Prof. Sally Keely

Another good read from @johndcook. Unfortunately I can't (won't) boost the toot because the image is missing ALT text.

This-Way-Up and Knuth's Up Arrow Notation:

Love playing with anything Donald Knuth ( The "up arrow" is a repeated exponentiation operator. One of the things that intrigued me early in my study of .

February 15, 2024
Prof. Sally Keely

Attn my II students studying integrals of arctrig functions this week: Fun integration problem whose answer involves both pi and the golden ratio, phi!

\[\int_0^{\pi}\arctan\left(1+\cos x\right)dx=arc\cot\left(\sqrt{\phi}\right)\]

February 09, 2024
Prof. Sally Keely

"Combining mathematics and art is an inevitable extension of their relationship." Art can help one visualize mathematics in a way where new patterns and relationships can be realized. This article shares some specific examples.

February 09, 2024
Prof. Sally Keely

@johndcook posted fun triangle problem yesterday appropriate for a pre-calculus course. I won't "boost" because the sketch is missing . Here is a re-toot instead w/ ALT.

Is this triangle a right triangle? How can you tell? If it is not, what's the measure of the largest angle?

SPOILER! It is not a right triangle (check Pythagorean Theorem). The largest angle measures approximately 90.000003 degrees (by Law of Cosines). Oh so close!

January 30, 2024
Prof. Sally Keely

Beyond the gorgeous photography of ice sheets, glacier covered mountains, blue seas, we get modeling ice sheet mathematics with equations involving partial derivatives explained in intuitive manner. Turn on CC because the audio is really low.

January 19, 2024
Prof. Sally Keely

James Bond - Agent 7? and other problems with leading zeroes. Good quick read, especially for programmers and coders, from John D. Cook.

January 12, 2024
Prof. Sally Keely

Grab some strips of paper as you work through this article from Quanta Mag.

"In 1977, two mathematicians conjectured that to be twisted in a Möbius strip, a rectangle of width 1 must be longer than √3, as in the strip on the lower right. In August 2023, Schwartz proved that they were correct: Any closer to a square than that, and there’s no way to twist the rectangle into a Möbius strip."

January 09, 2024
Prof. Sally Keely

2024 fun fact:

2024 = 9^3+8^3+7^3+6^3+5^3+4^3+3^3+2^3+1^3+0^3+(-1)^3

January 02, 2024
Prof. Sally Keely

Maths in Real Life: Fighting Big Ag Pollution with Maps and Math

"While geostatistics involves very complex math, illustrating spatial data and analytical findings on a map can help communicate the science by rooting the information in a personal sense of place, like showing how a factory-polluted river empties into a lake where people swim, boat, and fish."

"As a spatial statistician working on environmental policy, I use math to make sense of our environment and draw maps to communicate my research findings to policymakers and the general public."

December 16, 2023
Prof. Sally Keely

When is f(g(x)) ≤ g(f(x))? What conditions are necessary for f(g(x)) ≤ g(f(x))? Are there real-life applications for this inequality? Worthy questions to consider from intermediate algebra through calculus.

These questions are posed by John D. Cook. I look forward to comments being posted there.

SPOILER ALERT! Theorem below.

December 09, 2023
Prof. Sally Keely

Interesting piece of the Aperiodical's Carnival 222 ( hosted this month by John D. Cook ( is a conversation with ChatGPT about what happens when you ask ChatGPT to solve Elchanan Mossel’s dice problem.

"You throw a die until you get 6. What is the expected number of throws (including the throw giving 6) conditioned on the event that all throws gave even numbers?"

Try answering the problem on your own first before reading on. Careful, there is a common misconception important to avoid.

December 04, 2023
Prof. Sally Keely

From the book Lateral Solutions to Mathematical Problems by Des MacHale, try this problem called "Three cloves on an orange".

Given three points on the surface of a sphere, what is the probability there is a hemisphere on which they all lie?

After thinking about it, see answer courtesy of Alex Bellos at

November 28, 2023
Prof. Sally Keely

Great tips for actually learning and remembering what you just read. I'm a visual learner so having a visual representation of what I just read or listened to really helps me be a successful learner.

November 13, 2023
Prof. Sally Keely

"Ranked choice voting is an alternative electoral system that would mitigate the spoiler effect while giving voters more voice to express themselves at the polls."

November 10, 2023
Prof. Sally Keely

On Sunday Nov 5th most of us get to sleep in an hour when time falls back for winter. "Between 1941 and 1945, and again 1947, there was something call 'British Double Summer Time' in which clocks sprung ahead two hours." I never knew this. Did it ever spring forward to balance out?!

October 28, 2023
Prof. Sally Keely

RIP Eugenio Calabi, died on September 25, age 100. He was known for conceiving of novel geometric objects that later became fundamental to string theory. Good read from Quanta Mag. "Needless to say, the world does not appear to be 10-dimensional — there seem to be just three dimensions of space and one of time. By the mid-1980s, however, a group of physicists had realized that the six “extra” dimensions of the universe might be hidden in a minute Calabi-Yau manifold (less than 10−17 centimeters in diameter)."

October 17, 2023
Prof. Sally Keely

Fun little area problem from . Can you quickly see the area of the shaded triangle? No algebra needed!

October 08, 2023
Prof. Sally Keely

Hey new fall term students! Please read my "Why should college students study mathematics? Here are eight reasons. At least one will strike true for you."

September 20, 2023
Prof. Sally Keely

While not fully developed, a couple good ideas to think about. "The same is probably happening with the Apple Watch and AllTrails, and probably depends on the sampling frequency of both apps. The difference from the Coastline Paradox is that rather than talking about the scale of distance, we’re talking about the scale of time." I've noted extreme differences between AllTrail app and my Garmin watch tracking when hiking. Garmis is also pathetic at counting flights of stairs climbed. I wonder if it is the same "sampling" problem?

September 14, 2023