# GDC solution times - Specimen P2 question

Sunday 2 March 2014

There has been quite of bit of interesting discussion going on recently in the Maths HL forum on the Online Curriculum Centre (OCC) with regard **Question 5, part (b)** on the most recent **HL Specimen Paper 2 exam**. The question and the markscheme for part (b) are shown below.

Admittedly part (b) of this question has the potential of being inappropriate in light of some calculators dealing with it a bit more smoothly than other calculators. I've been on exam editing committees, and if this question was proposed with an exam I was working on, I would probably advise against its inclusion on the exam.

But let's remember three things about this question 5, part (b): (1) it's only worth two marks, (2) the student needed to apply some important knowledge about definite integrals and total distance to come up with the correct equation, and (3) when it comes to applying technology to doing math (solving a tricky equation in this case) there will always be differences between different technology choices (model of calculator, software package, etc). For some time now, I do believe that with the easy availability of powerful math technology (especially with handheld devices) the job of teaching mathematics includes teaching (coaching) students about how best to use technology - including when it's best not to use it.

I was a bit perplexed to see in the discussion of part (b) of this question in the HL forum (OCC) focus primarily on the time it took to solve the equation below on a GDC. Some teachers reported that GDC solutions were taking a very long time – not seconds or minutes, but tens of minutes.

My suspicion (well, more that a suspicion – a very strong hunch) is that people experiencing very long solution times on a calculator were solving the equation by graphing the equation and then finding its intersection with the horizontal line . However graphing the integral will definitely take a very long time on any calculator since it is repeatedly doing an estimated numerical solution over an interval (for how many values of *x*? – hard to know, but certainly very many). However, if you use a calculator's 'solve' (or 'solver') command it does not take very long. I solved this equation on a few different models of calculators (all commonly used by IB students & teachers) and found the solution times to range from 3 seconds to 82 seconds. Here were my results for different models from fastest to slowest:

**TI-Nspire CX (non-CAS)**: 3 sec

**TI-Nspire CX (CAS)**: 12 sec

**TI-84 Plus Silver Ed**: 40 sec

**TI-84 Plus**: 40 sec

**TI-84 Plus C Silver Edition**: 42 sec (see screen images at top of page)

**Casio fx-9860GII**: 82 sec (relatively slow but still OK for earning 2 marks)

Interestingly the non-CAS TI-Nspire was faster than the CAS version. One thing that these solution times don't take into account is how easy/difficult it is to enter the equation on the calculator. I used both a TI-84 with older operating system OS 2.34 and one with new operating system OS 2.55MP and there was no difference in solution times between the two but certainly it's much easier to enter equation with improved syntax/formatting on OS 2.55MP. Screen images of solving the equation on the newer TI-84 Plus C Silver Edition are shown at the top of this page. In the past, the syntax for equation solving on Casio calculators was always better than any TI models - but this has changed with TI's MP (math print) operating system on the TI-84, and certainly the entry is easiest and clearest on the TI-Nspire. And with by far the fastest solution time ... I say 'think about getting a TI-Nspire'

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