My Favorite One-Liners: Part 56

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.

This really awful pun comes from a 1980s special by the comedian Gallagher; I would share a video clip here, but I couldn’t find it. I’ll tell this joke the first time that the Greek letters \alpha, \beta, \gamma, or \delta appears in a course. For the discussion below, let’s say that \alpha appears for the first time.

Where does the symbol \alpha come from?

[Students answer: “The Greek alphabet.”]

Good. Now, where did the Greeks get it from?

[Students sit in silence.]

The answer is, ancient cavemen. The sounds in the Greek alphabet correspond to the first sounds that the caveman said when he first stepped out the cave, so you can tell a lot about human psychology based on the Greek alphabet.

The caveman stepped out of the cave, saw a nice bright, sunny day, and said, “Ayyyyy!”

[Students groan.]

So, “Ahhh.” What’s the second sound?

[Students: “buh” or “bee”]

Good, the second sound is “buh.” What’s the third sound?

[Students: “guh” or “cee”]

Don’t forget, it’s the Greek alphabet. “Guh.” What’s the fourth sound?

[Students: “duh”]

Good. Now let’s put these all together to see what the caveman was saying. “Ah buh guh day.”

“Have a good day!”

[Students laugh and/or groan deeply.]

One year, when I told this story, I had a student in the front row who was carefully taking notes as I told this story; she felt very silly when I finally reached the punch line.

My Favorite One-Liners: Part 54

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.

The complex plane is typically used to visually represent complex numbers. (There’s also the Riemann sphere, but I won’t go into that here.) The complex plane looks just like an ordinary Cartesian plane, except the “x-axis” becomes the real axis and the “y-axis” becomes the imaginary axis. It makes sense that this visualization has two dimensions since there are two independent components of complex numbers. For real numbers, only a one-dimensional visualization is needed: the number line that (hopefully) has been hammered into my students’ brains ever since elementary school.

While I’m on the topic, it’s unfortunate that “complex numbers” are called complex, as this often has the connotation of difficult. However, that’s not why our ancestors chose the word complex was chosen. Even today, there is a second meaning of the word: a group of associated buildings in close proximity to each other is often called an “apartment complex” or an “office complex.” This is the real meaning of “complex numbers,” since the real and imaginary parts are joined to make a new number.

When I teach my students about complex number, I tell the following true story of when my daughter was just a baby, and I was extremely sleep-deprived and extremely desperate for ways to get her to sleep at night.

I tried counting monotonously, moving my finger to the right on a number line with each number:

1, 2, 3, 4, ...

That didn’t work, so I tried counting monotonously again, but this time moving my finger to the left on a number line with each number:

-1, -2, -3, -4, ...

That didn’t work either, so I tried counting monotonously once more, this time moving my finger up the imaginary axis:

i, 2i, 3i, 4i...

For the record, that didn’t work either. But it gave a great story to tell my students.

 

My Favorite One-Liners: Part 48

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 techniques commonly taught in Algebra II or Precalculus is the Rational Root Test, which is a way of making a list of candidates of rational numbers that might (emphasis, might) be roots of the polynomial. This is a commonly taught method for finding the roots of polynomials whose degree is higher than 3. (Other techniques that are typically taught to students are Descartes’ Rule of Signs and (less commonly) the Upper and Lower Bound Rules.) For example, for the polynomial f(x) = 2x^3 + 5 x^2 - 2x - 15.

  • The factors of the constant term are \pm 1, \pm 3, \pm 5 and \pm 15, and so the numerator of any rational root must be one of these numbers.
  • The factors of the leading coefficient are \pm 1 and \pm 2, and so the denominator of any rational root must be one of these numbers.
  • In conclusion, if there’s a rational root, then it’s \pm 1, \pm 3, \pm 5, \pm 15, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2} and \pm \frac{15}{2}. In other words, we have a list of 16 possible rational roots. Not all of them will be roots, of course, since the cubic polynomial only has at most three distinct roots. Also, there’s no guarantee that any of them will be roots. The only way to find out if any of them work is by testing them, usually using synthetic division.

So, after a practice problem or two, I’ll ask my students,

What guarantee do you have that at least one of the possible rational roots will actually work?

After letting them think for a few seconds, I give them the answer:

The benevolence of your instructor.

In other words, there is no guarantee that any of the possible rational roots will actually work, except that the instructor (or author of the textbook) has rigged things so that it happens.

My Favorite One-Liners: Part 43

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. q Q

Years ago, my first class of students decided to call me “Dr. Q” instead of “Dr. Quintanilla,” and the name has stuck ever since. And I’ll occasionally use this to my advantage when choosing names of variables. For example, here’s a typical proof by induction involving divisibility.

