Re: 64 - is this the only number that is both a sq and a cube?

From: Dave Rusin (rusin_at_vesuvius.math.niu.edu)
Date: 09/21/04 

Date: 21 Sep 2004 17:15:06 GMT

In article <20040920213949.04433.00001144@mb-m05.aol.com>,
>>Hardly any of the students I see,
>>even the better ones, know (x+y)^2 = x^2+2xy+y^2 without multiplying
>>out the factors by "FOIL". These are all students who have been
>>through two years of high-school algebra, or more.
>
>You think memorizing (x+y)^2 = x^2+2xy+y^2 is more important than
>knowing FOIL? Isn't it better to know an algorithm than a fact?

Um, maybe. Be careful how you phrase that. The previous poster wanted the
students to know that (x+y)^2 = x^2+2xy+y^2 (for all real x,y), thinking
of this as an application of the binomial theorem. You want the students
to know that (a+b)(c+d)=ac+ad+bc+bd, and then to deduce the previous
identity as a special case; except you seem to prefer that the students
memorize the _algorithm_ S:=0; S:=S+ac; S:=S+ad; etc., which I would
claim is a different (probably weaker) level of understanding.

Personally I would prefer the students know two "facts": the distributive
and commutative laws, so that they don't have to memorize any of these
special cases. I suppose the thinking is that the FOIL instance is so
common it's worth spelling out specially, but then students start to
think that expanding e.g. (a+b+c)(d+f) requires a whole 'nother "fact".

Sigh. Well, I suppose anything is better than (x+y)^2=x^2+y^2.
My students live in a field of characteristic 2.

> Because I was never good at memorizing things, the only thing I
> memorized about trigonometry was sin(0)=0 from which I would derive
> the needed trig identity.

Yes, I'm the same way. I never did memorize the little formulas for
the equations for parabolas (there's a "4p" or something in there); it
was always easier for me to recreate the equation using the definition
of a parabola (the two-distances-equal one, not the slice-a-cone one).

dave

PS - Explanation for non-US readers: "FOIL" is a mnemonic taught in US
schools for the last few decades to cover the expansion of (a+b)(c+d). There
are four terms to add: the products of "First, Outer, Inner, Last" pairs.

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