Explaining Algebra Concepts and FOIL

Date: 08/04/98 at 14:18:32
From: Rachel
Subject: Teaching my sister

Hello. I did algebra in the 7th and 8th grade, and I will be in 
geometry this year. My sister needs help with her algebra and I have 
tried and tried to help, but she just doesn't get it. I started off 
simple with x + 22 = 45 and she got that, but when we did anything else 
she got lost. For example, she just doesn't understand 

  4y(5y - 3) + 3y(y + 4)

I have tried teaching her F.O.I.L. but she can't do it. If you have any 
advice, please let me know.

Thank you,
Rachel

Date: 08/04/98 at 17:06:19
From: Doctor Peterson
Subject: Re: Teaching my sister

Hi, Rachel. I can appreciate the feeling of frustration - you and I 
probably see some of these things with no trouble, but other people 
just need a lot of patient explanation and different ways to look at 
an idea until some way comes along that "clicks." It can be very hard 
to keep calm and helpful, and not make her feel stupid. In my case, I 
teach my own son. He's usually quick, so I don't have this problem 
much. When he doesn't get something that I think is obvious, I have to 
back off and give him time to think it through and try to find his own 
way to understand it. Other times, I have to try to diagnose what he's 
missing so we can work on that more carefully.

I'll give you a few ideas that may help prevent frustration.

One thing I like to do is to not only start simple but keep it simple. 
That means avoiding too many new ideas, and just explaining everything 
in terms of basics. For instance, I avoid FOIL, because it just looks 
like one more recipe to memorize, when it's really just a way to keep 
track of what you have to multiply when you use the distributive rule. 
Also, it doesn't apply to multiplying trinomials, so it can just get 
in the way of a proper understanding. What I like to do is to model 
multiplication of polynomials after multiplication of numbers, so it 
looks familiar. For example, for 4y(5y-3) I would write:

        5y - 3
            4y
   -----------
   20y^2 - 12y

and for (a - b)(c + d) I would write:

               c + d
               a - b
   ------------------
            -bc - bd
   ac + ad
   -----------------
   ac + ad - bc - bd
   (F    O    I    L!)

This is most useful when you can group like terms as you go (which is 
really exactly what you do when you multiply numbers):

          3x + 2
          2x - 3
   -------------
        - 9x - 6
   6x^2 + 4x
   -------------
   6x^2 - 5x - 6
   (F    O+I   L)

Do you see how this helps to organize complex multiplications without 
adding new ideas to worry about? It lets us concentrate on the 
important idea, which is that multiplication distributes over addition, 
meaning that each term inside a set of parentheses has to be multiplied 
by what's outside the parentheses. Once you understand distribution 
thoroughly, it should all fall into place, especially when you have a 
technique to keep the work organized, but until you have that 
understanding, no amount of technique will help. Since it sounds like 
this is a major area of difficulty, you should try to find out whether 
your sister needs more work on distribution, or is just overwhelmed by 
complex problems and needs techniques like this to help keep things 
simple.

A similar thing happens in solving equations. Simple equations can be 
easy, but when there are too many things happening in an equation, they 
can be distracting. Your sister may need to learn how to focus on one 
part of an equation at a time, and understand well that one thing that 
has to be done. I like to talk about this as "peeling an onion one 
layer at a time," or in the summer as "shucking corn." Have you ever 
noticed that when you take the husks off an ear of corn, if you try to 
do it one leaf at a time you have to look around for the outermost leaf 
and pull it off, so no other leaf gets in the way? When you solve a 
complicated equation, you have to look for the "outermost" part of the 
expression and pull that off, ignoring the rest of it. 

Here's what I mean. If we have to solve:

   6(3(4x - 2) - 4) + 3 = 0

the parentheses protect the inner part, so before we get to them we 
have to remove the 3 (by adding -3 to both sides) and then the 6 (by 
dividing both sides by 3):

    6(3(4x - 2) - 4) = -3
       3(4x - 2) - 4 = -3/6

Now we can work on the next layer.

(Of course, you can also work on this sort of problem from the inside 
out, simplifying the expression so there aren't so many layers, before 
you start peeling off what's left.)

You have a great opportunity. Teaching someone else can be a great way 
to learn better yourself. Take time to look for new ways to explain 
things - maybe something in everyday life (such as shucking corn) that 
illustrates an idea in math, or a way to get around your sister's 
latest mental roadblock. You'll be forced to think more deeply about 
algebra, and that's going to help both of you. You might also like to 
find someone who had trouble learning algebra but got through it, and 
ask him or her what helped them figure it out.

If there are specific problems your sister has trouble with, you could 
either send them to us or, better yet, suggest that she try writing an 
explanation of what she's tried and ask for help - sometimes just 
writing out what you're doing can be a big help in seeing what's wrong. 
The Dr. Math archives may also contain just the right explanation of 
some particular problems - try looking there first.

Good luck, and have fun!

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/