Predicate Logic and Popular Culture (Part 65): John Philip Sousa

Let $S(t)$ be the proposition “The Stars and Stripes wave at time $t$ .” Translate the logical statement

$\forall t (S(t))$ .

I tried to think of a fitting example for the Fourth of July, but the best that I could find was the closing line of the chorus of the Stars and Stripes Forever.

Which naturally leads me to this amazing version from the 1970s:

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 64): Abraham Lincoln

Let $F(x,t)$ be the proposition “You can fool $x$ at time $t$ .” Translate the logical statement

$\exists t_1 \forall x (F(x,t_1)) \land \exists x_1 \forall t(F(x_1,t)) \land \lnot(\forall x \forall t(F(x,t)))$ .

Of course, this is the famous quote commonly attributed to Abraham Lincoln: “You can fool all of the people some of the time, and some of the people all of the time, but you cannot fool all of the people all of the time.”

Predicate Logic and Popular Culture (Part 63): P. T. Barnum

Let $S(x) be the proposition "$ latex x$ is a sucker,” and let $B(x,t)$ be the proposition “ $x$ is born at time t.” Translate the logical statement

$\forall t \exists x (S(x) \land B(x,t))$ .

Naturally, this is the famous quote often attributed to P. T. Barnum: “There’s a sucker born every minute.”

Predicate Logic and Popular Culture (Part 62): George Strait

Let $X(x)$ be the proposition “ $x$ is my ex,” and let $T(x)$ be the proposition “ $x$ lives in Texas.” Translate the logical statement

$\forall x (X(x) \Rightarrow T(x))$ ,

where the domain is all people.

Naturally, this one of the great hits in the storied career of George Strait.

Engaging students: Using Pascal’s triangle

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 Jason Trejo. His topic, from Precalculus: using Pascal’s triangle.

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

After some research and interesting observations I came across while examining Pascal’s Triangle, I feel like I could create some sort of riddle worksheet that involves the Triangle. Once I have taught my students how to create Pascal’s Triangle, I could give my students riddles such as:

Once you go and strive in prime, belittling your neighbors isn’t a crime.
- Students might notice that each number (other than 1) in a prime number row is divisible by that prime number:
  - Row 7= 1, 7, 21, 35, 35, 21, 7, 1
  - Row 11= 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
- Naturally shallow slides aren’t much fun, but with a fib of raunchy, it is this one.
  - Given that I have gone over the Fibonacci sequence with my students prior to these riddles, I could include this one. The students should eventually see that if you take shallow diagonals on Pascal’s Triangle, the sum of those diagonals are the consecutive numbers in the Fibonacci sequence.
- In a game on blades, you can’t be a schmuck with a puck. Be nimble and quick to look for the stick.
  - This one is a little more straightforward compared to the last two so hopefully the students will make the connection to notice the hockey stick pattern on the diagonals of Pascal’s Triangle. When adding the numbers down a diagonal, then the number to the side and below will be the sum, thus looking like a hockey stick.
- What else is there? What else is in store? What patterns can you find when you know who to root four?
  - The “typo” is intentional to give a hint at another pattern the students might notice on Pascal’s Triangle. Now I am challenging the students to find more patterns within the Triangle such as:
    - Sum of rows are the powers of 2
    - Rows relate to the powers of 11 (get murky after the 4^th row)
    - Counting numbers, triangular numbers, etc.

The purpose of this activity would extend the use of Pascal’s triangle from what they already know. I could assign this at the beginning of the lesson and if no one understands what the riddles meant, we could come back as a class and figure them out together once the lesson was done. These riddles could be an assignment of their own if I introduce them after they are very familiar with Pascal’s Triangle.

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

I would say the primary use most students will get from Pascal’s Triangle would be to find the coefficients of binomials since it is much easier when working on binomial expansions, but there are also other ways they can use the Triangle as well. For one, it can be of great use in many courses that involve since it is a visual in seeing the number of combinations there are based on the number of items used. For example, say there are 6 different pieces of candy in a bowl and you need to know how many different ways can you choose 3 candies? Using Pascal’s Triangle, we look at the 6^th row and the 3^rd entry in that row (remembering the top row is Row 0 and the first 1 in each row is Entry 0), we can see that there are 20 possible combinations of 3 different pieces of candy. Other than that, even based on the riddle activity from above, students can use Pascal’s Triangle and its various patterns to help remember such things as triangular numbers, powers of 11, etc.

