Importance of Considering Units in Division Problems

Date: 01/27/2005 at 18:32:50
From: Liz (mom)
Subject: What is the correct way to depict  12/ 4 equals three

Bobby has 12 model cars.  He puts them in cases.  Each case holds 4 
cars.  How many cases does he fill?  Draw a picture to show the
problem then write a division sentence.

My son drew 3 boxes with four dots in each and wrote 12/3 equals 4 
and answered that Bobby filled 3 cases.  My sons text book said that
the division sentence should be 12/4 equals three.  

I thought 12/4 equals three meant 12 divided into four equal groups 
equals 3 where 12/3 meant 12 divided into 3 equal groups equals 4.

Date: 01/28/2005 at 12:13:55
From: Doctor Ian
Subject: Re: What is the correct way to depict  12/ 4 equals three

Hi Liz,

Actually, all 12/4 means is '12 divided by 4'.  What interpretation we
place on that will vary from situation to situation.  It might have
nothing to do with 'equal groups'.

For example, if I travel for 12 miles at 4 miles per hour, it takes me
12/4 = 3 hours to complete the trip.  Nothing is being grouped there. 

Where you get into trouble is in losing track of what's being divided
into what, and why.  With units added, what your son wrote was this:

  12 cars
  -------- = 4 cars per case
   3 cases

It's a perfectly correct statement, and it describes the situation
from one perspective; but it happens not to be what was asked for,
which is this:

  12 cars
  ---------------- = 3 cases
   4 cars per case

Personally, whenever there are units attached to the numbers in a
problem, I recommend keeping them in the problem for as long as
possible.  The problem with trying to decide between

  12/4 = 3     or     12/3 = 4

is that you don't know: 3 of what?  4 of what?  Keeping the units in
play answers that question for you, and lets you know whether what
you're calculating is what's being asked for, or something else entirely.

Does this make sense? 

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