Fun

• Open a new desmos graph and type "betchacant" into the equation/expression field. Do it again, and again...

• Rock-Paper-Scissors simulation found on reddit

• Mandelbrot Set fractal in Desmos found on reddit

• Christmas tree (3D projection) from  reddit

• Magic (Rubik's) Cube and Chess, playable games from Desmos Art page

• semi-polygon curves rotating from tweet (click "Animate" at bottom, try "random colors" curves)

• Crazy Circle Illusion inspired by video, by Mr. W. Re-creation of GeoGebra construction.

• sunset trig showing how sun "moves" relative to a location on Earth throughout the year

• Video of insanely-complicated Desmos animation that took itscreator 2 months to make.

• Polar Wrapping, showing how rectangular (x, y) graphs wrap around the pole/origin to create polar (r, θ) graphs, by Mr. W. Re-creation of GeoGebra construction.

• Rubber Pencil Illusion, using sinusoids to model how I believe the illusion works, by Mr. W. Re-creation of GeoGebra construction.

Constructions that Mr. W has shared on the GeoGebra site.  Or explore all materials posted on GeoGebra's site. Or go the the GeoGebra calculator and create your own content. While you may find Desmos easier to jump into and use, the more-expansive set of tools in GeoGebra make it worth checking out if you like creating Mathy constructions.

Somewhere in my childhood I acquired a red, plastic, asterisk-shaped toy. Loosely housed in the middle and at the extremities are five white dice and two black dice. I don’t know what the original intent of the toy was, but for years I’ve had fun rolling the dice and arranging the five numbers of the white dice into a mathematical expression that equals the sum of the two black dice. Many of my students have enjoyed this game too – at least, more than the lesson at hand. Click the image for a GeoGebra version of the game, or click here for an old html5 version. In the screen shot shown here, one solution would be 1*6*4+3+5 (white) =30+2 (black). Other solutions include 4*5+6(3-1)=32, or (6+5-3)*4*1=32, or (6+3)(5-1)-4=32, or (5*3+1)(6-4)=32, or (3-1)^(6+4-5)=32, just to provide a few.

Here are  the rules I play by:

The following ARE allowed:

• • four basic arithmetic operators (add, subtract, multiply, divide)

  • parentheses, brackets, or other grouping symbols

  • radical for square root

  • other roots, using the number on a white die as the index of the radical (for example, you may use a 3 from a white die to take the cube root of another quantity)

  • exponent, using the number on a white die (for example, you may use a 2 from a white die to square another quantity)

  • ! (factorial)

  • . (decimal point)

  • decimal repeating bar (for example, putting a repeating bar over .3 would make it .333333..., which is equal to 1/3)

The following ARE NOT allowed:

• • Using multiple white dice to form a multi-digit number (for example, using 2 and 4 to form 24). Allowing this would make the game too easy in many instances.

  • Ignoring a number on a white die. You must use each of the five numbers from the white dice exactly once.

  • Utilizing different bases (other than base-10)

  • Rounding off/up/down symbols. Cleverly used, such operations would make the game too easy in many instances.

  • Use of pi, e, or any other Mathematical/scientific constant.

  • More advanced operations not mentioned above, such as logarithms, trigonometry, summations, etc. Permutation/Combination notations and gamma function are about the only "advanced" operations that I could imagine sometimes being useful anyway, and I've always found a way to avoid needing it. (Admittedly rather arbitrary choice to allow factorials but disallow other combinatoric functions).

You may wonder: Does a combination of numbers ever come up that is impossible to solve? I'm sure there must be some possible combinations that would stump me, yet I've never come across one while playing this game. As an example of how one might solve a very difficult combination in a clever way, imagine you got 1, 1, 1, 1, and 1 on the white dice and 61 from the black dice. At first it seems impossible, right? But here's one possible solution.

But don't worry, most combinations won't require anything nearly this tricky. I'd estimate that 90% of the time I can solve the puzzle with nothing more than the four basic arithmetic operators and some parentheses.

For whatever it's worth, here's the original Flash version of this app from back in the day when Flash was the cool kid on the block.

Like Wordle, but with numbers. A new Daily Ooodle each day.

As a youth, Word Arithmetic was my favorite type of puzzle from Dell Puzzle magazines. This is the only computerized version of Word Arithmetic that I'm aware of. This puzzle is based on long division, but if you really wanna nerd out, here's a long multiplication version too. These html5 versions will play on a mobile device although the audio probably won't work.

Here is the original Flash version of the division game and Flash version of the multiplication game. Why I am I still linking to files created in a dead technology? I just can't let go of the past, man.

A bunch of great logic puzzles, some of which I use for extra credit in PreCalc Honors. Many of these puzzles types were originally published in Nikoli, the Japanese puzzle magazine that made Sudoku famous.

The first three are predominantly Mathy, and from there the Math tends to blend heavily with Science/Engineering. While the content often stretches beyond high school level, the success of these channels is the engaging communication skills of the presenters. Are there others that I should add to this list? Please share!

• Numberphile: The first educational/Math youtube channel that I (Mr. W) became a huge fan of. Started as mostly British Mathematicians sharing interesting facts about particular numbers, but expanded into other Mathy topics.

• 3blue1brown: "animated Math," and some combination of math and entertainment, depending on your disposition. The goal is for explanations to be driven by animations and for difficult problems to be made simple with changes in perspective.

• Vi Hart: Victoria Hart, commonly known as Vi Hart, describes themself as a "recreational mathemusician." Despite their "Math class is so boring" vibe, the playful doodling and sarcasm ultimately helps uncover the fun side of Mathy pursuits.

• Veritasium: Science and engineering videos featuring experiments, expert interviews, cool demos, and discussions with the public about everything science.

• (It's okay to) be smart: Deep answers to simple questions about science and the rest of the universe.

• brusspup: Optical illusions, science and more. A few of these are featured in my PreCalc lessons.

• Mark Rober: Former NASA engineer. Current CrunchLabs founder and friend of science. Like many, I first became aware of him from his first Backyard Squirrel Maze video.

• Smarter Every Day 

• Animation vs Math: Incredible animation that starts with simple counting numbers and graduates through Math of high school and beyond.

I first encountered infinite continued fractions as a way to find rational approximations for irrational numbers (e.g. rational numbers 4756/3363 and 355/113 approximate √2  and π accurate to six decimal places, respectively). It turns out there are multiple practical applications to them although I primarily enjoy them as a Mathy diversion that are fun to play with.  I've used this interactive UK university site to learn about continued  fractions, although the Wikipedia page is also worth checking out.

I saw OEIS cited many times in Mathy books before even knowing what it is or visiting the site myself. It's interface is about as low-frills as they come, but it's a hugely important resource in Math research and it's fun to just play with.  Type in a list of integers and see how many patterns have been documented that include that list. For example, if you enter 1, 2, 4, 8, 16, you'd probably assume (reasonably) that 32 must occur next in the list, continuing a pattern of a(n)=2^n. But it turns out there are 899 total results (as of this writing)!
For example, the next number in the list could be 30, continuing the pattern of a(n) = number of divisors of n!.

Or the next number could be 23, continuing the pattern of a(n) = sum of digits (in base 10) of all previous terms.

Or the next number could be 31, continuing the pattern of a(n) = maximal number of regions obtained by joining n points around a circle by straight lines (great puzzle, btw).

The pages might require some effort to sift through, but contained within them are some good rabbit holes to follow.

How could I not include the official site of Rubik's Cube, the most geometric of toys, and other Rubik's inventions?

Back in 2011, the "batman equation" intrigued the math nerd community. I became instantly intrigued with the activity of transforming basic functions into drawings/designs on my graphing calculator. Now it turns out there's a whole website devoted to this type of challenge. Given a design, can you come up with the equation(s) whose graph(s) create such an image? (By taking PreCalculus, your chances should increase dramatically!)

Hey all you artsy word-lovin' types, this site has HUNDREDS of poems that incorporate the language and concepts of math. Some of it is a little high-falutin' for my crass unsophistication, but perhaps it will resonate with those amongst you with more refined sensibilities (On the other hand, this site posted my "Love Triangle" poem in 2017, with my permission). Regardless, I absolutely love the existence of such a vast collection of works bridging math and language.

Oh hey, Mathematical Poetry offers another site with a similar mission!

Perhaps you've seen videos of savants who can instantly tell you the day of the week of any date in history (or other impressive mental feats). Chances are that neither you nor I will ever be able to do that so effortlessly. From what I read, psychologists and the savants themselves are typically baffled about how they do this.

However, with some basic math techniques, a bit of memorization, and a little practice, you can learn to do this trick mentally. Within a week of learning this, it takes me about 20 seconds to do it and I doubt my time will improve substantially. Maybe you can do even better?

I learned the trick from the book Secrets of Mental Math. The image and heading link to a web page that outlines the very same process from the book. It's interesting to check out other algorithms for performing this feat, such as those outlined here, although I believe the algorithm I'm highlighting is the easiest to grasp.

I sometimes use this "Day of the Week" calculator to check my results.  

If you don't know about Rube Goldberg machines, you should! Goldberg is most famous for his cartoons of fictional contraptions that are designed to perform mundane tasks in convoluted, intentionally-over-complicated ways. If you're interested, I have a couple books on Goldberg and his machines on my classroom bookshelf.

There are pleeeeeenty of awesome Goldberg Machine videos on the internet.

One of my fondest memories as a teacher was watching one of my students study Rube Goldberg and construct a series of contraptions of her own for her Senior Project.

Ambigrams are typically words written in such an artistic way that when viewed upside down, they read either the same word (exhibiting rotational symmetry) or a completely different word. To explore, I suggest clicking the included Google link and checking out (1) any of the top search hits, or (2) clicking "Images." The Wikipedia page lists numerous classifications of ambigrams. A variety of ambigrams can be found in my classroom (some of them that I designed myself), and I also have a couple books with more ambigrams if you're interested.

My interest in origami began in 4th grade when a classmate made an origami throwing star for some of his friends but refused to give me one. So, I stole one of his stars and took it home to figure out how to make one myself. Then out of spite I asked my mom to buy me some origami books so I could show this punk up in school.

