Engaging students: Using vectors in two dimensions

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 Zacquiri Rutledge. His topic, from Precalculus: using vectors in two dimensions.

Add vector A and vector B, what do you get? How about when we take the dot product of A and B? What is the magnitude of A, B, and A+B? All of these are basic questions that a teacher might ask their students during a basic high school pre-calculus class. However, how does the teacher respond when a student asks “Where am I ever going to see this again”? In mathematics, a student might never see vectors again unless they take higher math such as Calculus I through III, or possibly Linear Algebra. During the first two courses of Calculus students will continue to expand on the ideas of two dimensional vectors by talking about the path an object might take through the air after leaving a cannon or being thrown off a cliff. Calculus III (or vector calculus) however is a much stronger example of how vectors will be used in further education of mathematics. During this class students will not only look at two dimensional vectors and review simpler ideas, but they will expand these ideas into the three dimensional world creating three dimensional vectors. Here students will discuss what kind of shape or planes a combination of three vectors might create.

A scientific use for two dimensional vectors is in physics. During a physics class, students talk about forces that act on objects as they move or when an object hits another. To do this, students draw vectors to represent the magnitude of the force that is acting on the object and the direction the force pushes or pulls the object. For example, in the previous paragraph it was mentioned about an object being shot from a cannon and the students measuring the path the object might take. In physics, the students might do the exact same thing, but by looking more in depth at the forces acting on the object. Forces might include the force of the cannon firing the object at a certain angle into the air, gravity pulling that object toward the ground, and even the friction of the air on the object as it soars through the air. Each one of these forces is acting on the object as it moves, either helping the object move farther and faster or attempting to slow it down. However, two dimensional physics is not the end of vectors; just like calculus, physics goes on to discuss what happens to objects in a three dimensional world and the forces that act on them. So a very easy answer to give the student asking where he/she will see vectors again is in every day real life.

You have explained to your students that vectors are in everyday life and they still do not believe you. You have shown them countless examples on the board, drawing pictures of airplanes and the paths they fly through the air, objects being dropped from a cliff, objects being shot from a cannon, et cetera, and they still do not believe that vectors have any importance or use! Then you simply ask, has anyone ever seen a show called MythBusters on the Discovery Channel? Now MythBusters is a very well-known show, not only because it has been around for twelve years, but also for some of the crazy things that they test in the name of science. For example, some of my personal favorites include them making a boat out of pykrete, the many episodes on the uses of duct tape, and testing if a bullet dropped at the same time as a bullet shot from a gun will hit the ground at the same time. The great thing about this show is it is full of great examples of how physics affects things in real life. Also, not only do they test the myths, they explain how they are testing them, why they are testing them the way they are, and why it makes sense scientifically or does not. For example, during the bullet episode, they explain that once the bullet is shot from the gun, the only forces acting on the bullet are gravity and air friction. The only forces that would be acting on the bullet dropped would gravity and air friction as well. So in theory these two bullets should hit the ground at the exact same time if they are projected from the same height. By the end of the episode they had proven this by figuring out the best way to set up a live test and using a high speed camera to measure the time it took for each to hit the ground. For a high school class, this would be very easy to draw on a chalk board and walk the students through the thought process of why this happens using vectors to draw out the forces.

Finally, the Internet gives us access to a lot of videos. This would allow a teacher who is talking about Mythbusters and their amazing examples of vectors in motion the chance to display quick clips of some of their tests. Of course the teacher will need to have researched a few before class in order to make sure they can be used as vector examples, but after a video has been played the teacher could ask the students to explain why the test was either plausible or false. On a small scale this video, https://www.youtube.com/watch?v=BLuI118nhzc , works great to show how a truck moving at the same speed as a soccer ball being shot from the back cancels the two forces, leaving gravity as the only force acting on the ball. Using vectors, a teacher could explain how one vector is positive and the other is negative of the same magnitude, cancelling the other out. Then show how only one vector on the ball remains, pulling the ball in that direction.

