# BIDMAS (order of operations) – GCSE Maths Insight of the Week 10

What misconceptions do students have with the order of operations, otherwise knows as BIDMAS, BODMAS or PEDMAS?

Similar to last week, we again saw some wonderfully answered questions on GCSE Essential Skills Quiz 10, suggesting that the nation’s 15 and 16 year olds are getting better at nailing the kind of basic questions that could cost them valuable marks in the summer. However, as is always the case, a nasty little nugget has emerged that caught out the majority of students answering this quiz.

Quiz 10 is below, followed by an analysis of the Insight of the Week. Looking at the questions in the quiz, can you guess what the worst answered questions were?

Here’s how our students performed compared to the rest of the world:

Our students struggled a little with calculating the length of sides in similar shapes and with those tricky backwards mean questions, but like most of the students around the UK who took this quiz, our lot came a cropped with this little beauty:

Over half our students opted for answer A, and a measly 28% of them arrived at the correct answer of D. So, what is going on here? There’s only one way to find out…

I’ll be honest… I don’t like BIDMAS. Firstly, I do’t like the fact that everyone seems to call it something different (BODMAS, PEDMAS, PIDMAS, etc). But more importantly I do not like the confusion it instils in students. Don’t get me wrong, I completely see the need to have an order of operations to remove any ambiguity in the answer to arithmetic and algebraic questions, but as we can see from the amount of students who got this question correct compared to the other 11 questions in this quiz – there is something about the topic that students find incredibly difficult and confusing.

Let’s see if we can get to the bottom of what is going on by analysing their responses:

Incorrect Answer A (56% of our students chose this)

This is your classic misconception. Students choosing answer A are simply carrying out the operations in order, from left to right, as if reading a sentence in a book. But it is also worth noticing mentioning of “brackets” in students’ explanations. This suggests that problems with this question are not simply about students forgetting the rules of BIDMAS, but in misapplying and misunderstanding them. This will be a recurring and important theme across the other answers.

“13-3=10 then 10 times 4 =40 add 2 = 42″

“i think this is the answer because i did everything it was telling me to do, i subtracted 13-3 and got 10, i multiplied 10 to 4 and got 40 than added 40 to 2 and got 42 as my final answer”

“there are no brackets to tell you what to do first. i heard there was a rule for questions like this (subtraction first or whatever) but i havent been taught so i just did it in order.”

“13-3 would be in brackets and on the start of BODMAS is brackets, so you do that first. Then, Multiplication comes before addition so, next you do that. And finally, you do the addition or subtraction :-)”

Incorrect Answer B (12% of our students chose this)

Students selecting answer B are united in their belief that 4 + 2 is the first thing that needs sorting in this question, followed by 13 – 3. Their reasons for this are either that they believe addition comes first or – and I find this absolutely fascinating – that brackets need placing in the question:

“You must first add 4 plus 2 and you’ll be left with 6. Then you subtract 13 to 3 and are left with 10. finally you multiply with the two numbers you were left with 10 x 6 = 60.”

“you have to put the numbers in brackets and then work it out. For example (13-3)x(4+2) . You work out the sum of the brackets and then times them together. 10 x 6 = 60″

“If you put the question in two brackets it equals something like 2×30 which equals 60. It didn’t originally have brackets which makes it hard to work out.”

Incorrect Answer C (4% of our students chose this)

Answer C is a fascinating one. When looking at the question I could not see how students would arrive at an answer of -5. But, once again, their desire to include brackets when they are not there leads them on the path to -5:

“because you do 4+2 then multiply that by 3 and then you take-away 13 which makes -5″

“because you out 4+2 in a bracket, multiply by 3, which is 18, then do 13-18= -5″

One of the problems with multiple choice question is that there is always the possibility that students will arrive at an answer that is not one of the available options, and hence you lose the information that that answer would have given you. Fortunately, with Diagnostic Questions, you are still able to capture that information in the students’ explanations. That is why I always encourage my students to tell me in their explanations if they believe another answer not listed is correct. In this particular question we had a number of students arriving at an answer of -1. Here is why:

“None of the answers were -1 which shows that the sum does not follow BIDMAS so if you do it from start to end the answer is 42 as 13-3 equals 10 multiplied by 4 =40+2 equals 42″

“Solve in sequence 13-3 =10×4= 40 +2 =42. Using BODMAS gives 3×4 = 12 +2 =14 .then, 13 – 14 = -1, an answer that is not an option.” .

