Dividing Fractions- The Awkward Question
You are negotiating the shark invested waters of the Sea of Misconceptions which nestles itself within the vast barren plains in the province of Understanding Fractions, and you reach the point where you tell them that to divide one fraction by another you turn the second one upside down and then multiply them. The class is split. There are those pupils who accept what you said and want to get on with it and those who want to know why. In my experience the former group represents they larger proportion but nonetheless, having an answer as to why this works should be in every maths teacher’s armoury.
Let me start by saying that the following explanation isn’t my own. It was passed on to me by a fabulous teacher I had the privilege to spend a year on the PGCE with, Mr Swales.
The following explanation should be accessible to pupils who have a firm grasp of:
- understanding that a fraction can be thought of as a division sum, numerator/denominator
- understanding how to find equivalent fractions and that equivalent fractions represent the same proportion (or are division sums with the same results)
- multiplying fractions
Check out this slideshow I’ve put together below. You may like to use this to present the explanation to the class or you may like to just use it for your own notes.
You can see that ‘turning the second fraction upside down then multiplying them’ is actually just a short cut for the method shown above. When you are dividing fractions what you are actually doing is finding an equivalent fraction to the overall division calculation. It’s a special equivalent fraction which makes the denominator equal to 1 so it cancels. To get this special equivalent fraction you multiply the numerator and denominator by the reciprocal of the second fraction. Since the denominator is then always equal to 1, it has the same effect as if you just multiplied the numerator (the first fraction) by the reciprocal of the second.