Author: John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

From the Life of Fred

From the Life of Fred Pre-Algebra Course:

“A question for English majors: Suppose you wanted to say that the digit 9 followed by a decimal point is the same as a plain 9 without a decimal point. You write something like “9. is the same as 9.” To my eye, that seems a bit strange. Or, once, when I was in high school, I wrote in an essay the sentence: Rocky owed Sylvia $2.. The first dot was for the decimal, and the second dot was the period. The teacher marked it wrong.  Okay, English majors, here is your multiple-choice question: Is English harder than math? Here are your choices: ☐yes or ☐yes.”

Source: https://thehomelibraryonline.wordpress.com/2013/08/15/life-of-fred-pre-algebra-course/

Computer Cracks 200 Terabyte Math Proof

Here’s a cute problem, called the Boolean Pythagorean Theorem problem. Here are the first few Pythagorean triples:

3^2 + 4^2 = 5^2

6^2 + 8^2 = 10^2

5^2 + 12^2 = 13^2

9^2 + 12^2 = 15^2

8^2 + 15^2 = 17^2

12^2 + 16^2 = 20^2

7^2 + 24^2 = 25^2

15^2 + 20^2 = 25^2

10^2 + 24^2 = 26^2

20^2 + 21^2 = 29^2

18^2 + 24^2 = 30^2

16^2 + 30^2 = 34^2

21^2 + 28^2 = 35^2

12^2 + 35^2 = 37^2

15^2 + 36^2 = 39^2

27^2 + 36^2 = 45^2

9^2 + 40^2 = 41^2

27^2 + 36^2 = 45^2

OK, let’s have some fun with this. Let’s write every multiple of 5 (5, 10, 15, 20, 25, 30, 35, 40, 45) in boldface:

3^2 + 4^2 = {\bf 5}^2

6^2 + 8^2 = {\bf 10}^2

{\bf 5}^2 + 12^2 = 13^2

9^2 + 12^2 = {\bf 15}^2

8^2 + {\bf 15}^2 = 17^2

12^2 + 16^2 = {\bf 20}^2

7^2 + 24^2 = {\bf 25}^2

{\bf 15}^2 + {\bf 20}^2 = {\bf 25}^2

{\bf 10}^2 + 24^2 = 26^2

{\bf 20}^2 + 21^2 = 29^2

18^2 + 24^2 = {\bf 30}^2

16^2 + {\bf 30}^2 = 34^2

21^2 + 28^2 = {\bf 35}^2

12^2 + {\bf 35}^2 = 37^2

{\bf 15}^2 + 36^2 = 39^2

27^2 + 36^2 = {\bf 45}^2

9^2 + {\bf 40}^2 = 41^2

27^2 + 36^2 = {\bf 45}^2

For nearly all of these equations, there is one number that’s in boldface and one that’s not. However, there’s one that is all in one typeface: {\bf 15}^2 + {\bf 20}^2 = {\bf 25}^2.

So here’s a question: is it possible to divide the integers so that every Pythagorean triple (not just the small ones listed above) has at least one number in boldface and another that’s not?

This May, it was proved that it’s impossible. The proof is very brute-force (from https://cosmosmagazine.com/mathematics/computer-cracks-200-terabyte-maths-proof):

The team found all triples could be multi-coloured in integers up to 7,824. As soon as they hit 7,825, it became impossible.

But to prove a solution doesn’t exist, you need to try all possibilities. There are more than 10^{2300} ways to colour all those integers, so the scientists used a few mathematical tricks to reduce the number of combinations to trial to just under one trillion.

Two days later, with 800 processors at the University of Texas Stampede supercomputer crunching all possibilities in parallel, the team had their answer – no.

There is no way to colour the integers 1 to 7,825 in a way that leaves all Pythagorean triples multi-coloured, the team reported in arXiv.

I had to read this news article a couple of times to appreciate this: a supercomputer ran for two days on a supercomputer (without parallelization, computation time was 51,000 hours), producing an output file of 200 terabytes, comparable “to the size of the entire digitized text held by the US Library of Congress.” Wow.

Sphere Packing Solved in Higher Dimensions

I enjoyed reading this bit of mathematical news: https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions/

The opening paragraphs:

In a pair of papers posted online this month, a Ukrainian mathematician has solved two high-dimensional versions of the centuries-old “sphere packing” problem. In dimensions eight and 24 (the latter dimension in collaboration with other researchers), she has proved that two highly symmetrical arrangements pack spheres together in the densest possible way.

Mathematicians have been studying sphere packings since at least 1611, when Johannes Kepler conjectured that the densest way to pack together equal-sized spheres in space is the familiar pyramidal piling of oranges seen in grocery stores. Despite the problem’s seeming simplicity, it was not settled until 1998, when Thomas Hales, now of the University of Pittsburgh, finally proved Kepler’s conjecture in 250 pages of mathematical arguments combined with mammoth computer calculations.

Do’s, Don’ts for Parents to Help Teens Build Math Interest and Success

I really enjoyed reading this article: http://www.usnews.com/education/blogs/high-school-notes/2016/04/11/dos-donts-for-parents-to-help-teens-build-math-interest-and-success

A summary:

  • Don’t project negative feeling toward math onto teens
  • Do talk to teens and teachers about what’s being taught in math class
  • Don’t be too quick to hire a tutor for struggling students
  • Do support students with the right tools

I recommend the whole article and the references therein.

Statistics and percussion

I recently had a flash of insight when teaching statistics. I have completed my lectures of finding confidence intervals and conducting hypothesis testing for one-sample problems (both for averages and for proportions), and I was about to start my lectures on two-sample problems (liek the difference of two means or the difference of two proportions).

On the one hand, this section of the course is considerably more complicated because the formulas are considerably longer and hence harder to remember (and more conducive to careless mistakes when using a calculator). The formula for the standard error is longer, and (in the case of small samples) the Welch-Satterthwaite formula is especially cumbersome to use.

On the other hand, students who have mastered statistical techniques for one sample can easily extend this knowledge to the two-sample case. The test statistic (either z or t) can be found by using the formula (Observed – Expected)/(Standard Error), where the standard error formula has changed, and the critical values of the normal or t distribution is used as before.

I hadn’t prepared this ahead of time, but while I was lecturing to my students I remembered a story that I heard a music professor say about students learning how to play percussion instruments. As opposed to other musicians, the budding percussionist only has a few basic techniques to learn and master. The trick for the percussionist is not memorizing hundreds of different techniques but correctly applying a few techniques to dozens of different kinds of instruments (drums, xylophones, bells, cymbals, etc.)

It hit me that this was an apt analogy for the student of statistics. Once the techniques of the one-sample case are learned, these same techniques are applied, with slight modifications, to the two-sample case.

I’ve been using this analogy ever since, and it seems to resonate (pun intended) with my students as they learn and practice the avalanche of formulas for two-sample statistics problems.