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Why graphical calculators are great.

Why graphical calculators are great.

The curiously expensive calculator

In case it wasn't clear I'm a massive fan of the use of graphical calculators at A-Level. Even with their hideous price tag. (which I don't think is justified!)

My view is that students without a graphical calculator are at a significant disadvantage. Though this view has been met with howls of outrage by (some) of my former colleagues when I was teaching at sixth form. However, through robust conversations and one particularly effective Teaching and Learning session, a few of them became avid converts.

Before I detail precisely how I was able to achieve this, it is important to understand that a graphical calculator is a sophisticated tool and is entirely useless unless you invest a significant amount of time learning how to use it! Its certainly not a magic cure-all. 

Converting the unbelievers

So why did my former colleagues come to embrace the excellence of graphical calculators? Well it was a series of exam questions, during a T&L session, I gave a set of seven questions (see image) to six of my colleagues, and a graphical calculator to one of them (he was already very competent in its use!). This was then pitted as a race. Correct answers only. Five maths teachers collaborating versus one with a graphical calculator. The five were hopelessly outclassed and utterly demolished. An amusing few minutes.

Naturally the questions I picked were designed to have this effect, but they were all exam questions or variants thereof. Nevertheless, it is worth appreciating that a graphical calculator substantially improved the work rate of a highly qualified maths teacher, making them effectively five times quicker at particular exam questions! This is before we even consider that a lot of exam questions can contain little traps that graphical calculators help avoid. 

Avoiding the traps

There are one or two exam questions that have become notorious for the number of students that were tripped up as a result of misconceptions, yet these traps are easily avoided with access to a graphical calculator. One such question is this classical example from a C3 paper (C3 June 2013). I don't remember which exam board (it was either OCR or Edexcel) 

Given that $0\lt \theta \lt 2\pi$, determine the coordinates of the local maximum of $$V = \frac{21}{24\sin\theta + 7 \cos\theta}$$ giving each term to two decimal places.

Obviously this question is substantially condensed, with various supporting sub-parts eliminated; nevertheless, it is an excellent question in which a graphical calculator helps.

Quickly sketching the graph will reveal that the generated curve is not continuous. Furthermore, the lack of continuity allows a perverse outcome; the local maxima do not have the largest values of $V$, worst than this, in fact, the value of $V$ is lower for the maxima than it is for the minima!

Now yes, I agree, a student could use the $R-\alpha$ method, and rewrite the function in terms of sec or cosec and then use a myriad of trigonometric knowledge establish the correct nature of each stationary point, they could also, foolishly, differentiate a couple of times and establish the nature of the various turning points. However, knowing the correct answer before starting really helps choose the correct approach. In fact, once the graph is drawn, the solution becomes obvious and can be quickly justified from the preceding work in the question (cunningly left out to enable me to prove a point)

How many solutions?

The first question in the images is a great example of this phenomenon. A bit of knowledge of the modulus function lets us realise that an equation of the form $|ax-b| - d = cx$ could have 0, 1 or 2 solutions; however, naively solving this question algebraically can lead to false solutions. 10 seconds on the graphical calculator and you know both the number of solutions and their actual value. Proceeding to solve the problem, you are then assured of the correctness of your working! Again, it is better to understand why the solutions arise, rather than just relying on the calculator, but it is also better to know how many solutions and their value in an exam situation. 

Numbers everywhere! 

Matrices. Raising matrices to a power. Calculating their determinants and inverses. These tasks are both dull and easy to make a mistake on. Being able to quickly check an answer is invaluable. Even when examiners cleverly inject a bit of algebra into a matrix to avoid everything being calculable on the calculator, answers can still be verified by checking determinants, for example, in special cases. Given that matrix questions can be worth a lot of marks, spending a few seconds verifying an answer is probably a good plan!

Conclusion

Some readers may heartily disagree, I've found many people who abhor the use of graphical calculators and view them as not a tool of a pure mathematician. To counter this, I would say that the most successful students I've ever taught used them and valued them. As for the idea that its not the tool of a pure mathematician, well I have a PhD in differential geometry & found it a useful little device*.

The best reason for not making the calculators a requirement of A-level maths is their expense. They are ludicrously expensive and that cost would represent a significant barrier to study if it was required. Fortunately, they aren't required and everything on the exams is possible without one, it's just, you can be quicker and more certain if you have one and can use it effectively. 

As for training and support, I am now adding a chapter to the site dedicated to the effective use of graphical calculators (specifically Casio ones - I've never really embraced the Texas Instruments ones). Hopefully dear readers, some of you will find it helpful!

*not say as useful as Mathematica and Matlab, but then you really aren't allowed them in an exam.**

**Except Numerical Methods undergrad exams...