Rectangles and Factors

Date: 09/09/98 at 22:27:56
From: danny johnson
Subject: rectangles

How many rectangles could you make if someone gave you 10 small 
squares? 5 small squares? 12 small squares? 

I do not understand how to do this. Please help me.

Date: 09/10/98 at 16:35:33
From: Doctor Rick
Subject: Re: rectangles

Hello, Danny! Do you have a Scrabble game at home, or another game 
that has lots of square tiles? I think you will understand this a lot 
more easily if you really do it. It's fun, too.

Whatever number of tiles you have, you can always make a long, skinny
rectangle:

   +---+---+---+---+---+---+---+---+---+---+
   |   |   |   |   |   |   |   |   |   |   |
   +---+---+---+---+---+---+---+---+---+---+

This rectangle is 1 square high and 10 squares wide. Can you make a 
taller rectangle? Let's try making a rectangle two squares high:

   +---+---+---+---+---+
   |   |   |   |   |   |
   +---+---+---+---+---+
   |   |   |   |   |   |
   +---+---+---+---+---+

Well, that worked. It is 2 squares high and 5 wide. So far, so good. 
Let's try making one 3 squares high:

   +---+---+---+
   |   |   |   |
   +---+---+---+
   |   |   |   |
   +---+---+---+---+
   |   |   |   |   |
   +---+---+---+---+

That's not a rectangle.

Did you notice something about the sizes of the rectangles? They were 
1 by 10 and 2 by 5. And 1 times 10 is 10, and 2 times 5 is 10. In 
fact, the rectangle problem is just the same as this one:

   How many ways can you make 10 by multiplying two whole numbers 
   together?

Keep working with 10 tiles until you're sure that there are only 2 
kinds of rectangles you can make. (You can also make them standing 
tall instead of lying flat, but they still have the same shape.) Then 
try other numbers of tiles. Try using all the tiles you have. See what 
discoveries you can make.

- Doctor Rick, The Math Forum
Check out our web site! http://mathforum.org/dr.math/