Tuesday, December 3, 2013

Math is beautiful

The World of Mathematics” by Mathigon is a free online eBook that explores a diverse array of math ideas in an accessible and visually engaging way.  Taking advantage of the digital medium, Mathigon contains fun interactive components in each of its 30 sections, many of which are still in development. 

mathigon

I enjoyed the Symmetry and Groups page, which did a nice job of making the idea of group theory comprehensible.  Similarly, the Fractal page did a nice job showing what the Mandelbrot set actually is.

mandelbrot explanation

Tuesday, November 26, 2013

Angry Birds Calculator Program?

Here’s a little program I wrote for the TI-83 during the NW Math Conference last month, during the session just before my own.  It could serve a number of possible functions, including introducing programming to students, motivating a lesson on sine and cosine, or motivating a lesson on parabolas (though the program doesn’t use quadratic functions).
It’s not particularly rewarding to play, but maybe you can improve on it – right now it does three things:
  • draws a pig at a random distance
  • asks for angle and “force”
  • plots points until they go off the screen
There’s no collision detection or points system, and no end to the game, but what can you do?Here’s a look at the code so you can make your own improved version if you’re so inclined:

//FIRST, THE SETUP
  • Lbl B
  • ClrDraw
  • AxesOff
  • Degree
  • PlotsOff
  • FnOff
  • 10→Xmax:10→Ymax
  • -2→Xmin:-2→Ymin
//NEXT, THE SCENE
  • rand*4+4→P 
    (random number between 4 and 8 for the pig’s x-coordinate)
  • Line(-2,-1,10,-1) 
    (the ground)
  • Line(0,-1,0,0)
    (the slingshot)
  • Circle (P,0,1) 
    (this represents the pig. I made a more detailed pig face since I had some extra time, which I’ve included at the bottom).
//SET SOME STARTING VALUES
  • 0→X:0→Y 
    (starting values for plotting the bird path)
  • -0.07→G 
    (acceleration due to gravity… units are arbitrary, so I just tweaked this until it looked good)
  • Pause: ClrHome
  • Disp “ANGLE?”: Prompt T
  • cos(T)→D: sin(T)→E 
    (D and E will tell us the x and y increments for the bird’s path)
  • Disp “FORCE (1-10)?”: Prompt F
  • F/10*D→D: F/10*E→E
    (D and E are scaled by 1/10 of the “force” …also arbitrary)
//PLOT SOME POINTS
  • Lbl A
  • Pt-On(X,Y)
  • X+D→X: Y+E→Y
    (If D and E both remain constant, the bird will go in a line.  The curve is produced by changing E, the increment for y.)
  • E+G→E
    (You could think of E as velocity and G and acceleration)
  • If Y>0 and X<10: Goto A
    (Keep plotting points until you go off-screen to the right or hit the ground)
  • Pause
  • Goto B
    (New pig!)
//THAT’S IT!  If you want a more detailed pig, replace the Circle command with the following (I got rid of the circle because it takes so long to plot):
  • Line(P-1,0,P-0.5,1)
  • Line(P-0.5,1,P+.5,1)
  • Line(P+0.5,1,P+1,0)
  • Line(P+1,0,P+0.5,-1)
  • Line(P+.5,-1,P-0.5,-1)
  • Line(P-0.5,-1,P-1,0)
    (This forms a hexagon for the pig’s head)
  • Line (P+0.4,0.1,P+0.4,-0.7)
  • Line(P+0.4,-0.7,P-0.4,-0.7)
  • Line(P-0.4,-0.7,P-0.4,0.1)
  • Line(P-.4,0.1,P+0.4,0.1)
    (This forms a rectangle nose)
  • Pt-On(P+0.5,1.2,2): Pt-On(P-0.5,1.2,2)
    (Ears are points with “style 2” so they’re tiny squares)
  • Pt-On(P+0.5,0.5,3): Pt-On(P+-0.4,0.5,3)
    (Eyes have “style 3” so they’re tiny crosses)
  • Pt-On(P-0.15,-0.3): Pt-On(P+0.15,-0.3)
    (Nostrils!)

Monday, November 18, 2013

Growing plants

Leafing through a magazine the other day, I happened across this page:
1-IMAG1681

I wondered how many plants they’d have the next year.  With limited information, what would a best guess look like?  With only three points, you can use a variety of functions to predict the next one – which function makes the most sense?  The only other information we have is that the person quoted is evidently a professional gardener.

Monday, October 28, 2013

The nearsighted man

I heard this problem from a teacher a couple weeks ago -

A nearsighted man can see things only if they’re 3 feet or closer.  If he stands 2 feet in front of a mirror, can he see his reflection?