How has this topic appeared in high culture?

Within the past few years, the Shanghai-based design company, Super Nature Design, created the interactive art piece “Lost in Pascal’s Triangle”. This structure takes inspiration from Pascal’s Triangle and allows people to “explore the concept and magnification of the Pascal’s Triangle mathematics formula.” The following link takes you to the website that gives a bit more information behind the piece and shows how people can interact with the structure through a xylophone-type console: http://www.supernaturedesign.com/work/pascaltriangle#8

Another quick application that can be done through Pascal’s Triangle is by seeing the relationship between the Triangle and Sierpinski’s triangle (as shown below):

The pattern is by shading in every odd number on Pascal’s Triangle, you start creating Sierpinski’s triangle which is found in many works of art like these:

It might actually be a small but fun project to have the students create something like this at the beginning of the lesson and then explain the relation of the two special triangles.

References:

Pascal Triangle Information: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

Image of Pascal’s Triangle: http://mathforum.org/workshops/usi/pascal/images/pascal.hex2.gif

Lost in Pascal’s Triangle: http://www.designboom.com/weblog/images/images_2/andrea/super_nature_design/pascaltriangle01.jpg

Super Nature Design: http://www.supernaturedesign.com/work/pascaltriangle#2

Pascal and Sierpinski Triangle : http://mathforum.org/workshops/usi/pascal/images/sierpinski.pascalfrac.gif

Sierpinski Pyramid: http://www.sierpinskitetrahedron.com/images/sierpinski-tetrahedron-breckenridge.JPG

Sierpinski Art Project: http://fractalfoundation.org/wp-content/uploads/2009/03/sierpkids1.jpg

Conditional Statements

Source: http://xkcd.com/1652/

Engaging students: Recognizing equivalent statements

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 Tiffany Jones. Her topic, from Geometry: recognizing equivalent statements.

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

How has this topic appeared in pop culture?

The topic of recognizing equivalent statements appears in pop culture in Lewis Carroll’s “Alice’s Adventures in Wonderland”. English mathematician Reverend Charles Lutwidge Dodgson published the work in 1865 under the pseudonym Lewis Carroll.

Reverend Charles Lutwidge Dodgson was a mathematician, so it is not surprising that mathematical concepts appear in his writing. During the tea party scene, examples of logic statements are present. More specifically, how statements and their converse do not have the same meaning or truth-value. The March Hare asks Alice to say what she means, to which she responds with that she means what she says, thinking that to be the same thing. The Hatter disagrees and the conversation continues with three examples to show Alice that the statements are not the same. Here is the text:

Here is the excerpt from the text from chapter seven:

‘Do you mean that you think you can find out the answer to it?’ said the March Hare.

‘Exactly so,’ said Alice.

‘Then you should say what you mean,’ the March Hare went on.

‘I do,’ Alice hastily replied; ‘at least—at least I mean what I say—that’s the same thing, you know.

”Not the same thing a bit!’ said the Hatter. ‘You might just as well say that “I see what I eat” is the same thing as “I eat what I see”!

”You might just as well say,’ added the March Hare, ‘that “I like what I get” is the same thing as “I get what I like”!

”You might just as well say,’ added the Dormouse, who seemed to be talking in his sleep, ‘that “I breathe when I sleep” is the same thing as “I sleep when I breathe”!

”It IS the same thing with you,’ said the Hatter, and here the conversation dropped, and the party sat silent for a minute, while Alice thought over all she could remember about ravens and writing–desks, which wasn’t much.

The lesson on logical statements and truth-values would start with a reading of this section of text or viewing of a clip that keeps the original text. Take a simple statement and write its converse, inverse, and contrapositive. For example, “I like what I get” becomes “I get what I like”, “I do not like what I do not get”, and “I do not get what I do not like”, respectively. Discuss the truth-values of each of the statements show that the original and contrapositive are equivalent and that the converse and inverse are equivalent, to help the students see patterns when rewriting a statement.

