My Favorite One-Liners: Part 18

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 is a quip that I’ll use when a theoretical calculation can be easily confirmed with a calculator.

Sometimes I teach my students how people converted decimal expansions into fractions before there was a button on a calculator to do this for them. For example, to convert $x = 0.\overline{432} = 0.432432432\dots$ into a fraction, the first step (from the Bag of Tricks) is to multiply by 1000: How do we change this into a decimal? Let’s call this number $x$ .

$1000x = 432.432432\dots$

$x = 0.432432\dots$

Notice that the decimal parts of both $x$ and $1000x$ are the same. Subtracting, the decimal parts cancel, leaving

$999x = 432$

$x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}$

To make this more real and believable to them, I then tell them my one-liner: “I can see that no one believes me. OK, let’s try something that you will believe. Pop out your calculators. Then punch in 16 divided by 37.”

Indeed, my experience many students really do need this technological confirmation to be psychologically sure that it really did work. Then I’ll tease them that, by pulling out their calculators, I’m trying to speak my students’ language.

See also my fuller post on this topic as well as the index for the entire series.

My Favorite One-Liners: Part 17

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 it’s pretty easy for students to push through a proof from beginning to end. For example, in my experience, math majors have little trouble with each step of the proof of the following theorem.

Theorem. If $z, w \in \mathbb{C}$ , then $\overline{z+w} = \overline{z} + \overline{w}$ .

Proof. Let $z = a + bi$ , where $a, b \in \mathbb{R}$ , and let $w = c + di$ , where $c, d \in \mathbb{R}$ . Then

$\overline{z + w} = \overline{(a + bi) + (c + di)}$

$= \overline{(a+c) + (b+d) i}$

$= (a+c) - (b+d) i$

$= (a - bi) + (c - di)$

$= \overline{z} + \overline{w}$

$\square$

For other theorems, it’s not so easy for students to start with the left-hand side and end with the right-hand side. For example:

Theorem. If $z, w \in \mathbb{C}$ , then $\overline{z \cdot w} = \overline{z} \cdot \overline{w}$ .

Proof. Let $z = a + bi$ , where $a, b \in \mathbb{R}$ , and let $w = c + di$ , where $c, d \in \mathbb{R}$ . Then

$\overline{z \cdot w} = \overline{(a + bi) (c + di)}$

$= \overline{ac + adi + bci + bdi^2}$

$= \overline{ac - bd + (ad + bc)i}$

$= ac - bd - (ad + bc)i$

$= ac - bd - adi - bci$ .

A sharp math major can then provide the next few steps of the proof from here; however, it’s not uncommon for a student new to proofs to get stuck at this point. Inevitably, somebody asks if we can do the same thing to the right-hand side to get the same thing. I’ll say, “Sure, let’s try it”:

$\overline{z} \cdot \overline{w} = \overline{(a + bi)} \cdot \overline{(c + di)}$

$= (a-bi)(c-di)$

$= ac -adi - bci + bdi^2$

$= ac - bd - adi - bci$ .

$\square$

I call working with both the left and right sides to end up at the same spot the Diamond Rio approach to proofs: “I’ll start walking your way; you start walking mine; we meet in the middle ‘neath that old Georgia pine.” Not surprisingly, labeling this with a catchy country song helps the idea stick in my students’ heads.

Though not the most elegant presentation, this is logically correct because the steps for the right-hand side can be reversed and appended to the steps for the left-hand side:

Proof (more elegant). Let $z = a + bi$ , where $a, b \in \mathbb{R}$ , and let $w = c + di$ , where $c, d \in \mathbb{R}$ . Then

$\overline{z \cdot w} = \overline{(a + bi) (c + di)}$

$= \overline{ac + adi + bci + bdi^2}$

$= \overline{ac - bd + (ad + bc)i}$

$= ac - bd - (ad + bc)i$

$= ac - bd - adi - bci$

$= ac -adi - bci + bdi^2$

$= (a-bi)(c-di)$

$= \overline{(a + bi)} \cdot \overline{(c + di)}$

$\overline{z} \cdot \overline{w}$ .

$\square$

For further reading, here’s my series on complex numbers.

My Favorite One-Liners: Part 16

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 basic notions of functions that’s taught in Precalculus and in Discrete Mathematics is the notion of an inverse function: if $f: A \to B$ is a one-to-one and onto function, then there is an inverse function $f^{-1}: B \to A$ so that

$f^{-1}(f(a)) = a$ for all $a \in A$ and

$f(f^{-1}(b)) = b$ for all $b \in B$ .

If $A = B = \mathbb{R}$ , this is commonly taught in high school as a function that satisfies the horizontal line test.

In other words, if the function $f$ is applied to $a$ , the result is $f(a)$ . When the inverse function is applied to that, the answer is the original number $a$ . Therefore, I’ll tell my class, “By applying the function $f^{-1}$ , we uh-uh-uh-uh-uh-uh-uh-undo it.”

If I have a few country music fans in the class, this always generates a bit of a laugh.

See also the amazing duet with Carrie Underwood and Steven Tyler at the 2011 ACM awards:

My Favorite One-Liners: Part 15

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. Let me describe a one-liner that I’ll use when I want my class to figure out a pattern, thus developing a theorem by inductive logic rather than deductive logic.

Today’s one-liner is easily stated: “Gosh, I’ve seen that somewhere before.”

For example, In my statistics class, here’s the very first illustration that I show to demonstrate how to compute a standard deviation:

Find the standard deviation of the following data set: 1, 4, 6, 7, 8, 10.

The first step is finding the average:

$\overline{x} = \displaystyle \frac{1+4+6+7+8+10}{6} = 6$ .

We then find the deviations from average by subtracting 6 from all of the original data values:

Deviations from average = -5, -2, 0, 1, 2, 4

With these numbers, we can compute the standard deviation:

$s = \displaystyle \sqrt {\frac{ (-5)^2 + (-2)^2 + 0^2 + 1^2 + 2^2 + 4^2}{5} } = \sqrt{10}$ .

After asking if there are any questions of clarification about the nuts and bolts of this calculation, I’ll proceed to the next example:

Find the standard deviation of the following data set: 5, 8, 10, 11, 12, 14.

The first step is finding the average:

$\overline{x} = \displaystyle \frac{5+8+10+11+12+14}{6} = 10$ .

We then find the deviations from average by subtracting 10 from all of the original data values and then constructing the square root as before:

$s = \displaystyle \sqrt {\frac{ (-5)^2 + (-2)^2 + 0^2 + 1^2 + 2^2 + 4^2}{5} } = \sqrt{10}$ .

Then, in a loud obvious voice, I’ll declare, “Gosh, I’ve seen that somewhere before” and then wait a few seconds for the answer. Students can obviously see that the two answers are the same — which gets them thinking about why that happened.

Obviously, the two answers are the same. The real conceptual question that I want my students to figure out is why the two answers are same. Eventually, someone will come up with the correct answer — the second data set was made by adding 4 to all the values in the first data set, which may change the average but does not change how spread out the numbers are… so the standard deviation should be unchanged.

I love the “Gosh, I’ve seen that somewhere before” line after a couple of carefully chosen examples, as it cues my class that they really need to think a little harder than the dull and mechanical operations toward a deeper conceptual understanding of what’s really happening.

My Favorite One-Liners: Part 14

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 quip is similar to the “bag of tricks” one-liner, and I’ll use this one if the “bag of tricks” line is starting to get a little dry.

Sometimes in math, there’s a step in a derivation that, to the novice, appears to make absolutely no sense. For example, to find the antiderivative of $\sec x$ , the first step is far from obvious:

$\displaystyle \int \sec x \, dx = \displaystyle \int \sec x \frac{\sec x + \tan x}{\sec x + \tan x} \, dx$

While that’s certainly correct, it’s from from obvious to a student that this such a “simplification” is actually helpful.

To give a simpler example, to convert

$x = 0.\overline{432} = 0.432432432\dots$

into a decimal, the first step is to multiply $x$ by $1000$ :

$1000x = 432.432432\dots$

Students often give skeptical, quizzical, and/or frustrated looks about this non-intuitive next step… they’re thinking, “How did you know to do that?” To lighten the mood, I’ll explain with a big smile that I’m clairvoyant… when I got my Ph.D., I walked across the stage, got my diploma, someone waved a magic wand at me, and poof! I became clairvoyant.

Clairvoyance is wonderful; I highly recommend it.