He said he consistently finds people split on this, and hasn’t found a compelling proof one way or the other.  I humbly offer up the following “proof without words” –  but what does it show?

mirror focus

Wednesday, October 16, 2013

Pancake Toss

I made my kids some Angry Birds pancakes - what better way to serve them than with a high velocity launch?  Use the points to figure out if the second pancake lands on the plate or not!  First one to tell me wins!


Bonus points if you collect some data to find the equation of the first pancake's flight:

Sunday, October 13, 2013

Angry Nerds

Fellow teacher Casey Alexander and I attended the Northwest Math Conference this weekend – I’ll be posting some ideas about it in coming days.  We presented a session called “Angry Birds Teach Math,” in which we shared some Angry Birds math resources, and even shot actual angry birds at the audience. 
angry birds teach math

Tuesday, October 8, 2013

Finding the center of a circle

As I was working on a lazy Susan in my workshop the other day, I needed to find the center of a circle I had made (I had covered up the mark on the other side that I used for the compass).  Luckily I had paid attention in geometry (at least when I taught it), and used a couple of perpendicular chord bisectors.  Thanks, math!
find the center of a circle

Wednesday, August 28, 2013

Math Mystery: Water Challenge

Another video I made for MathMovesU.  Try it out before you check the solution:

Monday, July 15, 2013

Happy faces

1-IMAG0884
My parents still have a basket of baby toys from the early eighties, and I came across this one recently.  These happy faces are linked in such a way that they can rotate together and don’t come apart.  Here’s a question to ponder: Beginning with the orientation in the photo, if you hold one of the gears still and rotate the other one halfway around it, will the smiley be upside-down, right side up, or neither?

Monday, June 24, 2013

Math Mystery

MathMovesU, a multi-faceted math awareness program by Raytheon, recently posted some video puzzles for kids to think about or teachers to incorporate into lessons.  I did the drawing for these, the first puzzle being a reincarnation of the Monty Hall problem:

I don’t take credit for the script however; the solution could probably have been presented more clearly:

Friday, June 7, 2013

Noisemakers

Billions-of-cicadas-to-swarm-East-CoastThese lovable critters emerge from the ground in 13- or 17-year cycles, capitalizing on the power of prime numbers to keep away from predators with periodic population cycles and other cicada broods.  Here’s a cool interactive cicada map you can play with. 

Cicadas remind me of the white noise machine I used to have on my bedside table – and not because they sound alike.  As I would fall asleep, I would inevitably begin focusing on the repetition in the “babbling brook” sound pattern - the machine just played back a loop about 30 seconds in duration.

I always thought it would be a better use of the machine’s memory to store two shorter loops and play them together.  For example, a 13-second and a 17-second loop would get you 221 seconds (13x17) of noise before repeating.  

Ask your students: If you’re limited to 30 seconds of memory, are 17 and 13 the best choices for noise loop lengths?  See who comes up with the idea of non-integer loops… 

Monday, May 27, 2013

Dimensions



At this time of year I like to show some of the Dimensions series. Beautifully designed and distributed under a Creative Commons License, the series is free to download and show in class, and gives an excellent overview of projective geometry in two, three, and four dimensions! Incidentally, the animations were created using POV-Ray, a free ray-tracing program that generates photo-realistic images using precise mathematical modeling.

Tuesday, May 14, 2013

Right triangle days

On Mothers Day I wrote out the date and noticed 5/12/13 could be the sides of a right triangle.  That made it doubly special – but is it really that special? How many dates have that property?  Which years have the most?  Would they be evenly distributed throughout the century? 

5-12-13 triangle
1-Screen Captures3-001

Wednesday, May 8, 2013

Fibonacci date!

Well I should have seen that coming – but we’ll be prepared next time, right?  When’s the next Fibonacci date again?
1-Screen Captures2

Tuesday, April 9, 2013

Calendar

April is a good month.  We have the birthdays of Euler, Da Vinci, Gödel, and Gauss to celebrate, (not to mention my wife and son’s)!  Plus, the American Mathematical Society honored me with the inclusion of one of my fractal pancakes in their 2013 Calendar of Mathematical Imagery. 
calendar of mathematical imageryfractal pancake nathan shields
Admittedly, mine is the weakest of the lot, but it was nice to be included.
mathematical imagery calendar

Monday, March 25, 2013

Automata on the beach

I found this shell on the beach in Saipan last year.
1-DSC06416
Its pattern reminded me of the some of the cellular automata I had seen in Wolfram’s A New Kind of Science (chapter 3) – it turns out that this is no coincidence.  The process by which some species of mollusks color their shells can be modeled by cellular automata: as the shell grows, the coloring of each row is affected by the coloring of the previous row through a small set of rules. 

