Re: Why the FOIL Method is a failed method

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On 9/18/11, David Ginsburg wrote:
> The case for math teachers forgoing catchy but confusing
> mnemonics like FOIL, and instead stress the mathematics
> behind various concepts and procedures.

I never went through the FOIL method except to explain what it
was after covering the "handshake method" (and then only
because so many in class either already knew "FOIL" or had
heard about it and wanted to know that it was). The handshake
method is simply where each term in one polynomial "shakes
hands" (i.e. is multiplied by) with each term in the other
polynomial. Also, once P and Q shake hands, they don't shake
again, which you can emphasize by saying the rule as "each
term in the first polynomial initiates a handshake with each
term in the second polynomial".

There are two ways I like to justify this at the high school
level, one that could be classified as a left-brain method and
the other that could be classified as a right-brain method.

The left-brain method is to use the distributive law (the
ellipses below don't refer to infinite continuation):

(a + b + c + ...) * (A + B + C + ...)

= (a + b + c + ...)A + (a + b + c + ...)B + (a + b + c + ...)C
+ ...

= (aA + bA + cA + ...) + (aB + bB + cB + ...) + ...

Notice that each pairing of a lower case letter and an upper
case letter shows up, and no additional pairings show up (e.g.
ab doesn't show up, BB doesn't show up, etc.).

The right-brain method is to draw a rectangle and divide the
length up into sublengths a, b, c, ... and divide the width up
into subwidths A, B, C, ... and notice that the area of the
rectangle is (a + b + c + ...) * (A + B + C + ...), and that
this area is made up of subareas aA, aB, Ac, ..., bA, bB,
cB, ...

Again, it is clear from an appropriate picture that each
pairing of a lower case letter and an upper case letter shows
up, and no additional pairings show up.

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