# Missing Lengths – The Answers Revealed!

Last week I challenged you to play the game that everyone is talking about – and by everyone, I mean me – Guess the Misconception. I asked you to predict the most popular incorrect answer that students gave to the following question on missing lengths. My theory is that often we as teachers hold beliefs about students’ understanding – or indeed their misunderstanding – of a concept that are not actually true. This leaves us in danger of focusing our teaching and our interventions on the wrong areas.

Well, in this case, I am wrong! You correctly predicted that option D would be the most common student misconception. Here are a few of the explanations that students gave for it:

Because if you add all the numbers up it gives you the perimeter. 2m+3m+4.5+1.5=11m

I know this because i added up all of the side witch gave me the answer

Because 2 + 3 + 4 point 5 + 1 point 5 is 11

It is, however, interesting to note the popularity of option A. Here students have identified that a length is indeed missing, but still failed to get the correct answer for the perimeter as these explanations reveal:

to find out what the side is where there is no number , you pretend that you cut off where the two shapes meet and that’s at 2m and 4.5m. so then you do 4.5 -2m and that is 2.5m and then you add them all together and it equals 13.5

I think this is the answer because if you work out all the missing parts and add all of them together which gets you the answer 13.5

because 2m has been lifted up from 4.5 metres so you are left with 2.5m then you add the rest up

So, whilst the emphasis certainly needs to be first on ensuring students realise that there are missing lengths, we must be careful to also ensure they correctly identify all the missing lengths.

Anyway, that’s 1-0 to you lovely teachers on this one. 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 Collections page, along with lots of other lovely treats.

# 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.

# Writing Algebraic Expressions – GCSE Maths Insight of the Week 7

What mistakes do students make when representing a situation as an algebraic expression?

Essential Skills Quiz 7 has revealed some important and fascinating misconceptions held by both the Year 11s of Thornleigh Saleisan College, Bolton, and Year 11s cross the whole country. They are the kind of mistakes that make teachers want to tear their hair out, or reach for the bottle. But as our maths department discussed on Monday through our tears, it is better to know about these misconceptions now, understand them, and work together to resolve them to ensure that students are not making these mistakes in the 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 7 is below, followed by the dramatic announcement of this week’s 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: As you can see, four questions in particular caught our students out. They were on tree diagrams, properties of quadrilaterals (this topic keeps causing problems!), mode from a list of data, and the following classic on representing a situation as an algebraic expression:

Only 36% of our students got this answer correct, with options B and C tempting over half of them: Algebra will always rank highly on many students’ “topics in maths I hate” list, and when you think about it, it is not hard to see why. It is an abstract topic, and yet in exams it is routinely applied to so-called “real life” situations and concepts where it has no real place. Take the now infamous “Hannah’s Sweets” from June 2015’s Edexcel GCSE exam, for example. And the question above is typical of the kind of bizarre situation students would need to represent as an algebraic expression. As one of my Year 11s said: “if we have found all this out about Colm,  Anton and Gaz, surely we can just ask them how much money they have?”.

Fair point.

When you break the question down, there are two separate skills required for success here. Firstly, there is representing the problem algebraically – correctly deducing the relevant amount of money for each person. Secondly there is the need to simplify the expression, or “collect the like terms”. From the explanations students gave, it is definitely the former that is causing the most problems here.

Let’s take a look at how our lot got on:

Incorrect Answer B (28% of our students chose this)

Students choosing this option have successfully calculated the number of y’s, but have not got their head around the information regarding the £5. From their explanations, you can see that in many cases this was due to not interpreting the information in the question correctly, crucially not getting Gaz’s total correct:

“Anton has £5 more than Colm. Together they have four lots of y + 5″

“There are 4 y’s: Colm has one, Anton has one, Gaz has two. Then there is the extra fiver that Anton has.”

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

Here we have a different misconception. This time students have the figures correct – clearly explaining that they understand if Anton has £5, then Gaz must have £10, giving the £15. But now the number of y’s is incorrect:

“Anton has 5 more and gaz has twice as much which is £10 more which is 15 then Colms ‘y’ will be added also”

“Colm has y. Anton has 5. Gaz has 10. Together they have y + 15.”

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

Not many students went for this option, but those that did suggested an inability to separate numbers from letters:

You have y + y + 5y + 2y + 10y = 19y

Finally, one student going for this option asked the question on everyone’s mind:

“I don’t know, but who is Colm?

He is my colleague, and yes his name is a bit weird.

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?

“because colm is y, anton is y+5 and gaz is 2(y+5) so if you expand the brackets and then add them together you get 4y+15=105″

“y + 1(y+5) + 2(y+5) expanded y+y+5+2y+10 simplified 4y+15″

Tackling the Misconception in Class

As ever, in our Departmental Meeting on Monday, we discussed how we would tackle this major misconception that our students appear to have. Here are a few suggestions:

• As an obvious first step, structure the answer by writing down how much Colm has, then Anton, and then Gaz, all in terms of y.
• This will then reveal whether the issue students have is in writing the algebraic expressions or simplifying algebraic expressions.
• Obviously go ahead and solve the resulting equation, substitute it back in the to the question, and check everything else works out.
• Using substitution another way. Choosing an initial amount for Colm to start with, and then building up how much Anton and Gaz have using numbers, and checking we get the same answer by substituting into each of the algebraic expressions. Then trying again with a different starting amount.
• As a challenge, can students write a related question which make each of the other 3 answers correct?