As others have said, this is something that turns out to be immensely useful in a number of applications. As a computer scientist, polynomials play an integral role in analyzing time and space complexity (ie, how efficient an algorithm is). I’ve also done a fair amount of mathematical programming and sometimes just a few simple adjustments to a formula can make for fairly large improvements in performance or accuracy.
In a more generalized way, algebra is all about understanding the various algebraic relationships, so learning FOIL is generalizable to multiplying higher order polynomials. Sometimes it’s helpful to have a polynomial in a partially or completely factored form and sometimes it’s easier to have it multiplied out. For example, off the top of my head, (x - h)^2 + (y - k)^2 = r^2 is a conceptually simple way of understanding the equation for a circle, all the relevant information is right there in the equation (center coordinates and radius), whereas ax^2 + bx + c = 0 is important for using the quadratic equation. Those are both applications that ought to have shown up in high school level math, in or before Geometry. And the applications just get more common as you get into higher level math.
That all said, I do wish that math teachers particularly spent a little more time explaining the practical applications of some of the stuff they teach. I think a lot of students struggle with concepts they can’t tie back to the real world or other subjects they connect with better. Maybe that’s why so many people find math difficult or confusing, but that’s really a discussion for another time, I guess.
Anyway, TLDR, it’s important to teach both as a very common special case and for general frame of thought on more general cases and understanding manipulation of polynomials in algebra in general. It has numerous applications in higher levels of math as well as other fields.
Hmmm… This must be an example of that New Math I’ve long heard so much about :dubious:
By having two possible forms of the expression, as above, one gets the added flexibility of choosing whichever side is easiest to compute, or alternatively, whichever side gives me the more favorable answer!
Learning – memorizing and getting thoroughly familiar with – all those “special products and factoring” (as we called them), is important to help you with the pattern-matching to see more complicated examples. You may simply memorize that
( a + b )[sup]2[/sup] = a[sup]2[/sup] + 2ab + b[sup]2[/sup]
But can you recognize the pattern in this?
9x[sup]4[/sup] + 24x[sup]2[/sup]y + 16y[sup]2[/sup]
Yes, that’s also the square of a binomial: (3x[sup]2[/sup] + 4y)[sup]2[/sup]
Could you recognize that and factor it?
Here’s an even better one, that relies on the student’s ability to spot the patterns:
Factor: x[sup]4[/sup] + x[sup]2[/sup] + 1
Solution:
Add and subtract x[sup]2[/sup] to make it the difference of two squares:
x[sup]4[/sup] + x[sup]2[/sup] + 1
= x[sup]4[/sup] + 2x[sup]2[/sup] + 1 - x[sup]2[/sup]
= (x[sup]2[/sup] + 1)[sup]2[/sup] - x[sup]2[/sup]
= (x[sup]2[/sup] + 1 - x)(x[sup]2[/sup] + 1 + x)
= (x[sup]2[/sup] - x + 1)(x[sup]2[/sup] + x + 1)
This skill in recognizing patterns and making good use of them gets more and more important as you get more and more into mathematics. (Integration formulas, anyone?)
Many thumbs up to Wendell Wagner for this. How often have you sat in (or taught) class while a student tries, inarticulately, to ask a question and the teacher (and/or you) can’t figure out what the student is asking? If a student can’t speak a statement or question clearly, then that student most likely can’t think it clearly either. If I were teaching math – especially beginning algebra level – (a long-time fantasy of mind), I’d pay a lot more attention to vocabulary and clear-speaking, on the theory that it’s a pre-requisite to clear-thinking.
I have an old textbook: Algebra For College Students by Britton and Snively, first published in 1947. This is my go-to algebra book. Much more comprehensive and in-depth than anything you’ll find today that I know of! (The sum-and-difference-of-squares problem in my above post is from there.)
An excerpt from the introductory section to statement problems:
I’ve noticed that some modern beginning algebra books (even [Community]-College level) seem notoriously dumbed down, and this includes protecting students from all that incomprehensible mumbo-jumbo jargon. A few of the “Fundamental Laws” are shown, but the word axiom never appears. Words like “term” or “operand” are not given. (Okay, “factor” is always discussed.) The concept and logic behind “axiom” vs. “theorem” is not discussed.
