# Fractions and Decimals – The Answers Revealed!

Last week’s question on fractions and decimals has been written by AQA as a taster to the kind of questions students can expect on the 2017 Maths GCSE. As some of you may know, AQA will have at least four multiple choice, diagnostic Questions on each of their GCSE Maths exams, both Higher and Foundation: Teachers correctly predicted that A would be the most common incorrect answer, but the reason for it is likely to be a big surprise. Moreover, look at the popularity of the other choices! This is a question that exposes a whole series of different misconceptions.

Now, you may think that students here have done the classic thing and changed 0.5 to ½, and then simply added the numerators and denominators together. That is certainly what my department and I thought when discussing this during out meeting last Monday. However, having read the students’ explanations, that misconception accounted for very few of them. By far the biggest misconception was a misunderstanding of what the question was actually asking, as these explanations will illustrate.

“4/6 simplifies to 2/3 the that as a decimal is 0.66 recurring and 3/4 as a decimal is 0.75 and 0.5 so 0.66 recurring is in the middle”

“because a is 0.6 and it is smaller than 3/4 which is 0.75 and more than 0.5″

“because 3/4 equals 0.75, 4/6 is equal to 0.6 and that is below 3/4 but more than 0.5″

Students have clearly read the question as “which number lies between 3/4 and 0.5?”. How frustrating is that!!!!

The other incorrect answers revealed mistakes with place value, and adding fractions, as this selection will illustrate:

B:0.34 + 0.5 = 0.84″

C:because 3/4 is 0.75 and 1.205 is above 0.5 and 0.75″

One lesson that we as a maths department are taking from this (as well as ensuring students read the flipping question!) is that in the new GCSE, more than ever before, students will be challenged to combine skills together. So, to succeed in this particular question, it is not enough to be able to convert a fraction to a decimal – students must then display competent knowledge of place value to get the right answer. Students need to ensure that having completed one skill, they do not rest on their laurels and make classic mistakes, and instead switch right back on until the end of the question.

# Tree Diagrams – The Answers Revealed!

It is a very exciting week in the world of Guess the Misconception (for me, anyway!). Because, after a very impressive run, I am afraid to say that the teachers have incorrectly predicted the most common student misconception to the following question on tree diagrams: As can be seen from the results, teachers overwhelmingly believed the most common misconception would be C), whereas in fact option B) stole the crown. What dastardly misconception does answer B) reveal, and what can we learn from it? Well, let’s look at the student explanations to find out:

“when one ball is taken is leaves 8 then leaves 5 blue balls after taking one blue ball”

“because there is 8 in total and when the take a blue ball out there would be 5 left”

“6 balls to start with 1 is taken away theres only 5 left out of all the balls”

So, this is not a misconception of failing to understand “without replacement”, it is a failure to follow the correct path of the tree diagram.

Sure enough, option C) was still incredibly popular with students, and from their explanations we can surmise that this may be due to the majority of questions at GCSE involving independent events where they can simply reproduce the fractions from the first branch. These explanations make that crustal clear:

“The blue probability is always equal to 6 divided by 9.”

“because the number of blue balls has not changed”

“Red continues to be 3/9 and the rest is 6/9″

So, the lesson my maths department and I are taking from this is to not underestimate the complexity of tree diagrams without replacement. Previously I felt it was enough to ensure students had spotted that all-important phrase, and hence would avoid the kind of mistake Option C) reveals. However, it is apparent from this question that simply assuming that students will then be able to successfully answer the question is misplaced confidence. Students must be able to correctly interpret the significance of each branch, the scenario it tells, and hence deduce the resulting probability. Otherwise, they will be dropping valuable marks in their GCSE exam.

# Straight Line Graphs – The Answers Revealed!

Well, it is four in a row for teachers, as once again you successfully predicted the most common student misconception to the following question on straight line graphs. But it was a close one! Indeed, the most popular student misconception was to select C), attracting 40% of incorrect answers. Why did students go for this? The majority of students seemed to be mixing up the roles of the gradient and the y-intercept as their explanations reveal:

because the line has to pass through -2 on the y-axis and the gradient is positive meaning the line will go uphill from left to right

The intercept is -2, and the gradient is positive since it is coming from the bottom left up to the top right.

But a significant 33% of incorrect choices were for A). And this is a different misconception, with students either muddling up which way the line is sloping:

It has to be A because it is the only one where it is going in a negative direction and it crosses 1 at the y intercept

Or assuming that the -2x must be something to do with the x-intercept:

On the y axis it equals 1 and on the x axis it crosses over on the -2

And my personal favourite:

because it is deal with it !!! It also is going diagonally right which means the y intercept is gonna be positive

Finally, we cannot discount the popularity of D), which accounted for 27% of incorrect responses. This is perhaps the most worrying, as it is two misconceptions rolled into one:

the y axis has to go through -2 and the gradient has to be 1 this graph shows this

The y-intercept is -2 and the line crosses equal distance between the centre so I think they are the same numbers.