Theorem: If n \ge 1 is a positive integer, then 5^n - 1 is a multiple of 4.

Proof. By induction on n.

n = 1: 5^1 - 1 = 4, which is clearly a multiple of 4.

n: Assume that 5^n - 1 is a multiple of 4.

At this point in the calculation, I ask how I can write this statement as an equation. Eventually, somebody will volunteer that if 5^n-1 is a multiple of 4, then 5^n-1 is equal to 4 times something. At which point, I’ll volunteer:

Yes, so let’s name that something with a variable. Naturally, we should choose something important, something regal, something majestic… so let’s choose the letter q. (Groans and laughter.) It’s good to be the king.

So the proof continues:

n: Assume that 5^n - 1 = 4q, where q is an integer.

n+1. We wish to show that 5^{n+1} - 1 is also a multiple of 4.

At this point, I’ll ask my class how we should write this. Naturally, I give them no choice in the matter:

We wish to show that 5^{n+1} - 1 = 4Q, where Q is some (possibly different) integer.

Then we continue the proof:

5^{n+1} - 1 = 5^n 5^1 - 1

= 5 \times 5^n - 1

= 5 \times (4q + 1) - 1 by the induction hypothesis

= 20q + 5 - 1

= 20q + 4

= 4(5q + 1).

So if we let Q = 5q +1, then 5^{n+1} - 1 = 4Q, where Q is an integer because q is also an integer.

QED

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On the flip side of braggadocio, the formula for the binomial distribution is

P(X = k) = \displaystyle {n \choose k} p^k q^{n-k},

where X is the number of successes in n independent and identically distributed trials, where p represents the probability of success on any one trial, and (to my shame) q is the probability of failure.

 

 

My Favorite One-Liners: Part 42

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.

The function f(x) = a^x typically exhibits exponential growth (if a > 1) or exponential decay (if a < 1). The one exception is if a = 1, when the function is merely a constant. Which often leads to my favorite blooper from Star Trek. The crew is trying to find a stowaway, and they get the bright idea of turning off all the sound on the ship and then turning up the sound so that the stowaway’s heartbeat can be heard. After all, Captain Kirk boasts, the Enterprise has the ability to amplify sound by 1 to the fourth power.

My Favorite One-Liners: Part 40

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.

In some classes, the Greek letter \phi or \Phi naturally appears. Sometimes, it’s an angle in a triangle or a displacement when graphing a sinusoidal function. Other times, it represents the cumulative distribution function of a standard normal distribution.

Which begs the question, how should a student pronounce this symbol?

I tell my students that this is the Greek letter “phi,” pronounced “fee”. However, other mathematicians may pronounce it as “fie,” rhyming with “high”. Continuing,

Other mathematicians pronounce it as “foe.” Others, as “fum.”

My Favorite One-Liners: Part 32

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. Today’s story is a continuation of yesterday’s post. I call today’s one-liner “Method #1… Method #2.”

Every once in a while, I want my students to figure out that there’s a clever way to do a problem that will save them a lot of time, and they need to think of it.

For example, in Algebra II, Precalculus, or Probability, I might introduce the binomial coefficients to my students, show them the formula for computing them and how they’re related to combinatorics and to Pascal’s triangle, and then ask them to compute \displaystyle {100 \choose 3}. We write down

\displaystyle {100 \choose 3} = \displaystyle \frac{100!}{3!(100-3)!} = \displaystyle \frac{100!}{3! \times 97!}

So this fraction needs to be simplified. So I’ll dramatically announce:

Method #1: Multiply out the top and the bottom.

This produces the desired groans from my students. If possible, then I list other available but undesirable ways of solving the problem.

Method #2: Figure out the 100th row of Pascal’s triangle.

Method #3: List out all of the ways of getting 3 successes in 100 trials.

All of this gets the point across: there’s got to be an easier way to do this. So, finally, I’ll get to what I really want my students to do:

Method #4: Write 100! = 100 \times 99 \times 98 \times  97!, and cancel.

The point of this bit of showman’s patter is to get my students to think about what they should do next as opposed to blindly embarking in a laborious calculation.

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As another example, consider the following problem from Algebra II/Precalculus: “Show that x-1 is a factor of f(x)=x^{78} - 4 x^{37} + 2 x^{15} + 1.”

As I’m writing down the problem on the board, someone will usually call out nervously, “Are you sure you mean x^{78}?” Yes, I’m sure.

“So,” I announce, “how are we going to solve the problem?”

Method #1: Use synthetic division.

Then I’ll make a point of what it would take to write down the procedure of synthetic division for this polynomial of degree 78.