Engaging students: Fibonacci sequence

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 Taylor Vaughn. Her topic, from Precalculus: the Fibonacci sequence.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?
The article “Music and the Fibonacci sequence and Phi” talks about how the Fibonacci sequence correlates how the keys on the piano are laid out. Also talks about how the sequence also affects the frequency of chords. One thing I like about this article is that it doesn’t just talk about the sequence in one way, like how the instrument is made, but also different aspects like the chord. In other showings of the Fibonacci sequence they talk about pine cones and flowers. Personally, I think that music is something that is more relatable to students and a lot of people have seen a piano, but never just thought about the making. People who are that involved in music, probably noticed that there was some pattern to the keys, but didn’t think that by any chance that it was related to math.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
Since the sequence is named Fibonacci sequence you may think that the founder’s name is Fibonacci, but actually his name is Leonardo of Pisa. The nickname Fibonacci comes from the shortening of the Latin term “filius Bonacci”, which means son of Bonacci. Well why does that matter? That was his dad’s last name. Also, the Latin phrase is incorporated in the title of his book. One thing that I found cool, was that Leonardo actually had a North African education. When talking about mathematicians you never hear anything about Africa. So let’s look at the history of the sequence itself. After reading a few articles, some believe that he actually didn’t discover the sequence himself, but merely saw it during his travels and he was the one to actually write about it. Edouard Lucas is the person who named the sequence, the Fibonacci sequence. When Fibonacci wrote about the sequence it was in the 1200’s, Lucas wasn’t around until the 1800’s. That is 600 years that the sequence didn’t have a name. So during that time, what did people refer to it as? I really don’t know. Lucas is the person to look more into the sequence and noticed that the numbers have a common ratio, which is now called the golden ratio, he also discovered other patterns that lie in the sequence.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?
The video Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3] was an engaging video because it actually shows the sequence in different objects. For example, when she talks about the sequence in pine cones she actually gets glitter paint, and shows and counts the diagonals on the pine cone. Also I like that she uses pine cones of different sizes and shapes, and shows that the pattern still holds s that students don’t think that it was planned that she picked up that type of pine cone. I also like that she brings in relevant object like fruit. I think this is a good engage because it shows patterns of things that students see often, but never stopped and paused to think about. One thing I don’t like about Vi Hart is the speed that she talks. I normally have to watch the video multiple times to get all the information she gives. In a classroom, you really don’t have the time to allow students to watch the video multiple times. This video could also be given as homework before their lesson and it would allow students to watch it multiple times and could turn in their notes, or provide questions for them to answer. I definitely think that the video is cool and would spike some interest in entering sequences.

Citations
Meisner, Gary. “Music and the Fibonacci Sequence and Phi – The Golden Ratio: Phi, 1.618.” The Golden Ratio Phi 1618. N.p., 04 May 2012. Web. 15 Nov. 2015.
Knott, Dr. Ron. “Contents of This Page.” Who Was Fibonacci? Ron Knott, 11 Mar. 1998. Web. 15 Nov. 2015.
Knott, Ron, and The Plus Team. “The Life and Numbers of Fibonacci.” The Life and Numbers of Fibonacci. N.p., 4 Nov. 2013. Web. 15 Nov. 2015.
Hart, Vi. “Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3].” YouTube. YouTube, 21 Dec. 2011. Web. 15 Nov. 2015.

Engaging students: Graphing rational functions

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 Rory VerNesie. His topic, from Precalculus: graphing rational functions.

So one question may be how this topic can be used in your students future courses in math or science? The students start to learn about this concept in high school, and progressively builds on it until they are expected to know it in college. Courses that require this are: Complex Analysis, Numerical Analysis, Differential Equations, Abstract Algebra, Real Analysis and Meromorphic Functions. These classes deal with understanding what happens as we approach a limit or when the denominator approaches zero. In Abstract Algebra, they talk about a Field of Rational Expressions while Complex Analysis deals with a ratio of polynomials with complex coefficients. In Differential Equations, Rational Functions are seen in slope fields, Separable Equations, and Exact equations. Also in Real Analysis, the talk about convergence using 1/n. Also, Laplace Transforms and partial fractions in electronics and physics may need graphing along with partial fraction decomposition. All in all, graphing Rational Functions is a important part of math because they deal with division over zero or a singularity.