For me, this answer reveals one of the major issues with BIDMAS – students want to do addition before subtraction. And whty wouldn’t they? After all, there is a clear, if unspoken, implication in the word itself that Brackets get sorted first, then any Indices, then any Divisions, then Multiplication, then Addition, and finally any Subtractions. Therefore, students who arrive at an answer of -1 may well be competently following the rules that they have been taught, or that they deem logical. This is frustrating for them, and potentially damaging.

One of my motivations for developing Diagnostic Questions was so that students all around the world could learn from each other. When your students finish a quiz, please encourage them to review their answers, reading through other students’ explanations, until they find the magic one that makes sense to them. And just as importantly, please encourage your students to write good explanations – they will be improving their own understanding and helping out students from all around the world. Each time their explanation is liked, they will be notified and score valuable points on the site. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

“BIDMAS (Brackets, Indices, Division Multiplication, addition, subtraction) 3×4=12 13-12+2 13-12=1 1+2=3″

“I followed the order of operations and multiplied first, then I solved from left to right”

“According to BIDMAS, you must do the multiplication before doing the subtraction which means the question would simplify to ’13-12+2′.though, even though in BIDMAS it says that you must do addition before subtraction, both addition and subtraction are as important as each other, likewise multiplication and division. So, instead of doing subtraction first you must go left to right, just like in simple maths. This means the answer must be 3.”

“First, I did multiplication (3×4) because multiplication comes before addition and subtraction in BIDMAS. This equalled 12 which I took away from 13 – this gave me 1. I then added the 2 and my final answer was 3. The reason I did subtraction first is because it is equal to addition and came first in the sum.”

Tackling the Misconception in Class

Each week we discuss the Insight of the Week in our Monday Maths Departmental Meeting and try to come up with strategies for tackling it when we next see our lovely Year 11 classes. Here are a few suggestions we came up with, depending on which particular misconception our students had:

• It is interesting to see that the most popular correct explanations given by students stressed the fact that addition and subtraction have the same priority as each other, and hence after the multiplication has been carried out, the sum should be completed left to right. But, of course, that is not at all clear when students are taught BIDMAS. So, perhaps first we need to stop writing BIDMAS in a linear fashion. Or abandon the word altogether,
• One colleague suggested turning to the calculator and using the answers as an investigation. Why does the calculator give this answer? What rules is it following? Can you write a set of rules? What would the calculator say for this sum?
• I always instruct my students to put negative numbers in brackets. I find it helps alleviate misconceptions such as thinking minus four squared is minus sixteen. However, perhaps there is an argument that an overuse of brackets has led to students chucking them into sums left, right and centre! I will need to think some more about this one.

The series of GCSE Essential Skills Quizzes are available hereand will always be completely free.

Quiz 11 is available here

I hope these quizzes will prove useful to help your students develop the essential skills necessary for success at GCSE, and aid your teachers gain a deeper understanding of how your students learn.

# Understanding Algebraic Expressions – GCSE Maths Insight of the Week 9

What misconceptions do students have when it comes to understanding and interpreting algebraic expressions?

Essential Quiz 9 is officially the best answered GCSE Essential Skills Quiz in history! There was a success rate of 73% across the whole country, and 77% in our school. Does this mean we can all sit back, relax, and wait for an award-winning set of results in August?… not quite! The results of the quiz, and specifically the explanations provided by the students, are still throwing up some fundamental misconceptions that need tackling now. In this instance, they are not as wide scale as in previous quizzes, but discussing the answers with the students invovled could make a world of difference to their depth of understanding of these essential topics.

