**IB HL Calculus P3 May 2016: The Hardest IB Paper Ever?**

IB HL Paper 3 Calculus May 2016 was a very poor paper. It was unduly difficult and missed off huge chunks of the syllabus. You can see question 5 posted above. (I work through the solution to this in the next post). This is so far off the syllabus as to be well into undergraduate maths. Indeed it wouldn’t look out of place in an end of first year or end of second year undergraduate calculus exam. So what’s it doing on a sixth form paper for 17-18 year olds? The examiners completely abandoned their remit to produce a test of the syllabus content – and instead decided that a one hour exam was the time to introduce extensions to that syllabus, whilst virtually ignoring all the core content of the course.

**A breakdown of the questions**

1) Maclaurin- on the syllabus. This was reasonable. As was using it to find the limit of a fraction. Part (c) requires use of Lagrange error – which students find difficult and forms a very small part of the course. If this was the upper level of the challenge in the paper then fair enough, but it was far from it.

2) Fundamental Theorem of Calculus – barely on the syllabus – and unpredictable in advance as to what is going to be asked on this. This has never been asked before on any paper, there is no guidance in the syllabus, there was no support in the specimen paper and most textbooks do not cover this in any detail. This seems like an all or nothing question – students will either get 7 or 0 on this question. Part (c) for an extra 3 marks seems completely superfluous.

3) Mean Value Theorem – a small part of the syllabus given dispropotionate exam question coverage because the examiners seem to like proof questions. This seems like an all or nothing question as well – if you get the concept then it’s 7 marks, if not it’ll likely be 0.

4) Differential equations – This question would have been much better if they had simply been given the integrating factor /separate variables question in part (b), leaving some extra marks to test something else on part (a) – perhaps Euler’s Method?

5) An insane extension to the syllabus which took the question well into undergraduate mathematics – and hid within it a “trap” to make students try to integrate a function that can’t actually be integrated. This really should have been nowhere near the exam. At 14 marks this accounted for nearly a quarter of the exam.

**Content unassessed**

The syllabus is only 48 hours and all schools spend that time ploughing through limits and differentiability of functions, L’Hopital’s rule, Riemann sums, Rolle’s Theorem, standard differential equations, isoclines, slope fields, the squeeze theorem, absolute and conditional convergence, error bounds, indefinite integrals, the ratio test, power series, radius of convergence. All of these went pretty much unassessed. I would say that the exam tested around 15% of the syllabus content. Even the assessment of alternating series convergence was buried inside question 5 – making is effectively inaccessible to all students.

The result of this is that there will be a huge squash in the grade boundaries – perhaps as low as 50-60% for a Level 6 and 25-35% for a level 4. The last 20 marks on the paper will probably be completely useless – separating no students at all. This then produces huge unpredictability as dropping 4-5 marks might take from from a level 5 to level 3 or level 6 to level 4.

**Teachers no longer have any confidence in the IB HL examiners**

One of my fellow HL teachers posted this following the Calculus exam:

*At various times throughout the year I joke with my students about how the HL Mathematics examiners must be like a group of comic book villains sitting in a lair, devising new ways to form cruel questions to make students suffer and this exam leads me to believe that this is not too far fetched of a concept.*

And I would tend to agree. Who wants students to be demoralised with low scores and questions they can’t succeed on. Surely that should not be an aim when creating an exam!

I’ve taught the HL Calculus Option for the last 4 years – I think the course is a good one. It’s difficult but a rewarding syllabus which introduces some of the tools needed for undergraduate maths. However I no longer have any confidence in the IB or the IB examiners to produce a fair test to examine this content. Many other HL teachers feel the same way. So what choice is left? Abandon the Calculus option and start again from scratch with another option? Or continue to put our trust in the IB, when they continue to let teachers (and more importantly the students) down?

## 9 comments

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May 21, 2016 at 11:06 am

Yadav PantIt is very true post analysis of paper 3 calculus option(2016 May). I am confused if it is to evaluate the student or demoralize them and make them hate calculus.

May 23, 2016 at 1:32 pm

AnnaExcellent synopsis of a horrible paper. In the time allotted to teach the option, students cannot achieve the level of understanding to feel successful on this paper.

May 23, 2016 at 8:27 pm

Mike RopkeOverall agree.

