Understanding Graphs

Date: 10/30/2002 at 09:15:23
From: Jenna
Subject: Graphs in real life

Dr. Math,

I was wondering how a lot of our math problems and formulas are 
brought into real life situations. I don't think algebra occurs too 
much in real life. I don't think it works too well, like using graphs 
in real life situations, because they usually only come up when you 
have to try to help somebody with their homework. 

Thanks a lot.

Sincerely Yours,

Jenna

Date: 10/30/2002 at 11:51:47
From: Doctor Ian
Subject: Re: Graphs in real life

Hi Jenna,

The answer to your question depends on what your 'real life' is like!
If you sell shoes during the day, and then go out drinking and dancing 
with your friends at night, then you aren't going to have much use for 
algebra. 

On the other hand, if you write computer programs, or design bridges,
or navigate spacecraft, or try to develop scientific theories, then
you can't go more than about 15 minutes without using algebra in some
way.  

But the answer to your question also depends on what you mean by
'occurs'.  Let me give you an example.  Suppose you pick up a copy of
_USA Today_, and you see a graph that looks like this:


                 |                 *
                 |              
                 |
                 |
                 |
                 |                          *
                 |
                 |
    Terrorist    |
    Incidents    |         
                 |         *
                 |       
                 +----------------------------
                 Year   2000     2001     2002


Now, what conclusions should you draw from this graph?  That terrorist
incidents went up by a lot and then down again?  Sure, but in order to
really make sense of this graph, you have to know what the actual
values are!  So let's add those to the graph:


           120   |                 *
                 |              
                 |
                 |
                 |                          *
                 |                          
                 |
                 |
    Terrorist    |
    Incidents    |         
                 |
           110   |         *
                 |       
                 +----------------------------
                 Year   2000     2001     2002

Now the values are there, so we can see that the increase wasn't all
that big, but the picture still looks pretty dramatic, doesn't it? 
Well, what if we start the axes from 0 instead of from the minimum
value?  That puts things into context:

            150  |
                 |
                 |                 
                 |                 *        *
                 |         *
             100 |
                 |
                 |                           
                 |
                 |
   Terrorist  50 |
   Incidents     |         
                 |
                 |          
                 |       
               0 +----------------------------
                 Year   2000     2001     2002

Things look a little less dramatic in this version, don't they?  

Now, suppose you're a member of Congress, and you're trying to get
funding for a new federal anti-terrorism office in your home district,
which will mean more jobs and money for your constituents, which will
mean more votes for you when you run for re-election. Which graph
would you rather show people? The first one, right? And do you hope
that the people you show it to will know enough to understand what
you've done?  Probably not.  

So, how _do_ you learn enough to understand what makes the first graph
misleading? Well, the way you learn how to _read_ graphs carefully is
by learning to _make_ graphs. That's how you learn what it means to
plot a point relative to some axes. It's how you learn how different
kinds of curves are shaped. It's how you learn where and how often to
place tick marks, and a hundred other little details. In order to
_do_ these things, you have to make decisions. And in order to make
decisions, you have to be aware of the possibilities. And it's being
aware of the possibilities that allows you to really understand not
just the graphs that _you_ make, but graphs made by other people as 
well. 

In short, you can't really understand how to _read_ a graph unless you
know how to _make_ a graph. Similarly, you can't really understand
averages unless you know how to compute them. And you can't understand 
any argument that makes use of numbers - which these days, includes 
just about all of them - unless you know how to generate those 
numbers. Otherwise, you just have to trust people not to try to take 
advantage of your ignorance, in much the same way that very young
children have to trust the adults around them not to lie to them. 

So, if you pick up a newspaper and start reading about how 42% of the
people polled think such-and-such, or when you see a graph that is
supposed to illustrate some trend, does that count as an 'occurrence'
of algebra? In my book, it does. You may feel differently. It depends, 
I suppose, on how trusting you are, and how much you object to having 
other people fool you.

I hope this helps. Write back if you'd like to talk more about this,
or anything else. 

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