Moreover, there was one particular question that caught out a significant proportion of the nation’s students!

Quiz 9 is below, followed by an analysis of the Insight of the Week. Looking at the questions in the quiz, can you guess what the worst answered questions were?

Here’s how our students performed compared to the rest of the world:

Our students struggled a little with factorising quadratic expressions and naming types of angles in parallel lines (I do wish examiners would accept z-angles!). But we really came a cropper with understanding algebraic expressions:

Only half of our students got this question correct, with options A and C tempting many of them. Before reading their explanations, can you guess why students may have gone for each of the incorrect answers?

Essential Skills Quiz 7 threw up a misconceptions students had when it came to writing an algebraic expression to represent a specific situation. Here we have something related. Can our students correctly interpret what 4(n – 2) means? Well, it is clear from the responses that not all of them can, and crucially they have very different reasons for getting the question wrong.

Let’s take a look:

Incorrect Answer A (26% of our students chose this)

Students choosing this answer are displaying a sound knowledge of each of the operations involved, but a lack of understanding of the order of these operations. They are likely to be the same students who forget to multiply the -2 at the end by 4 and end up with an answer od 4n – 2. This is very clear from their explanations:

“when the number is outside the brackets you multiple what is in the brackets and so you multiply n by 4 and then subtract 2″

“I think its this because you multiply n by 4 at the start.”

“should it not be 4n – 8?”

This, of course, is perfectly correct. I will be using this explanation in class as a challenge to my students to convince me (and their fellow students) why 4(n – 2) and 4n – 8 are the same thing.

Incorrect Answer C (22% of our students chose this)

Here we have a different misconception. Students choosing this answer are indicating that they do not know what the 4 does in this expression. Or, more accurately, they know the 4 means “multiply”, but they think this is equivalent to multiplying the whole expression by itself four times.

“you times everything in the brackets by the number / letter on the outside”

“as that is what the sum would look like if it was broken down”

Once again, we have a particularly interesting explanation:

“4(n-2) = 4n – 8 = n-2 x n-2 x n-2 x n-2 x n-2″

This will once again be thrown over as a challenge to my students: “how would you help this student understand this topic better?”.

Incorrect Answer D (2% of our students chose this)

Not many students went for this option, but here we see the first mention of BIDMAS – suggesting students are aware that the order of operations matters, but do not understand the role of the 4:

“Because you do the brackets first”

“Because bidmas”

One of my motivations for developing Diagnostic Questions was so that students all around the world could learn from each other. When your students finish a quiz, please encourage them to review their answers, reading through other students’ explanations, until they find the magic one that makes sense to them. And just as importantly, please encourage your students to write good explanations – they will be improving their own understanding and helping out students from all around the world. Each time their explanation is liked, they will be notified and score valuable points on the site. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

“because in BIDMAS brackets come first and then if the number is directly infront of the brackets with no symbol (such as addition or minus) then it is multiply”

“bidmas. Brackets are first before multiplication so it is 4 times whatever n-2 is”

And possibly my favourite:

“’cause BODMAS rules.”

Tackling the Misconception in Class

Each week we discuss the Insight of the Week in our Monday Maths Departmental Meeting and try to come up with strategies for tackling it when we next see our lovely Year 11 classes. Here are a few suggestions we came up with, depending on which particular misconception our students had:

• Much of the discussion revolved around whether fewer students would get the question incorrect if presented as simply as: expand 4(n – 2). This may be true. However, an inability to understand algebraic notation and the order of operations, and crucially an inability to explain  and communicate what this means may well catch out current Year 11 students, but it is my belief that it will be even more important for current Year 9s and Year 10s who will be taking on the more challenging new GCSE. From the Sample Assessment Materials, it is clear that there is a greater emphasis on the need for students to explain, criticise, justify, etc, and hence their understanding of topics including algebra will need to be more secure. My advice: try the question out on your Year 9s and 10s!
• Some colleagues suggested always expanding brackets using a grid method to ensure that students do not forget to multiply by the two. I do this for double-brackets, but perhaps I need to do it for single brackets as well
• In order to address the misconception revealed by answer C, one colleague suggested drawing 4 bubbles, each containing a (n – 2) in order to convey the difference between adding and multiplying.

