Wednesday, September 28, 2011

Fun with Moebius Noodles

About 2 weeks ago, I stumbled across the Moebius Noodles group referenced on the Let's Play Math blog. Inspired by the idea of sharing cool math with my kids, we spent last Wednesday playing with Moebius noodles. For those of you who don't know, a Moebius noodle (also known as a Moebius band or strip) is a loop of a flat material (usually paper) that is twisted once before the ends are joined. Here, B demonstrates one.

Both P and E's fine motor skills are at a level where they can make their own, so they pasted together both regular loops and Moebius noodles. I had them each draw a line down the middle of "only one side" of each loop. One fascinating thing about Moebius noodles is that they only have one side. We then experimented with cutting our loops in half lengthwise. With a regular loop, of course, you get 2 loops. With a Moebius noodle, well... try it yourself if you've never done it. It's really fun, and surprising the first time.

Since we were still enjoying ourselves and it wasn't lunch time, I thought I'd try out another idea I stumbled upon. The gist is that you take two loops, glue them together, and then cut them both in half lengthwise. First, I took two regular loops and glued them at right angles. It looked like the beginnings of a chain you might put on a Christmas tree, only the connection was on the outside. I cut it in half down the middle of each loop, through the intersection point, and ended up with this:

P thought this was interesting, and tried it herself (by this time, E was busy attaching yellow loops to a piece of paper to make a "rocket", which seemed valuable in its own way). Meanwhile, I tried what happened when one of the loops is a Moebius noodle and one is a regular loop, which turns out to be the same as when both are regular loops. Finally, I tried both of the possible ways of joining two Moebius noodles: one when the loops are twisted the same direction, and one when they're twisted in opposite directions. I found it completely mind-bending! When the loops are twisted in the same direction, you get this:

But when they're twisted in opposite directions, this is what you get. (P tried this one as well, though I had to help her make the Moebius noodles so that they'd be even enough to cut in half neatly).

An instant "I LOVE MATH" logo!

Monday, September 12, 2011

Breakfast Math

Sometimes, marketing opens the door to a great educational experience.

Our box of Raisin Bran proclaimed, in large letters, "$100 CASH CARD INSIDE". In smaller letters, one realizes that in truth, "you could find up to a..." If you read the fine print, you find that your odds of finding that $100 cash card are 1 in 24,800. There are other prizes, though: a $50 cash card (1 in 20,667), a $25 cash card (1 in 17,714), a $10 cash card (1 in 15,500), and a $5 cash card (1 in 12,400). So I read these statistics to the kids and solicited guesses on just how many boxes of Raisin Bran one would have to buy to be likely to get any cash card at all. Then I hauled out my calculator and did the math: the total odds, obtained by adding the odds of each value of cash card, are 1 in 3444. If you bought 3444 boxes, though, you still wouldn't be guaranteed to get a cash card (though you'd have better than even odds). We discussed how long it might take to eat 3444 boxes of cereal (about a decade, if the whole family only ate Raisin Bran for breakfast every single day - we could probably polish off a box a day between us). And if you bought 3444 boxes of cereal, you'd most likely only get one cash card, which would most likely be a $5 cash card.

P and E - mostly E - have invented a useful number, the gi (hard g, rhymes with pea). One gi is generally defined as "a number larger than the one you were just talking about by a considerable margin". So we find it helpful to suggest, "even if you tried gi times, you wouldn't be able to throw an apple all the way up to the moon," or "do I have to ask you gi times to clean your room?" E announced, "If you bought a gi of cereal boxes, you'd probably only find 1000 cash cards. That's how big a gi is."

Okay, I'm curious. How large does gi have to be in order for 1000 to be the most likely number of cash cards when the odds of finding one is 1 in 3444? I think all the probability and statistics I was taught at Caltech has fallen out of my head. Anyone care to remind me?

Saturday, September 3, 2011


I want to post replies to comments that people have left. But every time I try, I get the following message, repeatedly:

"Your current account (my@email.address) does not have access to view this page."

What? I can't comment on my own blog? Does anyone who knows more about arguing with Blogger know how to fix this?

Reading Strategies

I had 2 good teaching ideas for E's reading lessons this week. One of them actually worked.

I mentioned that he was finding it hard to distinguish "b" and "d". So on Monday morning, having gleaned this idea from the Sonlight forums, I wrote the word "bed" in large lowercase letters. Then, I drew a stick figure lying horizontally on top of the word: "bed" looks a bit like a bed, with bedposts on either end. However, if you write "deb", there's no space for your stick figure to stretch out to sleep. So to figure out the difference, you find out if your letter would leave space for your stick figure to sleep if it was at the "B"eginning or at the en"D" of "bed". This seemed to work pretty well, and E started reading the first of his readers that contained both "b" and "d". After a few successful efforts at sounding out words, he looked up at me and said, "Do you know how I tell the difference?" I anticipated hearing something about how well my careful, clever explanation was working for him, but what he said was, "The 'b' doesn't have a tag at the bottom, and the 'd' does." In the font used in his readers, this is true. So it works reliably for him - he's had ZERO struggles since Monday morning when it comes to sounding out "b" and "d". I'm amused that he came up with something that didn't even occur to me, and it works great for him. Hopefully by the time he moves on to reading material that doesn't have that feature, he'll have internalized the other differences between the two letters.

The other great idea I had, which actually is working, was for helping E recognize and remember the word "the". Sounding it out each time was driving him nuts, and he really hated that he just had to remember that it didn't work that way. I'd just been telling him what the word was for over a week, when I had an idea. A while back, I heard a blonde joke (most of which I think are really stupid). A blonde is sitting on a bench with a book, and looking concerned, and saying, "ta-hee! ta-hee!" A brunette comes along, takes a look, and tells her, "'The', airhead!" Well, I didn't need to encumber E with the setup of the joke (both his siblings being blonde), but I told him, "When you sound out 'the', it sounds a bit like laughing: 'ta-hee!'." So now, whenever he encounters the word in his reader, he says "ta-hee...(giggle)...the!" What had been a roadblock for him is now something he finds amusing.