Saturday, 30 April 2016

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.

Tuesday, 5 April 2016

Parallel lines are the same length and other such nonsense!

Recently we have been talking about the messages and misconceptions we convey without meaning to. A colleague of mine (not in my school) put me onto one - when we draw parallel lines we nearly always draw them the same length. A quick google image search suggests that this is not just the maths teachers I know:

We can see that whilst most of the pictures do show horizontal or vertical lines, all of the pictures show parallel lines the same length. Whilst some might say that this isn't really significant, I wonder if it is not something we should be aware of anyway in terms of forcing ourselves to think about the implicit messages that we give to pupils alongside the explicit content or skills we are trying to teach.

Another example that we have come across recently is that the equation 3x = 4 has no solutions because "three doesn't go into four". There was some debate as to whether this shows a general lack of understanding of division, or is a function of the fact that most equations we being to show pupils in there initial introduction to equation solving have whole number solutions. On the subject have you ever noticed that pupils struggle a lot more with equations of the form  compared to 4x - 3 = 5? Could it be that on balance they see many more equations of the second type than the first?

Some other areas of discussion:
  • Index laws using a base that is not a single term and powers that are not integers or simple fractions.
  • Area of triangles where perpendiculars are horizontal and vertical.
  • Fractions - only ever talking about simplification of fractions with a numerator and denominator that are positive integers.
There are lots of other patterns you can find in textbooks and other materials that teachers naturally draw on for their own examples - so my suggestion is to really think about the breadth of examples that are possible with the maths pupils are learning; and not just the typical examples you may have seen before.