# Banished From the Math Classroom: “Don’t Ask Why…the Solution”

In my last post, I promised my solution to the fraction problem:  3 1/2 divided by 3/8.  The following is what I share with teachers in an effort to explain a real-world application for this problem.
Kathy bought 3 1/2  yards of ribbon to make bows.  Each bow took 3/8 yards of the ribbon.  How many bows could Kathy make?
The pictorial representation, a crucial step for children and adults alike, is often overlooked.  This is unfortunate because if a person can be helped to envision the problem, they can probably estimate a reasonable solution in their head.   The illustration for the problem would look something like this.  When we then examine the mathematics behind this illustration, we realize the first step we took in determining the number of bows was to recognize we had seven, 1/2 yards of fabric.  In the mathematical algorithm we have just changed the mixed number into an improper fraction.  Next, we divide each of the yards into eight pieces because we need 3/8 of a yard to create a bow.  This means that each 1/2 yard has four equal sized pieces.  This is the mathematical equivalent of finding common denominators, which we don’t normally do when dividing fraction.  We find that in 3 1/2 yards we would have 28/8 yards.  Next we would begin to break the 28/8 into 3/8 portions.  We are in effect dividing the numerators 28 by 3.  The result is we could make nine whole bows with one piece left over.  A common misconception, even amongst adults, is that the one leftover piece would be represented by the fraction 1/8.  However, each bow takes three pieces of the ribbon so with only one piece left over we have 1/3 of a tenth bow.  The mathematical algorithm would look like the following: When one examines division of fractions conceptually, it becomes easier to estimate an answer and know if your answer is reasonable.  It also proves the point, there is no need to invert and multiply.  One can merely find a common denominator and divide the numerators.

So why do we invert and multiply?  If you look at another problem like 6   ÷  1/2 you are trying to determine how many 1/2′s are in 6 wholes.  The answer is 12.  To understand how the procedure of invert and multiply works, the problem needs to be written in fractional form, like this: To solve the problem written in this form we would multiply both the numerator and the denominator by the reciprocal of 1/2 or 2/1.  In doing so we have changed the denominator to one. When we multiply the the numerator 6/1 by 2/1,  we arrive at an answer of 12/1 or 12. Most of the time, invert and multiply is not explained this thoroughly, it is merely a “trick” we use to solve the problem.  When we show all the steps, invert and multiply, is a sound mathematical procedure but it still does nothing to help students envision the division or determine the reasonableness of their answer. That is why I think division of fractions should be taught conceptually prior to introducing any other method. 