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### 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

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.
```
Associated Topics:
High School Basic Algebra
High School Polynomials
Middle School Algebra

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