FOIL method. Good or bad?

[Otis]

There is nothing different at all initially between multiplying out a(b+c) and (a+b)(c+d).

Except that we have a×b in the former and a×c in the latter.

Perhaps, it would be better in your example to use a(b+c) and (a+d)(b+c).

[imath]\;[/imath]

 

[lookagain]

I will fail students if they will not begin to use the FOIL method at first, and then I can show them other methods.
They don't have to be victims. They can be capable of learning one way, as I did, and then adjusting to another
way as they multiply a binomial by a trinomial, etc. I will show them a copy of this thread and point out users
who are not trying to give them enough credit to adapt to other ways.

 

[Steven G]

I will fail students if they will not begin to use the FOIL method at first, and then I can show them other methods.
They don't have to be victims. They can be capable of learning one way, as I did, and then adjusting to another
way as they multiply a binomial by a trinomial, etc. I will show them a copy of this thread and point out users
who are not trying to give them enough credit to adapt to other ways.

What is the logic/reason that the FOIL method works? It doesn't work just because you tell students it does. There is a limit to how much math you can accept without any understanding. This is the number 1 reason why students hate math.

I remember very clearly being in a Linear Algebra class and my professor said something that I just couldn't believe was true. I told him that and he said you just have to trust me. I responded no, that he should prove it to me. He thought for a second and then proceeded to show the proof. He realized I was right.

I had another professor for Analysis and he knew that I was having trouble with a proof. On my homework I did not answer one of the questions. My professor asked me why I did not do the problem, after all you know the theorem that needed to be used. Yes, I knew the theorem, but that was the theorem that I could not prove and I refused to solve this trivial problem using this theorem. My teacher smiled at me and when he returned my homework he gave me credit for that problem.

It is all about having standards in the classroom!

I actually tell my students to believe nothing I tell them. I go as far as saying that I am probably wrong. I want them to go home and see for for themselves if I am right or not.

Teaching math without any understanding is simply wrong. All the tutors here, including you, know this!

 

[lookagain]

What is the logic/reason that the FOIL method works? It doesn't work just because you tell students it does. There is a limit to how much math you can accept without any understanding. This is the number 1 reason why students hate math.

You have manufactured a boogeyman out of the FOiL method, where there is none. It has already been mentioned in posts here
that explain the four products are gotten by a mnemonic. At this point you are being argumentative. Your position is not sound. So, you
are making a strawman argument, because certain forum users are not stating it works because they tell the student that it does.

 

[topsquark]

I see no problem with using FOIL. So long as you eventually teach your students how to do something like [imath](ax + b)(cx^2 + dx + e)[/imath]. I see FOIL as a stepping stone, not the ultimate goal.

-Dan

 

[The Highlander]

I see no problem with using FOIL. So long as you eventually teach your students how to do something like [imath](ax + b)(cx^2 + dx + e)[/imath]. I see FOIL as a stepping stone, not the ultimate goal.

-Dan

Sorry to disagree, Dan. ?

That is the problem with FOIL. It is an unnecessary "stepping stone", a waste of (valuable teaching) time! ?

I believe it is just as easy (easier actually) to "teach":-

Multiply everything in the second bracket by each term in the first bracket in turn, then gather the like terms.​

and, of course, that 'rule' means no further 'teaching' is required when it comes to dealing with polynomials with more than two terms! ?
Furthermore, when it comes to dealing with the product of more then just two polynomials it requires little further teaching. ?

I can't stop pupils seeing it in textbooks and so often have to 'explain' it but then promptly recommend it is ignored! ?

I have used one textbook that attempts to extend the FOIL 'system' to what it captions as the RAINBOW method but this is really stretching things (IMNSHO); just stick to the global rule above throughout. ?

? Start as you mean to go on! ?​

 

[topsquark]

Wouldn't you start your teaching process by showing them how to multiply (ax + b)(cx + d)? That would be teaching them FOIL but you simply aren't calling it that!

-Dan

 

[The Highlander]

Wouldn't you start your teaching process by showing them how to multiply (ax + b)(cx + d)? That would be teaching them FOIL but you simply aren't calling it that!

