February 14, 2012

Small Satori: (a+b)²=a²+2ab+b² - But Why?

I was taught it until I knew it well enough to parrot it, but I never knew why. Until now.

HT: The American Scene

Posted by gerardvanderleun at February 14, 2012 9:55 PM
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"It is impossible to speak in such a way that you cannot be misunderstood." -- Karl Popper N.B.: Comments are moderated and may not appear immediately. Comments that exceed the obscenity or stupidity limits will be either edited or expunged.

I too was taught to parrot it back and, given my mathematical ability, a parrot would have been better at it.

Would it have killed my math teacher to have done this, oh, 35 years ago now? Perhaps he didn't know it either?

Posted by: Tom at February 15, 2012 1:39 AM

It works for (a+b)^3 also if you make a cube. After that you're stuck in hyperspace.

Posted by: bob at February 15, 2012 4:15 AM

He just presented a geometric version of the equation that does, indeed, make sense. It's more important, however, to understand how to extend the equation as that leads to the simplification of all formulae:

(a+b)2 = (a+b)*(a+b)

You would then use the (associative?) principal:

(a+b)*(a+b) = (a*a)+(b*b)+(a*b)+(a*b)
= a2 + b2 + 2(a*b)

Or something like that. It's been years since I had to do that, but I would not have been able to use these methods to simplify other equations far more complex than this one if the teacher had just shown me how to count squares.

Posted by: Daniel at February 15, 2012 4:31 AM

It's more than "counting squares". It shows a concrete relationship between the formula and reality. Those of us with a bit of a right-brain need this.

Posted by: Casca at February 15, 2012 7:24 AM

Really? I've never had trouble with math. I'm with Tom though,35 years ago this short video would have been helpful.

Posted by: Dave in PB at February 15, 2012 8:39 AM

That cute little long-haired hippy man with the funny accent seems pretty proud of himself on that one ...

Sure, his illustrative technique works for (a+b)^2 and even (a+b)^3. However, it does not work for powers greater than three. In cases greater than three, that operational method you/we learned years ago still works where Mr. Long-Hair's method does not.

Posted by: edaddy at February 15, 2012 10:03 AM

Any math teacher who DOESN'T show this to his students needs a serious ass-kickin'. Geometric intuition needs training, and is essential for lots of scientists and engineers.

Powers past the cube need not go into hyperspace, but do require a good half-hour explanation of Pascal's triangle, counting combinations, and the Binomial Theorem. All of which is readily accessible to any seventh-grader exposed to basic algebra.

Posted by: Mike Anderson at February 15, 2012 10:10 AM

I think that most of the troubles with math come from NOT being taught to read the formula as a sentence. And expand them rather than using the contracted form (a+b)2.

Posted by: Peccable at February 15, 2012 1:09 PM