Could I still edit, I would change “its height times its base” to “its base times its height” in the bolded instances, for better parallelism…
The FOIL method is nothing more than an organized application of the distributive property. There is nothing magical or special about it. If you distribute per the distributive law (without even thinking about it), you would arrive at the same answer. However, the FOIL method is taught to help people keep things straight as many would be intimidated when asked to expand. It makes people see it’s not so bad.
Now, personally, I’ve never understood completing the square. It seems like you’re cheating making a term that simply is not there. I’m not even sure when to recognize the right place(s) to use completing the square!?!?
Well, this takes us back to the very beginning of the thread, and the identity (a + b)[sup]2[/sup] = a[sup]2[/sup] + 2ab + b[sup]2[/sup].
Let’s say we want to solve the equation x[sup]2[/sup] + 8x = 41.
We want to add a third term to the left side, to make it fit the pattern of a[sup]2[/sup] + 2ab + b[sup]2[/sup]. In this case, the thing to add is 16 (take half the coefficient of x, and square it).
x[sup]2[/sup] + 8x + 16 = 41 + 16
We’re not cheating, because it’s legal to add the same number to both sides of an equation.
Hundreds of years ago, before people used symbolic algebra, people pictured this sort of thing in terms of completing a literal, geometric square, like this.
Now, thanks to (a + b)[sup]2[/sup] = a[sup]2[/sup] + 2ab + b[sup]2[/sup] (but in the “opposite direction”) the equation can be written as
(x + 4)[sup]2[/sup] = 57.
Then x + 4 = the positive or negative square root of 57,
and x = sqrt(57) - 4 and x = -sqrt(57) - 4 are our solutions.
Continued in next post…
One place completing the square comes in handy is when you have a second-degree equation in x and y and you’re trying to see what the graph looks like.
For example, a circle with center (h, k) and radius r would have an equation that could be written as
(x - h)[sup]2[/sup] + (y - k)[sup]2[/sup] = r[sup]2[/sup]
So if you saw (x - 7)[sup]2[/sup] + (y + 3)[sup]2[/sup] = 100, you’d know right away that the graph was a circle with center at the point (7, -3) and radius 10.
But this could be multiplied out as
x[sup]2[/sup] - 14x + 49 + y[sup]2[/sup] + 6y + 9 = 100
x[sup]2[/sup] - 14x + y[sup]2[/sup] + 6y + 58 = 100
x[sup]2[/sup] + y[sup]2[/sup] - 14x + 6y = 42
If the equation looks like that last when you first encounter it, completing the square lets you get to the version that makes it obvious what the center and radius are.
I don’t get the hatred for FOIL. I just see it as a way to systematically expand an expression without missing or duplicating anything.
Let’s say you have an equation:
2x + 3 = 9y + 2
In algebra, and to a certain degree even calculus, we tend to be solving for a variable, trying to get a given variable by itself. Thus we tend to think of manipulating equations as “moving” values around in specific ways. But that’s not the case, “moving” values from one side of the equals sign to another is a clever way of contextualizing adding (or subtracting) an arbitrary term.
When you manipulate 2x + 3 = 9y + 2 into the equation 2x = 9y - 1, what you’re doing is this:
(2x + 3) - 3 = (9y + 2) - 3
Now obviously the 3 - 3 on the left cancels to +/- 0, and 2 - 3 = -1, which gives us the equation above.
When you’re completing the square, you’re doing exactly the same thing, you’re just not immediately cancelling a term like you normally do for solving an equation.
Another way to do this is to accept an equation as an expression that asserts that two subexpressions are equivalent.
f(x) = g(x)
Means that for any value, x, f(x) and g(x) have the same value.
f(x) + 9 = g(x) + 9
Have the same value, what if f(x) is 3? Then g(x) is 3, adding 9 to make both sides 12 won’t change that they’re equivalent. Same with multiplying or dividing both sides by the same number, or whatever else.
There is an interesting sidenote, though, that a large part of the reason this works is because of functions. The reason we can square both sides of an equality, or take the logarithm (or, yes, even add and multiply) is because these functions are injective (also known as 1-to-1). You may remember this as the “vertical line test” – for each input there is exactly one output. If you have a mapping that’s not a function, it’s more difficult to invoke that mapping on both sides of an equals sign without extra constraints such as “for each input, always take one set of outputs” or “if you choose one output for one, whatever output you choose, choose the corresponding output on the other side”.
