Honestly, I wouldn’t mind year-round schooling. I really like teaching, I really like math, and I really like my students. But, it’s also nice to do some traveling, get some projects done around the house, and catch up on reading, too.
To all my students, HAVE A GREAT SUMMER! And we’ll see you in the fall. Seniors, come back and visit often, okay?
I’m always on the look out for nice java applets (or flash, or javascript, or whatever) that help visualize tough math concepts. The best applets are not merely gimmick, but truly make the mathematics more accessible and invite you to explore, predict, and play with the concepts.
Most recently, my dad pointed me to this website. At this site, you’ll find well over one hundred very nice applets that show things I haven’t seen around the web before. For instance, play around with your favorite polyhedra in 3d, then let the applet truncate or stellate the polyhedron, all with nice animation. And of course, you’ll want to mess with all the settings on the drop down menus, too! Additionally, you can see a mobius strip, klein bottle, cross cap, or a torus. Also, I was just recently blogging about complex functions. Some of these applets allow you to visualize with animation and color various complex functions. Having just completed a grad class in differential equations, I was also interested in these ODE applets. You may also have fun with fractals, like this one.
There are lots of other nice applet websites out there. As you know, I’m a huge fan of GeoGebra (free, open source, able to be run online without installation, easy to use, powerful). Someday maybe I’ll post more extensively about its merits. In the meantime, you can check it out here. Many of these applets use GeoGebra. This site has a lot of nice applets too. And here. This one’s rather elementary, but fun for educators. These are great, too. These are all sites I’ve used in class. Like I said, I think they’re more than just ‘cool’–they really do help elucidate hard-to-understand topics.
Feel free to share other applet sites you like.
Another math blog has featured one of my recent posts. Go check out the 66th Carnival of Mathematics at the Wild About Math blog.
In reading through the carnival, I was especially interested in (as a math teacher), JSXGraph, an interactive graphing application that you can easily embed in webpages using javascript. I’ve also been meaning to point you to Dave Richeson’s recent blog post about the volume of n-Dimensional balls. I would also be remiss to not mention the recent passing of world famous, and well-loved mathematician and puzzle-creator, Martin Gardner. The aforementioned carnival points us to a post here, which honors his memory (among hordes of other posts on math blogs around the web that have honored him recently).
Passing on a post from my dad…I think some of the math is accessible for my readers. In fact, for my precalculus students, it ties together some of the nice stuff we’ve studied this semester (infinite series, complex numbers).
Teaching History of Math this past semester gave me an excuse to read carefully two Dover Publications books that I have owned since high school, but only skimmed then. Imagine my delight to discover that if you are given a theorem that is hard to prove beforehand, you can prove that
is transcendental in just a couple of lines. The hard theorem gives many other corollaries too, corollaries that I’ve known in my gut but never had a handle on how to prove.
Here are the details, from p. 76 of Felix Klein’s book Famous Problems of Elementary Geometry. You can read it on-line at Google Books.
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Go check it out!
My precalculus classes just finished a unit on polar coordinates and complex numbers. When I teach about complex numbers, I mention functions of a complex variable in passing, but we don’t really give it much thought. We do complex arithmetic and that’s all; that is, problems like these:
Evaluate.
In our precalculus class, we also understand how to plot complex numbers. Complex numbers must be plotted on a two-dimensional plane because complex numbers are…well…two dimensional! The real number line has no place for them. For instance, we represent the complex number 
But we don’t ask questions about complex functions. This is sad! Because functions of a complex variable are fairly accessible. That is, we want to consider functions like
The first thing you’ll notice is that I’ve written these functions in terms of 
Notice we’re wired (because of schooling, perhaps) to understand the 
I’ve built another Geogebra applet to help you visualize this kind of function. Make sure you use it with purpose, rather than just dragging things around randomly. Try making predictions about what will happen before revealing the result. Read the directions.
Have fun, and I hope you learn something about complex functions! I’m sure to post more on them someday. There’s a lot more to say.
I posted a problem back in December that I never got back to answering. Sorry about that. The problem statement was:
Two jars contain an equal volume of soda. One contains Sprite, the other Coca Cola. You take a small amount of Coca Cola from the Coca Cola jar and add it to the Sprite jar. After uniformly mixing this concoction, you take a small amount out and put it back in the Coca Cola jar, restoring both jars to their original volumes. After having done this, is there more Coca Cola in the Sprite jar or more Sprite in the Coca Cola jar? Or, are they equally contaminated?
I have had the worked out solution for a while, just haven’t posted it until now. I’m relatively new with 
One of my recent blog posts has been highlighted at the Math Hombre blog. Check it out, along with all the other great posts in the most recent edition of Math Teachers at Play.
Passing on an interesting article. For the full article, go here.
A Math Paradox:
The Widening Gap Between High School and College MathBy Joseph Ganem
We are in the midst of paradox in math education. As more states strive to improve math curricula and raise standardized test scores, more students show up to college unprepared for college-level math. The failure of pre-college math education has profound implications for the future of physics programs in the United States. A recent article in my local paper, the Baltimore Sun: “A Failing Grade for Maryland Math,” highlighted this problem that I believe is not unique to Maryland. It prompted me to reflect on the causes.
The newspaper article explained that the math taught in Maryland high schools is deemed insufficient by many colleges. According to the article 49% of high school graduates in Maryland take non-credit remedial math courses in college before they can take math courses for credit. In many cases incoming college students cannot do basic arithmetic even after passing all the high school math tests. The problem appears to be worsening and students are unaware of their lack of math understanding. The article reported that students are actually shocked when they are placed into remedial math.
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This week I start complex numbers with my Precalculus class. This is a fun problem I found here (you can also go there for the solution when you’re ready).
Given 
Just as inspiration for those students of mine who are taking AP and IB exams, and soon to be taking finals:
![\sqrt[4]{40.5+40.5i\sqrt{3}}](https://s0.wp.com/latex.php?latex=%5Csqrt%5B4%5D%7B40.5%2B40.5i%5Csqrt%7B3%7D%7D&bg=ffffff&fg=333333&s=0&c=20201002)
Random Walks 
