Why FOIL?

Date: 11/21/2002 at 20:04:30
From: Stephanie
Subject: Why foil?

Hi - 

I am getting ready to student teach and am preparing lesson plans. I 
am doing one on FOILing, i.e. (2x + y)(x +2y) first, inside, outside, 
last. I'm wondering how to explain to the class the importance of 
doing this. I realize that the goal is to get them to be able to later 
factor expressions like x^2 + 4x + 3 (and I'm not sure the 
significance of being able to do that either except for help graphing 
and finding zeros). But the students want to know the reason for doing 
FOILing now. Please help me get them interested! 

Thanks, 
Stephanie

Date: 11/21/2002 at 23:18:13
From: Doctor Ian
Subject: Re: Why foil?

Hi Stephanie,

If you want to do your students a favor, skip FOIL altogether, and
make sure they understand the distributive property:

   When FOIL Fails
   http://mathforum.org/library/drmath/view/53241.html 

To answer your question, the reason you go from a nice, tidy
representation like 

  (2x + y)(x + 2y) 

to a messier one like 

  2x^2 + 5xy + 2y^2 

is usually so that you can add some other terms. Then you convert back 
to the tidy representation again, e.g., 

    (x + 2)(x - 3) = 24

       x^2 - x - 6 = 24

      x^2 - x - 30 = 0

    (x + 5)(x - 6) = 0

Standard polynomial form makes addition easy, and most other things
difficult. Factored form makes addition impossible, but lots of other
things easy.  

Part of the confusion that surrounds FOIL is treating it as if it's
something new or special, when it's not. It's just the typical use of
the distributive property, 

   Distributive Property, Illustrated
   http://mathforum.org/library/drmath/view/52842.html 

i.e., you un-factor so that you can combine like terms, and then turn
around and factor again.  

Does this make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/21/2002 at 23:28:53
From: Doctor Peterson
Subject: Re: Why foil?

Hi, Stephanie.

Let me first state that I dislike the FOIL approach. It's a crutch, 
and sometimes crutches are useful, but I'd much rather teach a method 
that is not restricted to multiplying two binomials. Here is what I 
have said in this area:

   Explaining Algebra Concepts and FOIL
   http://mathforum.org/library/drmath/view/52844.html 

But that wasn't your question; you want to know how to motivate 
students to care; perhaps you want to motivate yourself as well. I 
suppose that's actually another reason to avoid FOIL: it's just 
another formulaic method that has value only as a tool for one job. 
If you're not going to be doing that particular job, you don't need 
the tool. Here are some reasons for learning to multiply polynomials, 
which is the real task that FOIL helps with only in special cases:

1. We're learning about polynomials, and to do that, we need to know 
how to do things with them. Unless you can add, multiply, and so on, 
what good are they?

2. We'll see how polynomials are just an extension of numbers, because 
the method I teach looks exactly like multiplying two numbers. One of 
the key principles in math is to relate new ideas to old ones, and 
look for the familiar in the unfamiliar. As you learn this, be looking 
for connections to what you already know, and you will get some 
interesting surprises - both in things that look familiar, and aspects 
that are different from what you've seen before.

3. One side effect of that connection is that you will gain a better 
understanding of the arithmetic you've already learned, by seeing it 
in a different context.

4. It's an important application of the distributive property, so as 
you learn it you will be getting a stronger understanding of how 
numbers work.

5. Working with polynomials is an important aspect of algebra; if you 
go on to any kind of further math, you will have to do this task of 
multiplying over and over. It's not an end in itself, but an important 
basic tool of algebra.

6. There are many other areas of life where you will need to find a 
way to organize a task like this that involves finding all the ways to 
combine things (in this case, multiplying every term of one polynomial 
by every term of the other). Polynomial multiplication serves as a 
model that you can use elsewhere in problem solving. In fact, FOIL is 
an alternative method for the same thing, and makes a good 
demonstration of how there can be two ways to accomplish the task.

Whatever method you teach, it will probably help if you start by 
showing the task the hard way (applying the distributive property over 
and over), and then say that there is a way to organize the work so 
that it is so easy it becomes almost automatic. Then it becomes 
self-motivating - at least if they think the task of multiplying has 
any value in the first place. Perhaps you can look for a realistic 
problem in your text in which this task plays a part, to start the 
whole discussion.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/25/2002 at 23:33:40
From: Stephanie
Subject: Thank you (Why foil?)

Hi - 

Thanks for responding. I really appreciate it and your response 
triggered some thoughts to improve my lesson. Thanks.