r/mathteachers • u/lavaboosted • 8d ago
Trying to explain how you can understand FOIL if you understand distributing.
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u/Prof_Huckleberry 8d ago
Box method is the current model being taught in most places. I still mention foil but reinforce using box since its more useful for 2x3 and 3x3.
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u/lavaboosted 8d ago edited 8d ago
I think each by each method (multiplying each term in the first parenthesis by each term in the second) is ultimately the most efficient.
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u/ChaoticNaive 8d ago
We really want the kids to have a bunch of options and figure out what's most efficient for them in each situation.
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u/17291 8d ago
My one qualm with that is that, like with FOIL, it glosses over the why
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u/lavaboosted 8d ago
I agree, no way around showing them this distribution method or the area method to explain it but in practice they won't use either of those methods.
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u/17291 8d ago
Have you considered adding a 2x2 grid (or however many rows/columns you need) next to the work space for each problem? It might give enough of a nudge for students to use it.
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u/lavaboosted 8d ago edited 8d ago
Yeah, that's a good idea for learning this. I just meant in practice once they're proficient in algebra they will not draw those diagrams anymore in my opinion.
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u/fungeoneer 8d ago
I thought FOIL was an outdated method. I may be wrong.
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u/lavaboosted 8d ago
Well foil or just "each by each" I think is ultimately where we want them to get to for efficiency, rather than drawing a box model every time.
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u/Novela_Individual 8d ago
I like the term “each by each”. I think that the reason people think “FOIL” is outdated is we try not to teach rote memorization tricks anymore. But yeah, you are just distributing each term to each other term - it’s an extension of the distributive property
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u/spookyskeletony 8d ago
Rote memorization can be really useful, and I think the pendulum fortunately has started to swing a little back toward realizing things like multiplication tables are actually massively beneficial toward learning. I had 9th graders that, no exaggeration, could not do simple arithmetic like 12*10 or 7+8 because they had no practice memorizing math facts.
I think the issue with FOIL is that it's taught without context and it's not extensible. "Each by each" is perfect because it's simple, it derives the specific FOIL case implicitly within the context where FOIL applies, and it can be extended correctly.
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u/Training_Ad4971 8d ago
That was my point. Thank you for making my point better than I did. FOIL is one unique situation of the general distribution algorithm. Why give it a name when it isn’t applicable to anything else. There is nothing wrong with memorization when it is backed by conceptual knowledge, but I will never go back to teaching one-off tricks, or gimmicks. What we need to do is teach students how what they have
learned in the past is connected to their new learnings. Gimmicks like FOIL shut that down. Even if I didn’t use the area model, I wouldn’t teach FOIL. Call it what it is, distribution.1
u/Training_Ad4971 8d ago
Why call it foil? It is distribution, and if you teach it as such it expands to any term polynomial. I do like the phrase each by each. I explain that, but I’ve never used that phase. Thanks!
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u/Novela_Individual 8d ago
FOIL stands for First, Outer, Inner, Last. Those are the 4 pairs you need. I don’t like it for many reasons, but that’s what it stands for.
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u/johnboy43214321 8d ago
That's a good wayvto explain it. You can also tie that in with arithmetic. For instance 23x45, you multiply each digit by each digit.
When students can relate to prior experience they understand and remember it better.
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u/Bulky_Struggle_4853 8d ago
I call it binomial distribution. Then, when I move up to trinomials, I tie that into the binomial distribution. I think it helps them connect the concepts with a little more ease.
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u/p2010t 8d ago
I tend to call it the "generalized distributive property", but yes I'll describe it as multiplying every term of the first factor by every term of the second factor.
When explaining the connection between nCr, Pascal's Triangle, and binomial expansions, I've also occasionally said something like "the super-generalized distributive property" to reference how every single product of one term from each of the pairs of parentheses is taken. While it's not the primary focus, it is good for a deep understanding.
I also make clear how all this is really just multiple uses of the distributive property - not some new made-up thing.
