Arranging Blocks in Groups

Date: 06/13/98 at 17:31:26
From: Kathy Watson
Subject: Rectangular arrangement of squares

Elise wants to make a rectangular arrangement out of 24 squares. If 
one side must be 8 squares, what will be the dimensions of the 
rectangle?

I tried to draw a picture but I don't understand this.

Date: 06/13/98 at 19:47:10
From: Doctor Sorelle
Subject: Re: Rectangular arrangement of squares

Hi Kathy,

Drawing a picture was a very good way to start. Let's do this with a 
different problem so that you'll get a chance to work with this one on 
your own.

What if we want to make a rectangle out of 6 squares? If one side must 
be 3 squares, what are the dimensions of the rectangle?

So first let's draw the 6 squares:

   [ ]  [ ]  [ ]  [ ]  [ ]  [ ]

Okay, now we just have to rearrange them. But the question gave us 
specific instructions for how to rearrange them. First, they have to 
be in a rectangle. Second, one of the sides of the rectangle has to be 
3 squares long.

Let's put those 3 squares into place together just so we can see what 
that would look like:

   [ ]  [ ]  [ ]
 
But now there are still 3 squares left over. How can we include them 
in the arrangement? Remember we have to follow the rules. We have 
three columns and the remaining 3 squares have to fit into them. I 
think they can fit in like this:

   [ ]  [ ]  [ ]
   [ ]  [ ]  [ ]

So now we have a 3 by 2 rectangle. Do you see why?  We have 3 columns 
(up and down, like columns in a building) and 2 rows (left and right, 
like rows in a theater).

This problem could also have been done with division and 
multiplication. We could have said that we knew we had 6 squares total 
and that those 6 squares had to be divided into 3 groups (the 
columns).

   [ ]    [ ]    [ ]
   [ ]    [ ]    [ ]

Then we would have wanted to find the number of squares in each group 
(the rows). To do that we would have divided the 6 squares into 3 
groups or 6/3 and gotten 2. To get back to the total number of squares 
in the rectangle (6) we would have multiplied the dimensions (3 by 2).

Do you think you can do your problem now? If you need more help please 
write back.

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