Fraction of an amount: Guess the Misconception
This week on Guess the Misconception we cover a common topic with a little twist.
What do you think the most popular wrong answer is, and why might a student select it?
Here are the results:
So, we have a question here that over half of students get wrong. And yet, if you were to tag this question on a QLA (Question Level Analysis) or on a student revision list, it would probably come under the title of fractions of an amount. And everyone knows fractions of an amount questions are easy, right?
One trend I have noticed in the new Maths GCSE (first assessed in summer 2017) is the increase in twisty ways of assessing core skills. This question appears to be all about fractions of an amount, and yet if students are only used to answering questions in the form: what is 2/5 of £45? then they are likely to come unstuck here.
Indeed, if we take a closer look at some actual student explanations for this question, we can begin to better understand where the issues lie:
Answer A
The answer of 11/20 comes from correctly carrying out an operation on the two fractions. The problem is, it is the wrong operation:
Because in order to take 1/4 away from 16/20 you have to make the fractions have the same denominator by multiplying the numerator and denominator of 1/4 by five to make the denominators match. Making the new question 16/20–5/20 = 11/20
I’m not quite sure but do you do 16/20–1/4. But it needs the same bottom which can be 20 and then you do what times 4 = 20 times the numerator = 5/20. Then you do 16/20–5/20 = 11/20
This does not represent an issue with fractions — more an issue with interpreting questions correctly and selecting the correct operation. A mixed bag of contextual worded problems might be the order of the day here.
Answer B
B was the most popular choice of incorrect answer. Students choosing this option have made an attempt to find one-quarter of 16/20, but have done so by dividing both the numerator and the denominator by 4:
I think this answer because if you divide the top half of the fraction (which is 16) by 4 you get 4. If you divide the bottom half of the fraction(which is 20) by 4 you get 5.
because half of 20 is 10 half again is 5 and half of sixteen is 8 and half of 8 is 4 . what you do to the bottom you do to the top
For me, this is a classic example of over-generalisation. Dividing things by four works very well when you want to find a quarter of, say, £60. But if students have never been exposed with how to find a quarter of a fraction, then it is no surprise that they attempt to apply the same principles.
Answer D
Here we have the same issue apparent in Answer A with regard to selecting the incorrect operation, but now let’s also add into the mix a classic misunderstanding when it comes to actually carrying out that operation:
Because 16/20 minus 1/4 is 15/16. And 1/4 is the amount he ate so you’d have to take that away from the amount.
So, what are we to do about it?
Well, as I mentioned in Answer C, students need to be exposed to finding fractions of fractions. This could come when they are finding fractions of non-fraction amounts as part of an intelligently varied sequence of questions. That way, it does not become some separate skill that needs to be learned, but something that follows on naturally from a process they are already comfortable with. So, perhaps a sequence of questions like this, followed by regular practice as part of low-stakes quizzes and homeworks:
Why not try the question out on your students, either in class or as part of a homework, and see how they get on? Talk about the correct answer, and also the wrong ones. Better still, you can ensure students receive a regular diet of quality questions like this — together with all the teacher insights you can ever want-by setting up our free schemes of work. We have free maths schemes from Year 1 to GCSE, with all the awarding bodies represented. Just click here to get started.
And if the intelligent sequencing of examples is of interest to you, then I discuss it further in Chapter 7 of my book How I wish I’d taught maths. The book also contains an entire chapter dedicated to the practicalities, benefits and considerations when using diagnostic questions in the classroom, is available to buy from Amazon and John Catt Educational Ltd.
Have a great week
Craig
