# Borwein integrals

When teaching proofs, I always stress to my students that it’s not enough to do a few examples and then extrapolate, because it’s possible that the pattern might break down with a sufficiently large example. Here’s an example of this theme that I recently learned:

# Engaging students: Permutations

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 Sarah McCall. Her topic, from probability: permutations.

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

In high school math, word problems are essentially unavoidable. They can be a pain, but they do help students to be able to see applications of what they are learning as well as good problem solving skills. So, if we must make use of word problems, we might as well make them as engaging/fun as possible. Some examples of ones that I found and would use in my classroom:

1. Permutation Peter went to the grocery store yesterday and met a super cute girl. He was able to get her phone number (written on the back of his receipt), but today when he went to call her he couldn’t find it anywhere! He knows that it consisted of 7 digits between 0 and 9. Help Permutation Peter by figuring out how many combinations of phone numbers there are.
2. Every McDonald’s Big Mac consists of 10 layers: 2 patties, 3 buns, lettuce, cheese, onions, special sauce, and pickles. How many different ways are there to arrange a Big Mac?

How has this topic appeared in pop culture?

Many students are easily confused when they first learn the difference between permutations and combinations, because for most permutations is an unfamiliar concept. One way to show students that they have actually seen permutations before in everyday life is with a Rubik’s cube. To use this in class, I would have students pass around a Rubik’s cube, while I explained that each of the possible arrangements of the Rubik’s cube is a permutation. I would also present to them (and explain) the equation that allows you to find the total number of possibilities (linked below) which yields approximately 43 quintillion permutations. This means it would be virtually impossible for someone to solve it just by randomly turning the faces. Who says you won’t use math in the real world!

How can technology be used to effectively engage students with this topic?
In a day and age where a majority of our population is absorbed in technology, I believe that one of the most effective ways to reach high school students is to encourage the constructive use of technology in the classroom instead of fighting it. Khan academy is one of the best resources out there for confusing mathematics topics, because it engages students in a format that is familiar to them (YouTube); not to mention it may be effective for students’ learning to hear a different voice explaining topics other than their normal teacher. In my classroom, I would have my students use their phones, laptops, or tablets to work through khan academy’s permutation videos, examples, and practice problems (link listed below).

References

https://www.quora.com/How-are-permutations-applied-in-real-life

https://prezi.com/q3aaem0k2xie/permutations-in-the-real-world

https://ruwix.com/the-rubiks-cube/mathematics-of-the-rubiks-cube-permutation-group

Let $W(t)$ be the proposition “I try to walk away at time $t$,” and let $S(x,t)$ be the proposition “At time $t$, $latex x makes me turn around and stay.” Translate the logical statement $\forall t (W(t) \rightarrow \exists x (S(x,t)))$. This matches the opening lines of “I Can’t Tell You Why,” by the Eagles. 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 152): Stevie Wonder Let $Z(x)$ be the proposition “We are amazed by $x$,” let $A(x)$ be the proposition “We are amused by $x$, and let $D(x)$ be the proposition “$x$ is a thing you say you’ll do.” Translate the logical statement $\forall x (D(x) \Rightarrow Z(x) \land \lnot A(x))$. This matches the opening line of “You Haven’t Done Nothin'” by Stevie Wonder. 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 151): Carly Rae Jepsen Let $L(x)$ be the proposition “I can have $x$,” and let $D(x)$ be the proposition “You will do $x$.” Translate the logical statement $\lnot \exists x(\lnot L(x)) \land \lnot \exists x (\lnot D(x))$. This matches a line (complete with double negatives) from E-MO-TION by Carly Rae Jepsen. 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 150): Katy Perry Let $S(x)$ be the proposition “I stood for $x$,” and let $F(x)$ be the proposition “I fell for $x$.” Translate the logical statement $\forall x (\lnot S(x)) \land \forall x(F(x))$. This matches one of the lines in Katy Perry’s smash hit “Roar.” 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 149): Adele Let $L(t)$ be the proposition “At time $t$, it lasts in love,” and let $H(t)$ be the proposition “At time $t$, it hurts in love.” Translate the logical statement $\exists t_1 (L(t_1)) \land \exists t_2 (H(t_2))$. This matches part of “Someone Like You,” by Adele. 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 148): Miley Cyrus Let $M(t)$ be the proposition “At time $t$, there is another mountain,” let $A(t)$ be the proposition “At time $t$, I want to make it move,” let $B(t)$ be the proposition “At time$t\$, there is an uphill battle,” and let $L(t)$ be the proposition “At time $t$, I have to lose.” Translate the logical statement

$\forall t (M(t) \land A(t) \land B(t)) \land \exists t (L(t))$.

This matches the chorus of “The Climb,” by Miley Cyrus.

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 147): Hannah Montana

Let $M(x)$ be the proposition “$x$ makes mistakes,” let $D(x)$ be the proposition “$x$ has those days,” let $K(x)$ be the proposition “$x$ knows what I’m talking about,” and let $G(x)$ be the proposition “$x$ gets that way.” Translate the logical statement

$\forall x (M(x) \land D(x) \land K(x) \land G(x))$.

These are the opening lines to a Hannah Montana 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.

# Predicate Logic and Popular Culture (Part 146): Fallout

Let $C(t)$ be the proposition “War changes at time $t$.”Translate the logical statement

$\forall t (\lnot C(t))$.

This line was made popular by the video game series “Fallout.”

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.