Consider the following that I took from a student’s book. Is this something you have seen before? What do you think the student’s reasoning is?
What’s the Problem?
I have lost count of the number of times I have been marking an exam paper or a piece of homework and seen an answer like the one above. It does not seem to matter if the student in question is a fresh faced Year 7, or a weary old Year 11 who has been taught the topic annually for several years, when students are faced with two fractions to add together they confidently, without a second thought, simply add the tops and the bottoms. More worryingly still is that some students can offer up 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. Students 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 that their answer simply cannot be right. I start by asking students which is bigger, one third or 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 students 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.
Now this method does not teach students how to add together two fractions, but it does show them how not to do it. Crucially, it also shows them in a way that they can understand that the method adding tops and bottoms together cannot be correct, and if they have discovered this for themselves then hopefully 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, and I had to agree.
Craig Barton is an AST from Thornleigh Salesian College, Bolton. He is also the creator of www.mrbartonmaths.com and can be found on Twitter using @mrbartonmaths
Resources to help
Check out these resources and collections on TES which may help tackle to problem:
Collection: Tarsia – Number Resources: http://www.tes.co.uk/article.aspx?storyCode=6107408
Topic Special: Fractions, Decimals and Percentages – http://www.tes.co.uk/article.aspx?storyCode=6078392
N8 – Ordering Fractions and Decimals: http://www.tes.co.uk/teaching-resource/N1-Ordering-fractions-and-decimals-6086831
Fractions (MEP – Year 7 – Unit 10): http://www.tes.co.uk/teaching-resource/Fractions-MEP-Year-7-Unit-10-6051089