This can be seen by taking the square root of squares of both sides:
sqrt(x) = sqrt(y)
This only works if you remember that if you take the negative (or positive) result on the left, you must also take the negative (or positive) on the right. This is because sqrt is a mapping to a choice between the positive and negative square root. (You can also kind of deal with this by saying it’s a mapping that you change into a function by constraining the range of its outputs).
Squares and square roots in general play a lot into weird math psychology, where you tend to recognize that x = sqrt(y) is a parabola, but find it a bit alien that y = sqrt(x) is a horizontal parabola, because we’re so used to automatically constraining the range of sqrt(x) so that it’s a function on x and not a function on y.
No hatred for FOIL per se or any other mnemonic for that matter.
The hatred is for a system that enables students to regurgitate stuff that they have no real understanding of whatsoever – and this is for something that is conceptually straightforward and is practised by primary school students whenever they multiply 2digit numbers.
The hatred is reserved for those moments when facing a classroom of students with glazed over eyes who have no clue what they are doing and no interest because they believe that Mathematics is a disconnected set of arbitrary rules. The hatred is being unable to get students to progress into useful and beautiful math because they have no foundation. The hatred is for being unable to put together a straightforward assessment tool to gauge students’ understanding since a good portion of them are merely parroting without comprehension.
FOIL merely exacerbates these problems. There is a place for mnemonics and this aint it.
Somewhere recently (in this thread or the other one), I mentioned 5 “things” that I think Algebra Iz.
One of those “things” is: A collection of techniques for solving certain common types of problems. Completing the square falls into that bucket.
Anecdote: I once watch a carpenter building. . . . something.
He had a set of tools. Okay, anyone can buy a set of carpentry tools, like hammers and saws. But this guy also obviously went to carpentry school, and he learned a set of techniques for doing things with those tools.
He laid out a board, and using a carpenters square and pencil, he laid out a row of right triangles along one edge of the length of the board. Then he cut out those triangles, leaving a board with a saw-tooth edge. (Spoiler: See this photo, but it gives away what he was doing.) I watched for a while, trying to figure out what he was building. See that photo to find out!
The numbers systems of algebra, the notation, and the rules of how numbers behave all give us the tools to solve problems. But some common problems require non-so-obvious applications of the tools. So once some mathematician figures out some useful but less obvious technique, he writes it into a math textbook, for all math students to study and learn for the rest of eternity.
Completing the square is exactly one such technique.
Away back in Post #42, I showed the “worksheet layout” for multiplying two polynomials, of any number of terms each. This is the classical method, and IIRC was still being taught when I took Algebra I.
It seems clear, well-organized, and makes it easy to keep straight what terms you’ve multiplied by which, and helps you to get all the like terms collected easily from the partial products to the final result. I can’t see where a limited crutch like the FOIL rule has any advantage over this. Why not just teach the worksheet layout and consign FOIL to the dustbin of history?
Because FOIL has a snazzy name! Who doesn’t love tin foil!?
Slight nit-pick with the way you wrote that: It is possible, with a true function, that multiple inputs can produce the same output, and thus not be exactly “1-to-1”. Such a function would fail a “horizontal line test”. A truly 1-to-1 mapping would pass both a vertical and horizontal line test.
e: derp
My take on “cheating”: It’s actually just a result of what’s been beaten into your head.
We are taught from early on to simplify, simplify, simplify. Which usually means collecting like terms.
SIMPLIFY! SIMPLIFY! SIMPLIFY! Thus, we are taught to simply like a knee-jerk reaction. Whenever an opportunity appears to SIMPLIFY, just do that immediately before you even start to think of what to do next.
And, to be sure, this is usually good advice and the right, just, honest, and moral thing to do!!! Frequently, it even helps us solve the problem! Yay!
Occasionally, not. Sometimes you have to “un-simplify”, that is, introduce uncollected like terms, then re-arrange them differently, to get anywhere. (I like to say: Complify! The opposite of Simplify!) It’s like pushing your equation up-hill, against entropy, to get it over a hump.
But we’re so conditioned to SIMPLIFY! that it seems. . . well, wrongful, dishonest, and immoral to do. That’s why it seems like cheating.
ETA: More examples of complification:
See Post #22 (I think, or thereabouts) above, where I gave two polynomial factoring problems. The second one requires adding and subtracting a term.
See also the AC factoring method (Post #43), which I described earlier. That requires splitting a term into two terms, another example of complification.
