My Favorite One-Liners: Part 3

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

I always encourage students to answer occasional questions in class; naturally, this opens the possibility that a student may suggest an answer that is completely wrong or is only partially correct. Naturally, I don’t want to discourage students from participating in class  by blunting saying “You’re wrong!” So I need to have a gentle way of pointing out that the proposed answer isn’t quite right.

Thanks to a recent movie, I finally have hit on a one-liner to do this with good humor and cheer: “To quote the trolls in Frozen, I’m afraid your answer is a bit of a fixer-upper. (Laughter) So it’s a bit of a fixer-upper, but this I’m certain of… you can fix this fixer-upper up with a little bit of love.”

If you have no idea about what I’m talking about, here’s the song from the movie (you can hate me for the rest of the day while you sing this song to yourself):

See also Math with Bad Drawing’s excellent post with thoughts on responding to students who give wrong answers.

My Favorite One Liners: Part 2

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

When doing a large computation, I’ll often leave plenty of blank space on the board to fill it later. For example, when proving by mathematical induction that

1 + 3 + 5 + \dots + (2n-1) = n^2,

the inductive step looks something like

1 + 3 + 5 \dots + (2k-1) + (2[k+1]-1) =

~

~

~

~

~

~

= (k+1)^2

So I explained that, to complete the proof by induction, all we had to do was convert the top line into the bottom line.

As my class swallowed hard as they thought about how to perform this task, I told them, “Yes, this looks really intimidating. Indeed, to quote the great philosopher, ‘You might think that I’m insane. But I’ve got a blank space, baby… so let’s write what remains.’ “

And, just in case you’ve been buried under a rock, here’s the source material for the one-liner (which, at the time of this writing, is the fifth-most watched video on YouTube):

My Favorite One-Liners: Part 1

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

One of the most common student mistakes with logarithms is thinking that

\log_b(x+y) = \log_b x + \log_b y.

Whenever students make this mistake, I call it the Third Classic Blunder. The first classic blunder, of course, is getting into a major land war in Asia. The second classic blunder is getting into a battle of wits with a Sicilian when death is on the line. And the third classic blunder is thinking that \log_b(x+y) somehow simplfies as \log_b x + \log_b y.

Sadly, as the years pass, fewer and fewer students immediately get the cultural reference. On the bright side, it’s also an opportunity to introduce a new generation to one of the great cinematic masterpieces of all time.

One of my colleagues calls this mistake the Universal Distributive Law, where the \log_b distributes just as if x+y was being multiplied by a constant. Other mistakes in this vein include  \sqrt{x+y} = \sqrt{x} + \sqrt{y}  and  (x+y)^2 = x^2 + y^2.

Along the same lines, other classic blunders are thinking that

\left(\log_b x\right)^n  simplifies as  \log_b \left(x^n \right)

and that

\displaystyle \frac{\log_b x}{\log_b y}  simplifies as  \log_b \left( \frac{x}{y} \right).

I’m continually amazed at the number of good students who intellectually know that the above equations are false but panic and use them when solving a problem.

Predicate Logic and Popular Culture (Part 123): Willie Nelson

Let M(t) be the proposition “You were on my mind at time t.” Translate the logical statement

\forall t < 0 (M(t)).

Naturally, this matches the classic song by Willie Nelson (though Elvis did record it before him).

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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 122): Queen

Let p be the proposition “I cross a million rivers,” let q be the proposition “I rode a million miles,” and let r be the proposition “I still am where I started.” Translate the logical statement

(p \land q) \Rightarrow r.

This matches a line from this classic by Queen.

green line

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 121): OneRepublic

Let F(x) be the proposition “x is a right friend,” let P(y) be the proposition “y is a right place,” let I(x,y) be the proposition “x is located at place y,” and let H(x,y) be the proposition “They have x at place y,” and let $p$ be the proposition “We’re going down.” Translate the logical statement

\forall x \forall y(F(x) \land P(y) \land I(x,y) \Rightarrow H(x,y)) \land p.

This matches the chorus of this song by OneRepublic.

green line

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 120): Crossfade

Let C(t) be the proposition “At time t, I meant to be so cold.” Translate the logical statement

\forall t < 0 \lnot C(t).

This matches the echo of this song by Crossfade.

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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 119): Billy Joel

Let p be the proposition “I’m gonna try for an uptown girl,” let B(x) the proposition “x has hot blood,” let q be the proposition “She’s looking for a downtown man,” and let r be the proposition “I’m a downtown man.” Also, define the function f(x) to be how long x has lived in a white bread world. Translate the logical statement

p \land \forall x (B(x) \Rightarrow (f(x) \le f(\hbox{she})) \land q \land r.

Of course, this matches the first chorus of the Billy Joel classic.

 

green line

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 118): Bruno Mars

Let D(x) be the proposition “Today I am doing x.” Translate the logical statement

\forall x \lnot D(x).

This matches the closing line of the chorus of the Bruno Mars song.

green line

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 117): Kelly Clarkson

Let K(x) be the proposition “x kills you,” let S(x) be the proposition “x makes you stronger,” and let T(x) be the proposition “x makes you stand a little taller.” Translate the logical statement

\forall x( \lnot K(x) \Rightarrow (S(x) \land T(x))).

This matches the first line of this hit song by Kelly Clarkson.

 

green line

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.