When I was a kid, a good friend and I stumbled across an interesting idea while jumping on a trampoline (not relevant).  Our thought process went something like this: You know what’s cool?  “Word” is a word.  Yeah, and also “noun” is a noun. Is there a word to describe when words are an example of themselves? Wait!  If there is a word that means “a word that is an example of itself,” then is it an example of itself?!  Whhhhhoooooaaahhh! We thought about it a lot, for years, and would sometimes try to wow others, in a very nerdy way, with this paradox we could describe but not explain.  It was pre-Internet, and nobody had any idea if such[…]

This is a classic and popular puzzle that I vividly remember seeing for the first time at a friend’s house as a kid.  When we see this game in Precalculus, at the end of the year with discrete math, most of my students are already familiar with it. Starting with \(n\) disks on the first peg, the goal is to move all of the disks to the third peg. 1. You can move one disk at a time 2. You cannot place a larger disk onto a smaller disk. Ideally, you want to do this in the minimum number of moves \(m\).  If you have 30 Tower of Hanoi puzzles for your students, well good for you, but I have[…]

This is a brief overview of how I introduce volumes of known cross-section for my AP Calculus AB class (btw we use Larson 10e).  The sticky-note activity is a fun one that I stole from a stranger a couple years ago. I introduce the concept, using notes essentially directly out of the textbook along with an example using this visualization in GeoGebra.  Students attempt some of the book problems for homework but will usually find them difficult.  These volumes are difficult to visualize at first!  On the 2nd day we do arts & crafts to help build our conceptual understanding.  I have scissors, rulers, a huge stack of sticky notes, and printouts of two curves like this: [Desmos] [PNG] [PDF] IMPORTANT:[…]

My students are brilliant and teach me things.  I like to give them unsolved problems and garner insights.  Below is some work that Sophie did last year after the AP Calculus exam was over.  We were trying to see just why the Collatz Conjecture is so darn challenging.  What’s the hoopla all about? The Conjecture Choose any integer \(n\) greater than \(1\) and apply the following function: \[f(n) = \begin{cases} \frac{n}{2} & \text{ if } n \text{ is even} \\ 3n+1 & \text{ if } n \text{ is odd} \end{cases}\] Then iteratively apply this function to each new output.  For example, let’s start with \(n=13\).  It’s odd so we multiply by \(3\) and add \(1\) to get \(40\) which is[…]

It’s common for math teachers (and students) to question the topics we spend time on in class.  In certain subjects (*cough* Algebra *cough*), we tend to be particularly pressed for time while attempting to cram what seems like years worth of standards into one, often while stretching below grade level to reach our students.  I can feel myself already veering toward a rant, so I’ll try to ignore some legitimate questions and just address one about cubics. Are cubic polynomials really useful? Heck yes.  For example, robotic joints are controlled by cubic polynomials so they don’t wear out their motors. Consider a robot composed of a single link with one motor-powered joint at its base.  This isn’t just a hypothetical[…]