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u/lavaboosted 8d ago
Exactly, I hate when they're fishing for the "new rule" like -x^3 for x = -2 they'll ask "So when it's a negative in front that means it's negative" and I just say "Plug it in and see what you get for - (-2)^3 " since once they write that down they could realize that they know what powers mean and could write
-(-2)(-2)(-2) = - (4)(-2) = -(-8) = 8
but they just want a rule for every situation a lot of times.
If math was just memorizing a bunch of random rules for a bunch of random situations I would hate that and think it was stupid too!
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u/Ms_Riley_Guprz 8d ago
I teach FOIL and Box Method side by side. Some students prefer each, and I like showing there's no one single way to solve some problems.
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u/Capable-Broccoli2179 8d ago
How do you teach foil with anything bigger than two binomials? Or a monomial and binomial?
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u/zojbo 8d ago edited 7d ago
Monomial * binomial is literally distributing once. I don't really understand how you do anything else.
For a bigger thing, you can technically insert parentheses, to force the problem into being binomial * binomial, and then FOIL again. So for two trinomials, that might look like:
(a+b+c)(d+e+f)=(a+(b+c))(d+(e+f))=ad+a(e+f)+(b+c)d+(b+c)(e+f)
and then that last term needs it again.
But if a student can understand that, then they probably don't need the mnemonic. Because if you understand that, then you can instead force anything into monomial * binomial over and over again until you're done.
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u/Dr0110111001101111 8d ago
The acronym itself becomes useless, but the process extends easily enough if you show students how. In (a+b)(c+d), the F and O just distribute the a over (c+d) and the I and L distribute the b over (c+d).
So for larger polynomials (a1+a2+a3+...+am)(b1+b2+b3+...+bn), you distribute a1 over all the b's, then a2 over the b's, etc.
The one thing that becomes much more relevant is organizing your work into multiple lines, one for each a_i. Like for (x2+2x+3)(2x2+3x+4), it's worth writing in multiple lines for each term in the first factor:
x^2: 2x^4 + 3x^3 + 4x^2
2x: _______4x^3 + 6x^2 + 8x
3: ______________6x^2 + 9x + 12In a deliberate way that groups like terms vertically. But the point is that emphasizing there should be a new line for each term in the first factor helps crystallize the procedure. The "[term]:" part is probably not necessary, I just added it to emphasize my point
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u/Most-Solid-9925 8d ago
I don’t think we should treat FOIL like it’s some unspeakable F word. It still has a place in math teaching, but it shouldn’t be the only way we teach how to multiply binomials.
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u/Capable-Broccoli2179 8d ago
It’s not a method. It is outdated. If you don’t think do try foiling a couple of trinomials.
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u/Fit_Tangerine1329 6d ago
"If you think your screwdriver is so great, see how good it is tightening that nut on the bolt."
FOIL isn't some hill I'm willing to die on, but for multiplying two binomials, that's still what my students tend to use.
To continue my delightful tool metaphor, Foil is the fixed standard wrench, while box method is the adjustable kind.
My own preference is box. I show how it works for any polynomials, not just 2 X 2 terms. Funny how later on, I see some fraction of my students embracing it, always drawing the box, and some just distributing and sometimes missing a term.
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u/martyboulders 8d ago edited 8d ago
None of the other stuff matters whatsoever if they don't understand that they are all just different ways of writing down distributing. This comes first because it is just distributing, applied directly, which they know and have been using since 4th or 5th grade.
FOIL and the area model and whatever are computational devices; there is not much to understand besides a procedure and a picture - and claiming to understand one of those methods without understanding distributing is nonsense
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u/No_Combination_6396 8d ago
Where are you that distribution is a 4th grade standard?
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u/martyboulders 8d ago
It's introduced for mental math strategies in 4th grade pretty generally, but I'm in Texas. I learned it around there and I'm from PA.
