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Archive of posts filed under the Geometry category.

Math Photo: Spiky Symmetry

These cacti caught my. I can see both a dodecagon and a star in the 12-fold symmetry of the cactus in front. And to my surprise, the cactus behind it has thirteen sections!

I wonder about the range, and deviation, of the number of sections of these cacti. And what are the biological principles that govern these mathematical characteristics?

Regents Recap, August 2017: How Do You Explain that Two Things are Equal?

Sue believes these two cylinders from the August, 2017 New York Regents Geometry exam have equal volumes. Is Sue correct? Explain why.

Yes, Sue, you are correct: the two cylinders have equal volumes. I computed both volumes and clearly indicated that they are the same. Take a look!

Wait. Why did I only get half-credit? What’s the problem, Sue? You don’t think this is an “explanation”? The two volumes are equal. The explanation for why they are equal is that I computed both volumes and got the same number. I don’t know of any better explanation for two things being equal than that.

What’s that? You wanted me to say “Cavalieri’s Principle”? But if I compute the two volumes and show that they are equal, why would I need to say they are equal because of some other reason?  Oh, never mind, Sue. See you in Algebra 2.

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Symmetry, Algebra and the Monster — Quanta Magazine

I’m excited to announce the launch of my column for Quanta Magazine!  In Quantized Academy I’ll be writing about the fundamental mathematical ideas that underlie Quanta’s stories on cutting edge science and research. Quanta consistently produces exciting, high-quality science journalism, and it’s a tremendous honor to be a part of it.

My debut column, Symmetry, Algebra and the Monster, uses the symmetries of the square to explore the basic group theory that connects algebra and geometry.

You could forgive mathematicians for being drawn to the monster group, an algebraic object so enormous and mysterious that it took them nearly a decade to prove it exists. Now, 30 years later, string theorists — physicists studying how all fundamental forces and particles might be explained by tiny strings vibrating in hidden dimensions — are looking to connect the monster to their physical questions. What is it about this collection of more than 10^53 elements that excites both mathematicians and physicists? 

The full article is freely available here.

8/15/17 — Happy Pythagorean Triple Day!

Today we celebrate a rare Pythagorean Triple day!  And especially rare, as 8-15-17 is a primitive Pythagorean triple.  We won’t see many more of those!

Here’s an animation I made to celebrate.


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Regents Recap — June, 2017: Consistency and Precision

Two prominent themes of my critical review of the New York State Regents exams in mathematics are consistency and precision in language.  Here’s a pair of problems from the June 2017 Geometry exam that illustrates both issues.

First, the phrasing of the question “What is the number of degrees in the measure of angle ABC?” is awkward and somewhat unnatural.  Second, if we are going to ask for “the number of degrees” in the measure of an angle, then the answer should be a number.  The answer choices here are not numbers: they are degree measurements.

Why not simply ask for the measure of the angle, as was done in question 10 on the exact same exam?

While the issue in question 4 is minor, we know that imprecise use of language is deeply connected to student misconceptions in mathematics.  And we know that an important part of our job as teachers is getting students to use technical language correctly.  Our exams should model the mathematical clarity and precision that we expect of students in our classes.  Far too often, the New York State Regents exams don’t meet that standard.

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