For sure. There’s a study on high school misconceptions in algebra that looks at this, among others; it’s currently part of the University of Cambridge PGCE course. One of the questions begins with some words about vegetable shopping, then mentions that cabbages cost 57p and turnips cost 34p, then asks “what does 57c + 34t represent?”. Most pupils obediently write down “57 cabbages and 34 turnips”…
Agreed: There are better methods of doing polynomial products than FOIL; and the student needs to understand the Distributive Rule in any case.
I like the grid method better (as an instructional device) because it is very simple and intuitively straightforward, AND it very neatly illustrates the Distributive Rule. But actually drawing out a grid every time is still a crutch, that the student should outgrow with some practice.
The best method of polynomial products is still the old-fashioned worksheet layout that resembles ordinary multi-digit number multiplication:
3x[sup]2[/sup] + 5x - 7
4x[sup]2[/sup] - 2x + 3
9x[sup]2[/sup] + 15x - 21
- 6x[sup]3[/sup] - 10x[sup]2[/sup] + 14x
12x[sup]4[/sup] + 20x[sup]3[/sup] - 28x[sup]2[/sup]
12x[sup]4[/sup] + 14x[sup]3[/sup] - 29x[sup]2[/sup] + 29x - 21
@jsum_1: Care to check my work there? 
And while we’re at it, a related topic:
How many of you have learned, or even ever heard of, the “AC method” for factoring a trinomial?
When I was about 40 or so, I went back to Community College to finally get some edumacation (having dropped out of college years before that). There, a math teacher showed me. I had never heard of it before.
This is really slick. It entails much less “trial and error” than the usual technique.
Example: Factor: 6x[sup]2[/sup] + 29x + 35
Solution by the AC method:
[ul][li] Multiply the x[sup]2[/sup] coefficient by the constant term. (Hence, the “AC”.)[/li][li] Fully factor that product.[/li][li] Skim down the list of factor-pairs until you see a pair whose sum gives the middle term.[/li][li] Re-write the given trinomial as four terms, splitting the middle term into two separate terms with the coefficients you just found in the above step.[/li][li] Factor that by grouping. (Remember factoring by grouping? When was the last time that was taught to beginning algebra students?)[/li][/ul]
Try it!
Solution:
Factor: 6x[sup]2[/sup] + 29x + 35
AC product: 6 × 35 = 210
Fully factor that:
2 3 5 6 7 10 14
105 70 42 35 30 21 15
(Hint: It helps to factor the A and C coefficients first,
then use that to find the factors of AC.)
Look for factor-pair whose sum is 29: We see 14 × 15.
That is the only element of trial-and-error in this procedure!
Re-write the given polynomial:
6x[sup]2[/sup] + 14x + 15x + 35
Factor by grouping:
6x[sup]2[/sup] + 14x + 15x + 35
= (6x[sup]2[/sup] + 14x) + (15x + 35)
= 2x(3x + 7) + 5(3x + 7)
= (2x + 5)(3x + 7)
This is all true; but “bypassing reasoning” can actually be a good thing, in its proper time and place.
When teaching an important skill, ideally students will learn how to do it in such a way that they really comprehend what they’re doing and why. And they’ll know how to do it accurately, and be able to tell for sure that their result is correct, in cases where accuracy is important. And they’ll get to the point where they’re familiar enough with the procedure that they can do it automatically, quickly and facilely without having to think about what they’re doing.
This is, indeed, the “rule” I’d give for how to do polynomial multiplication: “Each term of the first polynomial gets multiplied by each term of the second polynomial.” But that’s not a procedure. If you gave me an eight-term polynomial and a twelve-term polynomial to multiply together, I wouldn’t have any trouble knowing how to do it, but I’d have to stop and plan out how I was going to approach all those term-by-term multiplications. “I’ll take the first term of Polynomial #1, and go through and multiply it by each term of Polynomial #2 in succession. Then I’ll come back and do the same with the second term of Polynomial #1; and so on.”
That’s exactly what FOIL is: a plan, a procedure, to actually carry out “everything in this set of brackets gets multiplied by everything in that set of brackets” in the case where both polynomials have exactly two terms.
That’s also exactly what the traditional worksheet layout is (see post #42 above): A plan, a procedure, , a canal, Polynomial! which neatly organizes the each-term-by-each-term process and the sum of the partial products, all while exactly and very visibly demonstrating the use of the Distributive rule .AND. it works for any number of terms in each polynomial. The FOIL rule does none of that, and is thus a seriously lame device.
I should emphasize that my suggestion of the Grid method is, I think, excellent for teaching all this, but still somewhat awkward once the student gets some proficiency, and should be abandoned once the student can use the worksheet layout.
I’m not against mnemonic devices per se as long as they are actually good devices. (I was never much into them myself, so I can’t think of any good examples.) FOIL isn’t one of them.
I’ve never heard of it, but it seems like it would only be useful if the problem is set up so that you end up with only integer factors. If the middle coefficient happened be 30 instead of 29, what do you do then?