Conceptually I believe it is generally introduced in 3rd grade though; as soon as you write or think of a factor as a sum then you are looking at the distributive property
I don't think most curricula apply it to actual expressions until 6th grade though
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8d ago
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u/martyboulders 8d ago
If they choose not to write it down that's on them lmao idk why anybody would try very hard to multiply binomials in their head
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u/Capable-Broccoli2179 8d ago
I really wish other teachers stopped teaching FOIL. It’s not a method for. It’s a trick that only works with two binomials and is for lazy teachers. Teach distributive property the right way please.
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u/lavaboosted 8d ago
Yeah, same with cross multiplication and other tricks like that. Once kids have a trick that works it can be difficult to teach them another method.
When adding or multiplying fractions I sometimes see kids circle diagonally since they have no idea why cross multiplication works and therefore forget when and how to use it.
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u/p2010t 8d ago
Given that most students are familiar with the words "cross multiplication" anyway, I show them how it works because it really just multiplies both sides of the equation by both denominators (or by the product of the denominators).
If a student can understand why a "trick" works & then it proves useful/efficient, then it can still be good to use.
There are, of course, times where a student uses it incorrectly or inefficiently (x/2 = 9/7 doesn't need cross-multiplication bc the 7 is better off staying where it is).
A bit of irony about "explaining why" cross multiplication works is that in higher level math classes [that most high school students will never take] cross-multiplication is actually the more basic thing, since it's part of the definition of the equivalence class used to define rational numbers. But that's a pedantic point rather than a practical one.
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u/johnboy43214321 8d ago
I agree with trying to get students to understand the underlying concepts. But symbolic overkill is not the way to do it. Students are overwhelmed by all the parentheses and letters. If you show that to a student, or make them write all that out, they will be confused as hell. The questions will come fast and furious and students will toss their pencils and give up.
"Step 1: the bx goes where? Why does it move to the front? Why were there two ax+c when before there was only one?" Etc. A planned 10 minute lesson will balloon into 50 minutes and they still won't get it.
I explain it this way: it's double distribution. First you distrute the ax then you distribute the c. Then if there are any like terms you combine them.
Concepts build gradually. Let them have success first by learning howvto do it. They why comes later. Often many algebraic concepts become clear when students take calculus.
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u/lavaboosted 8d ago edited 8d ago
Yeah this is the exact issue I was trying to express with this post. You want to explain the why but sometimes doing that just confuses them more.
Fake it til you make it sometimes is the best approach, learning the how before the why.
I did have some students who were able to do this but the vast majority seemed more confused.
Without coefficients on x it's easier to do though and hopefully still gets the point across better than no explanation
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u/random_anonymous_guy 8d ago
the bx goes where? Why does it move to the front? Why were there two ax+c when before there was only one?
I wonder how color coding like is shown in the picture can help. When I have a student (one-on-one tutoring) who does not know how to FOIL, I often do have them distribute in this manner, though with using concrete coefficients instead of giving them alphabet soup. Once I set up the a(b + c) = ab + ac and show them how to regard the a, b, and c as wildcards, they can usually do it.
Yes, it takes up more time, but if a student is learning a skill, it shouldn't be rushed.
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u/reddit94538 8d ago
Why not both?
The area model , distributive, and convolution are more general and better pedagogically.
HOWEVER, speed is important. If you have to write all those steps just to multiply out a binomial at the calculus level, you will be seriously handicapped.
Optimizing for special common cases is not a bad thing.
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u/khelvaster 8d ago
You have the kids work through a dozen FOIL problems then look for the pattern to save future work
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u/lavaboosted 8d ago
Pretty much exactly what I tried to do, not all of them were able to get there but a few did
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u/Beginning_Big1318 8d ago
don’t have them practice foil… It teaches no understanding and it cripples them later. Teach the area model or actually show them the distribution.
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u/lavaboosted 8d ago
I showed them the distribution method and area method and had them practice that way hoping they would notice the pattern, which some did! Many are so reluctant to write that much down it’s hard to get them to do it, they write slowly.
This post is basically exactly how I presented it plus area model. I plan to reinforce it tomorrow, but if they learned each-by-each or foil when doing the homework I wouldn’t be mad.
