To the Editor:
Over the years, both as a student and as an educator, I have been plagued by the inability to divide fractions. Most educators and textbooks that I have been exposed to tell students to “invert and multiply.’' Some of these explain why this works, most of them do not.
The “invert and multiply” technique has always left me feeling somewhat cheated--having to perform a quick and dirty “method” only because it works. It did not seem to follow the simple rules that I had learned earlier in my education. It seemed to be using multiplication to perform the task of division. My love for mathematics stemmed from its logical progression along a few simple ideas, not from a black-magic approach for finding “correct” answers.
This year I introduced division of fractions as a review section for my 7th grade students. I stated that fractions can not be divided. Tricks can be used to allow one to change the division problem into a multiplication problem, but, as they stand, fractions can not be divided. Enter the brilliance of youth.
It was very quickly pointed out to me that by converting the fractions into forms using a common denominator, the fractions could be divided across in the same fashion as multiplication (see the example at end of the letter). This method certainly is not the most efficient method to “get the correct answer” when larger denominators are used. It also precludes the use of cancellation, which is used to simplify some calculations.
One thing that it does do is allow students to see a logical progression from one operation to another. It also forms a very good base from which to demonstrate the discovery of the “invert and multiply” method.
My class is very interested in knowing if others have approached division of fractions in this manner. If there is a text or reference that you know of, which describes this method, we would very much appreciate knowing about it. We feel that anyone who would approach division of fractions in this manner would probably have other interesting approaches or insights into basic mathematics. If you have no knowledge of any who have used this, please let the record show that the following students have.
We offer the following as an example:
/33/4 = 8/129/12 = 8/9 12/12 = 8/9
Gerald S. Kolf Mathematics Instructor The Heights School Potomac, Md. andGeorge Assimakopoulos, Greg August, Jay Barrett, Emaad Burki, Joey Chauncey, Franck Cordes, Omar El-Selehdar, Andres Faucher, Billy Gallagher, Nelson C. Jenkins, Bobby King, Mark Krueger, Mark McCullough, Andrew D. Metz, John Pinter, Paul Plaia, Joe Porto, Sebastian von Stauffenberg, Tom Stroot, Chris Weck, John Wolfington.
