My Favorite One-Liners: Part 60

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’m a big believer using scaffolded lesson plans, starting from elementary ideas and gradually building up to complicated ideas. For example, when teaching calculus, I’ll use the following sequence of problems to introduce students to finding the volume of a solid of revolution using disks, washers, and shells:

Find the volume of a cone with height $h$ and base radius $r$ .
Find the volume of the solid generated by revolving the region bounded by $y=2$ , $y=2\sin x$ for $0 \le x \le \pi/2$ , and the $y-$ axis about the line $y=2$ .
Find the volume of the solid generated by revolving the region bounded by $y=2$ , $y=\sqrt{x}$ , and the $y-$ axis about the line $y=2$ .
Find the volume of the solid generated by revolving the region bounded by $y=2$ , $y=\sqrt{x}$ , and the $y-$ axis about the $y-$ axis .
Find the volume of the solid generated by revolving the region bounded by $x=\sqrt{2y}/(y+1)$ , $y=1$ , and the $y-$ axis about the $y-$ axis .
Find the volume of the solid generated by revolving the region bounded by the parabola $x=y^2+1$ and the line $x=3$ about the line $x=3$ .
Water is poured into a spherical tank of radius $R$ to a depth $h$ . How much water is in the tank?
Find the volume of the solid generated by revolving the region bounded by $y= x^2$ , the $x-$ axis, and $x=4$ about the $x-$ axis.
Find the volume of the solid generated by revolving the region bounded by $y= x^2$ , the $x-$ axis, and $x=4$ about the $y-$ axis.
Repeat the previous problem using cylindrical shells.

In this sequence of problems, I slowly get my students accustomed to the ideas of horizontal and vertical slices, integrating with respect to either $x$ and $y$ , the creation of disks and washers and (eventually) cylindrical shells.

As the problems increase in difficulty, I enjoy using the following punch line:

To quote the great philosopher Emeril Lagasse, “Let’s kick it up a notch.”

My Favorite One-Liners: Part 59

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 I’ll cover a topic in class that students really should have learned in a previous class but just didn’t. For example, in my experience, a significant fraction of my senior math majors have significant gaps in their backgrounds from Precalculus:

About a third have no memory of ever learning the Rational Root Test.
About a third have no memory of ever learning synthetic division.
About half have no memory of ever learning Descartes’ Rule of Signs.
Almost none have learned the Conjugate Root Theorem.

Often, these students will feel somewhat crestfallen about these gaps in their background knowledge… they’re about to graduate from college with a degree in mathematics and are now discovering that they’re missing some pretty basic things that they really should have learned in high school. And I don’t want them to feel crestfallen. Certainly, these gaps need to be addressed, but I don’t want them to feel discouraged.

Hence one of my favorite motivational one-liners:

It’s not your fault if you don’t know what you’ve never been taught.

I think this strikes the appropriate balance between acknowledging that there’s a gap that needs to be addressed and assuring the students that I don’t think they’re stupid for having this gap.

My Favorite One-Liners: Part 58

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 great annoyances is when I cover some topic in class, usually theoretical, and a student asks, “Will this be on the test?” Over the years, I’ve developed my standard tongue-in-cheek response to this question:

Put this on the test… that’s a really good idea.

[Students groan as I pull out a piece of paper and write a note to myself, talking out loud as I write:] “Put this on the test.”

[I then fold the paper and place it in my pocket, and then ask, with my best poker face:]

Are there any other questions?

After this, no one ever asks me again if something will be on the test.

My Favorite One-Liners: Part 57

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 will ask my procedure for grading their exams. So I’ll tell them, tongue in cheek, that I go home so that I could have some movie playing in the background that would get in the proper mood for grading… something like Braveheart, Gladiator, or The Godfather.

For some reason, students don’t find this terribly reassuring.

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 55

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.