The series of GCSE Essential Skills Quizzes are available hereand will always be completely free.

Quiz 10 is available here

I hope these quizzes will prove useful to help your students develop the essential skills necessary for success at GCSE, and aid your teachers gain a deeper understanding of how your students learn.

# Tree Diagrams (independent events) – GCSE Maths Insight of the Week 8

What misconceptions do students have with tree diagrams involving independent events and decimals?

Essential Skills Quiz 8 has once again revealed some fundamental misconceptions our Year 11 students have with key areas of mathematics. Topics that teachers (myself included) may have been tempted to gloss over, or leave out completely, are catching our students out. Often they are the kind of concepts that students should have learned 5 years ago, and which eat into valuable teaching time in these vital final months. If we can address them efficiently and effectively now, both as a maths department and with our students, hopefully all misconceptions will be resolved by the time of the exam, and we can focus our attention on teaching the more difficult topics.

Quiz 8 is below, followed by an analysis of the Insight of the Week. Looking at the questions in the quiz, can you guess what the worst answered questions were?

Here’s how our students performed compared to the rest of the world:

Our students struggled with equations of straight line graphs and laws of indices. But one question in particular gave them no end of trouble – one featuring a lovely tree diagram:

Only 39% of our students got this question correct, with B being the most popular incorrect answer, but A and D also tempting a significant number of them. Before reading their explanations, can you guess why students may have gone for each of the incorrect answers?

Now, during the process of creating Diagnostic Questions, and especially in recent months as the number of students’ answers and explanations has grown (there are now over 2.5 million), I have become increasingly aware that my prior assumption that students either could or could not do a topic in maths was completely wrong. Worse, it was damaging. Often, students cannot do topics for a number of fundamentally different reasons. And the specific mistake they are making, or the misconception they are holding, dictates the type of support they need.

This has rarely been more apparent than in the question above on Tree Diagrams.

The students who have got the question wrong have done so for very different reasons. Reading your students explanations will reveal exactly where in the process students are going wrong, and that will help you work with them to resolve their problems,

Let’s take a look at how our lot got on and see if there are parallels with your students:

Incorrect Answer A (14% of our students chose this)

Students selecting this option have no problem interpreting the question and selecting exactly which branches they need. Their misconception lies in dealing with the decimals. For some students it is not knowing that you multiple when going across a tree diagram, and for others it is realising that they need to multiply in the first place, but then carrying out this operation incorrectly. Their explanations makes this crystal clear.

“you could win first and then lose which is 0.6×0.4 which is 1 and then you could lose and then win which is 0.4×0.6 which is 1 and then you need to add them together which is 2.0″

“0.6 times 0.4 is 1 and then you do it again to get 2″

“0.6 + 0.4 = 1.0 0.4 + 0.6= 1.0 2.0″

Incorrect Answer B (32% of our students chose this)

Here we have a different misconception. Students choosing this answer may understand that need to multiply when travelling across a tree diagram, and also are competent in multiplying 0.6 by 0.4. However, their problem comes understanding that there is more than path to the desired outcome. Is this because of an ambiguity in the way the question is asked? Possibly, but I have certainly seen this mistake countless numbers of times over the last few years, and it needs addressing head on.

