Predicate Logic and Popular Culture (Part 240): AC/DC

Let P be the set of all people, let R(x) be the statement “x is about to rock,” and let S(x) be the statement “We salute x.” Translate the logical statement

\forall x in P (R(x) \Longrightarrow S(x)).

This matches the title and chorus of “For Those About to Rock We Salute You” by AC/DC.

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 239): Smashmouth

Let P be the set of all people, let T be the set of all times, and let W(x,t) be the statement “At time t, x told me the world is gonna roll me; I ain’t the sharpest tool in the shed.” Translate the logical statement

\exists x in P \exists t \in T (W(x,t)).

This matches the opening line from the hit song “All Star” by Smashmouth.

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 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.

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.

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.

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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.

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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.green line

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.

Engaging students: Compound interest

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 Lydia Rios. Her topic, from Precalculus: compound interest.

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

While this concept is tied with business which is something that started rapidly changing in the early nineteen hundreds, we have understand that there is an accrued interest on loans long before then. People would loan out seeds or cattle and the interest would be paid after a harvest or with the young of the cattle. Of course now we use this concept mathematically but the concept still holds. We understand that there is a base fee and you must return that fee along with a little more. We then started using this with loose change and then as our currency changed from the gold standard we adapted to a new understanding of compound interest. Today we use the equation A = P \left(1 + \frac{r}{n} \right)^{nt} , where A is the amount accumulated, P is the principal, r is interest rate, n is the compound period and t is the number of periods.

Compound Interest Is Responsible for Modern Civilization (businessinsider.com)

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What are the contributions of various cultures to this topic?

We have all experienced trade over the years. Native Americans would trade corn for other goods and offered payment plus interest with their corn harvest. The Silks Roads was a network of trading routes where China and other countries would trade textiles and other materials. They established the concept of payment and interest for purchases. Banks in America and other countries also have a set principal and a interest, whether this be in reference to your savings account or the billed interest on your credit card purchases. Even the invention of cars played a part on this and how our interest can decrease with the deterioration of the car. Over the years your interest payment can go down as the worth of the car goes down.

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How have different cultures throughout time used this topic in their society?

Native Americans used compound interest to create trade deals a maintain some status of peace. China had their silk roads where they turned a profit and tried to maintain a sense of livelihood. For some cultures this was their only source of income, if they didn’t’ make some sort of trade then they had nothing to bring home. For others, such as the Native Americans, the trade itself was to protect their lives and the interest was something they owed. We can even see reference to this with the trade markets in Disney’s Aladdin, Aladdin cannot make a trade as he is a peasant but we see other village people making trades and we grasp the concept of worth from each object.

Engaging students: Deriving the double angle formulas for sine, cosine, and tangent

