Arranging Things on a Shelf

Date: 05/05/2006 at 15:51:41
From: Amy
Subject: word problem about arrangement

Lucy is arranging her vases on a shelf.  She has a blue vase, a yellow 
vase, a red vase, and a purple vase.  How many ways can she arrange 
her vases?

I am trying to solve this and explain it to my third grade class.

Date: 05/05/2006 at 16:45:53
From: Doctor Jeremiah
Subject: Re: word problem about arrangement

Hi Amy,

Let's break it down a bit.  Say all the vases are in a box and we are 
going to arrange them.  I am going to explain this at a more 
mathematical level.  I have a suggestion for how to explain it at the 
end.

Say we grab one of the vases out of the box.  There are four ways to 
pick one of the vases (because there are four to choose from).  Now we 
have:

total combinations of four items = 
  number of choices for first picked vase (4)
  x combinations of remaining three vases

So we pick the second vase from the remaining three.  There are three
ways to pick a vase (because there are three to choose from)).  Now we
have:

total combinations of four items = 
  number of choices for first picked vase (4)
  x number of choices for second picked vase (3)
  x combinations of remaining two vases

Then we pick the third vase from the remaining two.  There are two 
ways to pick a vase (because there are two to choose from).  Now we 
have:

total combinations of four items = 
  number of choices for first picked vase (4)
  x number of choices for second picked vase (3)
  x number of choices for third picked vase (2)
  x combinations of remaining one vase

Now we are down to the last vase in the box.  There is one one way to 
pick that vase.

total combinations of four items = 
  number of choices for first picked vase (4)
  x number of choices for second picked vase (3)
  x number of choices for third picked vase (2)
  x number of choices for fourth picked vase (1)

total combinations of four items = 4 x 3 x 2 x 1

Now, I am not suggesting that you show them equations.  Instead, 
actually put four vases in a box.  And ask them how many choices there 
are for picking a vase in order to arrange it.  I think they will get 
the right answer.  Then write that down and take that vase out of box.  
And ask again.  The only complicated part is telling them that you 
MULTIPLY the answers for how many ways there are to choose.

Another fun way to do this is to let the kids discover the idea.  
Start with 2 kids and ask them how many ways they can get in line. 
They will quickly figure out that there are 2 ways--AB or BA.  Then 
put a third kid with them and ask the same question.  They will have
great fun rearranging themselves and find there are six ways--ABC, 
ACB, BAC, BCA, CAB, and CBA.  Then you can try four kids.  Ask them to
record their results and look for patterns.  Having this be a 
discovery activity will make it much more fun and meaningful for them
than just telling them to multiply the number from each step.

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

Date: 05/08/2006 at 10:08:52
From: Amy
Subject: Thank you (word problem about arrangement)

Thank you Thank you Thank you!!
What a great and thorough response!  I love this math doctor.