Archive for: May 2007

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I know I should be working on MathMojo and writing more books, getting podcasts of Eating Math for Breakfast out, writing the Math Mojo Monthly (”Comes out Quarterly, Mostly”) and making some videos about how to make your own abax, but I got a great distraction a few minutes ago.

My cousin, Jayne, who is about my age (only younger) (I am fifty) sent me an e-mail with a link to a website that was like a time machine. If you are a boomer like us, you may want to check out When Life was in Black and White.

Go ahead and check it out now. I’ll wait. (You kiddies can go back to your X-boxes while Mommy and Daddy go back in time for awhile.)

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This post is concerned with a very interesting problem, called “The Traveler’s Dilemma.” There is a very good article about it, written by it’s creator, Professor Kaushik Basu, in the June, 2007 issue of the Scientific American. The article begins:

“When playing this simple game, people consistently reject the rational choice. In fact, by acting illogically, they end up reaping a larger reward–an outcome that demands a new kind of formal reasoning.”

Please read the article before you read this post.

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Just had a great evening mulching and edging our apple tree. I ran out of mulch and ground paper, but I can get more tomorrow.

As I worked, I used a catch (see last post) to collect my thoughts. The one I used is my favorite - it’s a mnemonic device. I used the rhyming peg method, (”one is the sun, two is a shoe, three is a tree,” etc.) and only needed to get to “door” for the four Ideas I had been mulling over.

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I was out mulching our apple tree in the front yard, when some thoughts came to me. One of them was that I can see how gardening can be such great contemplative avocation.

I’m not a gardener, and have never been very good at things like that, but I can weed, I can mulch, and I can mow the lawn. All of which are sort of brain-dead activities which can lead you into a trance-like state.

If you had been contemplating something just before you get into that state, or have had something preying on your mind, it is easy to let thoughts about it come to you while you are in that state. They can wash over you in a pleasant way. The trick is to catch them, and get them recorded somehow, so you can recall them later, and work them into something and take advantage of them.

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I am thinking about an example from a GRE (graduate record exam) book
that was shown to me.
I think it was "Which is greater, x²+y² or (x+y)²?

Here is the poop on how to think about examples like that. When in doubt – substitute
(if you can) for whole numbers. (In the original post, I had written real numbers instead of whole numbers. See the comment below about this by astute reader Randall Jones for important information about the difference that makes in this equation.)

So, try, say,  "Which is greater, 5²+3² or
(5+3)²?"
In the first case, 5² = 25 and 3² = 9, so it would be
25+9, which equals 34.
In the second case, you would first do the 5+3 (because parenthesis come first
in the order of operations) and get 8. Then you would square that, and get
64, which is clearly greater than 34.
Therefore  (5+3)²  is greater than 5²+3².

For an easy substitution you can do in your head in seconds, substitute 1s for x and for y:
= x²+y² or (x+y)²
= 1+1 or 2 ²
= 2 or 4

What if the example had been a bit different, though? What if it had been:
"Which is greater, x²*y² or (x*y)² (using multiplication instead
of addition)?

This article is continued at Mathmojo.com.

Well, believe it or not, using the “math” they taught you in school, you can “prove” that is true.

Part of the math curriculum of schools is estimating, or rounding up. This is a legitimate and important concept, when it is taught by competent and interested teachers. Man, is that a big “when.”

This brings us to what is one of my main peeves about traditional math-ed. They never mention the consequences and “stuff” concerning what they are teaching you. They teach you how to estimate, and even sometimes what estimation is good for, but they never tell you the interesting stuff about it. In this case, it is the why not.

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I recently read a post on another blog concerning the two most important elements that children should master in math in order to succeed. The author suggests that basic skill with multiplication and basic mastery of fractions are the two essentials.

I am of the same opinion. The author also thinks that memorization and drills are the best way. On that, I’m not so sure. Yes and no.

Here’s my take:

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