Professors play many roles. I play the role of quality control when grading exams. However, I also play the role of head cheerleader… though I have standards that enforce, I want my students to meet those standards. At times, I have to play the role of motivational speaker when talking to a student who’s despondent about how he/she performed on a recent exam.

In these situations, if I’m sitting with a student to discuss a recent exam and explaining where he/she went wrong, I’ll use the following line to try to lighten the mood of the meeting as I’m looking over the test:

To quote Billy Crystal in the Princess Bride, I’ve seen worse.

Because the truth is, after over 20 years as a professor, I’ve almost certainly have seen worse.

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 53

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’ll often have my students pull out their calculators and assist me as we proceed through some big procedure… this hopefully keeps students engaged and also provides a deliberate pause so that (hopefully) what I’ve been teaching them has a little extra time to sink in before we move on to the next topic. Still, I hate dead silence, and so I’ll often throw in a little quip now and then to break up the silence.

For example, suppose that it’s necessary to find the first few digits of the decimal expansion of $34/97$ . As they pull out their calculators, I might say something like, “I’ll be nice and spot you the 0,” and write

$34/97 = 0.$

on the board. This usually gets a knowing laugh… clearly the answer is between 0 and 1, but getting even the next two significant digits requires a little bit more work. When students get the answer from their calculators, then we’ll fill in the next few digits after the decimal point.

Or, say that we have to compute $(34.234-46.615)/10.134$ . As they’re punching into their calculators, I’ll say “I’ll be nice and spot you the negative sign”:

$(34.234-46.615)/10.134 = -$ ,

and then write down the digits after the negative sign after their calculators return the first few significant digits.

My Favorite One-Liners: Part 52

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.

When I teach regression, I typically use this example to illustrate the regression effect:

Suppose that the heights of fathers and their adult sons both have mean 69 inches and standard deviation 3 inches. Suppose also that the correlation between the heights of the fathers and sons is 0.5. Predict the height of a son whose father is 63 inches tall. Repeat if the father is 78 inches tall.

Using the formula for the regression line

$y = \overline{y} + r \displaystyle \frac{s_y}{s_x} (x - \overline{x})$ ,

we obtain the equation

$y = 69 + 0.5(x-69) = 0.5x + 34.5$ ,

so that the predicted height of the son is 66 inches if the father is 63 inches tall. However, the prediction would be 73.5 inches if the father is 76 inches tall. As expected, tall fathers tend to have tall sons, and short fathers tend to have short sons. Then, I’ll tell my class:

However, to the psychological comfort of us short people, tall fathers tend to have sons who are not quite as tall, and short fathers tend to have sons who are not quite as short.

This was first observed by Francis Galton (see the Wikipedia article for more details), a particularly brilliant but aristocratic (read: snobbish) mathematician who had high hopes for breeding a race of super-tall people with the proper use of genetics, only to discover that the laws of statistics naturally prevented this from occurring. Defeated, he called this phenomenon “regression toward the mean,” and so we’re stuck with called fitting data to a straight line “regression” to this day.

My Favorite One-Liners: Part 51

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 I teach regression, I typically use this example to illustrate the regression effect:

Suppose that the heights of fathers and their adult sons both have mean 69 inches and standard deviation 3 inches. Suppose also that the correlation between the heights of the fathers and sons is 0.5. Predict the height of a son whose father is 63 inches tall. Repeat if the father is 78 inches tall.

Using the formula for the regression line

$y = \overline{y} + r \displaystyle \frac{s_y}{s_x} (x - \overline{x})$ ,

we obtain the equation

$y = 69 + 0.5(x-69) = 0.5x + 34.5$ ,

so that the predicted height of the son is 66 inches if the father is 63 inches tall. However, the prediction would be 73.5 inches if the father is 76 inches tall.

To make this more memorable for students, I’ll observe:

As expected, tall fathers tend to have tall sons, and short fathers tend to have short sons. For example, my uncle was 6’6″. His two sons, my cousins, were 6’4″ and 6’5″ and were high school basketball stars.

My father was 5’3″. I became a math nerd.