Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

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/   
    
Associated Topics:
Elementary Multiplication
Elementary Puzzles
Middle School Factoring Numbers
Middle School Puzzles

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/