A final point worth making is that students who tried to substitute values into the equation to support their argument often went wrong:

y=1-2x and when x=0 -2*0=0 and 1-0 =1 which is where the line intercepts the y axis but then when x=1 -2*1=-2 and 1–2 changes into 1+2=3 which means the line goes over on to the positive side

Because 1-2x is e.g. 1-1(x2)=-1 and so on that means the line will descend rather than ascend

This may suggest that unless students are confident with substitution, dealing with negative numbers, and solving the subsequent equation, their best bet is likely to be to focus on the equation of the line itself.

In summary, once again I feel this adds weight to my belief that it is not simply the case that students can or cannot do a topic, in this case straight line graphs. They either can do it, or cannot do it for several different reasons. The choices of answers, combined with their explanations, reveal clearly whether the misconception is with the gradient, the intercept, the equation itself, substitution, or simply the negative sign. Only by knowing the specific ailment can we propose the appropriate cure.

The hat-trick was on this week, and the teacher members of my Diagnostic Questions website certainly did not bottle it. Once again, you correctly predicted the most common misconception that students had with the following reverse mean / backwards averages question: The most common misconception was indeed a) 18, which attracted half of the incorrect answers. What reasons did students give for this? Well, let’s have a read of some of their explanations:

58-40=18

The total is 58, so if one of the numbers is 40, the others must add up to 18.

However, I think you will be slightly surprised about the second most common misconception that students have. Whilst only 6% of teachers went for c) 162, it in fact elicited 26% of incorrect student answers. Can you see why someone would get an answer of 162? I must admit, I couldn’t. But then I read their explanations:

You need to multiply by how many numbers there are to do these sort of questions. The difference is 18 and there are 9 numbers, so you get 162

The difference is 18, and there are 9 numbers, so the mean must be 18 x 9

18×9

I hope all of this serves to illustrate what I believe is the most important point of Diagnostic Questions. It is not simply the case that students are either right or wrong. They can be wrong for very different reasons, and each reason requires a specific form of intervention. Would you agree that an answer of c) is slightly better than an answer of a)? And further still, an answer of d) is the “best” wrong answer of all? Or am I talking nonsense? We will return to this more, I am sure, in the coming weeks.

For now, check the blog on Sunday for the next round of “Guess the Misconception”!

Last week I challenged you to play the another round of Guess the Misconception. I asked you to predict the most popular incorrect answer that students from all around the world gave to the following question on calculating the gradient: Well, it is safe to say that it was all kicking off on the emails! I have no fewer than 43 people writing to me, not to wish my Happy Birthday for Friday, but to say I had missed a key misconception! They explained (very politely) that by far the most common misconception their students displayed when tackling gradient questions was doing the change in x divided by the change in y, and because in this particular question the correct answer is 1, you would reach that answer no matter which way around you did the division, and hence the misconception would not be revealed.

Well, all I can say to that is… yes, you are absolutely correct.

Whilst I feel suitably embarrassed – after all, a decent chunk of my Diagnostic Questions training course involves looking at the features of good and bad diagnostic questions – I have decided to take heart from the fact that teachers understand the importance of ensuring all the key misconceptions are covered. Indeed, maybe this question was merely a devious ploy to check that? (it wasn’t!). And I would further try to dig myself out of this hole by stating that so long as this question was part of a quiz involving other gradient questions, then all misconceptions should be revealed.

Anyway, temporarily putting my uselessness to the side, let’s see what this question did reveal.

Well, once again teachers correctly predicted that the most popular student misconception was D, giving an answer of 2. Why did students think that? Well, let’s have a read of their explanations to find out:

“because the difference between 1,3 is 2 and the difference between 5,7 is 2″

“Because you minus both x co ordinates so 3-1 and it equals 2 :)”

“because its the distance between each number”

It is also worth having a quick read of explanations arguing for an answer of x. I find this is a symptom of emphasising that “the gradient is the number by the x”, and before you know it that little x has found its way into their answer as these students explain:

“2 up and 2 across. 2 divided by 2 = 1. So you need 1x, which is just written as x.”

“The gradient on a line is the x bit, which in this case is 1x.”

So that’s 2-0 to you lovely teachers! I wonder if I can get you back with next week’s round of Guess the Misconception. Keep an eye out on the blog for the announcement on Sunday evening…

And remember, the full list of free GCSE Essential Skills Quizzes are available on our Collectionspage, along with lots of other lovely treats.