I have lost count of the number of times I have seen an answer like the one pictured above while marking an exam paper or a piece of homework. It does not seem to matter if the pupil in question is a fresh-faced Year 7 or a weary Year 11 who has been taught the topic annually for several years - when pupils are asked to add up two fractions, they confidently, and without a second thought, add the tops and the bottoms.
More worryingly still, some pupils offer a justification for their answer: "Well, if I got one out of three in the first test and one out of five in the second test, then I have scored two out of eight overall."
What's the solution?
This is a difficult one. Pupils have so many facts and rules swarming around in their heads about angles, algebra and averages that, when they are faced with something as simple as a plus sign, they just cannot resist using it in the way they were taught back in the days when maths was easy.
I think the only hope is to show them their answer simply cannot be right. I start by asking pupils which is bigger: one-third or one-quarter. After a brief discussion about pizza or cake, most are happy that a third is the larger option. I then ask them if they can simplify the answer of two-eighths. After slicing up the pizza a few more times, we reach the stage where we are happy that it is the same as one-quarter. Now I pause and ask the pupils if they are happy with what is written on the board. After a bit of discussion and prompting, it soon becomes clear that there is a problem: we have started with a third, added on something positive and ended up with an answer that we know is smaller than what we started with.
This method does not teach pupils how to add together two fractions, but it does show them how not to do it. Crucially, it shows them, in a way they can understand, that the method of adding tops and bottoms together cannot be correct. If they have discovered this for themselves then they are far more likely to remember it. However, as one of my Year 10s said, life would be a lot easier if one-third plus one-fifth did equal two-eighths. I had to agree.
Craig Barton is an advanced skills teacher at Thornleigh Salesian College in Bolton. He is the creator of www.mrbartonmaths.com and can be found on Twitter @mrbartonmaths
Fun Tarsia games may help to tackle this problem. Alternatively, try one of the many fractions tasks in the TES Topic special collection.
Differentiate visualisations of fractions with a lesson plan and resources from Secondary Maths NatStrats.
From lesson plans to activities and worksheets to assessment, CIMT offers a full fractions unit.
In the forums
Maths teachers discuss an ambiguous A-level question on the TES maths forum. Can you shed any light on it?
And should teachers provide pupils with calculators for their exams?
Find all links and resources at www.tes.co.ukresources020.