You can actually take this one one step further and avoid the algebra. The way you have the triangles configured, the “area in the large square that is not in one of the triangles” comprises one square of area c^2. But if you move the triangles around – pair them up into two rectangles, and move the rectangles so that one if them is in the upper left corner with its long edge along the top, and the other is in the lower right corner with its long edge along the right – that area is now composed of two squares of area a^2 and b^2 respectively. The fact that moving stuff around doesn’t change areas means that the two quantities are the same, so a^2+b^2 = c^2.
Your proof with the algebra is the one I had memorized from my high-school days, but the “slide the triangles around” part takes it from “provable” to “intuitive”.
In fact, the way I had the triangles configured, the square of area c[sup]2[/sup] is the square comprising the whole diagram. I know the one you’re thinking of, though (no need to check the link) and you’re quite right that it lends itself very well to sliding triangles around and rearranging. Mine has a square of side (a - b) in the middle, when the area of this square plus the area of the four triangles can be shown to come out to a[sup]2[/sup] + b[sup]2[/sup]. There are, of course, others; I especially like the similar-triangles proof that I found out only a year or so ago (which AIUI is the one Pythagoras actually used).
I hate FOIL with a passion. I can see that it is an ok summary of the process and can serve as a mnemonic for those who just need the answer. But seriously, it is like needing a recipe book to operate a toaster.
I would much rather my students understand the principles underlying the process – to recognise the implied multiplication between the two sets of brackets and to be familiar with order of operations. If they need something a bit more concrete to hang their ideas from then they should reflect on the processes that they learned in primary school when they learned to multiply 12×13.
12×13 = (10+2)(10+3) = 100+20+30+6 = 156 (And I concede that this is non-standard setting out for this problem, but all of the same steps are involved.)
compares well with
(x+2)(x+3) = x[sup]2[/sup]+5x+6
If they are relying on FOIL to do that then I would hesitate to say they have a firm grasp of the actual process. The teacher is merely creating the illusion of understanding with the requisite pass grades. The student then comes unstuck later on because they don’t have a foundational understanding of what they are attempting to do.
I never heard of FOIL until many years after I originally learned algebra, and I immediately considered it awful. It only tells you how to multiply two binomials (as others above have noted), and says nothing about why it works. Students should really learn the distributive rule, and understand how polynomial multiplication follows from that.
Question: How many of you have heard of the grid method for polynomial multiplication? I’ve seen it mentioned, in the context of “Hating New Math” message boards where people who can’t do their children’s algebra thunder “THE FOIL METHOD WAS GOOD ENOUGH FOR MY GRANDFATHER AND IT WAS GOOD ENOUGH FOR MY FATHER AND IT’S GOOD ENOUGH FOR ME”, where the Grid Method was reviled as the spawn of Satan.
Now, it happens that I once devised a method of polynomial multiplication that happened to use a grid. I am certain it must be exactly the Grid Method that is so much reviled by the Hate-New-Mathers. The Grid Method has certain features that I find highly commendable:
(1) It generalizes nicely to multiplying any two polynomials, with any number of terms in each.
(2) It illustrates very nicely the extended Distributive Rule. The grid can be taken to be a rectangle whose area is to be found. The area can be found by adding the segments along the top (a + b + c + … ) and the side ( u + v + w + … ), then multiplying those. OR the area can be found by finding the area of each cell in the grid, and adding those up. ( Σ au, av, . . . cv, cw, . . . ). Which is, after all, just what the Distributive Rule says.
It’s like the difference between saying “water” and “H20”. One form, “H20” is a better description of what water is made of. The other form “water” is, to most of us, a more clear description of if something is safe to drink.
If you were trying to do a simple analysis of the effect of unsprung weight (the weight of the wheels) on driver comfort, you’d begin with measurements of the first form, then convert to the second form to see what they meant.
