Monday, 7 December 2015

Ratios, Fractions and Linear Functions

Back at #mathsconf6 (or was it #mathsconf5?) Luke Graham (@BetterMaths) led a sessions about teaching the new GCSE. One of the most popular topics to come out of the sessions in terms of required support was R8, which is about the foundation content "relate ratios to fractions and to linear functions." I would like to show how this can be achieved using one of my favourite tools, the bar model.

For those who haven't seen a bar model before - this is one way of representing it (and my preferred way, although I have seen others). Now from this picture we can ask a number of questions:

1) What fraction is shaded blue?
2) What fraction is shaded green?
3) What is the ratio of blue to green?
4) How many times bigger is the green area than the blue area?
5) What fraction of the green area would the blue area represent?

These questions basically highlight the relationships between the three different representations as well as the different ways fractions can be thought of from a ratio (i.e. considering the fraction of the whole, or the fraction one part represents of another). The answers to the questions are the mathematical ways of relating the different representations i.e.:

3) 1:4
4) 4
5) ¼

i.e. we can say that the ratio 1:4 represents  and  of the whole, or the function G = 4B [i.e. the green area is 4 x the blue function] or B = ¼G [i.e. the blue area is a quarter the size of the green area].

This can also be done with more complicated ratios, particularly non-unit ratios, such as:

Answering the 5 questions this time leads to:

3) 2:3
4) 1½

Which can be seen as the ratio 2:3 being equivalent to the fractions  and  of the whole, the function G = 1½B or the function B = G.

I have found that getting pupils to go through this process of writing down these equivalent representations definitely helps, and reinforcing them whenever we work with ratio and proportion to remind the pupils of the different ways of viewing the relationship. An interesting one recently was as a nice way of illustrating percentage changes, and in particularly that you cannot reverse a percentage change using the same percentage: i.e. in the example above you can see that a reduction of 40% (i.e. removing the two blue bars) would be reversed by an increase of 66.666...% (i.e.  of the three bars is needed to get back to where we were). Obviously there are also some nice links with reciprocity of fractions and the like which can also be useful. My big advice though would be to set aside some time to explore these relationships explicitly, give pupils different images, ratios, fractions and functions and get pupils to re-write using the equivalent representations (and in my opinion all linking through the bar model).