FOIL method. Good or bad?

Steven G

Steven G

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Can someone explain to me the benefit of learning the FOIL method, if any?
The FOIL method is restrictive to just multiplying two factors that each has two terms. Students still need to learn how to multiply factors that don't have two terms each. Since this last procedure also includes how to multiply two factors each with two terms, then I feel that teachers are showing an extra rule when they show the FOIL method.

Having students learn an extra rule when not needed is not beneficial to the students. They have enough to learn!
 
FOIL is great for outdoor cooking. In mathematics it has no place.
 
Since my students typically have already learned FOIL (or at least recall the name), I have to mention it; but I do so after teaching the full method, which I just call "each by each" (which is surely at least as memorable, and easier to carry out). Then I just point out that FOIL is nothing more than a way to describe the results that we got more easily, which happen to be in exactly the same order.

But I'd be quite happy if no one taught it. And I refuse to ever use it as a verb.

Well, it does have one advantage: Sometimes I need to remind a student I didn't teach, that they need to distribute, and if I just say, "Remember FOIL?" they usually know what I mean. Somehow, "Just distribute" or "Expand" isn't as familiar to them.
 
Foil is used for indoor or outdoor cooking. F.O.I.L. I recommend each day for the rest of my life
and the students' lives and the whole mathematical community, necessarily taken together with
the teaching of multiplying of two or more general polynomials. F.O.I.L. is another handy
mnemonic/tool, and it can help ease into the multiplication of different polynomials. You don't
have to see this as a competition of methods. I see them complementing each other. It's good.
 
I always have taught the F.O.I.L method for expanding. I find it useful to refer back to when teaching students how to factorise trinomials - how and why you have to turn the middle term into two terms. Then I go on to the quicker method for factorising, what I call the cross method.
 
Jomo said:
Can someone explain to me the benefit of learning the FOIL method, if any? … Having students learn an extra rule when not needed is not beneficial to the students …
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'FOIL' is a memory device, Jomo. One benefit is: giving beginning students a way to recall/identify the four multiplications needed, when using double distribution to write the product of two binomials.

?
 
Otis said:
'FOIL' is a memory device, Jomo. One benefit is: giving beginning students a way to recall/identify the four multiplications needed, when using double distribution to write the product of two binomials.

?
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I like your answer. I need to give this some thought.
 
I have never taught FOIL (& never will); it's far too restrictive! I have always taught: Multiply every term in the second bracket by each term in the first bracket in turn; then gather like terms. Doesn't matter how may terms are in either bracket then!
(Just my tuppenceworth. ?)
 
The Highlander said:
I have never taught FOIL ... it's far too restrictive!
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Good morning. Is that statement akin to claiming 'PEMDAS' is too restrictive because it doesn't cover all grouping symbols or situations?

I fully agree that students ought to understand multiple distributions by the time they reach the back end of beginning algebra. So, I don't have an issue with providing the 'FOIL' mnemonic upfront, when students first move beyond a(b+c).

I understand school districts vary (I once had a boss tell me to not discuss any special factoring patterns until later in Beginning Algebra II), but I believe in exposing the brain to layered perspectives over time, allowing it to reinforce (or dissolve) encodings on its own. In other words, I believe that generalizations are integrated sooner, when consistent, associated patterns have already been firmly established.

I still have memories of thinking in terms of FOIL, in beginning algebra, even after doing multiple-distributions. I don't have a clear memory as to when I stopped doing that, but it happened on its own, naturally.

No harm, no foul. :)

[imath]\;[/imath]
 
Otis said:
'FOIL' is a memory device, Jomo. One benefit is: giving beginning students a way to recall/identify the four multiplications needed, when using double distribution to write the product of two binomials.

