My Favorite One-Liners: Part 11

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.

Every once in a while, I’ll cover a theorem in class that looks utterly surprising to students at first glance. For example, in trigonometry, I might state that

$\sin^{-1} \left( \sin \pi \right) \ne \pi$ ,

so that the inverse function doesn’t quite behave like it’s supposed to (because of the restricted domain used to define inverse sine.)

Before explaining why $\sin^{-1} \left( \sin \pi \right)$ isn’t equal to $\pi$ , I’ll get the discussion started by saying, “Don’t believe me? Just watch.”… a tip of the cap to this recent hit song (at the time of this writing, the third-most watched video on YouTube).

While on this topic, I have to tip my cap to Kelli Hauser, a sixth-grade teacher in my city who made the following motivational video for students about to take their end-of-year high-stakes test (called, here in Texas, the STAAR exam).

One more parody concerning a recent spacecraft that visited Pluto:

For further reading, here’s my series on inverse functions.

My Favorite One-Liners: Part 10

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.

While I can’t take credit for today’s one-liner, I’m more than happy to share it.

A colleague was explaining his expectations for simplifying expressions such as

$\displaystyle \frac{\displaystyle ~~~\frac{2x}{x^2+1}~~~}{\displaystyle ~~~\frac{x}{x^2-1}~~~}$

Of course, this isn’t yet simplified, but his students were balking about doing the required work. So, on the spur of the moment, he laid down a simple rule:

Not simplifying a fraction in a fraction is an infraction.

Utterly brilliant.

My Favorite One-Liners: Part 9

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, I’d like to discuss a common mistake students make in trigonometry… as well as the one-liner that I use to (hopefully) help students not make this mistake in the future.

Question. Find all solutions (rounded to the nearest tenth of a degree) of $\sin x = 0.8$ .

Erroneous Solution. Plugging into a calculator, we find that $x \approx 53.1^o$ .

arcsine1

The student correctly found the unique angle $x$ between $-90^o$ and $90^o$ so that $\sin x = 0.8$ . That’s the definition of the arcsine function. However, there are plenty of other angles whose sine is equal to $0.7$ . This can happen in two ways.

First, if $\sin x > 0$, then the angle $x$ could be in either the first quadrant or the second quadrant (thanks to the mnemonic All Scholars Take Calculus). So $x$ could be (accurate to one decimal place) equal to either $53.1^o$ or else $180^o - 53.1^o = 126.9^o$ . Students can visualize this by drawing a picture, talking through each step of its construction (first black, then red, then brown, then green, then blue).

However, most students don’t really believe that there’s a second angle that works until they see the results of a calculator.

Second, any angle that’s coterminal with either of these two angles also works. This can be drawn into the above picture and, as before, confirmed with a calculator.

So the complete answer (again, approximate to one decimal place) should be $53.1^{\circ} + 360n^o$ and $126.9 + 360n^{\circ}$ , where $n$ is an integer. Since integers can be negative, there’s no need to write $\pm$ in the solution.

Therefore, the student who simply answers $53.1^o$ has missed infinitely many solutions. The student has missed every nontrivial angle that’s coterminal with $53.1^o$ and also every angle in the second quadrant that also works.

Here’s my one-liner — which never fails to get an embarrassed laugh — that hopefully helps students remember that merely using the arcsine function is not enough for solving problems such as this one.

You’ve forgotten infinitely many solutions. So how many points should I take off?

For further reading, here’s my series on inverse functions.

My Favorite One-Liners: Part 8

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.

At many layers of the mathematics curriculum, students learn about that various functions can essentially commute with each other. In other words, the order in which the operations is performed doesn’t affect the final answer. Here’s a partial list off the top of my head:

