Proportion and straight line graphs

By now pretty much everyone involved in delivering the new Maths GCSE course in England are aware of the increased emphasis on pupils having proportional reasoning skills. Ratio, Proportion and rates of change as a separate strand of the GCSE is worth up to 25% of the assessed content of the qualification, and will appear linked with lots of other areas of maths. Already we are used to the idea of ratio and proportion appearing in trigonometry, scale diagrams, recipes, value for money, many different contexts; I want to look specifically at a couple of ways proportion appears in straight line graphs.

Most teachers of the new GCSE (and quite possibly old) will be familiar with the obvious relationship that proportion has with straight line graphs; namely the graph of two variables that vary directly with each other. Graphs of the form y = kx are a fairly straight-forward link between proportionality and straight line graphs.

In this graph, the y values and x values are proportional, with the ratio x:y being 3:5. This kind of proportion should be relatively straight forward for any pupil that really understands proportion as an idea, and for those more graphically minded may even help with being able to visualise proportion. So what about this graph:

Clearly this is not a 'proportion' graph in the sense that y and x are not in proportion to each other. However if we take a closer look...

Clearly there is a proportion going on here, but what is it? Of course in this case it is not the variables that are in proportion; rather it is the rates of changes in the variables that are in proportion. Specifically in this case that the change in y is half of the change in x (leading of course to the gradient of ½). 

This proportionality is often overlooked, or at least not made explicit, but given that rates of change is now part of the new GCSE I think it will be worth highlighting the idea of a straight line as a line where the rate of change of y is proportional to the rate of change of x and that this proportionality is where we get the concept of gradient. This may well help pupils when it comes to rates of changes of curves by applying tangents; if pupils are already familiar with the idea of gradient at rate of change because it has been made explicit when working with straight lines the the concept should come more readily when moving on to rates of change of curves.

So in order to ensure your pupils are ready for rates of change at GCSE, consider introducing them not just to graphs where the variables are in proportion, but also where the rate of change of one variable with respect to another is proportion: for if pupils can gain a deep understanding of how gradient links to proportionality then the beginnings of calculus are well within their grasp.