Talk:FOIL rule

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[edit] Frowned upon?

The article says:

It is commonly taught, but is often frowned upon because the method does not work for higher order polynomials, and thus instead of actually teaching a general method, it is an example of learning by rules instead of concepts.

Hmm. I don't know about this. While the exact method may not be applicable, it is easily extended to polynomials with more terms. Besides, how else is one supposed to learn to muliply polynomials? What could this other "general method" be that would be useful to beginning algebra students? - dcljr (talk) 4 July 2005 10:10 (UTC)

Well that is a very common sentiment. I forgot to mark down any source that mentions it, but it is certainly common. I'll see what I can do to dig one up. - Taxman Talk July 5, 2005 15:26 (UTC)
I definitely agree with the article. FOIL avoids any kind of reasoning or understanding of what's going on. Extending it is confusing, because then it's no longer FOIL. To teach a general method without using FOIL, you could, for example, first explain why a(b_1 + b_2 + \ldots + b_k) = ab_1 + \ldots + ab_k.
Then show why (a_1 + a_2)(b_1 + \ldots + b_k) = a_1b_1 + \ldots + a_2b_k. The general principle follows. TMott 22:01, 2 September 2005 (UTC)
Okay, now that I've actually started teaching an algebra class, I understand: the "general method" is simply distributing, and is basically what TMott is talking about (and what I meant when I said FOILing is "easily extended"). Although... I guess "true" distributing requires two separate steps where you first distribute one polynomial (in parentheses) over the terms of the other, then distribute again to get rid of the parentheses (hence the term "double distributive property" mentioned in the article). What I was talking about is essentially a shortcut where you see the pattern and skip the first distributing step. TMott, it seems to me you were really talking about the same thing I was. BTW, a different way of looking at it, although completely equivalent of course, is called the "vertical format" and works the way most people learned to multiply numbers bigger than 10 on paper, by first lining them up vertically. - dcljr (talk) 06:51, 3 September 2005 (UTC)

Yeah, it seems you were talking about the same thing, but missing the part about it not being FOIL anymore. But the phrase could certainly use some attribution to a reputable source, and I haven't been able to locate one. Is there a well regarded textbook regarding teaching algrebra you know of? I may be able to go to the university bookstore and look for one. Also, I couldn't find any article that actually explains polynomial multiplication. Polynomial certainly doesn't, though we do have polynomial long division. Do you guys mind creating the article with a little more detail and middle steps than outlined above? - Taxman Talk 16:51, September 3, 2005 (UTC)

[edit] Stub?

Is there really much more to add to this article? I'm kind of new to the whole wikipedia thing and I was looking for something I could expand in and find the FOIL article. It looks fairly complete to me, any comments? —Preceding unsigned comment added by Mlw1235 (talk • contribs)

What about the process of "unfoiling"... —Preceding unsigned comment added by 69.234.165.36 (talk • contribs)

I assume you mean factoring. - dcljr (talk) 17:08, 23 May 2006 (UTC)
Possibly the history of the subject? A see also section? --TeaDrinker 19:22, 23 May 2006 (UTC)

I've upgraded the rating to "Start" since this is not a stub article. Is it B-worthy? I'm tempted to say so. - grubber 18:05, 25 April 2007 (UTC)

Contra: I believe this article should also show a reverse-FOIL method, such as how to revert x^2 + 12x + 35 \, to (x + 5)(x + 7). —Supuhstar Image:Supuhstar(SupuhSmall).gif
Again, "reverse-FOIL" is better known as factorization. If you want, add a comment to the article mentioning this, but do not attempt to include the whole process of factorization here. Lunch 01:45, 15 May 2007 (UTC)

[edit] Confused

I'm seriously confused: What is this rule about? Appears to be some weird kind of algebra with a non-commutative addition where one needs to remember the order in which the terms of the folded-out product comes. Could somebody please link to it? I fail to imagine a distributive law that could feel halfway natural and still dictate an internal order between the O and I terms. –Henning Makholm 20:27, 30 April 2007 (UTC)

From WT:WPM:

Hypnotize yourself and regress back to a time in your life when words like "commutative" and "distributive" meant nothing to you. The order of the terms in FOIL is irrelevant; the point of the mnemonic is to help students remember the four terms of the result. Apparently their brains would explode if asked to understand and use the distributive law twice, especially since the pieces are not numbers but symbolic monomials. --KSmrqT 20:54, 30 April 2007 (UTC)
No, I still don't get it. If the brains in question are that prone to exploding, what good does a rote like FOIL do at all? A word like "first" or "inner" is not meaningful unless you already know that you're supposed to take products of one term from each multiplicand. Indeed, if you didn't know that, FOIL of (a+b)(c+d) might as well mean ab+ad+bc+cd, because the first two things are a and b, and so forth. And once you do know what the point it, "FOIL" gives no additional information except for the order! And won't brains start exploding anyway the first time they are expected to follow an argument that happens to expand such a multiplication in FIOL order? –Henning Makholm 01:10, 1 May 2007 (UTC)
I think you do understand, you just don't like it. Fair enough; but we're not endorsing, we're reporting. Consider a news item that says "Between May 2000 and August 2005, Brazil lost more than 132,000 square kilometers of forest—an area larger than Greece"[1]; should we infer that the source recommends deforestation? As a search of the Web will confirm, the FOIL idea is discussed at many educational sites (some training teachers) and incorporated into textbooks. Two objections are: (1) this introduces a new isolated fact to memorize rather than relying on the already-learned distributive law, and (2) it only applies to a product of binomials. I find those compelling objections, and would like to see them mentioned in the article; however, my only sources are opinions stated on Web forums, which Wikipedia asks us to avoid. --KSmrqT 04:42, 1 May 2007 (UTC)
Oh, I'm not arguing for deletion. I'm just genuinely puzzled why this can help anybody remember anything relevant at all. The words "first" and "last" do not really tell what the terms they are supposed to imply are, unless you already know what the possible terms are, and then what do you need a mnemonic for? The objections are fair and I wouldn't mind them being in the article, but I can think them up myself and thus don't really need them. What I would have liked an encyclopedia to tell me, however, is how the proponents of the rule imagine that it helps. At the moment the article describes the rule in a way that invites ridicule ("The name comes from the order of multiplying terms of the binomials", when the order is immaterial? And "The FOIL dance"???) and it is quite likely that it deserves ridicule, but I would like to see the other side of the argument, too. Whatever it is. –Henning Makholm 16:21, 1 May 2007 (UTC)
I was taught in junior high to use the FOIL rule, and later in college I learned that when you tutor students in algebra (even college students), they understand much better when you use words like "FOIL", "cross multiply", "SOH-CAH-TOA" (which I still use :), and "Lo-dee-hi minus Hi-dee-lo over Lo-Lo" (for the quotient rule of derivatives). They may make pure mathematicians cringe, but these are mneumonics used by non-mathematicians to learn the rules. - grubber 15:35, 1 May 2007 (UTC)
I'm asking what it is the mnemonic is a mnemonic FOR. For (a+b)(c+d) "first" might just as readily mean ab as it might mean ac, and if you have another way of remembering which of the ones this is, what use is there left for "FOIL"? –Henning Makholm 16:21, 1 May 2007 (UTC)
A student who approaches this will know he is trying to multiply two factors together, so "first" means he will multiply the first terms of each factor together. When it comes to inside-outside... well, that's just a pictoral view of the other two things he needs to do to finish off the multiplication. It is mixing two different concepts, but I agree with KSmrg that the point is to remind students there will be 4 terms. A student has the four-letter acronym pounded into his head, and unraveling the meaning of each letter follows quickly. I wasnt ever confused by what "first" stood for, and I really dont remember it ever being an issue in junior high algebra class. (I do remember being annoyed that teachers would say complex multiplication and FOIL are entirely different processes, as though you had to then learn a completely new rule...) - grubber 18:07, 1 May 2007 (UTC)
If you weren't ever confused by what "first" stood for, it means that you understood what was going on, beneath the rote-learning. Congratulations. But, given that you possessed that understanding, what did you need a mnemonic acronym for, then? –Henning Makholm 22:09, 2 May 2007 (UTC)
How about "first summand in each multiplicand", "outer summands in each multiplicand", and so on. In a classroom setting, this is usually replaced with pointing with one's fingers. Lunch 20:44, 1 May 2007 (UTC)
That would be criminally misleading: it would teach the non-fact that the product of a1+a2+...+an and b1+b2+...+bm consists of terms of the form ai*bj where for each term there is a common principle that is applied uniformly to (1...n) and (1...m) to select i and j. That this "works" (after a fashion) for binomials is pure coincidence and does not generalize in any way. –Henning Makholm 22:09, 2 May 2007 (UTC)
It seems that for complex multiplication the rule "FLOI" is taught! That is (a+ib)(c+id) = (ac−bd)+i(ad+bc) = (F−L)+i(O+I). How sad for the teachers; how bizarre for the students. One teacher-of-teachers says this. --KSmrqT 20:34, 2 May 2007 (UTC)

[edit] "Generalizations"

KSmrq mentioned this article on mathforum.org as an example of student confusion. I think it's a good one. The student is confused because all of a sudden there are three multiplicands and no eight-letter acronym to help. But from the wording of the reply, I think "Doctor Ian" misunderstood the student's confusion (though I can only speculate). He simply launches into a discussion of his own understanding (but does offer some sage advice along the way that shouldn't be ignored).

FOIL is sufficient to expand this product if you creatively combine it with associativity and recursion. It might be instructive to include this tidbit to show how one could apply FOIL to expanding the product with two multiplicands with three summands each. (And, btw, thanks KSmrq for adding the tableau.) Lunch 00:21, 1 May 2007 (UTC)

[edit] Math ed

FOIL has been much debated, discussed, railed against, defended in the educational literature. This article, surprisingly, has none of that. --C S (Talk) 18:48, 7 May 2007 (UTC)

If you have concrete references to relevant parts of the educational literature, please do add them to the article and provide a summary of their arguments. –Henning Makholm 07:02, 8 May 2007 (UTC)

[edit] Childish term

Should the article mention that the name "FOIL" is a childish usage? I never heard of it until I'd been an undergraduate for more than three years and I attempted to help someone in an algebra course. It's clearly one of those mnemonics used only by those whose way of attempting to "learn" mathematics is phony: by memorizing. Adults (including, for example, 6-year-olds who wonder about these things) don't concern themselves with such things. Michael Hardy (talk) 18:27, 12 May 2008 (UTC)