Then the students will complete a worksheet by MathBits.com to ensure that they understand the process with simple English sentences and to introduce them to the idea with simple mathematical statements. The worksheet includes a portion of the text above for the students’ reference. The worksheet has the students take two simple sentence and write them in the form of “if…, then..”, then the students are to write their converse, inverse, and contrapositive. Next, the students compare the truth-values of each statement. Finally, the students are given two mathematical statements and are asked to determine the truth-values.

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

Recognizing equivalent statements appears in analysis courses and courses which proofs are often used. Being able to recognize equivalent statements adds another tool to the the tool box of proof writing. With Proof writing, sometimes the contrapositive form of the statement is easier to prove than the original statement itself.

For example, in Math 4050 Advanced Study of Secondary Mathematics Curriculum, the proof of the following theorem is easier to prove by contrapositive than just straight on.

If a prime p divides m*n with m and n composite, then p divides m or p divides n.

The contrapositive, if p does not divide m and p does not divide n with p, m, and n the same as before, then p does not divide m*n, follows easily (with a little clairvoyance) from another theorem for the class.

Carroll, Lewis. Alice’s Adventures in Wonderland. Lit2Go Edition. 1865. Web. <http://etc.usf.edu/lit2go/1/alices-adventures-in-wonderland/>. October 1, 2015.

Carroll, Lewis. “Chapter VII: A Mad Tea-Party.” Alice’s Adventures in Wonderland. Lit2Go Edition. 1865. Web. <http://etc.usf.edu/lit2go/1/alices-adventures-in-wonderland/17/chapter-vii-a-mad-tea-party/>. October 1, 2015.

Roberts, Frederick, and MathBits.com. Alice in Wonderland Worksheet. S.l.: Commission of the European Communities, 1993. Mathbits. Commission of the European Communities. Web. 1 Oct. 2015

University of North Texas course math 4950 Advanced Study of Secondary Mathematics Curriculum lecture Fall 2015 taught Dr. John Quintanilla

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. Here’s my series on the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, 2b, and 2c: The 1089 trick.

Part 3a, 3b, and 3c: A geometric magic trick (see also here).

Part 4a, 4b, 4c, and 4d: A trick using binary numbers.

Part 5a, 5b, 5c, 5d: Predicting a digit that’s been erased from a number.

Part 6: Finale.

Part 7: The Fitch-Cheney 5-card trick.

Part 8a, 8b, 8c: A trick using Pascal’s triangle.

Fun With Permutations and Asimov’s Three Laws of Robotics

I’m not a big fan of science fiction, but I know enough to know that Isaac Asimov was one of the great science fiction novelists of the 20th century. The following was written by him in the October 1980 issue of The Magazine of Fantasy and Science Fiction and was reprinted in his book Counting the Eons, which was published in 1983. (I’m now holding the battered and torn pages of my copy of this book; I devoured Asimov’s musings on mathematics and science when I was young.)

Robotics has become a sufficiently well development technology to warrant articles and books on its history and I have watched this in amazement, and in some disbelief, because I invented it.

No, not the technology, the word.

In October 1941, I wrote a robot story entitled “Runaround,” first published in the March 1942 issue of Astounding Science Fiction, in which I recited, for the first time, my Three Laws of Robotics. Here they are:

A robot must not injure a human being or, through inaction, allow a human being to come to harm.

A robot must obey the orders give it by human beings except where those orders would conflict with the First Law.

A robot must protect its own existence, except where such protection would conflict with the First or Second Laws.

Clearly, the order in which the Three Laws of Robotics matters. Shuffling the order leads to $3! = 6$ possible permutations, and xkcd recently had some fun about what the consequences would be of those permutations.

Source: http://www.xkcd.com/1613/

Predicate Logic and Popular Culture (Part 61): Taylor Swift

Let $S(t)$ be the proposition “We are in style at time $t$ ,” let $C(t)$ be the proposition “We crash down at time $t$ ,” and let $B(t)$ be the proposition “We come back at time $t$ .” Translate the logical statement

$\forall t (\lnot S(t)) \Rightarrow (\forall t(C(t) \Rightarrow \exists u>t(B(u)))$ .

The straightforward way of translating this into English is, “If we never go out of style, then whenever we crash down we come back at a later time. This approximately matches the second half of the chorus of one of Taylor Swift’s hit songs.