioms) are the assumptions used to define what we now call Euclidean geometry.” The fifth postulate is what we want to focus on because it is called the parallel postulate, “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” He also states how to construct a perpendicular in Proposition 12, “To draw a straight line perpendicular to a given infinite straight line from a given point not on it.”, this construction states that by a given line AB and a point C not on the line then it is possible to construct a perpendicular on line AB.

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

– A good group project for the topic of parallel and perpendicular lines would be to allow the students to create a town. The requirements would be for the student’s town to be no bigger than 100 square inches, the students would have the liberty to create any quadrilateral shape as long as it meets the 100 square feet requirement. Another requirement that the project would have is that there must be at least 4 parallel streets, one perpendicular street that is only perpendicular for one of the parallel streets and finally one diagonal street that intersects 3 parallel streets. A town obviously needs to have shops so the students would be required to put shops within the town but must have an explanation as to why the shops were chosen. Finally the students must bring a physical final product, the shops must be in 3D form, the town area may be made with cardboard, cardstock or any material that would sustain the shops on top of it, the streets and corners of streets must be labeled with the corresponding angles. Finally, when they bring their final piece as a class we will walk around and allow the groups to present their product. As an exit ticket for presentation day the students must turn in the definitions of parallel and perpendicular in their owns words and how it was shown in their project product.

Citations:

o Mondrian Returns to France (Figure 1)
https://worldhistoryproject.org/1919/mondrian-returns-to-france

o The Three Geometries
https://mathstat.slu.edu/escher/index.php/The_Three_Geometries

o Euclid’s Elements I-XIII
https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html#posts

Engaging students: Using the undefined terms of points, line and plane

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 Alec Bui. His topic, from Geometry: using the undefined terms of points, line and plane.

C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

This topic appears in a game that I play called League of Legends. To give context, in the game there are a total of 10 players with 5 players on each team. Within each team, they must work together to achieve the goal of destroying the enemy team’s Nexus, which is their base. Most games are not usually this straightforward. You must work with your team to take objectives such as towers, special buffs, and secure kills. This video provides a great summary as to what this game is https://www.youtube.com/watch?v=BGtROJeMPeE.

This topic appears in the game in a very dynamic way, specifically with the champion’s abilities. Each champion’s abilities have different interactions with the game and how it is used. It depends on your cursor placement, which decides what point in the plane you’re aiming at. Also the distance and position of your cursor from your champion can dictate the line in which the ability is casted towards. All of this is conducted on a plane in which the game is played on. These terms are very basic and is inherently understood when playing the game. These inherent concepts depends on the champions and their abilities. Basically the position of the cursor dictates the path and location in which the ability is used on the plane. In terms of the game, this is how you aim and move depending on your character.One specific champion that comes to mind is Thresh. This video (https://www.youtube.com/watch?v=Sv95nBi7ulQ) goes over the champions abilities. Death Sentence, Dark Passage, and Flay all clearly makes use of points (the position in which the cursor is related to the plane) and lines (in relation to where the cursor is respective to the champion, what path & direction will the ability follows).

My background with League of Legends:

●Played the game for about 3-4 years
●I played for our school at UNT (Esports Club)
●What role I played – ADC (Attack Damage Carry)
●In the U.S, I was ranked top 1.87% (in season 8)
●In the world top 2.1% (in season 8)
●Favorite champion: Lee Sin

D5. How have different cultures throughout time used this topic in their society?

This question and topic reminded me of a YouTube channel which I think uses the undefined terms without explicitly saying so. The channel is called “Primitive Tool” which you can search on YouTube. The channel has a multitude of construction videos of different structures such as pools and houses built back with primitive tools. It makes you think how basic knowledge of lines, points, and plane were naturally used without a mathematical explanation or background. It was very natural to consider this vocabulary in normal day to day life which was continuously used. You can see the progression and advancement of this simple vocabulary in our architecture over human history. Different cultures have used this topic expressed through art, architecture, etc. One that sticks out in my head are the Egyptians. It’s clear that basic mastery of the topic is needed to construct the phenomenal pyramids that stand today.

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

Geometer’s sketchpad would be a great way to engage this topic. It would be a great way for students to explore the different tools with little to no explanation needed. Student’s can play around with the tools up until guided practice is needed up to the discretion of the teacher. It would be far more engaging than simply explaining what these terms are and going through examples. It provides a dynamic way for the students to interact with all of the terms. This allows them to see the relationship between the terms. They can further their exploration by creating shapes and different polygons with the tools. Overall it’s great dynamic way to learn what the terms are instead of a static manner.

Engaging students: Defining the words acute, right, and obtuse

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 Johnny Aviles. His topic: how to engage geometry students when defining the words acute, right, and obtuse.

A1.) How could you as a teacher create an activity or project that involves your topic?