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 Morgan Mayfield. His topic, from Precalculus: deriving the double angle formulas for sine, cosine, and tangent. green line How could you as a teacher create an activity or project that involves your topic? I want to provide some variety for opportunities to make this an engaging opportunity for Precalculus students and some Calculus students. Here are my three thoughts: IDEA 1: For precalculus students in a regular or advanced class, have them derive this formula in groups. After students are familiar with the Pythagorean identities and with angle sum identities, group students and ask them to derive a formula for double angles Sin(2θ), Cos(2θ), Tan(2θ). Let them struggle a bit, and if needed give them some hints such as useful formulas and ways to represent multiplication so that it looks like other operations. From here, encourage students to simplify when they can and challenge students to find the other formulas of Cos(2θ). Ask students to speculate instances when each formula for Cos(2θ) would be advantageous. This gives students confidence in their own abilities and show how math is interconnected and not just a bunch of trivial formulas. Lastly, to challenge students, have them come up with an alternative way to prove Tan(2θ), notably Sin(2θ)/Cos(2θ). This would make an appropriate activity for students while having them continue practicing proving trigonometric identities. IDEA 2: This next idea should be implemented for an advanced Precal class, and only when there is some time to spare. Euler was an intelligent man and left us with the Euler’s Formula: e^{ix}=\cos x + i \sin x. Have Precalculus students suspend their questions about where it comes from and what it is used for. This is not something they would use in their class. Reassure them that for what they will do, all they need to understand is imaginary numbers, multiplying imaginary numbers, and laws of exponents. Have them plug in x = A + B and simplify the right-hand side of the equation so that we get: \cos(A+B)+i\sin(A+B)= a + bi where a and b are two real numbers. The goal here is to get \cos(A+B)+i\sin(A+B)= \cos \theta \cos \theta - \sin \theta \sin \theta + (\sin \theta \cos \theta + \cos \theta \sin \theta)i. All the steps to get to this point is Algebra, nothing out of their grasp. Now, the next part is to really get their brains going about what meaning we can make of this. If they are struggling, have them think about the implications of two imaginary numbers being equal; the coefficient of the real parts and imaginary parts must be equal to each other. Lastly, ask them if these equations seem familiar, where are they from, and what are they called…the angle sum formulas. From here, this can lead into what if x=2A? Students will either brute force the formula again, and others will realize x = A + A and plug it in to the equation they just derived and simplify. This idea is a 2-in-1 steal for the angle sum formulas and double angle formulas. It’s biggest downside is this is for Sin(2θ) and Cos(2θ).   IDEA 3: Take IDEA 2, and put it in a Calculus 2 class. Everything that the precalculus class remains, but now have the paired students prove the Euler’s Formula using Taylor Series. Guide them through using the Taylor Series to figure out a Taylor Series representation of e^x, sin x,  and cos x. Then ask students to find an expanded Taylor Series of to 12 terms with ellipses, no need to evaluate each term, just the precise term. Give hints such as i^2= -1 and to consider i^3=i^2 \cdot i = -i and other similar cases. Lastly, ask students to separate the extended series in a way that mimics a + bi using ellipses to shows the series goes to infinity. What they should find is something like this:
Look familiar? Well it is the addition of two Taylor Series that represent Sin(x) and Cos(x). This is the last connection students need to make. Give hints to look through their notes to see why the “a” and “b” in the imaginary number look so familiar. This, is just one way to prove Euler’s Formula, then you can continue with IDEA 2 until your students prove the angle sum formulas and double angle formulas.green line How does this topic extend what your students should have learned in previous courses? Students in Texas will typically be exposed to the Pythagorean Theorem in 8th grade. At this stage, students use a^2+b^2=c^2 to find a missing side length. Students may also be exposed to Pythagorean triples at this stage. Then at the Geometry level or in a Trigonometry section, students will be exposed to the Pythagorean Identity. The Identity is \sin^2 \theta + \cos^2 \theta = 1.  I think that this is not fair for students to just learn this identity without connecting it to the Pythagorean Theorem. I think it would be a nice challenge student to solve for this identity by using a right triangle with hypotenuse c so that Sin (θ) = b/c and cos (θ) = a/c, one could then show either c^2 \sin^2 \theta + c^2 \cos^2 \theta = c^2 and thus c^2(\sin^2 \theta + \cos^2 \theta) = c^2 or one could show (a/c)^2 + (b/c)^2 = (c/c)^2 = 1 (using the Pythagorean theorem). From here, students learn about the angle addition and subtraction formulas in Precalculus. This is all that they need to derive the double angle formulas.

\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta

\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta

\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta

\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta

\tan(\alpha + \beta) = \displaystyle \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}

\tan(\alpha - \beta) = \displaystyle \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}

This would be a good challenge exercise for students to do in pairs. Sin(2θ) = Sin(θ + θ), Cos(2 θ) = Cos(θ + θ), Tan(2θ) = Tan(θ + θ). Now we can apply the angle sum formula where both angles are equal: Sin(2θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ) Cos(2θ) = cos(θ)cos(θ) – sin(θ)sin(θ) =  (We use a Pythagorean Identity here) Tan(2θ) = \displaystyle \frac{\tan \theta + \tan \theta}{1 - \tan^2 \theta} = \frac{2 \tan \theta}{1-\tan^2 \theta} Bonus challenge, use Sin(2θ) and Cos(2θ) to get Tan(2θ). Well, if \tan \theta = \displaystyle \frac{\sin \theta}{\cos \theta}, then

\tan 2\theta = \displaystyle \frac{\sin 2\theta}{\cos 2\theta}

= \displaystyle \frac{2 \sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta}

= \displaystyle \frac{ \frac{2 \sin \theta \cos \theta}{\cos^2 \theta} }{ \frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta} }