Next, a very important man in the history of rational functions and their graphs was Charles Hermite. This man found a way or algorithm to integrate Rational Functions, while in early 19th century Ostrowski extended this idea and algorithm to Rational Expressions. The neat thing about Hermite is that he helped extend this idea to complex numbers and developed the idea of using interpolation to find the coefficients of rational functions. Without these contributions from Hermite and Ostrowski we would not be able to graph the derivatives of rational functions or the anti derivatives of rational functions. The methods discovered by these men were profound and in some ways led to the discovery of news ideas in math such as partial decomposition and other integration techniques that help integrate Rational Functions. Without these men, Rational Functions and there uses would be known about less.

A great activity involving graphing rational functions would be to have the kids get into groups and assign the each group a certain rational function. Every group would have a team leader who would be in charge of making sure everything about the function gets done. These responsibilities would include, graphing the function, finding the zeros of the function, the asymptotes(Horizontal and Vertical), Removable Discontinuities if any, and the y intercept. The students would then present what they found and would answer any questions for the class. This activity would be a good cooperative learning exercise for students who maybe are not the best at math. This could be a major confidence booster and fun activity for the students. Also they students are learning from each other so they are engaging in discovery learning.

All in all, graphing rational functions is a major part of mathematics and all of these statements mentioned above show how important rational functions are. They deal with division by zero and limits and are a great way to engage students in the novelty of singularities. Graphing polynomials also look really neat when you graph them.

Work Cited

http://integrals.wolfram.com/about/history/

http://www.sciencedirect.com/science/article/pii/0898122176900237

tutorial.math.lamar.edu

Slideplayer.com

Engaging students: Graphing with polar coordinates

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 Perla Perez. Her topic, from Precalculus: graphing with polar coordinates.

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

Graphing polar coordinates is usually taught in a Pre-Calculus class. Students have learned about the Cartesian Coordinates and extend their knowledge to polar coordinates. Unlike Cartesian Coordinates, which represent how to get from a specific point to the point of origin (or vice versa), the polar coordinate tells us the direction by the angle, and the distance from that point to the origin. Students will need to know how to take the measure of an angle and how to use the Pythagorean Theorem to solve for the distance which is considered the radius. Most students who are enrolled in a Pre-Calculus class have taken geometry where they have learned about the Pythagorean Theorem and what a radius is. This alongside their algebra 1 and geometry classes means they also know how to graph and plot points.

References:

http://ritter.tea.state.tx.us/rules/tac/chapter111/ch111c.htm

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

Polar coordinates use a different type of graph, rather than just an x and y coordinates plane. The polar coordinate plane includes symmetrical circles surrounding the center and is given a radius creating a graph that looks like a dart board. At this point students should know what a polar coordinate is. The next step is actually graphing it.

As an activity to get students excited for the wonderful world of polar coordinates, I have created a dart board game. Using an appropriate dart board, such as a magnetic one, have the students create groups of three to four student each. The point of the game is to have students create polar coordinates. The board must be properly labeled with the angles. There will be four rounds, depending on the number of members in a group. When a member throws a dart at the board it must land on a point. Wherever it lands the students must figure out the radius and the angle of the dart to the origin. This game enables students to practice finding the radius and the angle of the dart with only their previous knowledge, labels, and each other.

