Shikaku

Shikaku (四角), which means four corners in Japanese, is a fun puzzle game where you fill a puzzle board with quadrilaterals (squares and rectangles) using the clues on the board. Each quadrilateral must contain only 1 number and have the area given by that number.

It's a fun little game to help students connect area with multiplication (a quadrilateral with area 6 can be 1x6, 2x3, 3x2, or 6x1), discover properties of prime and composite numbers (all prime numbers n can only be drawn with 1xn, or nx1 and no prime number can be drawn as a square), and reasoning (some quadrilaterals will have to be redrawn as you encounter limitations).

Here's a video of me playing this Shikaku puzzle.



You can find more puzzles on the Nikoli Shikaku page.

Note: Shikaku was created by the same folks who came up with Sudoku.

(via Breedeen Murray)

Speeding Up Audio Podcasts with Winamp and PaceMaker

Here's a time-saving tip for those of you Winamp users who follow more podcasts than you have time to digest. There's a plugin called PaceMaker that allows you to speed up and slow down MP3s without changing pitch. This is useful if you want to listen to podcasts without feeling like you're listening to Alvin and Chipmunks. I first learned of the PaceMaker plugin in college when I was learning to play the guitar. It was helpful to be able to slow down songs so I could get some practice improving finger coordination as I strummed new chords along with the song.

Here's what you need to use it:

1. If you don't have it, grab a copy of Winamp from the Winamp download page. I usually skip the MP3 bundle that comes with the download.
2. Install Winamp.
3. Once Winamp is installed, you'll need to visit the PaceMaker Winamp plugin page and grab a copy of the installer.
4. Install PaceMaker.
5. To enable the PaceMaker plugin, go to "Tools" select the "Preferences" menu item.

6. Scroll down on the left and click on "DSP/Effect" then select the "PaceMaker tempo controller" option in the right pane.

7. Adjust the tempo to speed up the audio podcast. Then just play your MP3.

I find that for most podcasts +40% is a comfortable pace. This is true if the podcast is on a familiar subject or if it's easy to follow. Some podcasts can be played at even higher tempos but at higher tempos I spend more effort listening carefully to words than paying attention to the discussion.

There's some simple yet useful math in here, but that'll be another post.

3D Video of Asteroid Vesta

A stereoscopic anaglyph 3D video created by NASA's Dawn Spacecraft.

I think a commenter on Slashdot said it best:

Each time NASA releases images from some distant planet or asteroid, I'm floored. The number of things that have to go right, that have to not fail, millions of miles away, is immense. Kudos to the scientists and engineers who worked on imaging Vesta. Fantastic results!

(via Slashdot)

Mathy Conversations with Students

I use many non-verbal cues in my classroom. Often, I'll use hand signals or facial expressions to convey my requests/commands to students quietly. The signals and expressions help me avoid interrupting the class just to address a single student when other students are already working.

Today, I gave a student a judgmental look to hint that he wasn't doing what he was supposed to be doing. Usually students just go back to work, but today he decided to point it out.

Jimmy: You always give me that look!
Me feigning innocence: What look?
Jimmy: The one where your eyebrows are like this (student signs the letter v with his hands).
Me acting clueless while giving a subtler version of the expression I gave him: Huh?
Jimmy: Yeah. Your eyebrows look like a V. It's like they're perpendicular.
Me giving my best perpendicular brows: That's impossible! I'll even take a picture and show you that they aren't perpendicular! Eyebrows can't be perpendicular.
Jill: Some girls will draw 90 degree eyebrows using a Sharpie!
Me signing a V on my forehead: Like a 90 degree unibrow?
Jill: No, they draw sharp corners for eyebrows like this (she signals a circumflex).
Me: Are you saying I have a continuous function for an eyebrow?
Jimmy: No, you don't have a unibrow, it's discontinuous.
Jill laughs: Yea. You have a removable discontinuity!
Me laughing at their insight: Because the limit exists! My eyebrows meet from the left and right and it's undefined in the middle.
Me sensing more students getting off-task: Alrighty, back to work!

A spontaneous opportunity like this doesn't come up often which is why I'm mad I missed an opportunity for review here. I should have asked "what would my eyebrows have to look like for there to be a non-removable discontinuity?" instead of hurrying the kids back to work. Maybe someone would have responded with "you just raise one eyebrow so they don't meet in the middle." I need to be more open to opportunities like this one. I love it when students can apply a concept learned in class in unexpected ways. I love the fact that I get to have these conversations at all. I love my job!

A Function and its Inverse (Office Supply Edition)

A visual representation for the notation of a function and its inverse.

Doesn't work as well if you have the stapler on the outside and the staple remover on the inside. Would've been more useful if instead of x, there were a stack of papers. Maybe if I can find some time or if someone has the time...

(via MathFail)

PS: If you like this, also check out f(a bag of skittles) over at Sweeney Math.

Kaprekar's Constant (6174) in GeoGebra

While doing a little browsing, I came across Kaprekar's Constant. I remember reading about it a few years ago but I never did anything with it. I finally decided to play with it a little using GeoGebra. Here's a description from Wikipedia:

6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is notable for the following property:

1. Take any four-digit number, using at least two different digits. (Leading zeros are allowed.)
2. Arrange the digits in ascending and then in descending order to get two four-digit numbers, adding leading zeros if necessary.
3. Subtract the smaller number from the bigger number.
4. Go back to step 2.

The above process, known as Kaprekar's routine, will always reach its fixed point, 6174, in at most 7 iterations. Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 3524:

1. 5432 – 2345 = 3087
2. 8730 – 0378 = 8352
3. 8532 – 2358 = 6174

The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4:

Here's a screenshot of the GeoGebra applet:

If you'd like a copy just click on "File" then "Save As" and save a copy on your computer. Click on the "Play" button on the bottom left of the applet to begin the animation.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com