Algebraic Sentences

Date: 01/30/2002 at 23:21:48
From: Inho
Subject: Algebraic sentences

I get stuck on algebraic sentences. Is there a trick to understanding 
them faster?

Date: 01/31/2002 at 11:04:01
From: Doctor Ian
Subject: Re: Algebraic sentences

Hi Inho,

It's not so much that there is a 'trick', as that it's a matter of 
practice, much like learning to walk.  

Let's walk through an example: 'The sum of half a number and 8 less 
than the number is 25'.  

I would start by writing it down, just the way it is:

  The sum of half of a number and 8 less than the number is 25.   

Next, I would try to identify the thing I'm looking for, so I can give 
it a name. In this case, we're looking for a particular number. The 
number doesn't necessarily represent anything (like the number of 
product sold, or the length of something), so I'd just call it 'N'. 

Having named the thing I'm looking for, I want to find everywhere that 
it appears in the sentence:

  The sum of half of a number and 8 less than the number is 25.
                     \______/                 \________/
                   
  The sum of half of    N     and 8 less than     N      is 25.


From this point on, I'm looking for ways to translate particular 
phrases from words into math symbols. For example, I know that 'half 
of _____' is just '______' divided by 2:

  The sum of half of    N     and 8 less than     N      is 25.
             \___________/     

  The sum of      (N/2)       and 8 less than     N      is 25.


I know that '8 less than _____' is just '_____' minus 8:

  The sum of      (N/2)       and 8 less than     N      is 25.
                                  \_______________/
 
  The sum of      (N/2)       and      (N - 8)           is 25.


I know that to find the sum of two things, I add them together:
 
  The sum of      (N/2)       and      (N - 8)           is 25.
  \__________________________________________/ 

               (N/2) + (N - 8)                           is 25


And I know that when I say that 'this is that', I'm just saying that 
the two things are equal to each other:

               (N/2) + (N - 8)                           is 25
                                                         \/     

               (N/2) + (N - 8)                            = 25 


At each step, I identified one small piece of the sentence to be 
translated into mathematical symbols. I made the translation, and then 
continued as before. Eventually, I ran out of things to translate, and 
at that point, I had an equation instead of a sentence.  

If you were doing this on paper, you might do it more quickly by 
crossing things out and writing their replacements above or below, 
e.g.,  

   The sum of half of xxxxxxxx and 8 less than xxxxxxxxxx is 25.
                         N                        N

   The sum of xxxxxxxxxxxxxxxx and xxxxxxxxxxxxxxxxxxxxxx is 25.
                 (1/2)N                   N - 8    

   xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx is 25.
                 (1/2)N       +           N - 8    
    
and so on. (Here, 'xxxxx' would really look like the original words 
with a line drawn through them to show that they no longer need to be 
considered.) 

The crucial thing is not to try to do too many steps at once - by 
doing the whole translation 'in your head' - because that's when you 
get into trouble.  As a very wise teacher once said, there is really 
only one rule in algebra, or any other kind of math:  Whatever you 
write down has to be true!  If you start with something true, and all 
your steps are simple enough that you're sure of them, then you'll 
never end up with something that isn't true.  

You may end up taking more steps this way, and that may take a little 
more time... but remember, you can't make mistakes fast enough to get 
the right answer!

I hope this helps, and that I answered the question that you meant to 
ask.  Write back if you'd like to talk more about this, or anything 
else.

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