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

Throughout centuries and in all parts of the world, mathematicians and astronomers have come to shape our understanding of the polar coordinate system. Two Greek astronomers Hipparchus and Archimedes used polar coordinates in much of their work. Though they didn’t commit to the full coordinate plane, Hipparchus first begins by writing a table of chords where he was able to define stellar positions. Archimedes focused on a lot on spirals and developed what now known as the Archimedes spiral, in which the radius depends on the angle. Descartes also used a simpler concept of coordinates, but relating more to the x-axis. In 1671, Sir Isaac Newton was one of the biggest contributors to the elements used in analytic geometry. The idea of polar coordinates, however, comes from a man named Gregorio Fontana (1735-1803), centuries later. Astronomers now use his polar coordinates to measure the distance of the sky and stars.

References:

History of Mathematics, Vol. II:

https://books.google.com/books?id=uTytJGnTf1kC&pg=PA324&hl=en#v=onepage&q&f=false

https://en.wikipedia.org/wiki/Polar_coordinate_system

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 Loc Nguyen. His topic, from Precalculus: introducing the number $e$ .

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

To be able to understand where the number e is produced in the first place, students need to understand how compound interest is calculated. Before introducing the number e, I will definitely create an activity for the students to work on so that they can eventually find the formula for compounding interest based on the patterns they produce throughout the process. The compound interest formula is F=P(1+r/n)^nt. From this formula, I will again provide students a worksheet to work on. In this worksheet, I will let P=1, r=100%, t=1, then the compound interest formula will be F=(1+1/n)ⁿ. Now students will compute the final value from yearly to secondly.

When they do all the computation, they will see all the decimal places of the final value lining up as n gets big. And finally, they will see that the final value gets to the fixed value as n goes to infinity. That number is e=2.71828162….,

How has this topic appeared in the news?

To help the students realize how important number e is, I would engage them with the real life examples or applications. There were some news that incorporated exponential curves. First, I will show the students the news about how fast deadly disease Ebola will grow through this link http://www.npr.org/sections/goatsandsoda/2014/09/18/349341606/why-the-math-of-the-ebola-epidemic-is-so-scary. The students will eventually see how exponential curve comes into play. After that I will provide them this link, http://cleantechnica.com/2014/07/22/exponential-growth-global-solar-pv-production-installation/, in this link, the article talked about the global population rate and it provided the scientific evidence that showed the data collected represent the exponential curve. Up to this point, I will show the students that the population growth model is:

Those examples above was about the growth. For the next example, I will ask the students that how the scientists figured out the age of the earth. In this link, http://earthsky.org/earth/how-old-is-the-earth, the students will learn that the scientists used Modern radiometric dating methods to calculate the age of earth. At this time, I will show them radioactive decay formula and explain to them that this formula is used to determine the lives of the substances such as rocks:

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

To introduce to the students what the number e is, I will engage them with two videos. In the first video, https://www.youtube.com/watch?v=UFgod5tmLYY, the math song “e a magic number” will engage the students why it is a magic number. While watching this clip, the students will be able to learn the history of e. Also the students will see many mathematical formulas and expressions that contain e. This will give them a heads up that they will see these in future when they take higher level math. It is also pretty humorous of how Dr. Chris Tisdell sang the song.

In the second video, https://www.youtube.com/watch?v=b-MZumdfbt8, it explained why e is everywhere. The video used probability and exponential function to illustrate the usefulness of e, and showed how e is involving in everything. It gave many examples of e such as population, finance… Also the video illustrates the characteristics of the number e and the function that has e in it. Watching these videos will enhance students’ perception and understanding on the number e, and help them to see how important this number is.

Reference

https://www.youtube.com/watch?v=b-MZumdfbt8

https://www.youtube.com/watch?v=UFgod5tmLYY

http://www.math.unt.edu/~baf0018/courses/handouts/exponentialnotes.pdf

http://cleantechnica.com/2014/07/22/exponential-growth-global-solar-pv-production-installation/

http://www.npr.org/sections/goatsandsoda/2014/09/18/349341606/why-the-math-of-the-ebola-epidemic-is-so-scary

http://earthsky.org/earth/how-old-is-the-earth

Engaging students: Inverse trigonometric functions

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 Joe Wood. His topic, from Precalculus: inverse trigonometric functions.

