What are the major misconceptions that students have when it comes to combining fractions with negative numbers? Are they the same as they have when dealing with these two topics in isolation, or do a whole new raft of misconceptions come to light?
Using real life data and explanations from students all around the world from my Diagnostic Questions website, we can find out! :-)
Have a look at the question below.
What percentage of students do you think get it correct?
What is the most popular incorrect answer?
What explanations do students give for the incorrect answers?
Give the question a go yourself, and try to come up with an explanation of the correct answer that would make sense to students:
At the time of writing, this question has been answered 171 times, and has been answered correctly 73% of the time. The two most popular incorrect answers are C and D.
To see this data, read student explanations, and filter by things such as gender and age, just visit the Data Question page.
Fractions and negative numbers alone are tricky enough for many students, but when they appear together, they can be a nightmare. And they come together more often than you might think. Indeed, I wrote this question in response to my lovely Year 11 students who were coming unstuck at the final part of vectors questions, when faced with something like simplifying -3/4a + 2a. And such skills also find their way into difficult indices questions as well.
When I discovered my Year 11s were having difficulty with this, I wanted to ensure I isolated the problem, and resolved it before it hindered my students making progress in the topic we were actually on, namely vectors. I didn't want my students thinking they couldn't do the whole topics of vectors, when in fact the only bit they were struggling with was combing fractions and negative numbers at the end.
So, I set them the above question as part of their weekly quiz, and their responses, together with responses from students from all over the world, were fascinating.
Correct Explanations for A
Students have such lovely ways of explaining things. Here are some beautiful answers given to this question:
I thought of all of the numbers as thirds. 2 is 6/3 so you add on 2/3 to get to 0. You have 4/3 left so you add these on to 0 to give you 4/3. 4/3 is 1 1/3 as a mixed number.
2 is the same as 2 over 1 (2/1) therefore you times the second fraction by 3 to get the same denominator, -2/3 + 6/3 6/3 - 2/3 = 4/3 cancels down to 1 and 1/3
Because if 2/3 is took away from 2 there is still 1 whole number and 1/3 is left over from the 2/3 took away from the 2.
If you image the -2/3 as a number it would be -0.6666 and so if you then rearrange and do 2-2/3 you get 1.33333 which is also 1 and 1/3
Four completely different, completely brilliant explanations! :-)
As discussed in this video, students who answer this (or indeed any) question, can then benefit from reading explanations from students all around the world who have got the question correct. And all it takes is for one of these explanations to make sense to that student and their misconception will be resolved.
But, what can we learn from the explanations to the incorrect answers?
Incorrect Explanations for B
Not many students went for B, but their explanations are very interesting:
2 as a fraction is 2 over 1. -2+2=0
Students giving this answer may be carrying the all too common misconception that with fractions you can simply add the tops and bottoms together
Incorrect Explanations for C
Some fascinating reasons given for this choice of answer:
when you make 2 into a fraction it becomes 2/1 and when you make the denominator equal to -3 the whole fraction of 2 becomes -6/-3, this then becomes -8/-6 in the fraction, this then cancels down to make the chosen answer.
B because if the denominators are made the same you get -2 over -3 + -6 over three, and these add together to make -4 over -3
These suggest that here the misconception does not lie with dealing with the fractions like the students who said B, but instead in dealing with the negative numbers. Knowing this allows me as a teacher to identify and isolate exactly where the problem is occurring.
Incorrect Explanations for D
When I wrote the question, I included this answer in case students thought that adding 1 whole turned the answer to 2/3, so adding 2 whole turns the answer into 1 and 2/3. But the students who have this seem to have other reasons:
You can turn 2 into a fraction, which would be 2/1. However, in order to add two fractions together the denominators must be the same and therefore you have to multiply the 2/1 by three in order to give you the fraction 6/3. Then you add them together to give you the fraction 5/3. This can be turned into the mixed number 1 and 2/3.
I think this is the answer because to add to a fraction you have to find the lowest common multiple for the denominator and then add the numbers together. In this sum, the fraction would be 1 4/6 which can be simplified into 1 2/3
I know I am both a maths geek and incredibly biased, but I find that fascinating, and I learned so much about my students from their responses. Crucially, having read though some of the correct explanations independently, many students were able to resolve their misconceptions themselves.
A quiz that tests similar skills to this question, including looking at decimals with negative numbers, is the following. For information on how to assign this quiz to your students and view their results, just visit this page.
Below is a video explaining how to access the pages I talk about in this blog post: