Thursday, December 27, 2012

More interesting snowflakes

Looking for something to entertain semi-distracted minds after the break?  How about challenging your students to fold paper and cut out a seven-sided snowflake decoration? 
Of course you could differentiate the task by starting off easier with 4- or 8-sided forms, increasing the challenge with more realistic 6-sided snowflakes, and asking particular students to bust out a 5- or 7- or 9- or 10-sided model.  Keep a chart on the board with the number of folds required and see what patterns your class can uncover. 
Here are my window decorations -

Sunday, December 16, 2012

MoMath Opening

Check out this video about the new Museum of Math in Manhattan, which opened yesterday.

Incidentally, the video is by George Hart, whose daughter Vi makes some sweet math videos of her own.

Thursday, December 13, 2012


This is an older one, but I wanted to take it out for another spin - here's my model of the old Spirograph toy. Nothing but the beauty of trig functions. [Fullscreen/Download]

(Sorry if you don't have Flash.) 

Friday, December 7, 2012

Library of Babel

Jorge Luis Borges' short story Library of Babel conceives of a library comprised of all possible books.  Specifically, all possible permutations of 25 characters (22 letters, comma, period, space) within books with 410 pages.  (Each page has 40 lines; each line has 80 characters).  This set of books would contain all books written so far (spanning multiple volumes if needed), along with all books that could possibly be written in the future.  The majority of the books in the library would be gibberish, of course.

A few questions, ordered from easy to hard:

  • How many books are in The Library of Babel?

  • How much space would this set of books consume (let's ignore the rooms and shelves, and just stack the books, which measure, say, 6 in. x 4 in. x 1.5 in. )?

  • What proportion of the books would read coherently from start to finish?

In this library, there would be a book explaining the solutions to each of these questions!

Thursday, November 22, 2012

Here’s a tip

I usually end up doing some math to make the total amount nice and round, but this guy skipped the middle man.  So kids, was this a decent tip?
found on failblog.

Sunday, November 18, 2012

Have you noticed this?

Look at the number of characters... do you think this happens for any other sum?  Maybe a product?

Wednesday, November 7, 2012

Fractal mountains

Driving home yesterday I found myself admiring the Olympic mountains and wondering about the simplest way to model them.  Here’s a start (click the image if you have Java installed).
1-Fullscreen capture 1172012 93724 AM

I used a sequence of coin flips and the rule “Heads, go up, tails, go down”.  So a string of coin flips HTHHT would be translated into UP-DOWN-UP-UP-DOWN, and look like this:
To give the illusion of depth I increased the step length and darkness the farther down the screen you click.

Fractal mountain ranges and landscapes are used all the time in movies and video games, since they’re so easy to make and look just like the real thing. 

Thursday, November 1, 2012

Teacher fail

Recently this was posted in an email group:

A former student has e-mailed me that his current teacher marked wrong his interpretation that the origin is NOT located in any Quadrant.  This teacher told him that the origin is in all Quadrants.
I'd love to hear some of your takes on this.  I've pretty much tell my students that points in the coordinate plane can be classified in 7 positions - one of the quadrants, on an x- or y-axis, or at the origin.

I knew when my hackles raised that I should hold off on responding, but luckily Illinois teacher Paul Karafiol did the job for me:

On the pedantic side, "in" a quadrant means in the interior, which would clearly exclude the axes under the standard (metric/topological) definition of interior.  Whether we decide that those axes are parts of the quadrants (though not the interior) or are "special" is a different matter.  I could imagine this making for an interesting class discussion that would touch on big mathematical themes:  why we try to avoid making arbitrary distinctions (a reason for calling the axes special, rather than assigning them to quadrants), the importance of precise definitions, etc.  What I can't imagine is this making for a good reason to take a point off a test, especially in the absence of such a discussion.  What well-defined learning objective is it related to?  Why is this an important part of what students should know or be expected to do?  Why take this opportunity to reinforce kids' basic idea that math is about learning a set of arbitrary rules, especially when--as might have happened here--"learning" is really code for "divining by esp, ornithromancy, or whatever"?  (I'm not even raising--except in this parenthetical--the whole idea of "losing points" versus raising points, or the really awesome article I just read about other "toxic" grading practices:

Incidentally, the original teacher in this story may want to check out Paul’s diatribe on another stupid grading practice.

Tuesday, October 30, 2012

Pumpkin pi

Inspired by former student Jena’s tradition of carving up some pumpkin pi each year, I decided to do my own variation using the formula

π/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 …

Saturday, October 27, 2012

Will that tree hit my house?

Yesterday my dad and I set about sawing down a large alder tree that was leaning and looking precarious.  It was tricky to see whether it was tall enough to hit the house when it fell, but luckily some similar triangles came to the rescue.
I folded a 45º angle in a piece of scrap paper and walked to the side of the house closest to the tree. Sighting up the hypotenuse, I noticed that the topmost branches were still in view – so the tree was taller than the distance between its base and the house. 2-DSC04284
I used a guide rope attached to another tree to prevent it from falling too close the house, so my windows remained intact.  The main problem was that the chainsaw got stuck as we were cutting the notch, so we had to hack at it with an axe for awhile. Warning: the video below is not educational:

Friday, October 26, 2012

Similar parabolas

The arcs traced by projectiles are approximated by parabolas.  Parabolas come in all shapes and sizes - tall and thin, short and wide - or so I thought.

While looking at the graph of a parabola on Geogebra, I noticed that you could "flatten it out" by changing the x-scale on the axes or by zooming in  (changing x- and y-scales proportionally) on the vertex.  This is intuitive: if you zoom in on any smooth curve, it straightens out - in calculus you call that local linearity.


But this means that any two parabolas can be made to look identical just by zooming (as opposed to scaling only one axis), which is how two similar objects can be made to look identical.
We say two object are similar if their corresponding lengths form proportions (pairs of equal fractions); can we prove that two parabolas are similar in this way?

Say you have two parabolas, y = ax² and y = bx².  To show that two shapes are similar, we find a scale factor (k) so that any length on the first shape equals k times the corresponding length on the second shape.  For parabolas, let's use the pair of lengths x and y for convenience.

On the first parabola, this pair of lengths is x1 and ax1².On the second parabola, the pair of lengths is x2 and bx2².

We must find a k so that x2 = kx1 and bx2² = kax1².
Substituting for x2, we have b(kx1)² = kax1².
So bk²x1² = kax1².
Dividing, we have bk = a, or k = a/b.

So the scale factor between any two parabolas y = ax² and y = bx² is a/b, conveniently.

Saturday, October 20, 2012

Monday, October 15, 2012

Sextants, clocks, and logs

Congrats to my fellow teacher George Christoph, who just released a lesson at TedEd.  It gives an overview of nautical navigation and the development of logarithms.

Saturday, September 22, 2012

Double-check your napkin

The 2011 movie Contagion required viewers to appreciate the speed at which infection can spread.  The term “R-nought” is used to represent the reproductive rate: If R0 = 2, then each infected person infects two more.  So R0 is the base of the exponential function y = R0^x, where x is the number of time steps.  However, Jude here seems to have a different take on the function, spitting out the terms 2, 4, 16, 256, …

You might think about asking your students

  • In the sequence 2, 4, 16, 256,…
    What are the next several terms?  Is the 30th term close to a billion?  What is the recursive formula for the sequence?
  • If you start with 2 infected people, and each infected person will infect 2 others, give the number of new infections for the first few time steps.  What about total infections for the first few time steps?  Can you figure out how many people will have been infected after 30 time steps?

Friday, September 14, 2012

Harmonigraph revisited

1-Screen Captures1
A harmonigraph is still on my To-Build list (have you seen one yet?), so I’m still making do with a simulation.  I tweaked my first attempt to make it even awesomer. 

Now the color changes over time, and I narrowed down the possible periods for the pendulums, and it’s not so boring to watch because it’s super fast.  Try it out - click two places to get a new image (try two close to center).

1-Fullscreen capture 9142012 33851 PM

This simulation uses the parametric equations
x = sin (A t) + sin (B t)
y = cos (A t) + sin (C t)
with the period of each component limited to a value between 2 and 2.5. 

Saturday, September 8, 2012

Managing supplies

Now that school’s back in session, you may be noticing that your neatly arranged classroom supplies tend to fair better when students are at home.  After I noticed my scissors slowly dwindling in number, I began storing them in a 2x4.
I made some more holders like this for markers, and they worked pretty well.  The kids dubbed them “pen islands,” and the inappropriate name stuck.

Feeling more industrious, I came up with this industrial-sized tape dispenser.  The roll of tape was $5 at a hardware store and lasted at least 4 years.

Saturday, August 11, 2012

Stars and stripes ball

If you’ve ever been given the task of putting a star design on a sphere, you probably first thought of an icosahedron (or perhaps its dual, the dodecahedron).  Since five triangles meet at each vertex, the icosahedron is intimately linked with pentagons, stars, and the Golden Ratio.

But how do you find the side length, a, of an icosahedron when all you know is the diameter of the circumscribed sphere?  A quick trick is to use Wikipedia, which tells us r = 0.951a.

2-math blog

In my case, the sphere’s diameter was 3 9/16 inches, so I used the equation above to find a = 1.873 in.  Now set the compass to this length and get ready for some circular reasoning. 

This was a project of my dad’s so he took over and dressed it up real patriotic-like.


Tuesday, July 31, 2012

On rectangles and rhombi


Illinois math instructor John Benson recently came across something that caught him by surprise in some elementary mathematics materials aimed at teachers.  He posted it to a listserv, which is where I came across it.

..There was a careful definition of rhombus followed by a statement that was well intentioned but that caught me by surprise.
"While most rhombi do not have right angles, a rhombus can have right angles"
This took me by surprise and I wanted to know what other knowledgeable and thinking math teachers from all grade levels thought about it, so I asked the question on the list serve, out of context.

I had several thoughts myself and wondered what others would think.  First, I think this is a great question to ask kids.  Many of them do not understand that a square is a rhombus ( using the inclusive definition, which is what I hope everyone uses all the time for everything, including
trapezoids) This raises the issue in an indirect way, which I think is a good thing.

I learned that I did not state the question well enough in that some of us thought that the angles formed by the diagonals are "angles of the rhombus" so the answer is all of them. I had not thought of that. That is why I put the question out there. I intended the question to be about the vertex angles. What a cool thing if some kid thought of that. When I use it with students, I shall leave the question as is and hope a student mentions the diagonal angles, congratulate that kid, and redirect the question to the one I intended.

So, there are infinitely many squares, and there are infinitely many non-square rhombi so the answer is not obvious. One line of thought is that the probability that a randomly selected angle would be 90 is zero so most are not squares. That raises the question of infinity, which is always an interesting place to go in a math class. I think it also raises the question of random. From where are these quadrilaterals being selected? If the angles are being randomly selected by a computer, then it depends totally on the level of precision the computer is using. If it is selecting an integer between zero and 180, then the likelihood is 1 out of 179, I think. If it is rounding to the nearest whatever, less likely. Are these not great questions for students to think about?

If this question is being asked to a third grader, I suspect the immediate response will be that squares are not rhombi, so none. Once convinced, then they will conclude that most are squares, because likely is about the ones that one encounters, and I suspect that the average person encounters way more squares that any other sort of rhombus.

So, one silly question can lead to so much interesting mathematics. The best part is that each of these strands will feed into something bigger and more important.

Sunday, June 17, 2012

Harmonigraph – first attempt

1-Fullscreen capture 6172012 95530 AM

I’ve been thinking about building a harmonigraph once I have room for it, but in the meanwhile a simulation will have to suffice. 

This one is made with the parametric equations
x = sin (A t) + sin (B t)
y = sin (C t)

It still needs tweaking, but you can get some cool looking stuff.  You get a new configuration by clicking twice (the first click sets A and B by looking at your mouse’s x and y position; the second click sets C as the mouse’s x position).

Monday, June 11, 2012

Down the slide

As I took this picture of my son on a slide, I was reminded of the Mean Value Theorem for derivatives – that is, the curvy slide is exactly as steep as the straight slide in at least one location (probably three).
But which slide is faster?  What shape would the fastest possible slide have (starting and ending at given points)?  This is called the brachistochrone problem, posed by Johann Bernoulli in 1696.  I gave it a shot once, and didn’t have much luck.  Leave it to Newton to solve that bad boy in one day.  Some day, though, I’ll put a brachistochrone slide in my yard.

Wednesday, May 16, 2012

Toilet Paper Problem

What would it look like if you marked each perforation on a roll of toilet paper and rolled it all back up?  A striped pattern?  A spiral pattern?  I wondered about that in college, and wrote a program to answer the question - here's it's latest incarnation:
(The link above goes to a Processing app requiring Java - if it doesn't work, try this Flash version).

Of course, the pattern depends on the sheet thickness and the sheet length.  Try clicking different places on the screen - the farther your mouse is to the right, the thicker the sheet, the farther down, the longer the sheet.  I was surprised at what a difference small changes in these values had on the overall pattern.  My fractal sense is tingling - is yours?

At one point I found an article about this type of pattern (an Archimedean spiral marked at fixed intervals), but have since lost track of it - can you help me out? 

Monday, May 7, 2012

Four ways to start a high school programming course

When I graduated high school in the late nineties, I figured my school was behind the times because it didn’t offer a programming course.  I later found that programming in high school was, and unfortunately still is, somewhat of a rarity. 

After the AP test, I led my calculus classes through some programming concepts using their graphing calculators.  The unit was well-received, and culminated with some reasonable programs. 

But you don’t need calculus to learn to program.  In fact, programming may be the right context to teach math concepts to students in elementary and middle school. 

The relevance goes without saying.

But how to get started?  Here area  few options that come to mind:

  • Alice.  A free development environment for 2D and 3D animations.  Designed as a teaching tool, Alice has a drag-and-drop interface and is many a college student’s first experience with object oriented programming.
  • Processing.  Created as a tool to teach programming in a visual context, Processing allows users to quickly (i.e. simply) create graphics and animations.  Your curriculum is practically ready-to-go – click “learning”.  Did I mention it’s all free?
  • Lego Mindstorms.  Nice drag-and-drop software makes robotics programming a visual experience, and of course commanding a robot to do your bidding is always rewarding.  It’s pretty expensive though, at $280 per kit. 
  • Graphing calculators.  If your classroom has 30 of them, you’re in luck.  Otherwise, this might not work out.  The handheld aspect is convenient, and you won’t have kids checking Facebook on them.  Not object oriented, but hey, this is an introduction.

Thursday, May 3, 2012


I finally got around to trying out Processing a couple of days ago, a programming language for making… well, art.  It’s like a super-simplified Java geared toward graphics and animation.  Needless to say, I became an instant fan. 

Here’s my first program - a Pickover attractor.

1-Fullscreen capture 522012 93119 PM
//xn+1 = sin(a yn) + c cos(a xn)
//yn+1 = sin(b xn) + d cos(b yn)
//where a, b, c, d are variabies that 
//define each attractor. 

void setup(){
float x = 0; 
float y = 0;
float a = -1.4;
float b = 1.6;
float c = 1.0;
float d = 0.7;
float newx = 0;
float newy = 0;

void draw(){

  for(float z = 1; z < 100; z++){
    ellipse(170*x+width/2-40,170*y+height/2-20,3 ,3);
    newx = sin(a*y) + c*cos(a*x);
    newy = sin(b*x) + d*cos(b*y);

    x = newx;
    y = newy;

Monday, April 30, 2012

How does your computer see you?

1-Screen Captures
Well how do you look when you see a new math problem?  Facial recognition software uses a set of measurements (distance between eyes, width of nose, etc.) to create a numerical “faceprint” for a particular image, which can then be compared with a database. 

Mood Battle is an experiment to detect emotions by comparing measurements from a face.  If you have a webcam, try out some different emotions on it.  But if it mistakes your angry face for a happy one, remember: it’s only looking at numbers.