What are the contributions of various cultures to this topic?

Trig functions have a very long history spanning many countries and cultures. Greek astronomers such as Aristarchus, Claudius, and Ptolemy first used trigonometry; however, according to the University of Connecticut, these Greek astronomers were primarily concerned with “the length of the chord of a circle as a function of the circular arc joining its endpoints.” Many of these astronomers, Ptolemy especially, were concerned with planetary and celestial body’s rotations, so this made sense.

While the Greeks first studied trigonometric concepts, it was the Indian people who really studied sine and cosine functions with the angle as a variable. The information was then brought to the Arabic and Persian cultures. One significant figure, a Persian by the name Abu Rayhan Biruni, used trig to accurately estimate the circumference of Earth and its radius before the end of the 11^th century.

Fast-forward about 700 years, a Swiss mathematician, Daniel Bernoulli, used the “A.sin” notation to represent the inverse of sine. Shortly after, another Swiss mathematician used “A t” to represent the inverse of tangent. That man was none other than Leonhard Euler. It was not until 1813 that the notation sin^-1 and tan^-1 were introduced by Sir John Fredrick William Herschel, an English mathematician.

As we can see, the development of inverse trigonometric functions took quite the cultural rollercoaster ride before stopping some place we see being familiar. It took many cultures, and even more years to develop this sophisticated branch of mathematics.

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

Last Semester I taught a lesson on the trigonometric identities. I found this cool cut and paste activity for the students that allowed them to warm up to the trig identities by not having to do the process themselves, but still having to see every step of converting one trig function into another with the identities. Below, you will find the activity, then the instructions, and finally how to modify the activity to fit inverse trig identities specifically.

Directions:
1.) Begin by cutting out all the pieces.
2.) Students will take any of the four puzzle pieces with the black squiggly line.
3.) Find an equivalent puzzle piece by using some trig identity.
4.) Repeat step 3 until there are no more equivalent pieces.
5.) Grab the next puzzle pieces with the black squiggly line.
6.) Repeat steps 3-5 until all puzzle pieces have been used.
Ex.) Begin with cscx-sinx. Lay next to that piece, the piece that reads =1/sinx – sinx, then the piece that reads =1/sinx – sin²x/sinx. Contiue the trend until you reach =cotx * cosx. Then move to the next squiggly lined piece.

Modify:
This game can be modified using inverse trig functions. Start with pieces such as sin^-1(sin(30⁰)) in squiggles. Have a piece showing sin^-1(sin(30⁰)) with a line through the sines. Then a piece that just shows 30⁰. Next a piece in a squiggly line that is sin^-1(sqrt(2)/2) that connects to a piece of 45⁰, but make them write why this works.

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

Obviously, by this time students should know what trigonometric functions are and how to use this. Students should also know from previous classes what inverse functions are. Studying inverse trig functions then is a continuation of these topics. As I teacher I would begin relating inverse trig functions by refreshing the students on what inverse functions are. The class would then move into the concept that the trig expression of an angle returns a ratio of two sides of a triangle. We would slowly move into what happens then if you know the sides of a triangle but need the angle. From there we would discuss trigonometric expressions using the angles as variables. Finally, we would make the connection that that is a function, and on the proper interval should have an inverse function. That is when the extension into the new topic of inverse trigonometric functions would seriously begin.

Engaging students: Graphing exponential growth and decay functions

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 Irene Ogeto. Her topic, from Precalculus: graphing exponential growth and decay functions.

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.

The Legend of the Chessboard is a famous legend that illustrates exponential growth. A courtier presented a Persian king with the chessboard and as a reward the courtier asked the king for a grain of rice in each square of the chessboard, doubling the amount in each new square. The king agreed and gave the courtier 1 grain of rice in the first square, 2 grains of rice in the second, four grains of rice in the third and so on. The king didn’t realize how rapidly the amount of grain of rice would grow in each square. This video would be a great way to engage the students into the topic at the beginning of the lesson. The Legend of the Chessboard shows how rapidly exponential functions can grow. After watching the video the students can try to guess or calculate the total number of grains of rice the courtier would get in the end. Afterwards, the students can then graph the exponential function.