Dr. Coombes of the University of Nottingham gives a brief overview of some of the mathematics behind the structure and pigmentation of seashells here.

Wednesday, March 6, 2013

Your Life in Pi

Here’s a fun idea to talk about with your students this Pi Day:


What else can you do on Pi Day? As always, www.piday.org has some good ideas for you.  I like to have students do Bouffon’s needle experiment with toothpicks and compile the results with multiple classes. 

Discuss why an accurate value might be necessary, and some historical approximations:
•3.16045 (=16²/9²) Egypt, 2000 BC
•3.1418 (average of 3+1/7 and 3+10/71) Archimedes, 250 BC
•3.125 (=25/8) Vitruvius, 20 BC
•3.1622 (=√10) Chang Hong, 130 BC
•3.14166 Ptolemy, 150
•3.14159 Liu Hui, 263
•3.141592920 (=355/113) Zu Chongzhi, 480
•3.1416 (=62832/2000) Aryabhata, 499
•3.1622 (=√10) Brahmagupta, 640
•3.1416 Al-Khwarizmi, 800
•3.141818 Fibonacci, 1220
•3.14159265358979 Al-Kashi, 1430

A digit contest at the end of the day is always fun too.  Here’s the hand-out I give to kids a few days in advance if they want to try memorizing for fun (it has a spacer every 50 digits):
pi500

Wednesday, February 27, 2013

Bring coding to your students

I’ve griped before about the lack of computer programming opportunities that our students have today.  Here’s a free and painless way for kids to get started – something you could devote one period to (to hook the intrinsically motivated), or build an entire curriculum around:

www.code.org

Wednesday, February 13, 2013

Visualizing music


After besting me at snowflake making recently, Vi Hart came out with one her best video ideas to date, in my opinion.
 
Playing with the symmetries in music in a visual way, she reminded me of the visualization I made for Bach’s “Crab Canon,” which sounds the same forward as backward and “looks” like this:
crab canon2
 
I’ve always been a fan of music visualization videos because they allow you to appreciate some of the patterns, structures, and relationships (what we might call “math”) that you may otherwise miss.

Monday, February 4, 2013

Quilted geometry

A few years ago my mom began work on a quilt for my wife and me.  I had the opportunity to design it, and went with something less common than the right triangles and squares you see on a lot of quilts:
quilt design copy
This is an example of an Islamic tile pattern featuring 5- and 10-pointed stars. 
Getting the pieces cut out proved to be challenging.  After some puzzling, I found that the design could be created using two rhombus-shaped “stamps”:
quilt design copy2
You could estimate the lengths involved, but exact answers are so much more satisfying.  I can’t quite remember how I went about calculating the proportions, but I was happy to find the golden ratio Φ featuring prominently in them:
1-2-IMG_3689
It turned out pretty cool, don’t you think? 
1-DSC07077

But I hope you’re not in a rush to get yours – the hand quilting takes a long time.
2-DSC07083

Friday, February 1, 2013

One "type" of problem

Sometimes displaying math text on the web is a pain.  There's LaTeX, but it's a hassle to employ in Blogger.    You can render your expressions elsewhere and use images, but I think the MathML code provided by Web Equation works pretty well if you don't mind writing your expression on the screen.

It did glitch on me a couple of times and require a page reload.

Here's a quick example:

 x = - b ± b 2 - 4 a c 2 a

Tuesday, January 29, 2013

Math Team commercials

I just came across these videos I made to advertise our school's math team.  I don't know how effective they were, but they were fun to make.

Monday, January 21, 2013

Benford’s Law becomes more clear

You remember Benford’s Law?  Wolfram sums it up:

Benford's law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability ~30%, much greater than the expected 11.1% (i.e., one digit out of 9).

I think Wolfram meant to add that the law applies to the leading digits of the figures you’re using. In any case, the law applies when you have a large sample of “real” data, as opposed to computer-generated random numbers, and so it’s useful to forensic accountants trying to detect fraud (or so I hear). 

BenfordsLaw_800

It may seem counterintuitive (shouldn’t all digits occur with equal probability?) – but the last few minutes (start at 7:53) of this Numberphile video helped me get a grip on what’s going on:

Wednesday, January 9, 2013

Catenary out the window

I hung a rope between two trees for my kids to play on the other day, and when I looked out the window later I saw a catenary graph. 

1-DSC04143

The catenary is a curve describing the shape of a in idealized hanging chain, and looks a lot like a parabola (enough so to trick Galileo).  Its equation was determined independently by three people in response to a contest challenge.

I like to think of it as the sum of two exponential functions:
Here’s y = ex and y = e-x
1-Fullscreen capture 192013 95234 AM2-Fullscreen capture 192013 95250 AM
…and their sum
3-Fullscreen capture 192013 95315 AM