To have the students get engaged with the topic of Defining the terms acute, right, and obtuse, I will begin with having the classroom set up into groups of 4-5. Within their group they will create 10 examples of where each acute, right, and obtuse angles or triangles can be found in the classroom or in the real world in general. For example, the letter Y, end of a sharpened pencil, and the angle under a ladder can be used. They will be given about 10-15 minutes depending on how fast they can all finish. This is a great activity as the students can work together to try to come up with these examples and can familiarize themselves with amount of ways these terms are used in life. I will tell them before I begin the activity that the group that comes up with the most examples will be given extra credit in the next exam or quiz. This will give them extra incentive to stay on task as I am well aware that some groups may finish earlier than the rest and may take that extra time to cause disruptions.

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

In previous courses, students have learned had some exposure to these types of angles. Most students have been familiar with the use of right triangles and have learned methods like the Pythagorean theorem. When we extend the terms acute, right, and obtuse in geometry, it begins to be more intensified. These angles then extend in terms of triangle that will then have many uses. Students will then be expected to not only find missing side lengths but also angles. Students will then be exposed to methods later like, law of sines and cosines, special right triangles, triangle inequality theorem and triangle congruency in. This topic essentially is the stepping stone for a large part of what is soon to be learned. Other courses will use a variety of other was to incorporate the terms acute, right, and obtuse. Geometry, precalculus and trigonometry will essentially have a great deal of uses for these terms for starters and can then also be extended in many higher-level math courses in universities.

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

An effective way to teach this topic using technology and the terms acute, right, and obtuse would be games. There is a magnitude of game that involve angles and be beneficial in the understanding of these angles. I have found this one game called Alien Angles. In this game, you are given the angle of where the friendly alien at and you have to launch your rocket to rescue them. the purpose of the game is for students to be familiar with angles and how to find them. after you launch the rocket, you are given a protractor that shows the angles and I believe this is beneficial for students as they can also be more familiar with the application of protractors. I can post this on the promethean board and have students identify what the angle I need to rescue the aliens. I can then call for volunteers to go on the board and try to find the correct angle to launch the rocket.

https://www.mathplayground.com/alienangles.html

High-flying functions: Marching Jayhawks’ star twirler sees math in her pastime

As an internationally competitive baton twirler in Toronto, Canada, Nicole Johnson impressed audiences with the degree of difficulty found in her acrobatic routines. As a Jayhawk, she found a way to express that complexity using another talent — math. Nicole Johnson received an Undergraduate Research Award for “Baton Twirling into the Fourth Dimension.” Working with associate professor Estela Gavosto, she translated her astonishing performances into equally beautiful graphs. “Just being able to show that there’s this different side of math than what you normally see — that is one of the really cool outcomes of this project,” says Johnson, who graduated in May 2019 with honors in both mathematics and engineering physics.

Engaging students: Using the point-slope equation of a line

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 Johnny Aviles. His topic, from Algebra: using the point-slope equation of a line.

A2 How could you as a teacher create an activity or project that involves your topic?

On the 1987 NBA Dunk Contest, Michael Jordan won by dunking all the way from the free throw line. (I will play them a clip). Now can anyone tell me how high the hoop is from the ground? And how far is the free throw line from the base of the hoop? So, in total he went 10 feet in the air while jumping 15 feet! This is incredibly difficult and was why he won the contest. Now lets just compute that slope. With rise/ run we get that the slope was 2/3. Another example I can use is the time I took to get to school. I live 30 miles away and it took me 40 minutes to get to school. would anyone be able to find the average speed? (45 MPH) Then I will make it more complex and say I went 60 miles an hour for the first 20 minutes, how fast was I going the last 20 minutes?(30 MPH) Then I will have a round robin activity where I will give 5 min for my students to discuss amongst their groups where they can create a scenario where they can use point-slope equation of a line.

C3 How has this topic appeared in the news?

We all have many factors that interest us and the news’ job is to keep us updated. For many people, the stock market is a very serious subject of interest. Everything is shown in charts and done on points and percentages for simplicity reasons. This uses the concept of point-slope equation of a line to create this data. The news also covers may other topics like the rise of current temperature from given years to see if factors like global warming may have played a role to create the next leading story. The data from previous years can create point-slope equation that can predict the rain and snow fall amount for a given city or town. The weather initially can use point-slope equation of a line to predict all factors all data collected over decades. There is a copious amount of data that the news has to be used in all aspects of the news, one that has been shown is the rise of mass shootings. This is a very controversial matter as many people seek reform of the second amendment. Overall, point-slope equation of a line is widely used in many platforms of our news programs.

D4 What are the contributions of various cultures to this topic?

Architecture has been the biggest contribution that point-slope equation of a line and has to be applied. Various cultures have their own specific style of how they have their cities, towns and neighborhoods but all will apply the basics of point-slope equation of a line. For example, when creating a building, they use materials with large mass and need to be supported. If the slope of a beam is even slightly off, it can generally cause the building to collapse under its own weight causing the lives of many. Every aspect of the building needs to be measured in a precise way to create a solid structure. Styles then range from all cultures and can have tilted and rounded with elaborate beams to add more diversity. Overall, all cultures have their own specific style of houses that all require the same point-slope equation of a lines that contributes them to remain standing.