The students can use this website to check their guess:

http://britton.disted.camosun.bc.ca/jbchessgrain.htm

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

In order to explore graphing exponential growth and decay functions, the students could play a card sort matching game. The students will work in groups to play the card sort matching game. Some students will be given the graphs and have to use the points given to derive the equation. Some groups will be given the equations and have to create the graphs of the exponential functions. As a class, we will go over graphing exponential growth and decay functions and analyze the graphs. The students will be expected to identify the domain, range, asymptotes, y-intercepts and whether the graph is exponential growth or exponential decay. Also, we could explore how exponential functions compare to other functions that we previously studied. This is a great activity that can be used as review before an exam.

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

Exponential functions are used to model different real world scenarios involving population, money, finances, bacteria and much more. Students can use exponential functions in other courses such as Calculus, Biology, Chemistry, Physics, and Economics. In calculus, students explore differentiation and integration of exponential functions. Given the position of an object in exponential form, students can use Calculus to determine if the object will stop moving. Newton’s Law of Cooling is an example in physics that demonstrates exponential decay. Compound interest is a major application of exponential functions in finances. Exponential population growth, carbon dating, pH and concentrations of drugs are other examples in math and science that can be modeled by exponential growth and decay functions. In addition, students explore logarithmic functions, the inverses of exponential functions. Being able to recognize and graph exponential growth and decay functions is an important concept that can help students’ in their future courses in math or science.

References:

https://www.youtube.com/watch?v=t3d0Y-JpRRg

http://britton.disted.camosun.bc.ca/jbchessgrain.htm

http://www.shsu.edu/kws006/Precalculus/3.2_Applications_of_Exponential_Functions_files/3.2%20Applications%20of%20Exponential%20Functions%20(slides%204%20to%201).pdf

Engaging students: Using Pascal’s triangle

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 Jason Trejo. His topic, from Precalculus: using Pascal’s triangle.

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

After some research and interesting observations I came across while examining Pascal’s Triangle, I feel like I could create some sort of riddle worksheet that involves the Triangle. Once I have taught my students how to create Pascal’s Triangle, I could give my students riddles such as:

Once you go and strive in prime, belittling your neighbors isn’t a crime.
- Students might notice that each number (other than 1) in a prime number row is divisible by that prime number:
  - Row 7= 1, 7, 21, 35, 35, 21, 7, 1
  - Row 11= 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
- Naturally shallow slides aren’t much fun, but with a fib of raunchy, it is this one.
  - Given that I have gone over the Fibonacci sequence with my students prior to these riddles, I could include this one. The students should eventually see that if you take shallow diagonals on Pascal’s Triangle, the sum of those diagonals are the consecutive numbers in the Fibonacci sequence.
- In a game on blades, you can’t be a schmuck with a puck. Be nimble and quick to look for the stick.
  - This one is a little more straightforward compared to the last two so hopefully the students will make the connection to notice the hockey stick pattern on the diagonals of Pascal’s Triangle. When adding the numbers down a diagonal, then the number to the side and below will be the sum, thus looking like a hockey stick.
- What else is there? What else is in store? What patterns can you find when you know who to root four?
  - The “typo” is intentional to give a hint at another pattern the students might notice on Pascal’s Triangle. Now I am challenging the students to find more patterns within the Triangle such as:
    - Sum of rows are the powers of 2
    - Rows relate to the powers of 11 (get murky after the 4^th row)
    - Counting numbers, triangular numbers, etc.

The purpose of this activity would extend the use of Pascal’s triangle from what they already know. I could assign this at the beginning of the lesson and if no one understands what the riddles meant, we could come back as a class and figure them out together once the lesson was done. These riddles could be an assignment of their own if I introduce them after they are very familiar with Pascal’s Triangle.

