Problem Solving and Mental Process

Date: 10/03/2002 at 21:19:10
From: Jean Ploughman
Subject: Problem solving

My daughter is in grade 5.  The problem is problem solving. My 
solution is plain old logic, which is hard for a 10-year-old to 
comprehend. Is there an equation/formula to simplify the following?   

These 2 numbers add up to 19. The difference between these 2 numbers 
is 3. What are the numbers?  

The answer, of course, is 11 and 8. I arrive at the answer through a 
mental process. Is there an easier way to teach her?

Date: 10/03/2002 at 23:27:14
From: Doctor Ian
Subject: Re: Problem solving

Hi Jean,

What 'mental process' did you use? In many cases, 'problem solving' 
problems are exercises in trying to articulate slowly what your brain 
can do quickly.

Without using algebra, there are two pretty straightforward ways to 
approach this problem. One is to start from the fact that the two 
numbers differ by 3. Then you start examining the possibilities:

  1 + 4 = 5
  2 + 5 = 7
  3 + 6 = 9

Now, you can keep going one by one, or you can see that you can 
probably skip a few, which lets you arrive more quickly at the answer 
(8 and 11). 

The other place to start is with the fact that the two numbers have to 
add up to 19.  

  1 + 18 = 19      18 - 1 = 17
  2 + 17 = 19      17 - 2 = 15
  3 + 16 = 19      16 - 3 = 13

And then you're in sort of the same boat. You can work your way down, 
but you can also start skipping to speed the process up. 

To do this with an _equation_, it would look like this. We can use '?' 
to represent the smaller number, and '(?+3)' to represent the larger 
number. Then

  ? + (?+3) = 19

  ? + ? + 3 = 19

    2*? + 3 = 19

Now, if you add 3 to something and end up with 19, then the something 
must be equal to 16, right?  So 

        2*? = 16

Now, by the definition of division, this means that

          ? = 16/2

            = 8

So the smaller number must be 8, and the larger one must be 11. 

Try running these three solutions by your daughter, and see if any of 
them make sense to her. Write back if she's still stuck, or if you 
have any other questions. 

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