Predicate Logic and Popular Culture (Part 69): Whitney Houston

Let $L(t)$ be the proposition “I will love you at time $t$ .” Translate the logical statement

$\forall t\ge 0 L(t)$ ,

where the domain is all times.

Of course, this is the iconic song by Whitney Houston.

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 68): Justin Bieber

Let $M(x)$ be the proposition “My momma likes $x$ .” Translate the logical statement

$\lnot M(\hbox{you}) \land \forall x M(x)$ ,

where the domain is all people.

The straightforward way of translating this into English is, “My momma does not like you, and my momma likes everyone.” (An amusing consequence of this syllogism is the conclusion “You are nobody.”) This almost precisely matches the memorable line of the chorus in Justin Bieber’s recent hit song.

Predicate Logic and Popular Culture (Part 67): Janet Jackson

Let $T(x,t)$ be the proposition “I think of $x$ at time $t$ ,” and let $M(x,t)$ be the proposition “ $x$ seems to matter at time $t$ .” Translate the logical statement

$\forall t (T(\hbox{you},t) \Rightarrow \forall x \ne \hbox{you}(\not M(x,t)))$ .

The straightforward way of translating this into English is, “If I think of you at a time, then everything else does not matter at that time.” More lyrically, it’s the first line of the chorus to Janet Jackson first #1 single, released in 1986.

Predicate Logic and Popular Culture (Part 66): West Side Story

Let $L(x,t)$ be the proposition “We find a new way of living at place $x$ and at time $t$ ,” and let $F(x,t)$ be the proposition “We find a way of forgiving at place $x$ and $t$ .” Translate the logical statement

$\exists x \exists t (L(x,t) \land F(x,t))$ .

This matches almost perfectly one of the lines from the classic song “Somewhere” from the musical West Side Story.

Predicate Logic and Popular Culture (Part 65): John Philip Sousa

Let $S(t)$ be the proposition “The Stars and Stripes wave at time $t$ .” Translate the logical statement

$\forall t (S(t))$ .

I tried to think of a fitting example for the Fourth of July, but the best that I could find was the closing line of the chorus of the Stars and Stripes Forever.

Which naturally leads me to this amazing version from the 1970s:

Predicate Logic and Popular Culture (Part 64): Abraham Lincoln

Let $F(x,t)$ be the proposition “You can fool $x$ at time $t$ .” Translate the logical statement

$\exists t_1 \forall x (F(x,t_1)) \land \exists x_1 \forall t(F(x_1,t)) \land \lnot(\forall x \forall t(F(x,t)))$ .

Of course, this is the famous quote commonly attributed to Abraham Lincoln: “You can fool all of the people some of the time, and some of the people all of the time, but you cannot fool all of the people all of the time.”

Predicate Logic and Popular Culture (Part 63): P. T. Barnum

Let $S(x) be the proposition "$ latex x$ is a sucker,” and let $B(x,t)$ be the proposition “ $x$ is born at time t.” Translate the logical statement

$\forall t \exists x (S(x) \land B(x,t))$ .

Naturally, this is the famous quote often attributed to P. T. Barnum: “There’s a sucker born every minute.”

Predicate Logic and Popular Culture (Part 62): George Strait

Let $X(x)$ be the proposition “ $x$ is my ex,” and let $T(x)$ be the proposition “ $x$ lives in Texas.” Translate the logical statement

$\forall x (X(x) \Rightarrow T(x))$ ,

where the domain is all people.

Naturally, this one of the great hits in the storied career of George Strait.

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.