## All posts tagged **proof**

# My Favorite One-Liners: Part 119

*Posted by John Quintanilla on February 24, 2020*

https://meangreenmath.com/2020/02/24/prove-it/

# Borwein integrals

When teaching proofs, I always stress to my students that it’s not enough to do a few examples and then extrapolate, because it’s possible that the pattern might break down with a sufficiently large example. Here’s an example of this theme that I recently learned:

For further reading:

*Posted by John Quintanilla on February 5, 2018*

https://meangreenmath.com/2018/02/05/borwein-integrals/

# My Favorite One-Liners: Part 108

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 post marks the final entry in this series. When I first came up with the idea of listing some of favorite classroom quips, I thought that this series might last a couple dozen posts. To my surprise, it instead lasted for more than 100 posts. I guess that, in my 21-year teaching career, I’ve slowly developed my own unique lexicon for communicating mathematical ideas, and perhaps this parallels (on a decidedly smaller scale) what a radio talk show host (like local legend Randy Galloway, who was a sports reporter/commentator in the Dallas/Fort Worth area for many years before retiring) does to build rapport with his/her audience.

I’ll use this final one-liner near the end of the semester when it’s time for students complete their evaluations of my teaching. Back in days of yore, professors would take 10-15 minutes to pass out paper copies of these evaluations, students would complete them, and that would be the end of it. In modern times, however, paper evaluations have switched to electronic evaluations, which are perhaps better for the environment but tend to have a decidedly lower response rate than the old paper evaluations. Still, I value my students’ feedback. So I’ll tell them:

Please fill out the student evaluation; the size of my raise depends on this.

After the laughter settles down, I’ll tell them, “Who’s joking?” I can’t say this happens everywhere, but I can honestly say that my department’s executive committee does consider student evaluations of teaching when deciding on the quality of my teaching, and that partially determines the size of my annual merit raise. (The committee also considers other indicators of good teaching other than student evaluations.)

It’s important to note that I don’t tell my students to give me a good evaluation; I just ask them to fill it out and to be honest with their feedback. I also tell them, forgetting my raise, I also want to hear from them about how the semester went. If it went great, I want to know that. If it sucked, I also want to know that. However, if they think the class sucked, just writing “This class sucked” doesn’t give me a lot of information about how to fix things for the next time that I teach the course. So, if they have a criticism, I ask them to give me specific feedback so that I can consider their critiques.

A couple years ago, I served on my university’s committee for reconsidering the way that we conduct student evaluations of teaching. To my surprise, when I interviewed students in focus groups, there was a general consensus that students believed that their evaluations were a waste of time that didn’t actually contribute anything to the university — or if they did contribute something, they had no idea what it was. Ever since then, I’ve made a point of telling my students that their evaluations really do matter and can make a difference in future offerings of my courses.

*Posted by John Quintanilla on May 19, 2017*

https://meangreenmath.com/2017/05/19/my-favorite-one-liners-part-108/

# My Favorite One-Liners: Part 101

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 use today’s one-liner when a choice has to be made between two different techniques of approximately equal difficulty. For example:

Calculate , where is the region

There are two reasonable options for calculating this double integral.

- Option #1: Integrate with respect to first:

- Option #2: Integrate with respect to first:

Both techniques require about the same amount of effort before getting the final answer. So which technique should we choose? Well, as the instructor, I realize that it really doesn’t matter, so I’ll throw it open for a student vote by asking my class:

Anyone ever read the

Choose Your Own Adventurebooks when you were kids?

After the class decides which technique to use, then we’ll set off on the adventure of computing the double integral.

This quip also works well when finding the volume of a solid of revolution. We teach our students two different techniques for finding such volumes: disks/washers and cylindrical shells. If it’s a toss-up as to which technique is best, I’ll let the class vote as to which technique to use before computing the volume.

