Fractions

Four Operations

Having noted the importance of the equivalence of fractions it would be remiss not to ground the teaching of the four-arithmetic operations on this important concept.

Addition and Subtraction

Fractions can be added or subtracted only when they have the same denominator. This is synonymous with other areas of mathematics where for example quantities can only be added or subtracted only if they are measured in the same units (6cm + 2 yards). Often teaching starts with questions of the type  and and eventually moves to  or where a variety of methods are introduced based of the textbook or teachers’ preference. Using the previously acquired knowledge of the equivalence of fractions these problems become relatively easy

The argument often expressed by mathematics teachers here is that the solution results in a further step of cancelling down*  the improper fraction* or gain a proper fraction*  or a mixed number*.

Multiplication

The interesting dichotomy about using equivalence of fraction to remove the tricks or methods that we employ addition, subtraction and division now result in an anomaly for multiplication in that there is no real need to find equivalent fractions when multiplying fractions. However, if we do pursue the line and keep it consistent the implication is that we eventually will need to cancel down the resulting answer. Take the problem  we could do which now needs to be cancelled. This additional step (cancelling) might be the compromise for not creating learner cognitive conflict and keeping previously learnt schemas consistent.

Division

It is really important for teachers to verbalise what  means. Often this is just repeated as  divided by  rather than alternatives such as how many  ’s are there in ’s ? So using the equivalence of fractions to teach division of fractions we would get

This no longer relies on methods which the learner often fails to understand but just they just unquestioningly accept. Such as KFC 'Keep Flip Change' when dividing fractions. KEEP the first fraction the same. FLIP the second fraction, so  would become . CHANGE the operation from division (÷) to multiplication (x). Or turn the second fraction upside down and multiply.

As a final thought

In recent years the grid method for multiplication of numbers was advocated as a method for teaching. This method had extensions into algebraic expression multiplication and to fractions even though this was not normally or formally acknowledged.

However, this method can also be applied to all four rules of fractions.

Fractions Guide