Arithmetic/Algebra: $a \cdot (b + c) = a \cdot b + a \cdot c$ . This of course is commonly called the distributive property (and not the commutative property), but the essential idea is that the same answer is obtained whether the multiplications are performed first or if the addition is performed first.
Algebra: If $a,b > 0$ , then $\sqrt{ab} = \sqrt{a} \sqrt{b}$ .
Algebra: If $a,b > 0$ and $x$ is any real number, then $(ab)^x = a^x b^x$ .
Precalculus: $\displaystyle \sum_{i=1}^n (a_i+b_i) = \displaystyle \sum_{i=1}^n a_i + \sum_{i=1}^n b_i$ .
Precalculus: $\displaystyle \sum_{i=1}^n c a_i = c \displaystyle \sum_{i=1}^n a_i$ .
Calculus: If $f$ is continuous at an interior point $c$ , then $\displaystyle \lim_{x \to c} f(x) = f(c)$ .
Calculus: If $f$ and $g$ are differentiable, then $(f+g)' = f' + g'$ .
Calculus: If $f$ is differentiable and $c$ is a constant, then $(cf)' = cf'$ .
Calculus: If $f$ and $g$ are integrable, then $\int (f+g) = \int f + \int g$ .
Calculus: If $f$ is integrable and $c$ is a constant, then $\int cf = c \int f$ .
Calculus: If $f: \mathbb{R}^2 \to \mathbb{R}$ is integrable, $\iint f(x,y) dx dy = \iint f(x,y) dy dx$ .
Calculus: For most differentiable function $f: \mathbb{R}^2 \to \mathbb{R}$ that arise in practice, $\displaystyle \frac{\partial^2 f}{\partial x \partial y} = \displaystyle \frac{\partial^2 f}{\partial y \partial x}$ .
Probability: If $X$ and $Y$ are random variables, then $E(X+Y) = E(X) + E(Y)$ .
Probability: If $X$ is a random variable and $c$ is a constant, then $E(cX) = c E(X)$ .
Probability: If $X$ and $Y$ are independent random variables, then $E(XY) = E(X) E(Y)$ .
Probability: If $X$ and $Y$ are independent random variables, then $\hbox{Var}(X+Y) = \hbox{Var}(X) + \hbox{Var}(Y)$ .
Set theory: If $A$ , $B$ , and $C$ are sets, then $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ .
Set theory: If $A$ , $B$ , and $C$ are sets, then $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ .

However, there are plenty of instances when two functions do not commute. Most of these, of course, are common mistakes that students make when they first encounter these concepts. Here’s a partial list off the top of my head. (For all of these, the inequality sign means that the two sides do not have to be equal… though there may be special cases when equality happens to happen.)

Algebra: $(a+b)^x \ne a^x + b^x$ if $x \ne 1$ . Important special cases are $x = 2$ , $x = 1/2$ , and $x = -1$ .
Algebra/Precalculus: $\log_b(x+y) = \log_b x + \log_b y$ . I call this the third classic blunder.
Precalculus: $(f \circ g)(x) \ne (g \circ f)(x)$ .
Precalculus: $\sin(x+y) \ne \sin x + \sin y$ , $\cos(x+y) \ne \cos x + \cos y$ , etc.
Precalculus: $\displaystyle \sum_{i=1}^n (a_i b_i) \ne \displaystyle \left(\sum_{i=1}^n a_i \right) \left( \sum_{i=1}^n b_i \right)$ .
Calculus: $(fg)' \ne f' \cdot g'$ .
Calculus $\left( \displaystyle \frac{f}{g} \right)' \ne \displaystyle \frac{f'}{g'}$
Calculus: $\int fg \ne \left( \int f \right) \left( \int g \right)$ .
Probability: If $X$ and $Y$ are dependent random variables, then $E(XY) \ne E(X) E(Y)$ .
Probability: If $X$ and $Y$ are dependent random variables, then $\hbox{Var}(X+Y) \ne \hbox{Var}(X) + \hbox{Var}(Y)$ .

All this to say, it’s a big deal when two functions commute, because this doesn’t happen all the time.

I wish I could remember the speaker’s name, but I heard the following one-liner at a state mathematics conference many years ago, and I’ve used it to great effect in my classes ever since. Whenever I present a property where two functions commute, I’ll say, “In other words, the order of operations does not matter. This is a big deal, because, in real life, the order of operations usually is important. For example, this morning, you probably got dressed and then went outside. The order was important.”

My Favorite One-Liners: Part 7

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 my favorite one-liners is simply stated: “Mathematicians are lazy.” I’ll use this whenever I introduce my students a new piece of mathematical notation or lingo.

For example, in probability, a common notion is a sequence of independent and identically distributed random variables (say, rolling a die repeatedly). However, mathematicians will typically write “i.i.d.” instead of “independent and identically distributed.” Why? That’s when I break out the mantra: “Mathematicians are lazy.” It’s my quick way of saying, “Hey, this is new notation that you’re about to learn, but the whole point of new notation is to make writing mathematical ideas a little quicker.”

Mathematical notation like $\displaystyle \int_a^b f(x) \, dx$ can appear very intimidating when students first encounter them. Hopefully repeating this mantra a few dozen times each semester makes the introduction of new notation a little more palatable for my students.

My Favorite One-Liners: Part 6

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.