= \displaystyle \frac{2 \tan \theta}{1 - \tan^2 \theta}

The derivations are straight forward, and I believe that many students get off the hook by not being exposed to deriving many trigonometric identities and taking them as facts. This is in the grasp of an average 10th to 12th grader. green line What are the contributions of various cultures to this topic? I have included four links that talk about the history of Trigonometry. It seemed that ancient societies would need to know about the Pythagorean Identities and the angles sum formulas to know the double angle formulas. Here is our problem, it’s hard to know who “did it first?” and when “did they know it?”. Mathematical proofs and history were not kept as neatly written record but as oral traditions, entertainment, hobbies, and professions. The truth is that from my reading, many cultures understood the double angle formula to some extent independently of each other, even if there was no formal proof or record of it. Looking back at my answer to B2, it seems that the double angle formula is almost like a corollary to knowing the angle sum formulas, and thus to understand one could imply knowledge of the other. Perhaps, it was just not deemed important to put the double angle formula into a category of its own. Many of the people who figured out these identities were doing it because they were astronomers, navigators, or carpenters (construction). Triangles and circles are very important to these professions. Knowledge of the angle sum formula was known in Ancient China, Ancient India, Egypt, Greece (originally in the form of broken chords theorem by Archimedes), and the wider “Medieval Islamic World”. Do note that that Egypt, Greece, and the Medieval Islamic World were heavily intertwined as being on the east side of the Mediterranean and being important centers of knowledge (i.e. Library of Alexandria.) Here is the thing, their knowledge was not always demonstrated in the same way as we know it today. Some cultures did have functions similar to the modern trigonometric functions today, and an Indian mathematician, Mādhava of Sangamagrāma, figured out the Taylor Series approximations of those functions in the 1400’s. Greece and China for example relayed heavily on displaying knowledge of trigonometry in ideas of the length of lines (rods) as manifestations of variables and numbers. Ancient peoples didn’t have calculators, and they may have defined trigonometric functions in a way that would be correct such as the “law of sines” or a “Taylor series”, but still relied on physical “sine tables” to find a numerical representation of sine to n numbers after the decimal point. How we think of Geometry and Trigonometry today may have come from Descartes’ invention of the Cartesian plane as a convenient way to bridge Algebra and Geometry. References: https://www.mathpages.com/home/kmath205/kmath205.htm https://en.wikipedia.org/wiki/History_of_trigonometry https://www.ima.umn.edu/press-room/mumford-and-pythagoras-theorem

Engaging students: Computing the cross product of two vectors

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 Chi Lin. Her topic, from Precalculus: computing the cross product of two vectors.

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

I found one of the real-life examples of the cross product of two vectors on a website called Quora. One person shares an example that when a door is opened or closed, the angular momentum it has is equal to r \times p, where p is the linear momentum of the free end of the door being opened or closed, and r is the perpendicular distance from the hinges on which the door rotates and the free end of the door. This example gives me an idea to create an example about designing a room. I try to find an example that closes to my idea and I do find an example. Here is the project that I will design for my students. “If everyone here is a designer and belongs to the same team. The team has a project which is to design a house for a client. Your manager, Mr. Johnson provides a detail of the master room to you and he wants you to calculate the area of the master room to him by the end of the day. He will provide every detail of the master room in three-dimension design paper and send it to you in your email. In the email, he provides that the room ABCD with \vec{AB} = \langle -2,2,5 \rangle and \vec{AD} = \langle 5,6,3 \rangle. Find the area of the room  (I will also draw the room (parallelogram ABCD) in three dimensions and show students).”

Reference:

https://www.quora.com/What-are-some-daily-life-examples-of-dot-and-cross-vector-products

https://www.nagwa.com/en/videos/903162413640/

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

This topic is talking about computing cross product of two vectors in three dimensions. First, students should have learned what a vector is. Second, students should know how to represent vectors and points in space and how to distinguish vectors and points. Notice that when students try to write the vector in space, they need to use the arrow. Next, since we are talking about how to distinguish the vectors and the points, here students should learn the notations of vectors and what each notation means. For example, \vec{v} = 1{\bf i} + 2 {\bf j} + 3 {\bf k}. Notice that 1{\bf i} + 2 {\bf j} + 3 {\bf k} represents the vectors in three dimensions. After understanding the definition of the vectors, students are going to learn how to do the operation of vectors. They start with doing the addition and scalar multiplication, and magnitude. One more thing that students should learn before learning the cross product which is the dot product. However, students should understand and master how to do the vector operation before they learn the dot product since the dot product is not easy. Students should have learned these concepts and do practices to make sure they are familiar with the vector before they learn the cross products.

References:

https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/x786f2022:vectors-and-matrices/a/vectors-and-notation-mvc

https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/x786f2022:vectors-and-matrices/a/dot-products-mvc

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

Most people have the misconception that the cross product of two vectors is another vector. Also, the majority of calculus textbooks have the same misconception that the cross product of two vectors is just simply another vector. However, as time goes on, mathematicians and scientists can explain by starting from the perspective of dyadic instead of the traditional short‐sighted definition. Also, we can represent the multiplication of vectors by showing it in a geometrical picture to prove that encompasses both the dot and cross products in any number of dimensions in terms of orthogonal unit vector components. Also, by using the way that the limitation of such an entity to exactly a three‐dimensional space does not allow for one of the three metric motions (reflection in a mirror). We can understand that the intrinsic difference between true vectors and pseudo‐vectors.

Reference:

https://www.tandfonline.com/doi/abs/10.1080/0020739970280407