Actually, I think it is worse than that. When we reduce Mathematics to a meaningless and arbitrary set of rules that have to be followed for no particular reason on order to get [ta-da] The Answer [/ta da] then we create a real cultural problem within our classrooms. What I mean is this: Students get accustomed to producing work without really understanding what they are doing. But they feign understanding because they are getting the right answer and everyone else is too and they don’t want to appear dumb. This leads to a huge mental disconnect and a real discomfort in doing Mathematics. The next step is a lack of motivation. After all, who wants to spend their days feeling bad about producing screeds of stuff that has no particular meaning. This compounds over years as bad teacher succeeds bad teacher. The student develops acquired incompetence. Their confidence levels are so low that they fail to even recognise the obvious. I despair when I have to teach 17 year-olds how to multiply by 10 by shifting a decimal point. (I hate that construct too btw.) Through tedious explanation combined with countless mnemonics and hours of drilling skills in exercises that bear no resemblance to any real problem we battle with unmotivated students who lack any real comprehension of what they are doing much less an appreciation of why they are doing it and eventually squeeze out of them some kind of acceptable grade. Curriculum writers and textbook authors contribute their bit by dumbing down content while keeping it arbitrary and abstract enough to pass for Mathematics. The result is cohort after cohort of disenfranchised students who forever after complain, “I didn’t really get Math”.
In my observation these kinds of problems happen in Mathematics more than any other subject.
[end of rant]
Mathematics teaching has to be principle based and couched in a meaningful context*, grounded in sound pedagogy and carefully sequenced so that students are able to construct a coherent and integrated knowledge and skill set. There is no way that FOIL meets those criteria.
*I hasten to add that “meaningful context” does not equate with practical, topical or vocational. It can be entirely abstract or related to a puzzle or game that serves no immediate purpose. Meaningful simply means that there is some kind of connection point for the student so that they can relate what they are currently learning with what they already know.
When we reduce Mathematics to a meaningless and arbitrary set of rules that have to be followed for no particular reason on order to get [ta-da] The Answer [/ta da] then we create a real cultural problem within our classrooms.
Tired old complaint.
You don’t learn how to factorisation by learning what it means, you only learn factorisation by learning factorisation.
Disagree completely.
If I present someone with three unopened packets of widgets and 2 loose widgets and say that I need to triple this amount to get what I need, then I would expect any thinking person to return with 9 packets and 6 loose ones. This is equivalent to
3(3x+2)=9x+6
The meaning precedes the notation.
Any good notation is servant to the inherent meaning that is supported by logic. Once a notation is formalised it then becomes a tool to be used in deeper or more abstract problems.
Sure there is a place for drilling a skill. Rote memorisation of multiplication facts, algebraic manipulation, processing negative and positive numbers multiplying matrices – these all benefit from being practiced. But understanding does not inherently come with repetition. I have come across many bright students who can factorise pretty much anything you give them. But when faced with a larger problem for which factorisation is but one step, they are paralysed until someone gives the a cue so that they know what they should do. I don’t call that deep understanding and I am not going to blame the students.
To run an analogy.
It is entirely possible to have a basketball player who has learned to shoot hoops by shooting hoops but is completely lacking in the other necessary skills and has no real comprehension of how they fit into the game. If a player continually fouls, can’t maintain position, can’t intercept or control the ball, it doesn’t matter if they shoot 97%, they are a liability.
Fortunately this doesn’t really happen. Coaches are a lot more holistic. Interested players are highly motivated. (And those who aren’t have the option of dropping out which option just isn’t there in Mathematics classes.) Players shoot hoops for hours on end in the pursuit of excellence because they understand its implications in the wider scope of the game.
False dichotomy. Of course they should learn the distributive rule, and how polynomial multiplication works. Then they can learn the FOIL method, not as a method of multiplication, but as a method of remembering (or labeling) the steps involved in a particular, very common type of polynomial multiplication.
So that’s where so many incoming physics students get that ridiculous notion. Units are an inherent part of the quantization of any quantity. Carrying them through the calculation is always appropriate, and the only difficulty it ever introduces is the notion that they can be carried through the calculation (which, in turn, is probably only a difficulty at all because they’ve been told by books like that that they can’t). It’s also ultimately the only way to even know what the final units ought to be, and can very often reveal mistakes in one’s work.
