It is probably important to note two things at the outset: (a) I do not have direct knowledge as to whether Peter Keep's middle initial is indeed Q (but it's the funniest letter), and (b) Peter Keep is not dead.| Spencer's blog
This post is part of a three-post set (1 2 3) about groups of order $pqr$, where $p<q<r$ are prime. While designing final exam questions for my group theory class, I was playing around with groups of order 42, for no particular reason, and then a friend noticed something interesting that I wanted to try to prove more generally. Here's a complete writeup of this proof. Is it the most beautiful and elegant proof? Certainly not; you can directly quote some results about groups of square-free ord...| Spencer's blog
This post is part of a three-post set (1 2 3) about groups of order $pqr$, where $p<q<r$ are prime. While thinking about such $pqr$-groups, I've collected a bunch of theorems and lemmas into a little mental toolbox. An interesting commonality between a lot of these tools is that they tell us about the relationship between various different animals in the subgroup lattice; an interesting difference is that some of them go “upwards,” some of them go “downwards,”1 and some of them even g...| Spencer's blog
This post is part of a three-post set (1 2 3) about groups of order $pqr$, where $p| Spencer's blog
At the end of the term here, I'm making a few edits / updates to my syllabus policy on the use of LLMs etc., and I decided to post it in some place that's easily shareable.| Spencer's blog
Hi algebraists,| Spencer's blog
Yesterday I was scheduled to teach my Calculus 2 class about the Second FTC at 10am, so of course I had a great idea for an activity at approximately 9:48am.| Spencer's blog
Asides are a neat feature in PreTeXt books – in the web version, they show up as cute li’l notes floating semi-transparently in the margin. However, in LaTeX builds, asides get dumped into plain text with no particular styling. This is sad so I decided to fix it.| Spencer’s blog
Last time we met, we found a Taylor series for the arctangent function, and we used the fact that π = 4 arctan(1) to find approximations to π. Unfortunately, this representation had really poor convergence behavior, because 1 is way the hell out at the edge of the interval of convergence for this series. I’m loath to abandon a fun trick, though; is there a way we can iterate on this idea?| Spencer’s blog
A fun thing to do at the end of a Calculus II class is to use Taylor series to compute values of various functions. For instance, I have a whole problem set about the “68-95-99.7” rule, where we find a Taylor series for the normal distribution and then integrate it. Here’s another fun application:| Spencer’s blog
If I ask you what shape an hourglass is, there is a very specific shape that leaps to mind. (Maybe there is also theme music.)| Spencer's blog
The other day in class we were talking about something having to do with weighted averages1. As a familiar example of weighted averages, I had people calculate a weighted GPA for warmup (with weights given by credit hours). While we were discussing this calculation, I mentioned that I think GPAs are wack and we shouldn't use them, but that I wasn't going to get on my soapbox about this. So then a student asked about this on my daily exit quiz: The center of mass of a system of point masses in...| Spencer's blog
I had a great time marching in Westminster's entry in the pride parade the other weekend. The focus of our entry was our upcoming name change from Westminster College to Westminster University, so we got Westminster University shirts and drawstring bags and signs to wave around, and here's what all the pictures look like. Look at all these happy people happily waving happy signs with happy rainbow flags!| Spencer's blog
This is one of the most annoying things to teach in any functions-based class: Supposing $a, b > 0$, why does the transformation $y=f(x-a)+b$ move the graph of $f$ to the right by $a$ and up by $b$, even though the signs are different? Why does the thing that's happening horizontally work “backwards” from the thing that's happening vertically? Prompted by a nice activity from Matt Enlow, I finally have an explanation that feels persuasive. I road-tested this with my “Functions Modeling ...| Spencer's blog