Saturday, April 21, 2012

Fractal pancakes

What I cooked up for breakfast this morning:
Dragon curve, Apollonian gasket, Sierpinski sieve, Mandelbrot set, Pythagorean tree,
Koch snowflake, Star fractal, Hilbert curve, Lorenz attractor

Friday, April 20, 2012

Get rich with calculus

While demand and pay for STEM careers continues to rise, maybe your math students are feeling more like Alex here:
1-will solve calculus for food

The website is one way to make a few bucks answering math questions for students at various levels (though the upper math questions are most frequently asked, and least frequently answered).  The math notation editor is pretty slick and intuitive.  Answers pay $2 or $4, so you’re not going to get rich quick, but I made a fat $600 during the month I had nothing better to do.
1-Fullscreen capture 422012 70045 AM.bmp

Friday, April 13, 2012

Calculus cookies

Found these in Portland – they really do help you find dy/dx.  But not f’, though… for that you need Fig Newtons.  Sigh.

Wednesday, March 21, 2012

Thursday, March 1, 2012

Is your presentation accessible?

When a colleague and I team-taught our 9th-grade math classes for three years, we found out early on that we needed a way to make the presentations accessible to all students.  The document camera and projector were an improvement over the whiteboard (easier to read, but also easier to show students’ work and have them present).  But in our double-long classroom what we really needed was to synchronize our projectors.  Just running a VGA cable wouldn’t work by itself, since we needed a way to switch sending/receiving roles and switch between camera and computer display.  The solution was a VGA splitter box – we got two of these, each with two inputs and four outputs.


The outputs I have shown on mine are monitor (my screen), projector, 283 (the room next door), and student (another monitor I set up for near-sighted students).  The student monitor ended up being highly desirable (students seemed to like having their own personal screen), to the point where I installed a second one of those. 


Wednesday, February 22, 2012

Droppin’ the fractal beat!

I concluded my undergraduate computer science thesis on generating fractal music by postulating that “Western sounding” music would require the generator to impose a lot of predefined conventions on the musical structure.  My program wasn’t really making very musical music, in other words.  It was interesting enough, though:
Recursive string rewriting or L-Systems.
I noted that music written by humans oftentimes contains fractal structures within it, such as the beginning of Beethoven’s first Ecossaise, with successive bifurcations indicated:
(This structure is similar to the ABACABA song).
Daniel Levitin has spent some time researching fractal structures in existing music.  His recent work, involving the digital analysis of hundreds of scores, reveals fractals buried in the rhythmic signatures of history’s composers:

Sunday, February 12, 2012

Love, the Fibonacci way

Remember the Fibonacci sequence?  Start with 1.  Then another 1.  Now each subsequent number is the sum of the two previous numbers, so we have 1, 1, 2, 3, 5, 8, 13, … forever!  This sequence allegedly began with Fibonacci’s modeling of a rabbit population. 

Bonus: can you extend the sequence the other direction – what number would come before the first 1?  What family of function would you guess the sequence to be?

Thursday, February 2, 2012

Three quick ones

Along the lines of my last post, I came up with these for you to decipher immediately…


Monday, January 30, 2012

Pi designs

After seeing a cool “Pi-rate” shirt, I thought to myself, “Why didn’t I think of that?”  To prevent any more such occurrences, I have made a list of all the words containing “PI” and have begun an exhaustive search for pideas…


The list is long… feel free to help me narrow it down.


Saturday, January 21, 2012

Find 2 mistakes

I did a search for “volume formulas” and the second image returned was this one.  I almost provided these posters to a student until I noticed two mistakes…


Monday, January 9, 2012


We noticed a prominence of powers of two in our family this year.  My daughter will turn 2^1, my son will turn 2^2, my wife and I will turn 2^5 and celebrate anniversary number 2^3.  (This may just be a fluke of the Mayan calendar).

Given this information, will there be another numerically interesting year for us, or is this it?  We can't look forward to all being powers of 2 again, but perhaps there's something else in store for us?