I ban FOIL
We should teach factoring. But we should also make certain that the stuff we teach before factoring supports the method(s) of factoring that we will teach. Today we’ll look at multiplication of terms with like bases, and at polynomial multiplication. The next post will address the first part of how I teach students to factor.
Nothing amazing here, but beginning algebra students like to make two mistakes:
- they multiply the bases
- they multiply the exponents
There is nothing amazing in how I deal with this:
- emphasize ‘spreading out’ the multiplication, eg . At a certain stage they will just know the rule, but there should be no rush to that stage.
- Every student learns that eight times four is thirty-two, not sixty-four. So forget your rules? Think about this:
(FOIL/NOFOIL below the fold —>)
I see merit in several methods of multiplying polynomials.
My personal favorite is long multiplication. It looks just look the operation from arithmetic, with minor adjustment. Why not fall back on existing, solid knowledge? (I sometimes use this method)
Here are examples of long multiplication (arithmetic, left, and polynomial, right) from
Eric Weisstein’s Wolfram’s MathWorld.
However, while my students see long multiplication, we emphasize distribution, since our method of trinomial factoring rests on reversing the distribution process. The intermediate steps in multiplying by distributing will also be the intermediate steps we encounter in factoring.
Binomial times binomial:
Applying the distributive property to polynomials may be a bit novel, but kids don’t find it particularly challenging.
So, why, oh why, does so much of this country teach a mnemonic for multiplying binomial times binomial? and not touch long multiplication or distribution?
This is FOIL for (x – 8)(3x + 16):
F stands for First (first two terms)
O stands for Outer (outer two terms)
I stands for Inner (inner two terms)
L stands for Last (last two terms)
And then O and I need to be combined.
Look, I’m all for kids remembering things, and if a mnemonic helps, great. But here? No way! FOIL only works for binomials, a subset (albeit very important) of polynomial multiplication. Far worse, however, we can appeal to our knowledge of arithmetic (through long multiplication) or our knowledge of fundamental properties (through multiple distribution). FOIL appeals to nothing but cutesiness.
So I ban it. And the kids who already knew it fuss. And I explain why we don’t use it. And the kids who never knew it are curious how it goes, watch their friends do it, and decide to stay away. After a week I still have one or two grumbling at me under their breath, but they can handle other methods.
And a month or so later, when we begin factoring, the heavy emphasis on distribution pays off.