?
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I agree with Otis here. When I first learn binomial expansion, I had a difficult time remembering that 4 multiplications are needed, and only squared the inside terms. For me, FOIL reminds me to "watch out!" there are more than 2 multiplications involved when performing binomial expansion. I've never actually spelt out the pneumonic "First Out In Last". Once I do enough expansion and it's become natural, FOIL just faded away. I think it has the benefit for beginners, and at some point, it will stop. Moreover, it's catchy, that's why people remember it. As Otis said, no harm no foul.
:)
 
Otis said:
Good morning. Is that statement akin to claiming 'PEMDAS' is too restrictive because it doesn't cover all grouping symbols or situations?

I fully agree that students ought to understand multiple distributions by the time they reach the back end of beginning algebra. So, I don't have an issue with providing the 'FOIL' mnemonic upfront, when students first move beyond a(b+c).

I understand school districts vary (I once had a boss tell me to not discuss any special factoring patterns until later in Beginning Algebra II), but I believe in exposing the brain to layered perspectives over time, allowing it to reinforce (or dissolve) encodings on its own. In other words, I believe that generalizations are integrated sooner, when consistent, associated patterns have already been firmly established.

I still have memories of thinking in terms of FOIL, in beginning algebra, even after doing multiple-distributions. I don't have a clear memory as to when I stopped doing that, but it happened on its own, naturally.

No harm, no foul. :)

[imath]\;[/imath]
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Sure, chacun à son goût. ?
.
 
I agree with The Highlander as FOIL is far too restrictive.
Student need to learn that if they know that a(b+c) = ab +ac, then they could use that fact to handle (a+b)(c+d).
(a+b)(c+d) = (a+b)c + (a+b)d = ac+bc + ad + bd, thinking initially that (a+b) is just one term and use the fact that they know a(b+c) = ab +ac. That is how I teach the FOIL (the ability to completely understand rather than just saying this is the formula and move on).

This thinking come up again and again. We know that (a+b)^2 = a^2 + 2ab + b^2. We can use that to compute (a-b)^2 = (a + (-b))^2 = a^2 + 2a(-b) + (-b)^2 = a^2 - 2ab +b^2. Of course you only do this once to get the formula. Same pattern goes for computing the derivative of (f(x) + g(x) + h(x)).

You need to get the students to think and see things clearly. Sure some students can't do that and either they memorize that (a-b)^2 = .... or that multiply it out. This is not teaching to the better students, as it does not take that much time to derive the above formulas but it does show some students the beauty in math. Math is not a boring dry subject but rather it is a beautiful subject and this needs to be shown to students.

My daughter is in 11th grade and her teacher, I guess someone who has a degree in math education, told her that she is very good in math and should probably pursue a career in computer science. I do not have a problem with the computer science comment but what about a career in Mathematics or Physics. This country, America, never pushes these degrees.
 
Steven G said:
FOIL is far too restrictive
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So are stepping stones across a very muddy parking lot, if it's left like that. (Great for baby steps; not so good for riding bicycles.) The FOIL mnumonic wasn't designed as the final word. It's simply a memory device for very beginning students (some of whom won't see a trinomial factor until the next course).

Steven G said:
Math is not a boring dry subject but rather it is a beautiful subject and this needs to be shown to students.
Click to expand...
Agree. I think exposing students to multiple perspectives (over time versus dumping it all at once) is a good way to show that.

Did you tell your daughter before she was three that Santa isn't real. ;)

[imath]\;[/imath]
 
AvgStudent said:
I agree with Otis here. When I first learn binomial expansion, I had a difficult time remembering that 4 multiplications are needed, and only squared the inside terms.
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There is nothing different at all initially between multiplying out a(b+c) and (a+b)(c+d). If you know the 1st one then you can do the 2nd one. All it takes is to realize that (a+b) can be thought of as one value, like f.

One should build on what they know! If you remember how to expand a(b+c), then you should know how to expand (a+b)(c+d).

Once you see that it is the same thing, then you have it for life. Remembering to multiply each term in a bracket by each term in the 2nd bracket can easily be forgotten, but a(b+c) = ab+ac has been taught to us for so long that you problem will never forget it.
 
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