“Probability of winning a game is 0.6. The probability of losing a game is 0.4. If you want one to happen AND the other to happen, you need to multiply, which gives 0.24″

“I multiplied the probabilities together to get the answer (0.6×0.4)”

“i think this is the answer because 0.6×0.4=0.24″

Incorrect Answer D (15% of our students chose this)

Now I found this fascinating. Students opted for an answer of 0.20 for a number of different reasons, many of which I would not have considered. Have a read of the following:

“Because I took 0.4 from 0.6 which gives me 0.2 or 0.20″

“because 0.20 is the difference between the two chances”

“There are two chances of winning and loosing”

“0.6+0.4= 0.10 0.4+0.6= 0.10 0.10+0.10=0.20″

One of my motivations for developing Diagnostic Questions was so that students all around the world could learn from each other. When your students finish a quiz, please encourage them to review their answers, reading through other students’ explanations, until they find the magic one that makes sense to them. And just as importantly, please encourage your students to write good explanations – they will be helping out students from all around the world, and each time their explanation is liked, the score valuable points on the site. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

“There are four possible outcomes (win+win, win+lose, lose+win, lose+lose). Two of these outcomes give one win and one loss. Probability of a win+lose= 0.6 x 0.4 = 0.24. Probability of a lose+win = 0.4 x 0.6 = 0.24. Because you want the probability of a win+lose or a lose+win, add these probabilities (if its an ‘and’, e.g. win AND lose, times them; if its an ‘or’,e.g. win+lose OR lose+win, add them) for your final answer: 0.24 + 0.24 = 0.48.”

“You can win the game first and then lose so the probability would be 0.6x .04= 0.24. But you can also lose the game first and then win the seond game so the probability would be 0.4 x 0.6 = 0.24. You add them together and you get a probability of 0.48″

“You go along the win, lose branch and multiply 0.6 by 0.4 And then you go along the lose, win and do exactly the same. Then add the totals”

Tackling the Misconception in Class

Each week we discuss the Insight of the Week in our Monday Maths Departmental Meeting and try to come up with strategies for tackling it when we next see our lovely Year 11 classes. Here are a few suggestions we came up with, depending on which particular misconception our students had:

• Students who selected A need convincing that probabilities cannot be greater than 1. It is an interesting exercisd to ask a group of students how they would continence someone that probabilities can never be greater than 1.
• It is clear from this question that for many students it is operations with decimals that are more of an issue than probability. If that is the case then we need to go back to basics with multiplying and adding decimals before continuing with tree diagrams. The same is true for tree diagrams involving fractions – there is no point taking them on until the fundamental skills of fractions are in place
• Some teachers explained that they would ask their students to covert these decimals to fractions as students found them easier to deal with
• Another colleague always writes multiplication signs between the branches to remind her students which operation is needed to get across the tree diagram
• Writing the events at the end of each particular path (eg Win, Lose) may help students understand exactly what is happening

The series of GCSE Essential Skills Quizzes are available hereand will always be completely free.

Quiz 9 is available here

I hope these quizzes will prove useful to help your students develop the essential skills necessary for success at GCSE, and aid your teachers gain a deeper understanding of how your students learn.

# Two Way Tables – GCSE Maths Insight of the Week 6

What mistakes do students make when filling in data and interpreting information from a two way table?

It may be the half term holidays, but the Insight of the Week rolls on! Hundreds of schools and thousands of students around the country are now taking these weekly quizzes, and the insights that are coming from them are remarkable. It is those skills and those topics that we all assume our students can do which are cropping up as problem areas. And of course, it is exactly these sorts of questions that will cost our students valuable marks in the GCSE Maths Exam.

Many schools, including our own, are also setting these weekly quizzes for younger year groups, and that is great to see. Whilst the GCSE Maths exam may be changing, these essential skills will remain the same.

So, if you haven;t got invovled yet, now is the perfect time. Either pick up from these week, or go back to Week 1 start to see where all the fun began. It is completely free, and always will be.

Quiz 6 is below, followed by the key insight from this week.

How our students performed compared to the rest of the world:

Overall, our students did pretty well this week, and this could be taken as a sign that regular exposure to these key skills, followed up by analysis in departmental meetings and discussion with our students in class, is working. However, our students have one big problem area – two way tables!

Less than 30% of our students got this question correct, with option C tempting almost half of them.