For the people railing against FOIL because it’s rote learning rather than understanding: For some people, the choice isn’t between FOIL and a good understanding of the distributive rule, it’s between FOIL and never being able to expand the product of two binomials in subsequent homework. When deeper understanding is off the table, which would you prefer?
Well, it’s always the convention that “factoring a polynomial” means finding the factors with integer coefficients, unless otherwise stated. Whether you use the conventional trial-and-error methods or the AC method doesn’t change that. If the middle coefficient was 30, we would say the polynomial can’t be factored. What would you do with it in that case?
The FOIL rule may be usable by students who are never going to “get it” any deeper than that. Those students probably aren’t going to be taking Algebra 2. That’s okay. We don’t all need to be Algebra mavens. But shouldn’t we be addressing the instruction mostly toward those students who are likely to go on in their math studies?
In that case, I would factor it. All polynomials are factorizable, because all polynomials have roots (albeit sometimes complex or repeated roots). Sometimes it’s more difficult or impossible to explicitly express the factorization, but it still exists.
The homework is in service to the understanding, not the understanding to the homework. If a student can’t understand, then you try harder to help him understand. If full understanding is impossible, then you try for as much understanding as is possible. Teaching something to a student just so they can do the homework is a dead end.
I’ve never heard this criterion, certainly not as the default. Do you have a cite that this is always the convention unless otherwise stated? That seems very limiting. If the middle coefficient were 30, your example would still have real roots, and as Chronos stated, there will always at least be complex roots.
My son is currently in Algebra 2. It is a State requirement of all students. Even if it weren’t a state requirement, I’m sure many students who have a difficult time with some concepts in math would still take the class. My son is one of those students who’s just never going to get some things in math. If he can remember and apply FOIL, that’s a success. And to be sure, it’s not like he’s at the bottom of his class. Not everyone thinks the same way.
If FOIL is “as much understanding as is possible”, and it helps the student to continue learning other topics as the class progresses, instead of his learning coming to an end, I’d say FOIL is worthwhile.
I really don’t think FOIL is anything other than a time saver.
(a+b)(c+d)
So if I use FOIL I can do
ac+ad+bc+db (I actually had to double check that, for some reason mentally I do FLIO)
Or if I’m using the distributive property I can do this:
(a+b)c + (a+b)d = ac+bc + ad+bd
Neither is in any way more complicated than the other, and I’d argue that the latter is easier to understand because it’s a two-fold repetition of a one-step rule rather than one four-step rule. I use FOIL because it really does save some time, but from an educational perspective the latter is really better in every way IMO since it follows from the distributive property (which they learn anyway), and leads to fewer arbitrary things to remember.
I guess you could argue that FOIL is closer to the inverse of factoring a polynomial since you’ll basically never factor a polynomial into the intermediate stage (a+b)c+(a+b)d, but I’m not really convinced. At the very least teach the distributive method before FOIL and introduce FOIL as “hey, guess what, some of you may have noticed these but these two always do the same thing and this one is faster once you master it!”
Thanks for this. I’ve been doing it wrong for years.
You didn’t pay close attention to that AC method I mentioned above, complete with a worked out example! On the surface, the beauty of the AC method is that it reduces the amount of trial-and-error work. (Yes, I’m specifically talking about factoring a trinomial with integer coefficients into factors with integer coefficients.) But taking a close look, the process has another neat facet: As you go through the steps, you can actually see the distributive multiplication process happening in reverse! Take a look at the worked-out example I wrote above. Yes, the trinomial IS factored into the intermediate stage (a+b)c + (a+b)d before being further factored into (a+b)(c+d)
As for factoring stuff: Students typically first learn to factor using integers only, with trial-and-error methods (factor the x[sup]2[/sup] coefficient and the constant term, then try all combinations of those factors until you find the pair that gives the right middle term), then later learn about completing squares and the quadratic formula, and then learn that you can factor all polynomials with this and other advanced techniques.
But I was only referring to all-integer factoring. I was talking about the actual process of finding the factors. Using the quadratic formula and other advanced techniques will get you the factors, but not by the process of factoring, if you get the distinction. The AC factoring method just accomplishes exactly what the regular factoring method accomplishes, neither more nor less – all-integer factoring – and seems much simpler as long as you remember how to do factoring by grouping (that intermediate factoring stage).
Actually I didn’t, I was in a rush :). But that’s pretty cool (for integral examples at least).
I’m reviewing for the GREs tomorrow and it actually amazes me how much math classes expect you to take things on faith or by rote. I can’t believe nobody ever mentioned to me that the somewhat obvious (in hindsight!) fact that the area of a circle is just a special case of the sector area where the angle is 2pi (or 360deg) – aka the whole circle! Same for the arc length. They were just treated as separate formulas for doing separate things, despite the fact that once you know what you’re doing the relationship is blindingly obvious. I did a lot of mental maintenance and now I have to remember far fewer formulas because of stuff like this that nobody ever mentioned. I see this a bit in FOIL vs distribution, where FOIL is some great unknowable process that is alien and weird, when there’s really no need to memorize it when you can just point out the obvious that it’s a shortcut for doing a just-as-simple two-step calculation that students already know.