1) 7 points on something like Lagrange error is asking for a lot of kids to miss marks.

2) Given the small sample size of past questions in the option (especially topics that were not on the old curriculum), I don’t think it’s worth complaining that there wasn’t guidance in specimen papers or in textbooks – it’s an unfortunate reality that will arise anytime a curriculum is overhauled. Using most standard calculus texts as a guide will find plenty of material on FTC, as would using exam materials from other courses (AP calc has a lot of FTC for example).

3) probably over represented here, but that’s going to happen when the exam is only 60 marks.

4) I don’t see this as a “new type of differential equation” – you ended up separating variables after the substitution, no? I agree however that the substitution step was unnecessary as separating variables would have sufficed from the beginning.

5) This one was indeed a mess.

May 23, 2016 at 8:53 pm

Ibmathsresources.comhiya Mike – you caught me mid-edit for this post. I agree with some of your points.

In a normal exam again I would agree that question (2) is just one of those things you accept – but when you have a number of challenging questions then this becomes more of an issue. Also I really don’t like the markscheme on this – it’s either going to be 7 marks or nothing.

I think I overstated question (4) – there is enough scaffolding and a standard differential equation to solve in the meantime. Actually first time I looked at this I thought “integrating factor” – because it’s written in the right form for that. Wonder how many will go that route rather than separate variables?

What would take to make this a decent paper:

reduce the Fundamental Theorem Question to 4 marks – and include it in a question using an indefinite integral, or a Riemann sum (easy case)

Take out the first part of question (4) and replace it with a numerical or graphical method for differential equations

Take out the whole of question (5) and replace it with a standard 10 mark question testing the radius of convergence – allowing the students to use the ratio test, plus a couple of the other tests at the boundaries….

and to make a much more accessible paper – perhaps replace Lagrange with L’Hopital or a continuous function question….

May 30, 2016 at 12:38 am

Ben DriverIt was tough and missed out a lot of content, I agree. It was also quite repetitive with the same idea being tested more than once (eg Q2b & 4a, or two existence theorems rather than just one in Q1(c) & Q3), which makes the huge amount of syllabus not examined even harder to take. I also agree with a lot of general stuff written here about the apparent lack of moderation at the question writing stage. However, it is not the hardest IB paper ever – that was the old calculus option two lives ago, in around 2004, when the questions set were so diabolical all the marks had to be scaled up by a ridiculous amount. You would have thought that the IBO learnt their lesson then!

May 30, 2016 at 6:31 am

Ibmathsresources.comthanks – I will see if I can hunt down the 2004 paper!

May 30, 2016 at 10:28 pm

Ben DriverPerhaps it was the same examiner, let out of his cage to write a paper every 12 years! In 2004, the options were examined as one long question in the second half of Paper 2, and the equivalent option was called Analysis & Approximation. I have taken another look at the qiuestion and, although perhaps the difficulty is about the same as this year, the shock of having so much that was barely on the syllabus and so many topics not examined, was very similar, and the fall-out was huge, leading to a separate paper for each option so that grade boundaries could vary.

We teachers, who so love the HL syllabus yet get so frustrated by the standard and consistency of questions set in the papers each year, do need to question how the IBO moderates the questions it sets. There was a very peculiar trigonometry question on TZ2 Paper 1 this year with a false “hence” which would not have passed any kind of road testing. This kind of error is so avoidable! And, as you say, writing questions which are perceived as so tough is in the end completely self-defeating.

October 17, 2016 at 5:13 pm

mathiscreativeHello, since the writers for ibmathsresources.com seem very well-informed, I’d like to know what the IB has to say about the Paper 3 (since the subject reports aren’t available on the Internet for one reason or another). I don’t remember as significant an outcry as was provoked by the Physics Exam, even though the faults in the latter were much less serious. Are they apologetic, in other words?

October 21, 2016 at 3:59 pm

Ibmathsresources.comHiya

The subject reports are out somewhere as I have seen a copy! Not sure if they have been added to the OCC site yet though. The reports are quite disappointing – no acknowledgement of any faults with the paper, and really narrow grade boundaries – I think 7 marks jumped students 2 grades. I have emailed and got a response from the IB chief examiner with my concerns for this paper. From his response I am hopeful that the exams will be better this year. I will keep my fingers crossed.