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.

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 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

Engaging students: Adding, subtracting, and multiplying matrices

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 Perla Perez. Her topic, from Algebra: adding, subtracting, and multiplying matrices.

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

“Cryptography is the study of encoding and decoding messages. Cryptography was first developed to send secret messages in written form.” Cryptography also uses matrices to code and decode these messages by multiplication and the inverse of them. This, however, can be done by using any operations. By using the worksheet below as a foundation for an activity, teachers can have students act like hackers to engage students in computing different operations with matrices. In this activity, prepare the classroom by dividing it into four sections each with one of the phrases separated on the worksheet. Display the message (numerically) that is to be coded. Display the alphabet with corresponding number somewhere visible for students to have references throughout the activity. The instructions given are:

Students are to get into four groups (more groups can be added for larger classrooms by making the phrase longer).
Students are given an index card with the matrix [2, 7; 13, 5]
Students are to add the matrix on each station to the the matrix on the card.
When completed students must go change the message on the broad with the code.

When the students finish coding the message they can continue developing their skills by having them do this in the beginning of class throughout the lesson plan period. As the lesson progresses the teacher can change the phrase and require different operations to be made to either code or decode or even come up with their own message. With this activity the teacher gets the opportunity to see how the students choose to add the matrices together.

Click to access using_matrices_in_cryptography_intro.pdf

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.

In today’s society we have access to a plethora of technology that can aid us in our everyday lives. There are so many ways one can learn something with different methods and from different people. The best part about the technology that we have access to is we can be manipulative to fit the needs of our students. When students get to the topic of adding, subtracting and soon multiplying matrices, they should be familiar with what a matrix is, the dimensions of one, and how to solve linear system with them. At this point it is a good a time to bring in a game into play like this one:

http://www.intmath.com/matrices-determinants/matrix-addition-multiplication-applet.php.

In this game the player chooses an operation such as adding, subtracting, multiplying by another matrix or scaler, and its dimensions. When a certain operation is chosen such as multiplication, it only allows the player to choose any size matrix but then spits out one with specific number of rows to multiply it with. The teacher can play this game with their students in any way they sit. The purpose is to get students thinking why and how the operations are working. From there the teacher can introduce the new topic.

Resources:

http://www.intmath.com/matrices-determinants/matrix-addition-multiplication-applet.php

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

So many times students don’t understand that what they learn in class is used in everyday life, but teachers can give students the resources and knowledge to see applications of their work. In the video below, it shows different ways matrices can be applied. For instances the operations of matrices are used in a wide variety of way in our culture.

The main one can be in computer programming and computer coding, but they are also seen in another places such as dance and architecture. “In contra dancing, the dancers form groups of four (two couples), and these groups of four line up to produce a long, two-person-wide column” and where each square that is created is a formed by two pairs. Like the video had said, matrices can be used to analyze contra dancing. This can be done by having squares and multiplying them creating different types of configurations. By creating different groups and formations, essentially it is using different operations to create different matrices to.

Resources:

References:

“Common Topics Covered in Standard Algebra II Textbooks.” Space Math @ NASA. NASA, n.d. Web. 18 Sept. 2015.

Knill, Oliver. “When Was Matrix Multiplication Invented?” When Was Matrix Multiplication Invented? Harvard, 24 July 2014. Web. 18 Sept. 2015.

Smoller, Laura. “The History of Matrices.” The History of Matrices. University of Arkansas at Little Rock, Apr. 2001. Web. 18 Sept. 2015.