I mean, it’s still bullshit that I have to memorize formulas for the test, but still.
Is “teaching by rote” coming back into fashion? Is it part of the “teaching to the test” fad? Is it part of the new Common Core Curriculum or something?
Lately I’ve occasionally been helping a neighbor teen with his Algebra I. His textbook is written by Larson, one of the long-time luminaries of math textbooks. (I learned Calculus I from his book, 2nd edition, back in 1982 or so.) I leafed through this Algebra book. Holy Cthulhu, I couldn’t believe me peepers!
The chapter on “Special Products and Factoring” is way towards the end of the book. WTF? It’s even a few chapters AFTER the chapter on the Quadratic Formula. How in the f*ck do you teach the Quadratic Formula (completing the square, remember?) without teaching special products and factoring first?
So I took a closer look. You guessed it: Forget the process of completing the square. He just presents the Quadratic Formula, without showing its development, for the student to memorize I guess. (He promises to show the development later. In fact, after teaching about products and factoring, he goes back and develops this.)
What kind of a scrambled way is that to teach beginning Algebra? Please tell me this was just a manufacturing error, that they put together the chapters in the wrong order when the pages were bound into the book.
Yes! Or, you could think of the formula for the area of a circle as being just a special case of the formula for the area of an ellipse.
And did they ever mention to you that the Pythagorean Theorem — c[sup]2[/sup] = a[sup]2[/sup] + b[sup]2[/sup] — is just a special case of the Law of Cosines?
c[sup]2[/sup] = a[sup]2[/sup] + b[sup]2[/sup] - 2ab cos γ
where γ = π/2 or 90°
I think a WHOLE LOT of the formulas you know are just “special cases” of more general formulas.
ETA: And 0.999… is just a special case of 1 
By the way, f.coli, are you still with us?
All you higher-math types: Do any of you have (or have you ever seen) the Algebra text I cited earlier, Algebra for College Students by Jack R. Britton and L. Clifton Snively, Publ. Rinehart & Company, Inc., 1947?
There must be some more copies of that still floating around — I see it in used bookstores now and then, and I see there are a few listed at Amazon.
Much more thorough and in-depth coverage than anything you see in algebra textbooks today. (Although the choice of topics to cover is a bit antiquated. There’s a chapter with extensively detailed instruction on computation with logarithms, for the benefit of slide-rule users, for example. And utterly nothing about sets.)
Quick! Find the cube root of:
(567.90)(0.79023)
—————————————————
91.346𝜋
Well, I mean to a certain degree most areas are just special cases of integration of some function :p. Obviously at some point memorizing things becomes a necessity unless you want all your students to derive things from first principles. It makes sense to teach area formulas before integration, for instance. I’m just arguing for sane levels of memorization. It makes sense to teach the area of squares and circles as different, less so to teach that the section area and circle area are different formulas.
And I was taught ellipse = circle, but you really don’t get to see ellipses that often so I don’t really use that property much.
And I figured out the Pythagorean theorem = law of cosines thing by myself a while ago, it may have been mentioned when I first learned it, I don’t remember. I don’t use the law of cosines much intentionally, I usually (accidentally) rederive it on the fly using the vector dot product, usually in computer graphics.
I wouldn’t say rote memorization is “coming back”, but that janky disjointed thing is definitely a problem. I know for a fact (since it was your example) we learned the quadratic formula first and completing the square later, for instance. Yes, we did prove the QF eventually, but it was well after we learned it. A teacher may have mentioned the section area = circle area thing at some point, but it would have been so far after being forced to memorize both separately it probably didn’t stick. I just noticed it because I’m more educated in formal proofs and purely symbolic math now. Having to make XOR functions linearly separable in 5D space with an order 2 polynomial kernel (specifically a hyperbola) does things to your math sense and ability to recognize function relationships that can never be reversed.
Edit: Oh, and don’t get me started on conics. We learned that all conics are, well, intersections with a cone but it was never mentioned that they’re all ultimately permutations of the formula Ax^2 + Cy^2 + Dx + Ey + F = 0, where the coefficients determine whether you’re getting a hyperbola/parabola/ellipse/circle.
Wow, when I was a kid, we were taught the sector and arc-length formulas by deriving them from the corresponding circle formulas-- Is that really not done any more? I think we also showed why the pi in the circle area formula is the same as in the circumference formula, though I might have picked that one up from a book on my own.
To be fair, it is still necessary sometimes to just plain memorize some formulas-- There’s no easy way to derive the formulas for the volume or surface area of a sphere without calculus, for example (it can be done, but it’s not easy).
You can take Pythagoras’s Theorem as a special case of the cosine rule, for sure, but you can also derive the Theorem without ever going near any cosines or even any awareness of what one is.