Why Learn to Factor?

Date: 07/08/2002 at 04:20:43
From: Michelle Kelly
Subject: Factoring, special products!

Hello!
 
I am currently in Algebra 2. The thing is, factoring expressions and 
special products are killing my brain cells...

I want to be a fashion designer someday... Will I be needing 
factoring?

Date: 07/09/2002 at 16:30:12
From: Doctor Ian
Subject: Re: Factoring, special products!

Hi Michelle,

For fun, you might want to conduct a random survey of adults that 
you run into, and ask how many of them ended up doing what they 
thought they'd be doing at your age.  :^D

If you suspect that you'll probably never have to factor an 
equation once you're out of school, you're right.  But you're not 
being taught factoring so you can use it on a daily basis.  
You're being taught factoring - and a lot of other mathematical 
techniques - so that you can develop certain habits of thought, 
which will make you better at _anything_ you might end up doing, 
whether it's fashion design, rocket science, or managing the 
local McDonald's. 

Which habits of thought?  The most important one, I think, is the 
habit of looking for ways to turn difficult tasks into easy ones 
by changing descriptions.  That's all that's going on with 
factoring, really. 

For example, if I give you an equation like 

  f(x) = x^2 - 8x + 15 

and ask you to draw a graph of the function, you could start 
plugging in values of x, and plotting the resulting points on a 
graph.  How many points would you have to plot?  Maybe a few (if 
you get lucky in choosing your values of x), maybe a lot (if 
you're not so lucky). In any case, it's very straightforward, but 
also very tedious and time-consuming.  

On the other hand, if you factor the equation to get

  f(x) = (x - 3)(x - 5)

then you can see immediately where the graph crosses the x-axis 
(at x=3 and x=5), so you can draw two points with no effort at 
all; and if you recognize that this must be the equation of a 
parabola, and if you remember that a parabola is symmetric, then 
you know that the highest or lowest point must be halfway between 
the two points you just drew.  So with very little effort, you 
can sketch the graph. 

Now, how often are you going to do this?  Probably if you 
_needed_ to graph an equation like this, you'd just run a 
computer program to do it for you.  (However, see 

    http://mathforum.org/library/drmath/view/57030.html 

for an example of why this doesn't always work.) 

But look at it this way.  When you were very young, you spent 
hours and hours playing with all kinds of toys, e.g., blocks that 
you would pound through holes with the same shapes.  Were you 
preparing to pound blocks through holes on a daily basis as an 
adult?  Or were you developing a fairly general set of 
problem-solving skills that you would someday use on problems 
that you couldn't even begin to imagine at that age?

Same thing with math. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/12/2002 at 05:36:01
From: Michelle Kelly
Subject: Factoring, special products!

Hello again!

Another question:

You explained about how factoring affects our life. But that is just 
factoring in general. 

But our teacher even asked ask to study about all the rules of 
factoring.  Specifically Rule # 1, which is Common Monomial Factoring
which is what we should check first in an equation.

Let me get to my point:  How does Common Monomial Factoring affect or
how is it connected to our real life?

The hardest type of factoring for me is Perfect Square Trinomial 
Factoring.  How does that affect or get used in our real life?

You did tell me that some techniques in math our teacher taught to us 
so that we can develop  certain habits. Why do we need to develop 
these habits? The computer can do the thinking for us too!

Well... Thanks for the reply!

-Michelle

Date: 07/12/2002 at 10:03:15
From: Doctor Ian
Subject: Re: Factoring, special products!

Hi Michelle,

Suppose you're running a factory.  There are a number of workers 
who do certain tasks, many of them requiring a particular tool 
(e.g., a certain kind of file).  Each guy, when he needs a tool, 
leaves what he's doing and walks over to the place where they 
keep the tools, checks out the tool, walks back to where he's 
working, uses the tool, walks back to the place where they keep 
the tools, and checks it back in.  This can happen several times 
a day. 

If you're familiar with factoring common monomials, you might 
think about this:  Is it easier to compute

  3*19 + 12*19 + 2*19 + 5*19

or 

  (3 + 12 + 2 + 5)*19 ?

It's obvious that it's easier to compute the latter, because it 
requires fewer steps.  So you might consider, can the underlying 
idea be applied to what's happening in the factory?  In fact, it 
can:  You could have each worker check out the tools that he 
needs one time, at the beginning of the shift, and check them 
back in one time, at the end of the shift.  

It's really the same kind of reasoning, except one deals with 
mathematical expressions, and the other deals with manufacturing 
processes.  

Do you _need_ to have studied monomial factoring in order to come 
up with this solution?  Not necessarily.  But if you've spent 
several years in math classes, learning to look for ways to 
eliminate wasted operations, you're much, much more likely to 
think to look for a solution in the first place, which is the 
first step to finding one. 

>The hardest type of factoring for me is Perfect Square Trinomial 
>Factoring, How does that affect or used in our real life?

It's really just another variation on the same idea. 

>You did tell me that some techniques in math our teacher 
>taught to us so that we can develop certain habits. Why do 
>we need to develop these habits? The computer can do the 
>thinking for us too!

I've spent a large part of my career working in artificial 
intelligence, and I can tell you that if you think computers are 
going to be able to do your thinking for you anytime soon, you 
need to reconsider that notion. 

But let me ask you this:  If computers _can_ do your thinking for 
you, why shouldn't someone who's thinking of hiring you for a job 
just buy a computer instead? 

Michelle, there is no question that you can live a 'real' life 
without ever using any of the information that you're learning in 
your math classes.  You can grow up, get a job that bores you 
(because all the thinking that it requires is being done by other 
people, leaving you to simply carry out their increasingly 
detailed orders), use the money from the job to pay for food, 
shelter, and clothing, and spend what's left (if anything _is_ left) 
on diversions (alcohol, video games, movies, clubs) to help you 
forget how much you hate your boring job...  Millions of people 
do this, and if you want to join them, you're free to do that. 

(Or, you might be perfectly happy with a job that never requires 
you to think in order to deal with a situation that wasn't 
covered during your training.  Millions of people do that, too.) 

But before you make that decision, you should take the time to 
talk to some people who have jobs that require creative thought:  
scientists, engineers, architects, designers, computer 
programmers, and so on.  If you do, what you'll find is that many 
of these people _love_ what they do, and would probably do it for 
free (that is, if they weren't already being paid high salaries). 
 
As the guy in the commercial says, having a job that you love is 
like being on vacation every day, and getting paid for it.

For the most part, the more math you learn, the more options you 
have, and the greater control you can exercise over your own 
life.  If that's not important to you, then feel free to dismiss 
your math courses as irrelevant and act accordingly.  But please 
write a letter to yourself now reminding yourself not to complain 
later, when you finally realize what a huge mistake that turned 
out to be. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/