*Posted by John Quintanilla on May 12, 2017*

https://meangreenmath.com/2017/05/12/my-favorite-one-liners-part-101/

# My Favorite One-Liners: Part 99

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 a light-hearted one-liner that I’ll use to lighten the mood when in the middle of a complex calculation, like the following limit problem from calculus:

Let . Find so that whenever $|x-2| < \delta$.

The solution of this problem requires isolating in the above inequality:

At this point, the next step is dividing by . So, I’ll ask my class,

When we divide by , what happens to the crocodiles?

This usually gets the desired laugh out of the middle-school rule about how the insatiable “crocodiles” of an inequality always point to the larger quantity, leading to the next step:

,

so that

.

Formally completing the proof requires starting with and ending with .

*Posted by John Quintanilla on May 10, 2017*

https://meangreenmath.com/2017/05/10/my-favorite-one-liners-part-99/

# My Favorite One-Liners: Part 98

I’ll use today’s quip just after introducing the methodology of mathematical induction to my students:

Induction is so easy that even the army uses it.

*Posted by John Quintanilla on May 9, 2017*

https://meangreenmath.com/2017/05/09/my-favorite-one-liners-part-98/

# My Favorite One-Liners: Part 92

This is one of my favorite quote from Alice in Wonderland that I’ll use whenever discussing the difference between the ring axioms (integers are closed under addition, subtraction, and multiplication, but not division) and the field axioms (closed under division except for division by zero):

‘I only took the regular course [in school,’ said the Mock Turtle.]

‘What was that?’ inquired Alice.

‘Reeling and Writhing, of course, to begin with,’ the Mock Turtle replied; ‘and then the different branches of Arithmetic — Ambition, Distraction, Uglification, and Derision.’

*Posted by John Quintanilla on May 3, 2017*

https://meangreenmath.com/2017/05/03/my-favorite-one-liners-part-92/

# My Favorite One-Liners: Part 38

When I was a student, I heard the story (probably apocryphal) about the mathematician who wrote up a mathematical paper that was hundreds of pages long and gave it to the departmental administrative assistant to type. (This story took place many years ago before the advent of office computers, and so typewriters were the standard for professional communication.) The mathematician had written “iff” as the standard abbreviation for “if and only if” since typewriters did not have a button for the symbol.

Well, so the story goes, the administrative assistant saw all of these “iff”s, muttered to herself about how mathematicians don’t know how to spell, and replaced every “iff” in the paper with “if”.

And so the mathematician had to carefully pore through this huge paper, carefully checking if the word “if” should be “if” or “iff”.

I have no idea if this story is true or not, but it makes a great story to tell students.

*Posted by John Quintanilla on March 10, 2017*

https://meangreenmath.com/2017/03/10/my-favorite-one-liners-part-38/

# My Favorite One-Liners: Part 37

Sometimes, I’ll deliberately show something wrong to my students, and their job is to figure out how it went wrong. For example, I might show my students the classic “proof” that :

After coming to the conclusion, as my students are staring at this very unanticipated result, I’ll smile with my best used-car salesman smile and say “Trust me,” just like in the old Joe Isuzu commercials.

Of course, the joke is that my students shouldn’t trust me, and they should figure out exactly what happened.

*Posted by John Quintanilla on March 9, 2017*

https://meangreenmath.com/2017/03/09/my-favorite-one-liners-part-37/

# My Favorite One-Liners: Part 34

Suppose that my students need to prove a theorem like “Let be an integer. Then is odd if and only if is odd.” I’ll ask my students, “What is the structure of this proof?”

The key is the phrase “if and only if”. So this theorem requires two proofs:

- Assume that is odd, and show that is odd.
- Assume that is odd, and show that is odd.

I call this a blue-light special: Two for the price of one. Then we get down to the business of proving both directions of the theorem.

I’ll also use the phrase “blue-light special” to refer to the conclusion of the conjugate root theorem: if a polynomial with real coefficients has a complex root , then is also a root. It’s a blue-light special: two for the price of one.

*Posted by John Quintanilla on March 6, 2017*

https://meangreenmath.com/2017/03/06/my-favorite-one-liners-part-34/