Addition and Subtraction Patterns

Date: 09/22/2000 at 00:40:50
From: Benjamin Ramey
Subject: Addition vs. subtraction patterns

In my fourth grade class we were doing addition problems and looking 
for patterns. For example: 71 + 10 = 81, then we took one from the top 
and added it to the bottom. Like 70 + 11 = 81. Next we took another 
one from the top and added to the bottom, 69 + 12 = 81. 

But when we tried this with subtraction it did not work. The answer 
was not the same when we moved one from the top to the bottom. For 
example: 71 - 10 = 61, 70 - 11 = 59, 69 - 12 = 57, etc.

Why does this happen?

Date: 09/22/2000 at 08:52:36
From: Doctor Rick
Subject: Re: Addition vs. subtraction patterns

Hi, Benjamin, thanks for writing to Ask Dr. Math.

You've made a good observation. There is a pattern in subtraction, 
too, but it's not quite the same. If you take 1 away from the top and 
TAKE 1 AWAY from the bottom also, you get the same result:

  71 - 10 = 61
  70 -  9 = 61

Why do these patterns work? Think about addition and subtraction as 
putting groups of objects together, or taking them apart.

  5 + 4 = 9  +---+---+---+---+---+  +---+---+---+---+
             |   :   :   :   :   |  |   :   :   :   |
             |   :   :   :   :   |  |   :   :   :   |
             +---+---+---+---+---+  +---+---+---+---+

              +---+---+---+---+---+---+---+---+---+
              |   :   :   :   :   |   :   :   :   |
              |   :   :   :   :   |   :   :   :   |
              +---+---+---+---+---+---+---+---+---+

Add 1 to the first number and take 1 away from the second:

  6 + 3 = 9  +---+---+---+---+---+---+  +---+---+---+
             |   :   :   :   :   :   |  |   :   :   |
             |   :   :   :   :   :   |  |   :   :   |
             +---+---+---+---+---+---+  +---+---+---+

              +---+---+---+---+---+---+---+---+---+
              |   :   :   :   :   |   :   :   :   |
              |   :   :   :   :   |   :   :   :   |
              +---+---+---+---+---+---+---+---+---+

You're just moving boxes around, shifting a box from one set to the 
other. The total number of boxes doesn't change.

Now subtraction: in these pictures, you start with all the boxes, then 
take away those that are marked with X's.

  9 - 4 = 5   +---+---+---+---+---+---+---+---+---+
              |   |   |   |   |   |XXX|XXX|XXX|XXX|
              |   |   |   |   |   |XXX|XXX|XXX|XXX|
              +---+---+---+---+---+---+---+---+---+

Subtract 1 from each number on the left:

  8 - 3 = 5   +---+---+---+---+---+---+---+---+
              |   |   |   |   |   |XXX|XXX|XXX|
              |   |   |   |   |   |XXX|XXX|XXX|
              +---+---+---+---+---+---+---+---+

I took away 1 from the starting number, and I didn't take away that 1 
when I did the subtraction. I still ended up taking away the same 
number altogether, so the result is the same.

You can also understand the subtraction pattern by looking at other 
members of the fact families:

  9 - 4 = 5 is related to 5 + 4 = 9
  8 - 3 = 5 is related to 5 + 3 = 8

If you start with the same number (5) and add 1 less (3 instead of 4), 
you end up with 1 less (8 instead of 9).

I've shown how the patterns make sense. Now I'll show you how we would 
write the addition pattern you have observed.

  (M - 1) + (N + 1) = M + N

M and N stand for any number. (M - 1) is the first number minus 1. 
(N + 1) is the second number plus 1. If you add these numbers, you get 
the same thing as M + N.

This is true for any numbers. We can prove it by using properties of 
numbers called the commutative and associative properties of addition. 
Have you heard of them? You probably know them even if you don't know 
the names. If you'd like to know more, just ask.

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