The joke, of course, is that the only reason that I multiplied by 1000 is that someone figured out that multiplying by 1000 at this juncture would actually be helpful. Subtracting $x$ from $1000x$ , the decimal parts cancel, leaving

$999x = 432$

$x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}$ .

In my experience, most students — even senior math majors who have taken a few theorem-proof classes and hence are no dummies — are a little stunned when they see this procedure for the first time. I learned this procedure when I was very young; however, in modern times, this procedure appears to be a dying art. I’m guessing that this algorithm is a dying art because of the ease and convenience of modern calculators. As always, I hold my students blameless for the things that they were simply not taught at a younger age, and part of my job is repairing these odd holes in their mathematical backgrounds so that they’ll have their best chance at becoming excellent high school math teachers.

For further reading, here’s my series on rational numbers and decimal expansions.

My Favorite One-Liners: Part 13

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 a story that I’ll tell my students when, for the first time in a semester, I’m about to use a previous theorem to make a major step in proving a theorem. For example, I may have just finished the proof of

$\hbox{Var}(X+Y) = \hbox{Var}(X) + \hbox{Var}(Y)$ ,

where $X$ and $Y$ are independent random variables, and I’m about to prove that

$\hbox{Var}(X-Y) = \hbox{Var}(X) + \hbox{Var}(Y)$ .

While this can be done by starting from scratch and using the definition of variance, the easiest thing to do is to write

$\hbox{Var}(X-Y) = \hbox{Var}(X+[-Y]) = \hbox{Var}(X) + \hbox{Var}(-Y)$ ,

thus using the result of the first theorem to prove the next theorem.

And so I have a little story that I tell students about this principle. I think I was 13 when I first heard this one, and obviously it’s stuck with me over the years.

At MIT, there’s a two-part entrance exam to determine who will be the engineers and who will be the mathematicians. For the first part of the exam, students are led one at a time into a kitchen. There’s an empty pot on the floor, a sink, and a stove. The assignment is to boil water. Everyone does exactly the same thing: they fill the pot with water, place it on the stove, and then turn the stove on. Everyone passes.

For the second part of the exam, students are led one at a time again into the kitchen. This time, there’s a pot full of water sitting on the stove. The assignment, once again, is to boil water. Nearly everyone simply turns on the stove. These students are led off to become engineers. The mathematicians are ones who take the pot off the stove, dump the water into the sink, and place the empty pot on the floor… thereby reducing to the original problem, which had already been solved.

My Favorite One-Liners: Part 12

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.

Often in mathematics, one proof is quite similar to another proof. For example, in Precalculus or Discrete Mathematics, students encounter the theorem

$\sum_{k=1}^n (a_k + b_k) = \sum_{k=1}^n a_k + \sum_{k=1}^n b_k$ .

The formal proof requires mathematical induction, but the “good enough” proof is usually convincing enough for most students, as it’s just the repeated use of the commutative and associative properties to rearrange the terms in the sum:

$\sum_{k=1}^n (a_k + b_k)= (a_1 + b_1) + (a_2 + b_2) + \dots + (a_n + b_n)$

$= (a_1 + a_2 + \dots + a_n) + (b_1 + b_2 + \dots + b_n)$

$= \sum_{k=1}^n a_k + \sum_{k=1}^n b_k$ .

Next, I’ll often present the new but closely related theorem

$\sum_{k=1}^n (a_k - b_k) = \sum_{k=1}^n a_k -\sum_{k=1}^n b_k$ .

The proof of this would take roughly the same amount of time as the first proof, but there’s often little pedagogical value in doing all the steps over again in class. So here’s the line I’ll use: “At this point, I invoke the second-most powerful word in mathematics…” and then let them guess what this mysterious word is.

After a few seconds, I tell them the answer: “Similar.” The proof of the second theorem exactly parallels the proof of the first except for some sign changes. So I’ll tell them that mathematicians often use this word in mathematical proofs when it’s dead obvious that the proof can be virtually copied-and-pasted from a previous proof.

Eventually, students will catch on to my deliberate choice of words and ask, “What the most powerful word in mathematics?” As any mathematician knows, the most powerful word in mathematics is “Trivial”… the proof is so easy that it’s not necessary to write the proof down. But I warn my students that they’re not allowed to use this word when answering exam questions.

The third most powerful phrase in mathematics is “It is left for the student,” thus saving the professor from writing down the proof in class and encouraging students to figure out the details on their own.

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.