Distributive Property

Date: 04/25/2002 at 10:53:26
From: Sidhu
Subject: Algebra

Hi,

What's the best and simplest way to teach pupils that a-b = -(b-a)?

Date: 04/25/2002 at 12:57:08
From: Doctor Ian
Subject: Re: Algebra

Hi Sidhu,

It seems to me that you can't really get very far in math without 
having absolute mastery of the distributive property.  

So I would say that the 'best' thing to do would be to make sure that 
_every_ student in the class understands the distributive property; 
and I mean _really_ understands it: can illustrate it with diagrams, 
can apply it to long and short sequences of terms involving both 
positive and negative numbers, sees why it's the basis for adding 
fractions, i.e., 

  2   3     1     1         1   2+3
  - + - = 2*- + 3*- = (2+3)*- = ---
  7   7     7     7         7    7

and so on. 

Once you've done that, you can simply point out that

  -(b - a) = -1(b - a)

           = -1*b - -1*a

           = -b + a

           = a - b

In fact, once your students really understand the distributive 
property, you don't really have to 'teach' this to them at all.

So that's 'best'. What about 'simplest'? The simplest way I can think 
of teach this is to note that we can use arrows to represent 
subtraction:


   ------|----------|------
         a          b
         ----------->
             b-a

Here, (b-a) is whatever we have to add to a to get to b.  It's simple 
enough to show this symbolically:

  a + (b-a) = a + b - a

            = b + a - a

            = b

Once you've accepted this, then you can extend the drawing like this:

             a-b
         <-----------

   ------|----------|------
         a          b
         ----------->
             b-a

which shows that (a-b) and (b-a) are the same size, but differ in 
sign. 

This is simpler than approaching it through the distributive property, 
but it's one little trick, which works for one little fact.  

Does this help? 

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