High-pointing a football?

Today is one of the high points of the American sports calendar: the AFC and NFC championship games to determine who plays in the Super Bowl.

A major pet peeve of mine while watching sports on TV: football announcers who “explain” that a receiver made a great reception because “he caught the ball at its highest point.”

Ignoring the effects of air resistance, the trajectory of a thrown football is parabolic, and the ball is the essentially the same height above the ground when it is either thrown or caught. (Yes, there might be a difference of at most three feet, but that’s negligible compared to the distance that a football is typically thrown.) Therefore, a football reaches the highest point of its trajectory approximately halfway between the quarterback and the receiver.

And anyone who can catch the ball that far above the ground should be immediately tested for steroids.

Preparation for Industrial Careers in the Mathematical Sciences: Building a Better Filter

The Mathematical Association of America recently published a number of promotional videos showing various mathematics can be used in “the real world.” Here’s the third pair of videos describing how mathematics is used for certain problems in materials science. From the YouTube descriptions:

Dr. Sumanth Swaminathan of W. L. Gore & Associates talks about his career path and the research questions about filtration that he considers. He works to understand the different waste capture mechanisms of filtration devices and to mathematically optimize the microstructure to create better filters.

Prof. Louis Rossi of the Department of Mathematical Sciences of the University of Delaware presents two introductory mathematical models that one can use to understand and characterize filters and the filtration processes.

New world record for largest prime number

As of this week, we have a new world record for the largest known prime number:

$2^{74,207,281}-1$

The adjective known is important, because there are an infinite number of prime numbers (but not all of them are known). A good video describing this finding is below.

A good article is here:

https://www.washingtonpost.com/news/speaking-of-science/wp/2016/01/20/the-newest-prime-number-is-more-than-22-million-digits-long/?tid=sm_tw

Mathematics of Juggling

Quite accidentally, I recently stumbled on the following video. The speaker is Allen Knutson, who is a Professor of Mathematics at Cornell. Hope you enjoy it.

The antiderivative of 1/(x^4+1): Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the computation of

$\displaystyle \int \frac{dx}{x^4+1}$

Part 1: Introduction.

Part 2: Factoring the denominator using De Moivre’s Theorem.

Part 3: Factoring the denominator using the difference of two squares.

Part 4: The partial fractions decomposition of the integrand.

Part 5: Partial evaluation of the resulting integrals.

Part 6: Evaluation of the remaining integrals.

Part 7: An apparent simplification using a trigonometric identity.

Part 8: Discussion of the angles for which the identity holds.

Part 9: Proof of the angles for which the identity holds.

Part 10: Implications for using this identity when computing definite integrals.

Diamond Rio and Proofs

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

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

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

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

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

$= ac - bd - adi - bci$

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

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

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

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

$\square$

Meat-A-Morphosis: An Introduction to Functions

The video below was posted by my colleague Jason Ermer, founder of Collaborative Mathematics. From the YouTube description:

A cartoon about the proverbial mathematical “function machine”.

I was part of the creative team of mathematics teachers (including Patty Hill and Michael Word) who created this cartoon. It was originally (and still is) a component of the Honors Algebra 1 curriculum at Kealing Middle School in Austin, Texas.

I take full responsibility for the spelling error. (Can you find it? 🙂

Enjoy.

CBE and computer programming

Saving school math

What hope can there be for Competency Based Education, by computer of course, if commercial programmers continue to come up with this sort of stuff?

computer program oops

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Another Reasoning Puzzle From ChefMongoose

I enjoyed this challenge.

nebusresearch

My friend ChefMongoose had another reasoning problem come to him, and I’m happy to share it further. It’s rather like that famous Singapore Birthday Problem that drove people crazy a couple of months ago. Here’s the problem:

I have a combination lock at work. There are three digits, all in the range 1 – 40; they’re all prime numbers. They’re X+Y, X+2Y, X+3Y — where X and Y are positive integers.
If I told you what X was but not Y, you wouldn’t be able to tell me the combination. If I told you what Y was but not X, you wouldn’t be able to tell me the combination. Now, what’s the combination?

I did work out the puzzle. It did make me notice a couple of strings of uniformly-spaced prime numbers I hadn’t done before, too, such as 3-13-23. (However, 3-13-23 isn’t one of the possible answers, because of…

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