Predicate Logic and Popular Culture (Part 209): The Office

Let $p$ be the statement “I am superstitious,” and let $q$ be the statement “I am a little stitious.” Translate the logical statement

$\lnot p \land q$ .

This matches a quote from the popular TV show “The Office.”

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 208): Masashi Kishimoto

Let $R(t)$ be the statement “I run away at time $t$ ,” let $G(t)$ be the statement “I go make on my word at time $t$ ,” and let $T$ be the set of all times. Translate the logical statement

$\forall t \in T (\lnot R(t) \land \lnot G(t))$ .

This matches a quote from the main character Naruto in Masashi Kishimoto’s anime (also named Naruto).

Engaging students: Solving one-step algebra problems

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 Alizee Garcia. Her topic, from Algebra: solving one-step algebra problems.

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

As stated in the topic, one-step algebra problems can also lead up to two-step, three-step, and so on and so forth. Being said, as students’ move on to future courses, the knowledge they have over one-step problems is what will get them through more complex equations. Throughout algebra courses, the basis of problems will be to solve an unknown variable. Without the understanding of the base of algebra, things will not be smooth. Also, solving one-step algebra problems will help students’ even in science classes. For example, chemistry classes contain a lot of variables and unknowns and it is up to the student to solve for them. The amount of solution a student has to put into another solution may need to be figured out by a simple one-step algebra problem and without this knowledge, it can lead to a ruined lab or maybe even an explosion. Solving one-step problems and understanding how to will help students tremendously from the time they learn it to the end of time.

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

When solving any algebra problem, or solving for an unknown, it allows students to incorporate order of operations. As for just one-step algebra problems, it gives students the opportunity to practice addition, subtraction, multiplication, and division. It also gives them to opportunity to practice setting up an equation when solving for the unknown. There are many things that one-step algebra problems extends for students but as they have more practice, they should not have to think about it much. Furthermore, when solving algebra problems one of the most important things is doing the same application on both sides of the equality. Sometimes students may have done one-step algebra problems in the past but have not set it up in an equation. This also will extend the topic of addition, subtraction, multiplication, and division. Although the students may already have a lot of experience with those applications, it gives them more practice to decide what application to use when solving a one-step algebra problem.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Recently, I have discovered that when appropriate, using websites such as Quizziz, Kahoot, and online games as such helps students engage in the topic. Especially for one-step algebra problems that can be done mentally or quickly on paper, it lets students become more active in the lesson. Students will want to be their peers high score and get the questions right. Using such technology will enable students to have more practice and wanting to do it correctly as well. Making topics a friendly competition for students will make things more exciting for them. Also, these website will allow for an untimed quiz so they do not feel rush and are able to accurately solve problems. Although this can be tricky for some math topics, with simpler things such as one-step algebra problems, it definitely will be a very good opportunity for students to learn material and have fun with it as well.

Predicate Logic and Popular Culture (Part 207): Patrick Rothfuss

Let $p$ be the statement “You tell a story the right way,” and let $q$ be the statement “You are a bit of a liar.” Translate the logical statement

$p \Rightarrow q$ .

This matches a line from the novel “Name of the Wind” by Patrick Rothfuss: “You have to be a bit of a liar to tell a story the right way.” https://www.goodreads.com/quotes/155428-you-have-to-be-a-bit-of-a-liar-to

Engaging students: Dot product

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 Haley Higginbotham. Her topic, from Precalculus: computing a dot product.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

For the dot product of vectors, there are lots of word problems regarding physics that you could do that students would find more interesting than word problems self-contained in math. For example, you could say that “you are trying to hit your teacher with a water balloon. Your first try had a certain velocity and distance in front of the teacher, and your second try had a certain velocity and distance behind the teacher. In order to hit the teacher, you will need half the angle between the vectors to hit the teacher. Figure out what angle and velocity you would need to hit the teacher with a water balloon.” This could also turn into an activity, where the students get to test their guesses to see if they can get close enough. There would be need to be something they could use to accurately catapult their water balloon, but that’s a different problem entirely.

