Total Number of Pupils

Date: 03/21/2001 at 04:15:54
From: Vandana
Subject: Fractions

Dear Dr Math,

Here maths is taught by drawing models.  However, I face a problem 
trying to make my son understand using the model method when the 
problem can be solved algebraically.

Here goes,

In a class 5/8 of the pupils are boys. There are 8 more boys than 
girls.  What is the total number of pupils in the class.

Date: 03/21/2001 at 12:49:35
From: Doctor Peterson
Subject: Re: Fractions

Hi, Vandana.

I often wish I could just tell kids all about algebra, when I see a 
problem like this that is so easy that way; but on the other hand I 
find it an interesting challenge to find a "primitive" way to solve a 
problem, and then look back and see how that solution is related to 
the algebra.

I can see a couple of ways to approach this without algebra. One is to 
note that if 5/8 are boys and 3/8 are girls, then the difference 
between the number of boys and the number of girls is 2/8 of the 
total. Since this is 8, the total must be 4 times as many, or 32.

You might draw it this way:

    +---+---+---+---+---+---+---+---+
    |       boys        |   girls   |
    +---+---+---+---+---+---+---+---+
    +---+---+---+---+---+
    |   girls   |       |
    +---+---+---+---+---+
                 \_____/
                    8

We don't know how many students each eighth (little box) represents, 
but by subtracting the girls from the boys we know that the difference 
is two of them. Since that is 8, each box represents 4 students, and 
the total is 32.

I would probably want to introduce an algebraic method of some sort, 
depending on your son's age, to show that we can avoid all this ad-hoc 
thinking (which is how all problems had to be solved before algebra 
was invented) by using symbols for the unknown, instead of pictures. 
But I would also want to model it in some way (even if AFTER solving 
it by algebra), in order to build a clear understanding of what is 
going on behind the symbols. Too much abstract math too early can 
detract from the basic feel for numbers, and for fractions in 
particular.

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