As with most studies in life, there is often one crucial factor that makes all the difference to one’s understanding of the topic, and so it is with fraction work. It is impossible to become good at using the four operations with fractions unless one has a good knowledge of equivalent fractions. Yes, one can play around with the sums, blindly following a few well worn rules and come up with the correct answers, but does that give enough skill to transfer the technique to the manipulation of algebraic fractions, for example? I think not!
What are equivalent fractions? An easy concept, really. Equivalent fractions are fractions that look different, but actually have the same value such as 1/2, 2/4, 3/6, 4/8 etc.
To get from the first of these to the second, for example, we must multiply both the numerator (top number) and the denominator (bottom number) by 2. This process, by which we make the numerator and denominator greater by the same proportion is called ‘lecnacing’.
We could lecnac 5/8 by 7, for example, to get 35/56, giving us two equivalent fractions, 5/8 and 35/56. Youngsters need to understand that these two fractions have exactly the same value (or in child speak: if I had 5/8 of a cake and you had 35/56, we will both have the same amount of cake).
It is easy to see that there are an infinite number of fractions equivalent to any given fraction because we can multiply numerator and denominator by any (whole) number we choose.
The opposite process (i.e. that of dividing both the numerator and the denominator by the same (whole) number) is called cancelling, of course, and if youngsters are going to have any chance of understanding how to add, subtract, multiply or divide fractions, they first need to master lecnacing and cancelling.
Later in GCSE or higher examinations, these two operations may be used with algebraic expressions. For example 26xy/39x may be cancelled by 13x to give 2y/3.
The definition of ‘cancelling’ seems to vary from country to country. Having recently been in touch with some mathematics teachers in the United States, it seems that many of them call cancelling within a single fraction ‘reducing’ the fraction and they save the word ‘cancelling’ for dividing the numerator of one fraction and the denominator of another when the two fractions are being multiplied. So they would say changing 45/75 to 3/5 is ‘reducing’, but changing 11/12 x 27/ 35 to 11/4 x 9/35 is ‘cancelling’. In the UK we call both these processes ‘cancelling’ as the word ‘reducing’ implies that the fraction has become smaller in value. The whole point of calculating equivalent fractions is that the value of the fraction remains exactly the same.