Method #2: (As my students anticipate the real way of doing the problem) Use long division.

Understanding laughter ensures. Eventually, I tell my students — or, sometimes, my students will tell me:

Method #3: Calculate f(1).

 

My Favorite One-Liners: Part 28

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. Today’s quip is one that I’ll use when simple techniques get used in a complicated way.

Consider the solution of the linear recurrence relation

Q_n = Q_{n-1} + 2 Q_{n-2},

where F_0 = 1 and F_1 = 1. With no modesty, I call this one the Quintanilla sequence when I teach my students — the forgotten little brother of the Fibonacci sequence.

To find the solution of this linear recurrence relation, the standard technique — which is a pretty long procedure — is to first solve the characteristic equation, from Q_n - Q_{n-1} - 2 Q_{n-2} = 0, we obtain the characteristic equation

r^2 - r - 2 = 0

This can be solved by any standard technique at a student’s disposal. If necessary, the quadratic equation can be used. However, for this one, the left-hand side simply factors:

(r-2)(r+1) = 0

r=2 \qquad \hbox{or} \qquad r = -1

(Indeed, I “developed” the Quintanilla equation on purpose, for pedagogical reasons, because its characteristic equation has two fairly simple roots — unlike the characteristic equation for the Fibonacci sequence.)

From these two roots, we can write down the general solution for the linear recurrence relation:

Q_n = \alpha_1 \times 2^n + \alpha_2 \times (-1)^n,

where \alpha_1 and \alpha_2 are constants to be determined. To find these constants, we plug in n =0:

Q_0 = \alpha_1 \times 2^0 + \alpha_2 \times (-1)^0.

To find these constants, we plug in n =0:

Q_0 = \alpha_1 \times 2^0 + \alpha_2 \times (-1)^0.

We then plug in n =1:

Q_1 = \alpha_1 \times 2^1 + \alpha_2 \times (-1)^1.

Using the initial conditions gives

1 = \alpha_1 + \alpha_2

1 = 2 \alpha_1 - \alpha_2

This is a system of two equations in two unknowns, which can then be solved using any standard technique at the student’s disposal. Students should quickly find that \alpha_1 = 2/3 and \alpha_2 = 1/3, so that

Q_n = \displaystyle \frac{2}{3} \times 2^n + \frac{1}{3} \times (-1)^n = \frac{2^{n+1} + (-1)^n}{3},

which is the final answer.

Although this is a long procedure, the key steps are actually first taught in Algebra I: solving a quadratic equation and solving a system of two linear equations in two unknowns. So here’s my one-liner to describe this procedure:

This is just an algebra problem on steroids.

Yes, it’s only high school algebra, but used in a creative way that isn’t ordinarily taught when students first learn algebra.

I’ll use this “on steroids” line in any class when a simple technique is used in an unusual — and usually laborious — way to solve a new problem at the post-secondary level.

 

 

My Favorite One-Liners: Part 27

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.

Here’s an anecdote that I’ll share when teaching students about factorials:

1! = 1

2! = 1 \times 2 = 2

3! = 1 \times 2 \times 3 = 6

4! = 1 \times 2 \times 3 \times 4 = 24

5! = 1 \times 2 \times 3 \times 4 \times 5 = 120

The obvious observation is that the factorials get big very, very quickly.

Here’s my anecdote:

Many years ago, I was writing lesson plans while the TV show “Wheel of Fortune” was on in the background. And the contestant solved the puzzle at the end, and Pat Sajak declared, “You have just won $40,320 in cash in prizes.

So I immediately thought to myself, “Ah, 8 factorial.”

Then I thought, ugh [while slapping myself in the forehead, grimacing, and shaking my head, pretending that I can’t believe that that was the first thought that immediately came to mind].

[Finishing the story:] Not surprisingly, I was still single when this happened.

My Favorite One-Liners: Part 20

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.

Perhaps the world’s most famous infinite series is

S = \displaystyle \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots

as this is the subject of Zeno’s paradox. When I teach infinite series in class, I often engage the students by reminding students about Zeno’s paradox and then show them this clip from the 1994 movie I.Q.

This clip is almost always a big hit with my students. After showing this clip, I’ll conclude, “When I was single, this was part of my repertoire of pick-up lines… but it never worked.”

Even after showing this clip, some students resist the idea that an infinite series can have a finite answer. For such students, I use a physical demonstration: I walk half-way across the classroom, then a quarter, and so on… until I walk head-first into a walk at full walking speed. The resulting loud thud usually confirms for students that an infinite sum can indeed have a finite answer.

For further reading, see my series on arithmetic and geometric series.