B1. How can this topic be used in your students’ future courses in mathematics or science?

The dot product (and vectors in general) can be seen in physics, calculus 3, linear algebra, vector calculus, numerical analysis, and a bunch of other upper level math and science courses. Of course, not all students are going to be taking upper level math and science courses. However, out of the students going into STEM majors, they most assuredly will see the dot product and by seeing how vectors work earlier in their math careers, they will be more comfortable manipulating something they have already seen before. Also, the dot product and vectors are very useful as a tool to use in upper levels of math and in many different applications of engineering and computer science. In the game design, the dot product can be used to help engineer objects movements in the game work more realistically as a single unit and in relation to other objects.

E1. How can technology be used?

Geogebra is a great site to use since it has a tool https://www.geogebra.org/m/PGHaDjmD that will visually show you how the dot product works. It’s awesome because you get multiple different representations side by side, so that students who understand at different levels can all get something from this visual, interactive program. They can see how changing the position of the vectors changes the dot product and how it relates to the angle between the two vectors. Also, students will most likely be more engaged with this activity than just doing a bunch of examples with no real concept of how all of these pieces relate together which is not good in terms of promoting conceptual understanding. I think you could also use Desmos as an activity builder to make something similar to the above tool if students find the tool confusing to either use or look at.

References:
https://hackernoon.com/applications-of-the-vector-dot-product-for-game-programming-12443ac91f16

Happy Pythagoras Day!

Happy Pythagoras Day! Today is 12/16/20 (or 16/12/20 in other parts of the world), and $12^2 + 16^2 = 20^2$ .

Engaging students: Powers and exponents

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 Andrew Cory. His topic, from Pre-Algebra: powers and exponents.

B1. How can this topic be used in your students’ future courses in mathematics or science?

Exponents are just an easier way to multiply the same number by itself numerous times. They extend on the process of multiplication and allow students to solve expressions such as 2*2*2*2 quicker by writing them as $2^4$ . They are used constantly in future math courses, almost as commonly as addition and multiplication. Exponential functions start becoming more and more common as well. They’re used to calculate things such as compounding interest, or growth and decay. They also become common when finding formulas for sequences and series.
In science courses, exponents are often used for writing very small or very large numbers so that calculations are easier. Large masses such as the mass of the sun are written with scientific notation. This also applies for very small measurements, such as the length of a proton. They are also used in other ways such as bacteria growth or disease spread which apply directly to biology.

C2. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Any movie or TV show about zombies or disease outbreaks can be referenced when talking about exponents, and exponential growth. The rate at which disease outbreaks spread is exponential, because each person getting infected has a chance to get more people sick and it spreads very quickly. This can be a fun activity to demonstrate with a class to show how quickly something can spread. A teacher can select one student to go tap another student on the shoulder, then that student also gets up and walks around and taps another student. With students getting up and “infecting” others, more and more people stand up with each round, showing how many people can be affected at once when half the class is already up and then the other half gets up in one round.

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Euclid discovered exponents and used them in his geometric equations, he was also the first to use the term power to describe the square of a line. Rene Descartes was the first to use the traditional notation we use for exponents today. His version won out because of conceptual clarity. There isn’t exactly one person credited with creating exponents, it is more of a collaborative thing that got added onto over time. Archimedes discovered and proved the property of powers that states $10^a * 10^b = 10^{a+b}$ . Robert Recorde, the mathematician who created the equals sign, used some interesting terms to describe higher powers, such as zenzizenzic for the fourth power and zenzizenzizenzic for the eighth power. At a time, some mathematicians, such as Isaac Newton, would only use exponents for powers 3 and greater. Expressing things like polynomials as $ax3+bxx+cx+d$ .

References:

Berlinghoff, W. P., & Gouvêa, F. Q. (2015). Math through the ages: A gentle history for teachers and others.

Wikipedia contributors. (2019, August 28). Exponentiation. In Wikipedia, The Free Encyclopedia. Retrieved 00:24, August 31, 2019, from https://en.wikipedia.org/w/index.php?title=Exponentiation&oldid=912805138

Engaging students: Absolute value

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 Haley Higginbotham. Her topic, from Pre-Algebra: absolute value.

