“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.
Tuesday, November 26, 2013
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
//FIRST, THE SETUP
- Lbl B
(random number between 4 and 8 for the pig’s x-coordinate)
- 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).
(starting values for plotting the bird path)
(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)
- Lbl A
- 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.)
(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)
- Goto B
(This forms a hexagon for the pig’s head)
- Line (P+0.4,0.1,P+0.4,-0.7)
(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)
Wednesday, November 20, 2013
Monday, November 18, 2013
Leafing through a magazine the other day, I happened across this page:
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
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?
Wednesday, October 16, 2013
Bonus points if you collect some data to find the equation of the first pancake's flight:
Sunday, October 13, 2013
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.
Tuesday, October 8, 2013
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!
Wednesday, August 28, 2013
Monday, July 15, 2013
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
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
These 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
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
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?
Wednesday, May 8, 2013
Tuesday, April 9, 2013
Admittedly, mine is the weakest of the lot, but it was nice to be included.
Monday, March 25, 2013
I found this shell on the beach in Saipan last year.
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
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):
Wednesday, February 27, 2013
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:
Wednesday, February 13, 2013
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:
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
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”:
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:
It turned out pretty cool, don’t you think?
But I hope you’re not in a rush to get yours – the hand quilting takes a long time.
Friday, February 1, 2013
It did glitch on me a couple of times and require a page reload.
Here's a quick example:
Tuesday, January 29, 2013
Monday, January 21, 2013
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).
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
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.
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
…and their sum