Sometimes, I’ll expect students to learn and master operations that cancel. For example, in Precalculus, I want my students to know the sum-to-product trigonometric identities

$\sin u + \sin v = \displaystyle 2 \sin\left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right)$ ,

$\sin u - \sin v = \displaystyle 2 \cos\left( \frac{u+v}{2} \right) \sin\left( \frac{u-v}{2} \right)$ ,

$\cos u + \cos v = \displaystyle 2 \cos\left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right)$ ,

$\cos u - \cos v = \displaystyle -2 \sin \left( \frac{u+v}{2} \right) \sin\left( \frac{u-v}{2} \right)$ .

These can be helpful for solving trigonometric equations. For example, to solve $\cos 3x + \cos 7x = 0$ , we have

$\cos 3x + \cos 7x = 0$

$\displaystyle 2 \sin\left( \frac{3x+7x}{2} \right) \cos \left( \frac{3x-7x}{2} \right) = 0$

$2 \sin 5x \cos (-2x) = 0$

$2 \sin 5x \cos 2x = 0$

$\displaystyle x = \frac{n\pi}{5} \qquad \hbox{or} \qquad \displaystyle x = \left( \frac{n}{2} + \frac{1}{4} \right)\pi$ for integers $n$ .

However, I also want my students to know the product-to-sum trigonometric identities

$\cos x \cos y = \displaystyle \frac{1}{2} [\cos(x+y) + \cos(x-y) ]$ ,

$\sin x \sin y = \displaystyle \frac{1}{2} [\cos(x-y) - \cos(x+y) ]$ ,

$\sin x \cos y = \displaystyle \frac{1}{2} [\sin(x+y) + \sin(x-y) ]$ .

These are useful when computing certain definite integrals (especially related to Fourier series). For example, if $m \ne n$ are both integers, then

$\displaystyle \int_0^{2\pi} \cos mx \cos nx \, dx = \displaystyle \int_0^{2\pi} \frac{1}{2} \left[\cos([m+n]x) + \cos([m-n])x) \right] \, dx$

$= \left[ \displaystyle \frac{\sin([m+n]x)}{2(m+n)} + \frac{\sin([m-n]x)}{2(m-n)} \right]^{2\pi}_0$

$= \displaystyle \frac{\sin(2[m+n]\pi) - \sin 0}{2(m+n)} + \frac{\sin(2[m-n]\pi) - \sin 0}{2(m-n)}$ .

$=0$

This integral and other similar integrals are necessary to find the formula for the coefficients in a Fourier series.

In other words, sometimes I’ll want my students to convert a product into a sum. Other times, I’ll want my students to convert a sum into a product.

To help this sink in, I’ll tell my students, “To quote the great philosopher: Sometimes you gotta know when to hold ’em, know when to fold them.”

However, when I made this joke recently, a student innocently asked, “What great philosopher said that?” I turned the question back to my class, but not one of my class of millennials knew the answer. One person came close with his answer of “Willie” — wrong answer but correct genre and time frame. (Somebody else answered Socrates.)

So that my students actually learn something important in my class, here’s the cultural reference:

My Favorite One-Liners: Part 5

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.

Every once in a while, students encounter a step that seems far from obvious. To give one example, to evaluate the series

$\displaystyle \sum_{n=1}^{100} \frac{1}{n^2+n}$ ,

the natural first step is to rewrite this as

$\displaystyle \sum_{n=1}^{100} \left(\frac{1}{n} - \frac{1}{n+1} \right)$

and then use the principle of telescoping series. However, students may wonder how they were ever supposed to think of the first step for themselves.

Students often give skeptical, quizzical, and/or frustrated looks about this non-intuitive next step… they’re thinking, “How would I ever have thought to do that on my own?” To allay these concerns, I explain that this step comes from the patented Bag of Tricks. Socrates gave the Bag of Tricks to Plato, Plato gave it to Aristotle, it passed down the generations, my teacher taught the Bag of Tricks to me, and I teach it to my students.

Sadly, there aren’t any videos of Greek philosophers teaching, so I’ll have to settle for this:

My Favorite One-Liners: Part 4

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.

During in-class discussions, students often but inadvertently make the same mistakes over and over again… say, thinking that $\sqrt{a^2+b^2} = a + b$ or $\displaystyle \frac{d}{dx} (uv) = \displaystyle \frac{du}{dx} \cdot \frac{dv}{dx}$ . Naturally, such mistakes need to be corrected, but hopefully politely and in a memorable way.

After the third or fourth such repetition of the same mistake during a semester, I’ll try to lighten the mood by saying, “You think that you can do these things, but you just can’t, Nemo!”

Pop culture reference:

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):