A2. How could you as a teacher create an activity or project that involves your topic?

A fun activity to do would be to have a graph on the floor taped out and divide people into pairs and give them sets of points to graph. Then, they would have to measure how far away they were from origin. I would ask if it mattered that the x and y values were sometimes negative, and why or why not. Hopefully they’ll respond that since they were measuring distance, and distance isn’t negative, then it didn’t matter if the x and y values were negative. And that would lead into the idea that absolute value refers to the distance from origin, and it doesn’t just “make the negative a positive number.” If I were to teach absolute value, I would very much want to emphasize this point because even though it seems like the absolute value just magically gets rid of negative signs, it is important to know what it actually is.

D2. How was this topic adopted by the mathematical community?

Originally, the term absolute value came from Jean-Robert Argand’s term ‘module’ (unit of measure in French). The term wasn’t commonly used in English until about 1857. The standard notation of vertical bars came from Karl Weierstrass in the time intermediate time. Now, the notation of vertical bars is used for different purposes in other areas of mathematics, like determinants and cardinality, which don’t relate to distance. However, the idea of absolute value (or magnitude) extends to the realm of physics, and science in general. Generally, when you want to know how far an object has traveled, but it has returned to its original position, you take the magnitude of the distance. In physics, you often want to find the magnitude of a vector, in order to know the distance. It’s also helpful because you can extend this idea into multiple dimensions, even though the calculations can become longer than just removing the negative sign.

E1. How can technology be used?

GeoGebra’s graphing calculator is fantastic for math in general because it has a wide range of functionality besides just graphing. In terms of absolute value, you can graph the absolute value function easily and it will actually pop up with the vertical bars next to it and not just abs(x) which is good since then student can get more familiar with the notation. GeoGebra allows you to measure distance between points, which is really the important tool in this case. You can easily plot different points and measure the distances to verify more accurately that the distances are the same regardless of sign. GeoGebra is also fairly intuitive to use, which is good if you have students who aren’t very familiar with using technology. Plus, it’s just plain fun to play with and students will love the fact they don’t have to graph a bunch of points and functions by hand.

References:
en.wikipedia.org/wiki/Absolute_value
geogebra.org/graphing

Engaging students: Arithmetic series

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 Eduardo Torres Manzanarez. His topic, from Precalculus: arithmetic series.

A1) What interesting (i.e., uncontrived) word problems using this topic can your students do now?

One interesting word problem to ask students to get them thinking about the idea of an arithmetic series, specifically a finite arithmetic series, is to have students come up with the total sum of the first 100 positive integers larger than 0 (i.e., 1 to 100) without actually adding all the integers up. Students will probably not figure out the total sum without adding the integers up one by one but if students are shown these numbers physically as cards labeled then a few might notice that the numbers taken at each end form pairs that add to the same sum. Turns out that the total sum is the number of pairs multiplied by 101. It can be explained to students that the 101 results from taking the first term and the last term (i.e., 1 and 100) and seeing that the sum is 101. This is true when we add 2 and 99, 3 and 98, 4, and 97, and so on. Hence, we will have 50 pairs since we have 100 numbers and so we have 50*101 as our sum. This problem can be extended to the story Gauss and how he apparently solved this problem as a child relatively fast and the teacher pointed out this question to them because he was apparently lazy. Now, this can be extended to adding all the integers from 1 to 200 and so on and having students come up with a general formula. Students can then think about an odd number of integers and see if that formula holds. Lastly, the connection between adding a number of terms with the same difference between each term is defined as an arithmetic series and so all the problems they have been doing are arithmetic problems in disguise.

B2) How can this topic be used in your students’ future courses in mathematics and science?