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

I would say the primary use most students will get from Pascal’s Triangle would be to find the coefficients of binomials since it is much easier when working on binomial expansions, but there are also other ways they can use the Triangle as well. For one, it can be of great use in many courses that involve since it is a visual in seeing the number of combinations there are based on the number of items used. For example, say there are 6 different pieces of candy in a bowl and you need to know how many different ways can you choose 3 candies? Using Pascal’s Triangle, we look at the 6^th row and the 3^rd entry in that row (remembering the top row is Row 0 and the first 1 in each row is Entry 0), we can see that there are 20 possible combinations of 3 different pieces of candy. Other than that, even based on the riddle activity from above, students can use Pascal’s Triangle and its various patterns to help remember such things as triangular numbers, powers of 11, etc.

How has this topic appeared in high culture?

Within the past few years, the Shanghai-based design company, Super Nature Design, created the interactive art piece “Lost in Pascal’s Triangle”. This structure takes inspiration from Pascal’s Triangle and allows people to “explore the concept and magnification of the Pascal’s Triangle mathematics formula.” The following link takes you to the website that gives a bit more information behind the piece and shows how people can interact with the structure through a xylophone-type console: http://www.supernaturedesign.com/work/pascaltriangle#8

Another quick application that can be done through Pascal’s Triangle is by seeing the relationship between the Triangle and Sierpinski’s triangle (as shown below):

The pattern is by shading in every odd number on Pascal’s Triangle, you start creating Sierpinski’s triangle which is found in many works of art like these:

It might actually be a small but fun project to have the students create something like this at the beginning of the lesson and then explain the relation of the two special triangles.

References:

Pascal Triangle Information: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

Image of Pascal’s Triangle: http://mathforum.org/workshops/usi/pascal/images/pascal.hex2.gif

Lost in Pascal’s Triangle: http://www.designboom.com/weblog/images/images_2/andrea/super_nature_design/pascaltriangle01.jpg

Super Nature Design: http://www.supernaturedesign.com/work/pascaltriangle#2

Pascal and Sierpinski Triangle : http://mathforum.org/workshops/usi/pascal/images/sierpinski.pascalfrac.gif

Sierpinski Pyramid: http://www.sierpinskitetrahedron.com/images/sierpinski-tetrahedron-breckenridge.JPG

Sierpinski Art Project: http://fractalfoundation.org/wp-content/uploads/2009/03/sierpkids1.jpg

Engaging students: Graphing Sine and Cosine Functions

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 Emma Sivado. Her topic, from Precalculus: graphing sine and cosine functions.

D.1: What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

First, I would pose the question “how did the Egyptians build the pyramids without calculators without measuring tapes and without the advanced mathematics we have today?” After a short discussion I would ask them if we want to build a pyramid that is 250 meters high and the base is 360 meters long how long would we need to make the hypotenuse? Already knowing the Pythagorean Theorem the students would be able to answer the question. Then, I would tell them that historians have found Egyptian scribes asking questions such as these in order to build the pyramids, and systems of ropes with knots were used to measure lengths. These relationships in right triangles created the sine and cosine functions we know today. Sine and cosine date back to 1900 BC where they were used to calculate angles in order to track the motion of the planets and stars. However, the definition of sine and cosine in terms of right triangles was not recorded until 1596 AD by Copernicus.

http://www.math.ucdenver.edu/~jloats/Student%20pdfs/40_Trigonometry_Trenkamp.pdf

A.2: How could you as a teacher create an activity or project that involves your topic?

