Predicate Logic and Popular Culture (Part 238): Among Us

Let $P$ be the set of all people, let $I(x) be the statement “ $x$ is an imposter,” and let $A(x)$ be the statement “ $x$ is among us$.” Translate the logical statement

$\exists x \in P (I(x) \land A(x))$ .

This matches a line from the video game “Among Us.”

Context: 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 237): Psycho

Let $P$ be the set of all people, let $T$ be the set of all times, and let $M(x,t)$ be the statement “ $x$ goes mad at time $t$ .” Translate the logical statement

$\forall x \in P \exists t \in T (\sim M(x,t))$ .

This matches a line from the movie Psycho.

Decorative Constants

Engaging students: 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 Fidel Gonzales. His topic, from Precalculus: vectors in two dimensions.

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

When a student learns about vectors in two dimensions, they worry about the magnitude of the vector and the direction that it goes. The direction is kept within its limitations which are up, down, left, and right. A student might be curious as to how this topic can be extended further. The way it extends further is by extending vectors into higher dimensions. It is even possible to extend vectors to the sixth dimension! However, for the sake of showing how vectors in two dimensions extend to future courses in math, we will stick to three-dimensions. Learning about vectors in the second dimension creates groundwork to learn about vectors in the third dimension. With the third dimension, vectors could be seen from our point of view compared to seeing it in the two dimensions on paper. The new perspective of the third dimension in vectors includes up, down, left, right, forward, and backwards. Having the new dimension to account for will give students a bigger tie into how mathematics applies into the real world.

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

Vectors in the two dimension is used all around our everyday life and we as people rarely notice it. The most common use of vectors in our culture is a quantity displaying a magnitude and direction. This is normally done on a x and y graph. Now you might be asking yourself, I do not play any types of games that sound like this. I am here to tell you that you do. One game that iPhone users play without noticing this would be a game on gamepigeon called knockout. The game appears to be an innocent game of knocking out your friends’ penguins while keeping yours in the designated box. However, math is involved, and you probably didn’t notice. First you must anticipate where the enemy is going. Then you must decide how strong you want to launch your penguin troopers without making them fall out of the ring. Does that sound familiar? Having to apply a force (magnitude) and direction to a quantity. Congratulations, you have now had fun doing math. Next time you are playing a game, try to see if there is any involvement of vectors in two dimensions involved.

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

Vectors in two dimensions has many ways to be incorporated in the classroom. A way to do so while connecting to the real world would be having an activity where the students tell a robot where to go using vectors. The students will have a robot that can walk around and in need of directions. The students will be given maps and asked to create a path for the robot to end up in its destination. Essentially, programming the robot to navigate though a course solely using vectors. If the robot falls or walks too far, then the student will realize that either the magnitude was wrong or the direction. Some students might seem to think this would be impractical to the real world, however, there is always a way to show relevance to students. Towards the end of the activity, the students will be asked to guide me to around the class using vectors. Then to sweeten the deal, they will also be asked to show me on a map being projected to them how to get to McDonald’s. Students will realize that vectors in the second dimension could be used to give directions to somewhere and can be applied to everyday life. They will walk outside of the classroom seeing math in the real world from a different perspective.

References:

https://www.khanacademy.org/science/physics/two-dimensional-motion/two-dimensional-projectile-mot/v/visualizing-vectors-in-2-dimensions

In Memoriam: Harry Lucas, Jr. (1932-2022)

I was saddened to recently read of the passing of Harry Lucas, Jr., who was a great proponent and benefactor of inquiry-based learning (IBL). To remember his contributions to the mathematical community, I certainly won’t be able to surpass the eloquent words of Michael Starbird in the June-July issue of MAA Focus.

Instead, I’ll share a little bit about my own interactions with Mr. Lucas. My first administrative position at my university was the founding co-director of Teach North Texas, a UTeach replication of the pioneering program UTeach program at the University of Texas for preparing teachers of secondary mathematics and science. I first met Mr. Lucas at the annual UTeach conference, and I don’t remember how it came up, but he personally encouraged me to submit a proposal to the Educational Advancement Foundation for the funding of equipment typically found in physics labs to get our university’s new Functions and Modeling course off the ground. Thanks to his generosity, hundreds of UNT students have experienced IBL firsthand early in their mathematical studies, often giving them an eye-opening new perspective on the way that mathematics “should” be taught. At future conferences, Mr. Lucas always had a keen interest in how Teach North Texas was progressing and seemed delighted to hear of our successes.

In the words of Dr. Starbird, “Mr. Lucas is one of very few individuals whose personal vision, decisions, persistence, and encouragement have clearly improved the lives of thousands of students and teachers across the country.” Thank you, Mr. Lucas.