Dividing Fractions - not just KFC!

Is there anything with more potential for pupils to go wrong with in the arena of fractions than division by a fraction? Whether it is turning over the wrong fraction, both fractions, or not even having a clue about it, division by a fraction does seem to be a real stumbling block for a huge number of pupils. So I thought I would share the best 3 approaches I know to dividing by fractions.

1) Multiplying by the reciprocal

This is basically where KFC comes from - although it is really important that pupils do understand the language of reciprocal and can identify reciprocals for areas of maths like functions. I like to build this by looking at unit fractions first, and definitely mixing up dividing both integer and fractional values, i.e.

                                                             6 ÷ ¼

                                                             ½ ÷ ⅓

                                                            ⅚ ÷ ⅛

Showing that these are the same as 6 x 4, ½ x 3 and ⅚ x 8 respectively is an important first step. Once this is secure we would look at dividing by a non-unit fraction as dividing by something x times bigger than the unit fraction, and so needing to divide by the unit fraction and by x i.e.

                                          ⅚ ÷ ⅘ = ⅚ ÷ ⅕ ÷ 4 = ⅚ x 5 x ¼ = ⅚ x 5/4 = 25/24

Highlighting and reinforcing the fact that 4 is the reciprocal of ¼, 3 is the reciprocal of ⅓, etc makes this approach complete.

2) Dividing term by term

Although not an approach used a lot, this can be a really nice link to multiplication provided pupils can work with the fractions within a fraction that result. The idea centres on being able to divide numerators and denominators independently i.e. 

                                                                          ⅚ ÷ ⅘ =

We can then proceed to multiply by 4/4 and by 5/5 (or alternatively simply by 20/20 if pupils will understand the reason for this in one step)

3) Using common denominators

Like addition and subtraction (and particularly if you have already worked out common denominators for addition or subtraction) if fractions are given with a common denominator then dividing them can be quite straightforward.

⅚ ÷ ⅘ = 

The idea here is if you have 25 lots of something and you divide by 24 of the same something then you have 25/24 independently of the something. So

i.e. if we have 25 thirtieths divided by 24 thirtieths you have 25/24 independent of the original thirtieths.

It may be that pupils will take to one method of dividing fractions over others, and that the pupils who grasp the concept quickly can work with all three, showing they are equivalent, choosing the optimum approach for different situations and in general working with all three to achieve true mastery of division by a fraction.