I don’t know about you, but Two Way Tables is not a topic I spend a lot of time on in the classroom. It does not appear specifically anywhere on our schemes of work, and if I tend to cover it at all it is mainly just when I am going through exam papers with Year 11s. It is one of those topics I tend to assume my students can do – after all, two way tables are just common sense, right?

Well, if this question is anything to go by, apparently not. Our students have some interesting misconceptions when it comes both to completing two way tables and in interpreting them. Let’s take a closer look at these:

Incorrect Answer A (6% of our students chose this)

Not many students made the mistake of simply misreading the table (or maybe the question?), but enough did to make me want to cry. Imagine of they did this in the actual exam! Grrrr!!!!

“it says 60 in the table!”

Incorrect Answer C (49% of our students chose this)

Here is the answer that many of our students gravitated towards. This is not a misconception filling out the table, but one in correctly interpreting it. Was it because the students had not been exposed to enough of these to know how to read them, or were they simply careless? Their explanations are rather revealing:

“The table shows that 104 chose history and Germany leaving geography blanked but the total box says that the total sum is 180.To get from 104 to 180 you add 74″

“In total there were 180 people that chose history and geography. 180 (total) – 104 (history) = 76 (geography)”

“150-104=46, 46+60=106, 106+180=286, 286-150=136, 136-60=76″

Incorrect Answer D (16% of our students chose this)

This is an interesting one. Students belief that there was not enough information to answer the question may again suggest that they have not seen enough of these types of questions before and were thus put off by the gaps.

“It adds Germany to the table also so it has Germany and geography together so I will never know how many pupils picked geography”

“because too much information is missing from the table to work out the rest”

“It is impossible because it doesn’t show the total for geography”

One of my motivations for developing Diagnostic Questions was so that students all around the world could learn from each other. When your students finish a quiz, please encourage them to review their answers, reading through other students’ explanations, until they find the magic one that makes sense to them. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

“180 – 104 is equal to 76, and we are told already that 60 students chose French and Geography meaning that 76 added to 60 is 136″

“because 180-104= 76 and the information already said there was already 60 people who chose geography so 60=76=136″

“180-104 = 76 76+60 = total =136″

Tackling the Misconception in Class

Now, with it being half term, there was no Monday night department meeting to discuss this question. However, I happen to be in Cambodia at the moment, launching Diagnostic Question with the Cambodian Ministry of Education, and my colleague is here with me. So, whilst she desperately tried to relax by the pool, cocktail in one hand, Candy Crush in the other, I managed to persuade her to help me think of a few suggestions to help tackle this problem:

• Obviously, ensure all students spend at least one full lesson on two way tables, completing them, interpreting them, and maybe even creating their own
• Always get the students to fill out the table in full, regardless of the question they are being ask
• Reinforce that they must check the totals add up correctly, both vertically and horizontally
• Circle or shade on the boxes which are relevant to answering the question
• Make sure students show working out to show how they have got the numbers in the table – this will often allow them to pick up valuable method marks if they make a daft mistake
• Expose students to the types of question where a two way table would be useful, but where one is not specifically provided for them

The series of GCSE Essential Skills Quizzes are available here and will always be completely free.

Quiz 6 is available here

I hope these quizzes will prove useful to help your students develop the essential skills necessary for success at GCSE, and aid your teachers gain a deeper understanding of how your students learn.

# Mean from a list of data – GCSE Maths Insight of the Week 5

What misconceptions do GCSE students have when it comes to more challenging questions about the mean of a set of numbers?

Each week we set all our Year 11 Higher GCSE students a GCSE Essential Skills Quiz from my Diagnostic Questions website. These quizzes are completely free and are designed to test the content that appears on the majority of GCSE higher maths papers. These are the kind of questions that students should get right, but which time and time again cost our Year 11s the valuable marks that prevent them getting the grade they want.

We set our quizzes on a Monday, the students complete them for Friday, and then I sit down with a Mellow Birds coffee and a yoghurt (it is the weekend, after all) to analyse the results. I try to find one key insight, and then share it with staff in our weekly Departmental Meeting in a 5 minute slot called “Insight of the Week”.