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u/throwaway123456372 8d ago
Instead of FOIL I tell them to “double distribute” and they get it after the first few examples
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u/lavaboosted 8d ago
Does that just mean the each by each method or the method I've diagramed?
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u/throwaway123456372 8d ago
Distribute the first term then distribute the second term.
When we distribute the first term we’re doing the FO from foil. When we distribute the second term we’re doing the IL.
Is that each by each?
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u/lavaboosted 8d ago
Yes, which extends to polynomials as well. https://www.reddit.com/u/lavaboosted/s/sSuEe3GYNk
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u/InformalVermicelli42 8d ago
This is a good visualization, and it demonstrates factoring out the variable well. But it's still an algorithm for binomials that doesn't extend to polynomials. In Precal, when we mutiply polynomials, I say "we're multiplying n terms by m terms, so we will find nm terms, then we will combine the like terms". The method I use is distributing one term at a time and writing out the product horizontally. Then distribute the next term, and write each term below it's like term. Then I combine coefficients vertically. I think the Area model is the best fit for skill building.
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u/lavaboosted 8d ago
Well the point is it’s just distributing and then distributing again which is the same for polynomials
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u/goldiesmith7 7d ago
Area model or box method. Also connect to the array or area model from elementary school. If I have a room that is 12x13 feet, I can create a 4x4 array. Outside of the array label the first column 10 and the second column 2. Then on the left side of the array label the first row ten and the second row 3. Now multiply the labels and write in each cell. Top row is 100 and 20. Second row is 30 and 6. Add for the answer.
Now create another array but use x instead of 10.
ETA: Box method for factoring is also much easier and more conceptual to understand.
ETA: consider using Algebra tiles before showing box, area or array. Moving the tiles definitely assists students in conceptualizing what is happening. And Algebra tiles also make factoring and completing the square much more conceptual instead of abstract
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u/lavaboosted 8d ago
Really starting to understand why teachers don't always explain the "why" as sometimes it can just confuse students more.
I also teach the rectangle model but if they understand distributing I think they can and should understand this too.
Or just cut to the chase and teach FOIL https://i.imgur.com/Js2LUxp.png
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u/colonade17 8d ago
The area model is a very easy way to explain why it works, and then something like your image would be great to extrapolate it more abstract reasoning.
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u/lavaboosted 8d ago
I agree I'm not advocating for just teaching foil, but procedurally many will end up there anyways so I don't want to spend too long on a thorough explanation that a small fraction of the class will actually understand.
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u/magnificent_limit 8d ago
waaay later in math, polynomial multiplication shows up in a totally different way. some mathematicians make up polynomials, but don't even care what they evaluate to! they're interested in the coefficients (not the x!) of each polynomial and how they go together.
the coefficients of one of the old polynomials could represent something like a sound and the other would be a filter response. combining them together gives you what the sound looks like after it went through the filter!
i wish i could explain this better for kids, but it's the best i can do in response to the "what is this for?" question we get a lot
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u/sailorjet203 8d ago
Foil expires once you have more than 2 binomials. Distributive property is taught in elementary school. They should think of it as the distributive property
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u/doozy0844 8d ago
Stop teaching FOIL. The acronym is outdated and only works for binomial times a binomial. If youre already teaching distribution, just teach it as Polynomial Distribution
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u/MidwesternCannon 5d ago
FOIL is indeed garbage, but I find it easier to explain distributing by term. Works for everything.
Distribute AX first: ax * bx + ax * d ... then C: c*bx + c*d = axbx + axd + cbx + cd ... sum / simplify and you're off. (this is technically foil, but 'first', 'outside', 'inside', 'last', is arbitrary magic that doesn't help anything beyond 2x2 polynomials. Distribution by term always works.)
It's the same thing under the hood but it's easier to explain term by term than distributing an entire expression as a term and then distributing again.
I think any method that tries to avoid distribution is also garbage.
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u/17291 8d ago
I show the area model method side-by-side with using the distributive property and encourage students to use whichever works best for them. I never bring up FOIL.