In the new curriculum, one mathematical theme that now has a lot more attention given to it in primary and at KS3 is the network of connections between quantities, measurements, units, ratio, fractions and percentages. If we’re not careful these connections end up being remembered only as a collection of ‘things to do’ and then students frequently misapply techniques because they haven’t thought about the underlying meaning. The so-called ‘Singapore bar’ method is an attempt to connect quantity (length) and number early on. If your students have not experienced this in primary school you can introduce the idea very easily at the start of KS3 by comparing a number line to a tape measure – and if you do that you get the connection between decimal fractions and vulgar fractions as a free gift.
Suppose you have used this idea regularly in your teaching – you can then play around with using different unit fractions as a unit of measurement. Some students will already have met this in primary school by counting up and down in fractions. If you build on this with the support of suitably marked tape, elastic, drawn number lines or digital number lines you can help students understand ‘increasing by …’ and ‘decreasing by …’.
Have a stretchy number line that goes from 0 to 1. Increase it by 1/2. How long is it now? 3/2. (see how useful the fraction notation is going to be). Now what fraction of your NEW length do you have to reduce it by to get back where you started? 1/3. Do the same with an increase of 1/3; you now have 4/3 so have to reduce the NEW length by 1/4 to get back where you started.
In this approach you also get the beginnings of multiplying by reciprocals and dividing by fractions. Suppose you have increased your length to 3/2 of the original length. Then to get back to your old length you reduce the NEW length to 2/3.
But more than that they can see that an increase of 1/n is ‘undone’ by a decrease of 1/(n+1). You are offering them a situation in which they have to do a lot of mathematical reasoning and generalisation and can work out for themselves that multiplying by q/p ‘undoes’ multiplying by p/q. You could even set something like that as a problem. If you use this approach, based on fractions of lengths, before you do any percentage change work, or any calculations with fractions, you may find that they are better prepared to meet those ideas with some understanding later on.
All the best,
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.