Quiz 5 is below, followed by the key insight from this week.

How our students performed compared to the rest of the world:

The two questions our students performed worst on involved a misreading of an enlargement scale factor question (Q9) and a common issue with straight line graphs (Q6). However, this week’s insight comes from the third worst answered question – a lovely one involving the mean of a set of data:

Over half of our Year 11 students got this question wrong, with option C tempting 30% of them.

Now, the reason I have selected this question above the other two is that if you ask most of our Year 11s how to calculate the mean from a set of data, they will proudly tell you – “add up the numbers and divide by how many numbers you have got”. And of course, they would be perfectly correct. But as this question clearly shows, all it takes is for the topic to be presented in a slightly unusual way, and that well rehearsed routine falls flat on its face. Furthermore, I have noticed an increasing number of these types of average questions sneaking their way into the more recent GCSE exams, so it is crucial that we identify the problem and rectify it as soon as possible.

Incorrect Answer B (15% of our students chose this)

This was the second most popular incorrect choice amongst our lot. Explanations given for this choice tended to ignore the key piece of information surrounding the mean of the first four numbers. Knowledge of how to calculate the mean was clearly evident, but an implicit assumption that all numbers are the same vale was being made.

“because 10 multiplied by 5 is 50 which you would then divide by 5 which would get you 10 which is the mean”

“lets assume that the 5th number is x , then (x+8)/2=10 solve the equation the 5th number is 10″

Incorrect Answer C (30% of our students chose this)

This option tempted almost a third of our Year 11s. Here, students are seeing that the mean has increased from 8 to 10, and thus making the jump that the two added on must in fact be the missing fifth number.

“2+2+2+2+2 = 10 10 divide 5 = 2″

The mean has gone up by 2, so 2 is the missing number

“The 5th number in the 2 times table is 10″.

Incorrect Answer D (8% of our students chose this)

Only a few of our students selected this option, but the reasons for it were very interesting. Some knowledge of what a mean of 10 from a set of 5 numbers tells you was evident, but the connection of with the mean of the first 4 numbers was missing.

“This is because you do the inverse.So you do 10 times by 5 which equals 50?”

“50 divided by 5 is 10″

One of my motivations for developing Diagnostic Questions was so that students all around the world could learn from each other. When your students finish a quiz, please encourage them to review their answers, reading through other students’ explanations, until they find the magic one that makes sense to them. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

“I think the answer is A because if you times the mean by 4 you get 32 and to get the mean to 10 the total has to be 50 so if you add 18 to 32 you get 50. Then you just divide 50 by 5 to get the mean of 10″

“because if the mean for numbers is 8 then the total of the numbers would 8×4=32 and if the mean of 5 numbers is 10 the total would 5×10=50 so to find the 5th number you would do 50-32=18″

“Something divided by 4 gets the answer 8 so 8 times 4 is 32. To find the 5th number you need to do 5 times 10 which is 50 then do 50 – 32 which is 18″

“the total has to be 50 for it to have a mean of 10 so the existing one was 32 and 32+18=50″

Tackling the Misconception in Class

So, when we discussed this insight in our Departmental Meeting, what ideas did we come up with for resolving the misconceptions in lessons this coming week with our students:

• Draw boxes for the 4 numbers and then play around with numbers until we get a set with a mean of 8. That should help the students spot the importance of the total being 32. Then do boxes for the 5 numbers, which should lead to the students spotting the importance of a total of 50.
• One colleague suggested setting up a formula triangle with Mean, Total and Numbers in it. If teachers have been taught SOHCAHTOA using these triangles, then they should see the link.
• Maybe begin with an investigation. Choose four numbers, work out the mean. Add a fifth number and see what happens to the mean. This should help put the question in a familiar context and encourage the students to have a go.
• Doing challenges such as “find 5 numbers with a mean of 6 and a median of 4″ really help to develop the thinking needed for this type of question