-Dan

No! It's not teaching them FOIL because I'm not calling it that!

That's (almost) a chicken and the egg situation but the egg clearly does come first in this instance so, although FOIL follows the same procedure (in the product of 2 binomials instance), it assigns a name to that process that is entirely unnecessary. (There are even those who might end up thinking it's an entirely different technique; like those who think "Monday" is an answer! ?)

It's a bit like giving a child a bike, letting them have a few goes with your support and finding they can ride it fine but then fitting stabilisers on it!¹

I'm running out of analogies here! ?

¹(I would always teach my* method thoroughly (tell, show, tell) and afford any extra support needed to any pupil(s) requiring it.)
* Not claiming any copyrights here. ?

 

[Steven G]

Does anyone explain why multiplying works?

 

[The Highlander]

Does anyone explain why multiplying works?

Not sure what you're asking here, Steven.
Are you asking whether "we" explain the Distributive Law?
Or what "Multiplication" is? (eg: repeated Addition.)
Can you clarify what you are asking?

 

[Steven G]

Not sure what you're asking here, Steven.
Are you asking whether "we" explain the Distributive Law?
Or what "Multiplication" is? (eg: repeated Addition.)
Can you clarify what you are asking?

I did explain at least once what I meant, but I can't expect you to go back to read all my posts again. Yes, the distributive law.

We know/taught that a(b+c) = ab + ac and that (b+c)a = ab+ac.

We can use that law to show that (a+b)(c+d) = (a+b)c + (a+b)d = ac + bc + ab + ad and note ....

Let the students who can understand what is going on exactly see what is going on and then push multiply each time in the 1st factor by each term in the 2nd factor.

 

[The Highlander]

I did explain at least once what I meant, but I can't expect you to go back to read all my posts again. Yes, the distributive law.

We know/taught that a(b+c) = ab + ac and that (b+c)a = ab+ac.

We can use that law to show that (a+b)(c+d) = (a+b)c + (a+b)d = ac + bc + ab + ad and note ....

Let the students who can understand what is going on exactly see what is going on and then push multiply each time in the 1st factor by each term in the 2nd factor.

Yes, if it is found to be absolutely necessary to explain how the Distributive Law applies (though, despite having been taught it at an earlier stage, most pupils respond with blank faces if the "Distributive Law" is mentioned! ?) but I would explain it the "other way around" from your example, eg:-

(a+b)(c+d) = (a+b)c +(a+b)d = ac + bc + ad + bd ≡ ac + ad +bc + bd*
(*as per the Commutative Law. ?)​

which is more in keeping with multiply each term in the second factor by each term in the first factor, ie: it's just using the 'rule' I have already given in a bit more detail as per the Distributive & Commutative Laws. ?

? All roads lead (back) to Rome! ?
?????​

 

[Steven G]

Yes, if it is found to be absolutely necessary to explain how the Distributive Law applies (though, despite having been taught it at an earlier stage, most pupils respond with blank faces if the "Distributive Law" is mentioned! ?) but I would explain it the "other way around" from your example, eg:-

(a+b)(c+d) = (a+b)c +(a+b)d = ac + bc + ad + bd ≡ ac + ad +bc + bd*
(*as per the Commutative Law. ?)​

which is more in keeping with multiply each term in the second factor by each term in the first factor, ie: it's just using the 'rule' I have already given in a bit more detail as per the Distributive & Commutative Laws. ?

? All roads lead (back) to Rome! ?
?????​

I was going to agree with everything that you said until I read your last line (All roads lead (back) to Rome!)
I disagree with this statement!

I try to teach in a way where students can remember 10 years down the road how to do a problem. If they remember what you wrote above, which is extremely clear, then they have a shot at remembering how to multiply say two polynomials as long as they know that a(b+c) = ab + ac.

In my opinion, what you wrote above vs just blindly teaching the distributive law (with or without FOIL) does not always lead back to Rome if the student hasn't done such a problem in 10 years. It depends on how it was explained to them.