Angle Sums and Pythagorean Triples

I’ve always found it cool that if you double the smaller acute angle in a 3-4-5 triangle you get the larger acute angle in a 7-24-25 right triangle. You can see this as a consequence of the double angle formula for sine. If \alpha is the smaller acute angle in a 3-4-5 triangle, then

\sin (2\alpha) = 2\sin\alpha\cos\alpha=2\frac{3}{5}\frac{4}{5}=\frac{24}{25}

In fact, if the sine and cosine of an angle are both rational, then so will be the sine and cosine of twice that angle. This gives a way to turn Pythagorean triples into new Pythagorean triples!

For example, suppose \alpha is an acute angle in a right triangle with a^2 + b^2 = c^2 . Then

\sin 2\alpha = \frac{2ab}{c^2}
\cos 2 \alpha = \frac{a^2-b^2}{c^2}

By the Pythagorean identity

\left(\frac{2ab}{c^2} \right)^2 + \left(\frac{a^2-b^2}{c^2} \right)^2 = 1

And so

\left(2ab \right)^2 + \left(a^2-b^2 \right)^2 = \left(c^2\right)^2

which of course also follows directly from algebra.

For example, using this process

(3,4,5) \mapsto (7,24,25) \mapsto(336,527,625)

Originally posted on Mastodon.

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2025 — Year in Review

Here are some highlights from what has been another busy professional year.

In September I published a new series in the New York Times that turns Steven Strogatz’s wonderful “Math, Revealed” essays into teaching and learning resources. I got to write about favorite topics like taxicab geometry, triangular numbers, the golden ratio, and packing problems. Best of all, I got to collaborate directly with Strogatz himself! It was an honor to work with him, even if this doesn’t officially make my Erdős number four.

I’ve been thinking and writing about AI in education and math this year. I wrote about how students are using AI, how it is impacting the college admissions process, about experts calling AI-errors “sophisticated”, and how it’s affecting me as a book author. I’ve been through a few hype-bubbles in my time, and am generally skeptical by nature, but there’s no denying the impact these technologies will have in how we learn, teach, and even do math.

I’m wrapping up my 20th, and final, year as a Math for America Master Teacher, and I was proud to run one more workshop for teachers through MfA. “A One-Problem Tour of Statistics” was the story of what I learned by writing Painless Statistics, but also an homage to the kinds of math problems that you keep going back to. It was a fun and satisfying end to a string of nearly 30 workshops I’ve run in my time at MfA.

The applied mathematics and modeling program I run at my school had unprecedented success in 2025. We had a team win the NCTM Award in the 2024 High School Mathematical Contest in Modeling (HiMCM) and finish in the top ten worldwide in the 2025 International Mathematical Modeling Challenge (IMMC). Another team won the MAA Award in the 2025 MCM, finishing among the top of over 12,000 entries worldwide, nearly all of which were college teams! We also had a team earn an “Outstanding” designation — the highest honor — in the 2025 SCUDEM competition, a college-level differential equations modeling competition. I was very proud to be profiled on the COMAP website for “teaching, modeling, and mentoring at the highest level”!

This past summer I was invited to present to the Alliance for Decision Education on my work with student forecasting, after our students made impressive showings in two individual forecasting competitions in 2024 and 2025. I also appeared on MoMath’s QED and spoke with mathematician-in-residence David Reimann about math and math education.

It’s been a good year professionally, and I’m looking forward to more to look back on in 2026!

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AI-Generated Letters of Recommendation

I write around 30 recommendation letters a year. These are mostly for students applying to college, but increasingly I’m asked to write recs for competitive summer programs, private schools, scholarships, even internships. It’s a lot of work. I estimate that I spend around 100 hours a year on it

And it’s uncompensated work. Almost all of these hours come directly from my personal time, which colleges treat as a free resource. There is nothing to stop them from making me fill out one more form, complete one more ranking, respond to one more school-specific prompt. I often feel like collateral damage in the school admissions arms race.

Some teachers simply refuse to do it. I have come to empathize with that position, but ultimately these recommendations are important to my students, so I put in the time and effort, even though the process is frustrating.

What’s most frustrating is that I’m not sure all this effort makes any difference. Do letters of recommendation really matter in college applications? I find it hard to believe they do. Last year 30,000 students applied to MIT. Who reads those 60,000 letters of recommendation?

I’ve long assumed that these letters just get passed through some kind of sentiment analysis software, where a large language model produces a score, appends it to the student’s profile, and the admissions process grinds on, one automated step at a time. I even recently speculated that colleges were feeding my letters to LLMs without my consent. What’s to stop them?

So when I logged into Naviance, the now-universal portal for college admissions, I shouldn’t have been surprised to see a new feature under the “Letters of Recommendation” tab: a compose-with-AI button.

But I was surprised. Isn’t this an admission that letters of recommendation aren’t that important? If colleges will accept an algorithmically-generated, averaged-out narrative as a substitute for whatever I might have said, how could they possibly value what have I say? Why shouldn’t I just click “Compose”, fill in a couple of blanks, and reclaim my time?

I guess there’s a part of me that still believes a good letter of recommendation can have an impact. Maybe that’s naïve, but if it’s true, then anything less than my full effort would put my students at a disadvantage. I respect them too much to do that, even if the process doesn’t respect me.

For now, I’ll hope that my carefully considered letters will give my students an edge in a world of AI-powered chatbots processing AI-generated recommendations. But I’ll be watching this AI-powered arms race closely.

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Workshop — A One-Problem Tour of Statistics

I was excited to run my workshop “A One-Problem Tour of Statistics” for teachers at Math for America in New York City this past week. When I was writing “Painless Statistics” a few years ago, there was one simply-stated problem I kept going back to that continually helped illuminate important statistical ideas for me. I returned to that problem frequently, and each time I came away with a better understanding of sampling distributions, estimators, confidence intervals, and even the debate between frequentism and Bayesianism. This workshop was years in the making, and I was happy to finally be able to share my story, and what I learned, with teachers!

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Strogatz, the NYT, and Triangular Numbers

My latest article for the New York Times Learning Network turns Steven Strogatz’s wonderful “Math, Revealed” essay on triangular numbers into a teaching and learning resource. Learn about how a favorite number pattern connects algebra, geometry, and calculus, and even extends into CAT scans the Fab Four!

The article is freely available here, and as with the articles in the series, include free access to Strogatz’s original New York Times essay.

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