This topic is heavily used when discussing convergence in calculus. It provides insight into the validity that every series has a total sum that can be written as a number. Turns out this is true for all series that are finite but when discussing infinite series, it can be true of false that it converges to an actual value. So, students will have to ponder this idea for infinite arithmetic series in the future. Also, arithmetic series can be used to model certain situations in science within biology and physics. Thinking about arithmetic series provides information in tackling other types of series such as geometric in terms of behavior and solution. How does a geometric series behave? Well, each term increases with a common ratio instead of a common addition. Does the finite series converge? Yes, we know that every finite series does and this one basically behaves like the arithmetic in which we can easily find the total sum using a formula. Does the infinite series converge? Well, just like an arithmetic series it depends on the situation and the terms within the problem.

C1) How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

This topic has appeared in a particular movie called “All Quiet on the Western Front” which was released in 1930 and is an adaption of the novel that was published in 1929 by Erich Remarque. Within this movie, there is a scene in which a soldier states the formula for finding the sum of an arithmetic series. The soldier specifically states the formula S = A + N*(L / 2) and this corresponds to arithmetic series in accordance with the area of a rectangle and the area of a triangle. This is in a way a longer version of the short-hand formula we use today. One particular statement made from the soldier is that he mentions how beautiful the formula is. For some students, they can probably relate to the idea that something so complicated as adding 100000 terms that have a constant difference can be found using a short formula. Many problems in mathematics seem complicated at first in accordance with doing “grunt work” but many of them have beautiful solutions to them.

Engaging students: Computing logarithms with base 10

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 Andrew Sansom. His topic, from Precalculus: computing logarithms with base 10.

D1. 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 slide rule was originally invented around 1620, shortly after Napier invented the logarithm. In its simplest form, it uses two logarithmic scales that slide past each other, allowing one to multiply and divide numbers easily. If the scales were linear, aligning them would add two numbers together, but the logarithmic scale turns this into a multiplication problem. For example, the below configuration represents the problem: $14 \times 18=252$ .

Because of log rules, the above problem can be represented as:

$\log 14 + \log 18 = \log 252$

The C-scale is aligned against the 14 on the D-scale. The reticule is then translated so that it is over the 18 on the C-scale. The sum of the log of these two values is the log of their product.

Most modern students have never seen a slide rule before, and those that have heard of one probably know little about it other than the cliché “we put men on the moon using slide rules!” Consequently, there these are quite novel for students. A particularly fun, engaging activity to demonstrate to students the power of logarithms would be to challenge volunteers to a race. The student must multiply two three-digit numbers on the board, while the teacher uses a slide rule to do the same computation. Doubtless, a proficient slide rule user will win every time. This activity can be done briefly but will energize the students and show them that there may be something more to this “whole logarithm idea” instead of some abstract thing they’ll never see again.

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

Computing logarithms with base 10, especially with using logarithm properties, easily leads to learning to compute logarithms in other bases. This generalizes further to logarithmic functions, which are one of the concepts from precalculus most useful in calculus. Integrals with rational functions usually become problems involving logarithms and log properties. Without mastery of the aforementioned rudimentary skills, the student is quickly doomed to be unable to handle those problems. Many limits, including the limit definition of e, Euler’s number, cannot be evaluated without logarithms.

Outside of pure math classes, the decibel is a common unit of measurement in quantities that logarithmic scales with base 10. It is particularly relevant in acoustics and circuit analysis, both topics in physics classes. In chemistry, the pH of a solution is defined as the negative base-ten logarithm of the concentration of hydrogen ions in that solution. Acidity is a crucially important topic in high school chemistry.

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Many word problems could be easily constructed involving computations of logarithms of base 10. Below is a problem involving earthquakes and the Richter scale. It would not be difficult to make similar problems involving the volume of sounds, the signal to noise ratio of signals in circuits, or the acidity of a solution.

The Richter Scale is used to measure the strength of earthquakes. It is defined as

$M = \log(I/S)$

where $M$ is the magnitude, $I$ is the intensity of the quake, and $S$ is the intensity of a “standard quake”. In 1965, an earthquake with magnitude 8.7 was recorded on the Rat Islands in Alaska. If another earthquake was recorded in Asia that was half as intense as the Rat Islands Quake, what would its magnitude be?