Predicate Logic and Popular Culture (Part 193): Randy Travis

Let $T$ be the set of all time, and let $L(t)$ be the proposition “I am going to love you at time $t$.” Translate the logical statement

$\forall t \in T (L(t))$.

This matches a chorus of the famous Randy Travis song.

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.

I enjoyed this video about Madeleine Gates, who is both a middle blocker for the Stanford women’s volleyball team (ranked #2 in the country at the time of this writing) and also a graduate student in statistics. There aren’t a whole lot of graduate students who play NCAA sports (which would necessarily mean finishing their undergraduate degrees in three years or less), let alone play at an exceptionally high level, which pursuing an advanced degree in a field as demanding as statistics.

The IRS Uses Geometric Series?

I recently read the delightful article “The IRS Uses Geometric Series?” by Michelle Ghrist in the August/September 2019 issue of MAA FOCUS. The article concerns a church raffle for a \$4000 ATV in which the church would pay for the tax bill of the winner. This turned out to be an unexpected real-world application of an infinite geometric series. A few key quotes: According to the IRS rules at the time,

…winnings below a certain level [were] subject to a 25% regular gambling withholding tax…

My initial thought was that the church would need to pay $0.25 \times \4000 = \1000$ to the IRS. However, I then wondered if this extra $\1000$ payment would then be considered part of the prize and therefore also subject to 25% withholding, requiring the church to give $0.25 \times \1000 = \250$ more to the IRS. But then this $\250$ would also be part of the prize and subject to withholding, with this process continuing forever.

I got quite excited about the possibility of an infinite geometric series being necessary to implement IRS tax code. By my calculations… [gave] an effective tax rate of 33-1/3%.

I then read more of the instructions, which clarified if the payer pays the withholding tax rate for the payee, “the withholding is 33.33% of the FMV [Fair Market Value] of the noncash payment minus the amount of the wager.” It was satisfying to discover the behind-the-scenes math leading to that number…

In any event, I am glad to know that the IRS can properly apply geometric series.”

Here’s a link to the whole article: http://digitaleditions.walsworthprintgroup.com/publication/?m=7656&l=1#{%22issue_id%22:606088,%22page%22:%2214%22}

Note: The authors notes that, in January 2018, the IRS dropped the two above rates to 24% and 31.58%.

Solving a Math Competition Problem: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on an interesting math competition problem. This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)

Part 1: Statement of the problem.

Part 2: Initial thoughts on getting a handle on the problem.

Part 3: Initial insight.

Part 4: Geometric insight with a Riemann sphere.

Part 6: Getting past the roadblock.

Part 7: Insight.

Part 8: Proof of insight.

Part 9: Alternate solution (now that we know the answer).

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on what I’m calling the Facebook birthday problem, a simple variant of the classic birthday problem in probability.

Part 1: Statement of the Facebook birthday problem.

Part 2: Solution for expected value.

Part 3: Finding the variance (a).

Part 4: Finding the variance (b).

Part 5: Finding the variance (c).

My Favorite One-Liners: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on my favorite one-liners.

Mathematical Wisecracks for Almost Any Occasion: Part 2Part 7, Part 8, Part 12, Part 21, Part 28, Part 29, Part 41, Part 46, Part 53, Part 60, Part 63, Part 65, Part 71, Part 79, Part 84, Part 85, Part 100, Part 101Part 108, Part 109, Part 114

All-Purpose Anecdotes: Part 38, Part 50, Part 64, Part 70, Part 92, Part 94

Addressing Misconceptions: Part 3Part 4Part 11, Part 14, Part 15, Part 18, Part 30, Part 32, Part 33, Part 37, Part 45, Part 59

Tricky Steps in a Calculation: Part 5, Part 6

Greek alphabet and choice of variables: Part 40, Part 43, Part 56

Homework and exams: Part 39Part 47, Part 55, Part 57, Part 58, Part 66, Part 77, Part 78, Part 91, Part 96, Part 97, Part 107

Inequalities: Part 99

Simplification: Part 10, Part 102, Part 103

Polynomials: Part 19, Part 48, Part 49, Part 81, Part 90

Inverses: Part 16

Exponential and Logarithmic Functions: Part 1, Part 42, Part 68, Part 80, Part 110

Trigonometry: Part 9, Part 69, Part 76, Part 106

Complex numbers: Part 54, Part 67, Part 86, Part 112, Part 113

Sequences and Series: Part 20, Part 35, Part 111

Combinatorics: Part 27

Statistics: Part 22, Part 23, Part 36, Part 51, Part 52, Part 61, Part 95

Probability: Part 26, Part 31, Part 62, Part 93

Calculus: Part 24, Part 25, Part 72, Part 73, Part 74, Part 75, Part 83, Part 87, Part 88, Part 104

Logic and Proofs: Part 13, Part 17Part 34, Part 44, Part 89, Part 98

Differential Equations: Part 82, Part 105

Decimal Approximations of Logarithms: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the decimal expansions of logarithms.

Part 1: Pedagogical motivation: how can students develop a better understanding for the apparently random jumble of digits in irrational logarithms?

Part 2: Idea: use large powers.

Part 3: Further idea: use very large powers.

Part 4: Connect to continued fractions and convergents.

Part 5: Tips for students to find these very large powers.

Another Poorly Written Word Problem: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series poorly written word problem, taken directly from textbooks and other materials from textbook publishers.

Part 2: Estimation and rounding.

Part 3: Probability.

Part 4: Subtraction and estimation.

Part 5: Algebra and inequality.

Part 6: Domain and range of a function.

Part 7: Algebra and inequality.

Part 8: Algebra and inequality.

Part 9: Geometric series.

Part 10: Currently infeasible track and field problem.

Part 11: Another currently infeasible track and field problem.

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the Wason Selection Task.

Part 1: Statement of the problem.

Part 2: Answer of the problem.

Part 3: Pedagogical thoughts about the problem, including variants.

Engaging students: Area of a trapezoid

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 Lissette Molina. Her topic, from Geometry: finding the area of a trapezoid.

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

I believe most students in America all discovered finding the area of a trapezoid with one very easy and simple activity. Students are to receive a trapezoid of some different sizes. They are then asked to find area by cutting off the triangular sides. The student then finds that all trapezoids are composed of triangles and a rectangle. This is a very quick activity that requires students to come up with a formula that works across all trapezoids. Learning about finding the area of a shape with hands-on discoveries keeps the formula and how it became embedded into students’ memories. This activity may also work with most polygons.

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

Find the area of a trapezoid does not require much information from previous courses. One major topic the student should be able to have learned before coming into a geometry class should be area. However, very rarely, students do not know what area is already. So, the student should be able to apply what they know about area into finding the area of a trapezoid. This involves finding the area of a rectangle and a triangle. It is important that a student understands exactly where a formula is derived, so it is also important that students know that the trapezoid contains two shapes and that finding the area of those two shapes will help them find the area of the resultant trapezoid.

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

One helpful website or program is Desmos (desmos.com). There are usually modules made for students often made by teachers. I have not yet come across one already made, but here is what I have in mind. Desmos is primarily made for graphing, but there are so many functions in this website that it can be manipulated to perform other things such as the unit circle. One very helpful idea would be to make a shape of a trapezoid by combining two triangles of different sizes off each of a rectangle’s sides. Since these shapes are placed on top of a graph, students would be able to calculate the area by counting the square units. WIth triangles, students can count the number of half, quarter, etc. square units. This way, students can find the area of a trapezoid by counting the squares, and realize that it would be easiest to find the area of those two triangles and one rectangle and combine them.