Regardless what students are taught in general and whether they can use it in general, it’s obvious to me that f.coli learned no general principles of algebra, just some techniques that he (for the purposes of simplicity, I’ll refer to f.coli as “he,” although I don’t know his/her sex) memorized without understanding them. He refers to the two sides of the equations as each being an equation. He talks about the FOIL method without mentioning the distributive law, which makes me suspect that he never understood what the distributive law was. He talks about “solving polynomials.” You don’t solve a polynomial. You solve an equation. You can factor a polynomial or multiply out a polynomial or simplify a polynomial, but you can’t solve it. Despite his claim to understand “working these problems out,” he apparently didn’t actually know what he was doing in algebra.
Look, I don’t think that some of the mathematics generally taught in high school is as useful as it’s supposed to be. I think that some of the geometry taught isn’t very useful. I think that instead more probability and statistics ought to be taught at that level. I don’t like the idea that people read or hear about polls without understanding that the margin of error on them is two standard deviations, which means that they are 95% likely to be within that interval. I don’t like the fact that they don’t even know what a standard deviation is.
Understanding factoring polynomials and solving equations though is important, and students ought to really comprehend them, not just memorize some techniques.
To be fair, the terminology “solving a polynomial” is often used as shorthand for “finding the roots of a polynomial”, or solving the equation that consists of setting the polynomial equal to zero.
And the primary virtue of high school algebra is that it’s the first math class that most students ever encounter. Though of course you could instead introduce proofs in any other context: I suspect that it’s just tradition (dating back to Euclid) that the concept is first introduced in geometry.
I think you missed the point. The point isn’t that one is supposed to forget the units – as you say, you shouldn’t. The point is to state clearly what your variables represent, which is something students generally don’t do in my experience. Give them a problem about apples and oranges and they’ll say “let x be apples and let y be oranges.” I want to scream at them, “What about apples does x represent? Their weight, their volume, their luminance…?” Because when they next have to translate “there are twice as many apples as oranges” they’re lost. Or if they have a problem that requires them to consider both the number of apples and the total weight of the apples, which one did they use x to represent?
Well, you it is good practice to represent your system in terms of dimensionless parameters (e.g. if you have a length x, try to find another natural length L in the system and parameterize in terms of x/L). Just because it can both reduce the number of parameters in the system and make the dynamics clearer.
I think it is safe to say that Senegoid writes a better mathbook than me. Even if he chooses to leave the units out of his working.
As for my glitch, once it is on the net it can’t be erased.
But seriously, this kind of illustrates my point. It is possible to have a good understanding of the process and make an error. And it is possible to get the correct answer without real comprehension. Comprehension does not always correlate with correctness. And that is why I dislike mathematical mnemonics so much. They facilitate the correctness of an answer while bypassing reasoning. It makes it more difficult for me to gauge what my students do and do not know.
I model incorrect answers routinely in my classroom – usually, as above without trying. They become a discussion point of “What is wrong with this?” and “Why?” Students then have to articulate how the process works. It also serves to reinforce the notion that it is the process that we are learning and it is not really about the answer. It is ok to make mistakes.
As for a memory device – whatever is wrong with “everything in this set of brackets gets multiplied by everything in that set of brackets”? It is a colloquial restatement of the underlying principle (the distributive law). It is no more difficult to remember. It is specific to the two multiplicand case but generalises to more than two terms in the brackets. It can easily be extended to three or more multiplicands. (“Everything in each set of brackets gets multiplied by everything in all the other sets of brackets in every possible combination.”) (Now let’s look for a systematic way of doing that.)
If students have a grasp of what multiplication actually means and have algorithms for manually multiplying two digit numbers and have an understanding of what pronumerals are, how they simplify and some basic manipulation skills: if they have all that then there is absolutely no need for FOIL. If they don’t have these concepts then what an earth are we hoping to accomplish by having them expand binomial expressions? It just doesn’t make sense.
In short, Although mnemonics such as FOIL might not harm the learning of some students I can see no good reason why they should be taught. And I can see a large number of reasons why they should be avoided.