I found a great activity that encompasses all of the aspects of graphing sine and cosine on the University of Arizona website. Depending on how transformations in the linear and quadratic functions were introduced, this activity could follow the same pattern; allowing the students to explore the ideas themselves and having them put the content into their own words. The activity begins by giving an example of a bug walking on an upright loop. The instructor asks the students what the graph would look like of the bug’s distance from the ground vs. time. I would probably use a different, more concrete example because there are plenty of things the students know that go around in circles. The best example I think is a Ferris wheel. So after the students are able to tell you what the graph would look like you relate that to the unit circle and how the sine and cosine functions follow the same pattern of going around the circle counterclockwise. Next, you let the students plot points from the unit circle onto the Cartesian plane showing them that their prediction was correct; the sine and cosine functions make a wave. Now that they have drawn the parent function you let them explore the functions f(x) = asinx or f(x)= acosx, then f(x) = sin(bx) or f(x) = cos(bx), then finally f(x) = sin(x+c) or f(x) = cos(x+c) to let them discover how a, b, and c change the amplitude, period, frequency, and starting point of the graphs.

This is a great activity because the students use multiple examples to see how a, b, and c affect the parent graph of sine and cosine. The activity promotes inquiry based learning and will help deepen the understanding of the graphs of sine and cosine.

http://ime.math.arizona.edu/g-teams/Profiles/JC/Graphing_Sine_and_Cosine_2013.pdf

E.1: 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.

Math can be seen in many forms of art from music to painting. I remember one of my favorite activities from math in high school was creating pictures with sine and cosine functions. We were able to draw flowers, clovers, and hearts simply with only the sine and cosine functions. After the students understand the parent function you can give them an exploration activity on their graphing calculator where they plug in various sine and cosine functions to draw flowers, clovers, and hearts. After that challenge the students to draw their own picture using the patterns they see from the examples. These same ideas can be used in computer graphics and animation to draw similar figures, and a lot of students are interested in computers and especially video games so this should be a fun activity for them.

ftp://arts.ucsc.edu/pub/ems/DANM%20220-2012/Drawing%20with%20trig.pdf

Engaging students: Using radians to measure angles instead of degrees

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 A’Lyssa Rodriguez. Her topic, from Precalculus: using radians to measure angles instead of degrees

D1. How did people’s conception of this topic change over time?

Babylonians came up with the degree system. For their number system they liked to use the number 60 and multiples thereof. Therefore they decided on the number 360 and each number represented a degree in a circle. This number was completely arbitrary and was simply a matter of preference by the Babylonians. Although this makes handling circles and angles seem easier, due to it being an arbitrary number, it makes degrees unnatural. So the deeper concepts in math needed a more natural number. Radians are that more natural measurement we needed. Using the length of the radius of any circle and wrapping around the outside of that circle, one can see that it almost completely goes around the entire circle 6 times. To make up for what is left we multiply the radius by 2pi. Thus the equation for the circumference of a circle is C = 2 πr. This is the reason and the change over time for the use of radians instead of degrees.

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?
Pure tones are found in music. Regardless of other musical properties such as amplitude or the time relation to other sound waves (phase), these tones will have a consistent sinusoidal sound wave. The sine function used to measure these waves use radians. Although degrees are technically possible, this function is most accurate when using radians. According to Mathematics and Music: Composition, Perception, and Performance by James S. Walker and Gary W. Don, the formula that can be used to determine the oscillation for a tuning fork is y= Asin(θ) where θ is measured in radians and is equal to 2 πvt+ θ₀and θ₀ is the initial value of θ when t=0. So y = Asin(2 πvt+ ^π/_2).

B2. How does this topic extend what your students should have learned in previous courses?

In previous courses, students use degrees to measure angles and to refer to circles. Even activities outside of the classroom, such as snowboarding, use degrees. This was easiest and best for learning purposes, up until this point that is. Now that trigonometric functions will be introduced, the circle will be studied more in depth, and more real life situations will be given, it is necessary to use radians instead of degrees. The calculations will become more accurate in some cases, some even easier, and it is essential to use a more natural number. This topic merely adds on to what the students already know about angles but also makes them think about it in a different way. One way their previous knowledge of degrees will be extended is by learning to convert from degrees to radians and back again.