You’re right that I totally botched the definition. For some reason I got it into my head that all functions are injective mappings. That’s not the case, functions are defined as mapping one input to one output. A function is only injective if each possible output* has exactly 0 or 1 inputs that can produce it ("horizontal line) .
- Also called the co-domain, but who’s counting?
And, if a given prospective “possible output” has exactly 0 inputs that can produce it, then it’s not even a possible output, is it? 
Example of an impossible output: That guy named Charlie who got stuck riding the MTA for the rest of eternity! 
I was actually going to make a followup post ranting about “co-domain”, because it is fucking impossible to describe. It’s one of those math terms that sounds really scary, but using anything other than it makes no sense if you think about it unless the person you’re explaining it to has some (however small) background in set theory. In which case it’s obvious.
I fall back on “possible output” because it’s the most intuitive way to phrase it, but you’re right that it’s silly. It’s more like “an output of the same type as the other outputs that could be produced in another world where this function produced similar-but-distinct outputs”.
Strictly speaking, the set of outputs is the “range”, which is a subset of the codomain which is (more or less) the set of things the range is allowed to be generated from. And the presence of injectivity and surjectivity effectively control 1. what inputs product what element in the range and 2. which elements of the codomain are in the range in the first place. If a function is both injective and surjective, then it’s bijective and the range and the codomain are the same thing.
I haven’t learned math at the level you are talking here, so lemme see if I am more-or-less wrapping my mind correctly around this.
Your usage of “range” is just as I have always understood it.
Your description of codomain here sounds like: The Universal set from which the range may be taken. That is, the entire set of “prospective” outputs (as I put it above), some subset of which may actually be possible outputs (the range).
ETA: When I took Algebra I in 9th grade (circa 1962), there was much emphasis on sets and in fact the very first chapter of the book was introductory set stuff. When we got into solving equations, much fuss was made on identifying the “replacement” set for the variable in question, and then finding the “solution” set, as a subset of the replacement set.
Let’s say we have the set X={a,b,c}. We’ll call this the domain. Now we have a set Y={d,e,f}. We’ll call this the co-domain.
Now let’s define a few functions:
f(x in X) =
if x = a; return d
if x = b; return e
if x = c; return e
What’s the range of this function? {d,e}
g(x in X)=
if x = a; return d
if x = b; return e
if x = c; return f
What’s the range of this function? {d,e,f}
So effectively the co-domain is what you select from when making your function’s outputs. In high school and under math, this is usually integers, rationals, or reals. By high school, it’s almost always reals. This is why math classes usually only teach the range, because otherwise the codomain is pretty much always “all the numbers you’re familiar with”.
It’s also true that in most practical cases at the high school level, it tends to be irrelevant whether the co-domain is the range or a superset of the range.
An interesting example of domains and co-domains is in programming languages. If your function returns an int, then its co-domain is all values that can fit into an int in your language/compiler, but its range may be a smaller value like “0 or 1” (for bools in C). You also get examples that show off that the domain and co-domain aren’t always the same like a function that takes an int and returns a string.
I’m not saying anything that hasn’t been said already, but my bemusement at students resorting to “the FOIL method” is in bizarrely focusing so much attention on what seems to me a not particularly dominant special case when the more general principle really deserves the attention: why focus on (a + b) * (c + d), when you can just as well think in terms of plain distributivity, from which everything else follows naturally? A student who feels it necessary to walk through the FOIL mnemonic is one whose ability to handle (a + b + c) * d * (e + f) I am forced to wonder about, when there’s no reason this should be understood through any different a conceptual path.
[Granted, distributivity also follows from FOIL, if you take the route of a * (c + d) = (a + 0) * (c + d) = ac + ad + 0c + 0d = ac + ad + 0 + 0 = ac + ad, but that seems rather less than straightforward…]
Also, all that having been said, I have often seen students use the word “FOIL” simply to mean “distribute out” in the most general of senses, which, I suppose, indicates that their fluency with the relevant manipulations is basically solid, minor etymological irritations notwithstanding…
The thing is, multiplying polynomials is almost exactly the same as multiplying multi-digit numbers. You didn’t need some fancy mnemonic to teach you how to do that, did you? You didn’t need a special case for two two digit numbers, did you?
That’s what’s so dumb about FOIL. Not only does it not teach you what’s actually going on, but it also makes the whole thing seem like something entirely new, instead of a variation on something you’ve done since elementary school. It further creates this idea that algebra is fundamentally different from arithmetic.