I've since gone on the straight and narrow path of paper-folding.

There must be thousands of instructional web pages out there with models of all different levels. There is definitely a learning curve to this art form, and I enjoyed watching one of my recent students choose origami as the topic of his senior project. While it may be a bit intimidating for someone looking to get started in origami, I suggest checking out the web page of Robert Lang, who is the most famous American origami artist. The picture to the left is one of Mr. Lang's models. He adheres to the practice of always starting with a square or rectangular piece of paper and allowing no cutting or gluing during the creation process. Lang, who has an accomplished background in physics, uses advanced math principles and computer programming to help develop the patterns for his models.

One day while browsing a bookstore for origami books as a wee lad, I found a book called "Paper Magic" by Masahiro Chatani. At first glance I viewed it merely as a book showing how to make "pop-up-cards," but closer inspection revealed that the designs in this book were far more intricate and beautiful than the typical pop-up. I was surprised at first, because most origami books frown upon cutting and gluing in the creation process, yet each of the models in this book featured a multitude of cuts. However, putting aside any rigid rules of one art form, I saw this was an incredible art form unto itself. I ended up buying almost a dozen books by Mr. Chatani over the years, and now since the emergence of the internet one can easily find hundreds if not thousands of Origamic Architecture patterns online.

Ethiopian Multiplication, also called Egyptian Multiplication, reminds us that the way we were taught to multiply numbers in Western schools is certainly not the only way to accomplish the task. This method is based on the same Mathematics as binary code, yet it was developed long before computers were invented.

Every so often a student will forward me a video for Japanese, or Chinese, Multiplication (I don't personally know how much this method is really taught in any Asian countries). Can you see how this is actually just a visual way of generating the same numbers that appear in the multiplication method that you learned as a kid?

Here's an image comparing four multiplication methods that I once sent to a student who had just discovered the Japanese/Chinese method.

These paper fascinating paper creations fold flat, which would lead one to conclude that they have two sides, right? However, when you fold or "flex" them, the original faces disappear while different hidden faces are revealed. Flexagon.net has a lot of information and patterns, but it might be a bit overwhelming at first. A better place to start might be this page, which includes a simple pattern for a hexagonal shaped flexagon and a short video showing how it works. This video also shows how to make a very simple square shaped flexagon with no tape/glue required.

Topology is the branch of math behind flexagons, and similar principles are used in the creation of toys Jacob's Ladder and Rubik's Magic.

Another good video about hexaflexagons here from recreational math darling Vi Hart.

Wow, how does one pick just a few cool pages to launch an exploration into fractals?

While the subject of fractals doesn't fit neatly into any math category that's taught in high school, fractals bring to mind some of the more interesting facets of Geometry as well as the "recursive sequences/series" that we study at the end of PreCalculus.

Furthermore, when you start exploring the areas and perimeters of fractals, the topic of Limits that bridges PreCalculus to Calculus is of great use to us.

As for any subject on this page, I'll just pick a few pages to list here and invite you to explore further on your own:

• • Try this paper-folding exercise to explore the "Dragon Curve" fractal

  • The Numberphile video on the "Dragon Curve"

  • I encourage you to try Fractal Grower Java simulation on this page

  • The Koch Snowflake is one of the best-known fractals

  • The Sierpinski Triangle is another famous fractal

What happens when you start swinging a bunch of pendulums of varying lengths side by side? I bet you wouldn't anticipate the results.

The mathematics which describes the motion of a swinging pendulum is called "simple harmonic motion," which we study briefly in the midst of our Trigonometry studies. In a Physics class you might also get some exposure to how pendulums behave.

• • The first video I saw on pendulum waves

  • A similar video with glow-in-the-dark effects

One can find plenty of attempts to sing or rap about Math on the internet, but this is one of the best I've come across. This project is quite prolific, and most of the content is at Algebra 2, PreCalculus, or Calculus level. I wish we could bring their stage production of Calculus: The Musical! down to New Orleans.

Albums include:

• Calculus: The Musical!

• Log: A Rhythm

• Be Rational

• C-Sections

This site's titled as "Interactive Mathematics Miscellany and Puzzles." Yeah, that sums it up pretty well. Great variety of topics and skill levels represented. Some of the activities require some perseverance and willingness to just play around in order to understand the point, but those aren't bad qualities to exercise in math.

This is one of the coolest blog posts I've ever seen. Fourier transforms are a very weighty math topic, but this site gives a very accessible and fascinating summary. While the post itself is good reading, what really makes it outstanding is the collection of links to other videos and interactive explorations including:

• • a little insight into the math behind music compression (MP3 format) and picture compression (JPEG format)

  • great ways to visualize what happens when you "add" sinusoids of different frequencies

  • how such addition of sinuosoids can be used to describe the orbits of planets

  • how such orbits can trace out just about any arbitrary path, including portraits of famous figures

• • "A game about love and graphing, built and maintained by teenagers at @hackclub!" Enter/edit equations for graphs that form hills for a sled to slide down with the goal of hitting various targets. Beautifully and artistically created.