Why I Hate FOIL

Let's use our imagination a bit. Picture yourself in math class (Algebra I to be exact), minding your own business, having fun playing with the axioms (aka rules) of algebra, and then one day your teacher drops this bomb on you:

"Expand (x+3)(x-1)"

And you might be thinking, "woah now, where did come from?"

It makes sense that this would shock you. You were just getting used to the idea of expanding 3(x-1), and you probably would have been fine with x+3(x-1), but (x+3)(x-1) is a foreign idea all together.

Well, before you have much time to think about it on your own and discover anything interesting, your teacher will probably tell you that even though you don't know how to solve it now, there is a "super helpful", magical technique that will help you…

For those of you lucky enough never to have heard of FOIL, I will explain. FOIL stands for First Outside Inside Last and is a common mnemonic device used to confuse children about a fairly easy concept.

If you remember the distributive property a(b+c) = ab+ac, then it might seem odd that all of a sudden we put two grouping next to each other and now we are doing something "new".
But is FOIL really new?

The answer is of course no, what we are actually doing is just a short cut for the distributive property. And if you were aloud to try and solve it before being told what to do, you might actually have figured that out. For example, if we have (a+b)(c+d), we could distribute (a+b) as if it was a whole quantity. So (a+b)(c+d) = c(a+b) + d(a+b) and then we distribute again and get ac + bc + ad + bd.

To me this seems much simpler than having to learn a mnemonic device, and remember how to draw our "rainbow lines" and remember where to put a plus and a minus, and so son and so forth. We merely follow a simple rules we already know.

Another useful reason not to teach FOIL is because in only works for expressions similar to (a+b)(c+d). But what about expressions that look like (a+b)(c+d+e), or even (a-b+c)(d+e)(f-g-h+i+j)(k-l+p)? You can't use FOIL for these, but of course, you use the distributive property.

So please, if you are a math teacher, the next time you have a chance to teach FOIL… don't. Spare your students the confusion and teach them what is really going on. FOIL might be quicker, but math isn't about the destination, it's about the journey.


I totally agree.  I was just about to write a blog entry about FOIL and searched for it first.  FOIL set me back significantly back in Algebra1,  for the reasons you stated (a new rule that didn't make any sense).     
This is how I had planned to describe it, FWIW:
=     (a+b)(H)      (let H = c+d)
=    (aH + bH)      (distribute the H)
=   a(c + d) + b(c + d)    (replace H with c+d)
=  ac + ad + bc + bd       (distribute a and b)


Devin F.

Tutor on the cutting edge of Math Education

5+ hours
if (isMyPost) { }