Predicate Logic and Popular Culture (Part 100): Toto

Let $L(t)$ be the proposition “Love is on time at time $t$ .” Translate the logical statement

$\lnot \forall t L(t)$ .

This matches part of the chorus of this hit song of the 1970s.

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 99): R.E.M.

Let $W(t)$ be the proposition “The world exists at time $t$ .” Translate the logical statement

$(\forall t<0 W(t)) \land (\forall t \ge 0 \lnot W(t))$ .

This approximately matches the title of one of R.E.M.’s greatest hits.

Predicate Logic and Popular Culture (Part 98): Carly Rae Jepsen

Let $T(x,y)$ be the proposition “I’d trade $x$ for $y$ ,” let $L(x)$ be the proposition “x was looking for this,” and let $W(x)$ be the proposition “ $x$ is now in my way.” Translate the logical statement

$T(\hbox{my soul}, \hbox{a wish}) \land T(\hbox{pennies and dimes}, \hbox{a kiss}) \land \lnot L(\hbox{I}) \land W(\hbox{you})$ .

This matches some of the opening lines from the summer hit of 2012.

And I can’t resist also sharing this incredible mashup:

Predicate Logic and Popular Culture (Part 97): U2

Let $L(x)$ be the proposition “I am looking for $x$ ,” and let $F(x)$ be the proposition “I have found $x$ .” Translate the logical statement

$\forall x (L(x) \Rightarrow \lnot F(x))$ .

This is a wonderful song by U2.

Predicate Logic and Popular Culture (Part 96): Roy Orbison

Let $K(x)$ be the proposition “ $x$ knows the way I feel tonight,” let $L(x)$ be the proposition “ $x$ is lonely.” Translate the logical statement

$\forall x (K(x) \Rightarrow L(x))$ .

This matches the opening line from this classic from 1960.

Predicate Logic and Popular Culture (Part 95): The Police

Let $B(t)$ be the proposition “You take a breath at time $t$ ,” let $M(t)$ be the proposition “You make a move at time $t$ ,” let $N(t)$ be the proposition “You break a bond at time $t$ ,” let $S(t)$ be the proposition “You take a step at time $t$ ,” and let $W(t)$ be the proposition “I watch you at time $t$ .” Translate the logical statement

$\forall t ((B(t) \lor M(t) \lor N(t) \lor W(t)) \Rightarrow W(t))$ .

This matches the opening lines of this iconic song by The Police.

Predicate Logic and Popular Culture (Part 94): Sweeney Todd

Let $p$ be the proposition “I am around you,” and let $H(x)$ be the proposition “ $x$ will harm you.” Translate the logical statement

$p \Rightarrow \lnot \exists x H(x)$ .

This matches the opening lines of this wonderful song from the musical Sweeney Todd.

Predicate Logic and Popular Culture (Part 93): Annie Get Your Gun

Let $B(x)$ be the proposition “ $x$ is a business,” and let $L(x,y)$ be the proposition “ $x$ is like $y$ .” Translate the logical statement

$\lnot \exists x (B(x) \land x \ne \hbox{show business} \land L(x, \hbox{show business}))$ .

This matches the showstopper from Annie Get Your Gun:

Engaging students: The quadratic formula

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 Storm Boykin. Her topic, from Algebra II: the quadratic formula.

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

Learning the Quadratic Formula can be quite a tedious process for students. They have to learn the why and how, but still need to be able to recall the Quadratic Formula fairly quickly. Students cannot use it as a tool to solve quadratic equations if they cannot remember it. The chart below is to be a card sort from a mnemonic device that was passed down by a former teacher. The idea is that the students will have these cards mixed up, and then they will have to match the story with its’ mathematical counterpart. Then, using the mnemonic device, the student can put the story in order, and will have the quadratic formula before their eyes.

The negative boy	-b
couldn’t decide	±
if he wanted to go to the radical party	√
but the boy was square	b^2
and missed out on four awesome chicks	-4ac
it was all over by two am	/2a

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

Quadratics began to come about around 2000 BCE because mathematicians needed a way to make “architectural floor and wall plans.” Quadratics have been worked on by various cultures since then. While working on quadratics, Pythagorus found that the square root of a number did not have to be an integer. However, he didn’t believe that imaginary numbers existed, and had one of his associates killed for even suggesting to the public that “non- integers” existed. They idea of a quadratic formula was touched on by hindu and islamic mathematicians, but did not come into form as our modern day formula until 1637, when Rene Descartes published it in his geometric works.

https://musingsonmath.files.wordpress.com/2011/08/history-quadratic-formula.pdf

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

All algebra and calculus courses will have quadratic equations in them. Students will have a much easier Physics experience if they have the equation at their disposal. Quadratics are used to calculate velocity and height of objects in the air. Real world examples involving velocity and height are the perfect blend of physics and science! Students will also need the quadratic formula for the SAT and PSAT.

Engaging students: Dividing polynomials

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 Sarah Asmar. Her topic, from Algebra/Precalculus: dividing polynomials.

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

Many high school students are introduced to Polynomials in Algebra I. They are taught how to factor and to even graph Polynomials. In Algebra II, students are asked to add, subtract, multiply and divide Polynomials. Dividing Polynomials is challenging for many students because they are not only dividing numbers, but now they have added letters to the mix. There are two ways to divide Polynomials: Long Division and Synthetic Division. Since this is a topic that most students find difficult to grasp, I would split the students into groups of about 3 or 4 and provide each group with Algebra tiles. I would then provide each group with an index card with a specific Polynomial for them to divide. The index card will have a dividend and divisor for the students to use in order for them to create find the answer using the Algebra tiles. First, they will need to create a frame. Then, the dividend should be formed inside the frame while the divisor is formed on the left hand side outside of the frame. The answer will be shown with the tiles on the top line outside the frame. I will do an example with them first and then have them do the problem provided on their index card with their group. This activity will provide the students with a visual representation on how dividing polynomials would look like in order for it to be easier for them on paper.

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

Students are introduced to dividing Polynomials in Algebra II. Most would never like to see this topic again, but unfortunately that is not the case. Dividing Polynomials is revisited in a Pre-Calculus class. However, it is taught at a much deeper level. Students are required to divide using long and synthetic division. Synthetic division is taught as a short cut for dividing Polynomials, but it doesn’t always work and students would have to divide using long division. Synthetic substitution is taught as well to find the solution of the Polynomial given. Synthetic substitution is as easy as just plugging in the given number for the variable provided in the Polynomial. Dividing Polynomials is also used in Binomial Expansion in Pre-Calculus. Along with all of these topics in Pre-Calculus, dividing Polynomials appears in all future basic Math courses such as Calculus. A real life example that uses Polynomials is aerospace science. These equations are used for object in motion, projectiles and air resistance.

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

I was searching the Internet and I came across this video. I thought that this video would be an amazing tool to help the students understand how to divide polynomials without me just lecturing to them. It is sung to the tune of “We Are Young” which is a very popular song in the pop music culture. Using something like this would show a visual representation, but it will also drill the steps in their head. Our brains can easily remember songs even after listening to a song just once. The fact that dividing polynomials is put into a song makes it more likely for a student to remember the steps they need to take in order to perform the indicated operation.

References:

http://www.doe.virginia.gov/testing/solsearch/sol/math/A/m_ess_a-2b_1.pdf

http://polynomialsinourlives.weebly.com/polynomials-in-the-real-world.html