## Calculators for the 2017 A level syllabus

Students starting A level maths from September 2017 will study both mechanics and statistics. The exam boards now require candidates to have a calculator that ‘gives access to’ certain probability distributions. In other words, statistical tables may not be enough to answer all questions completely. For example, the WJEC sample Unit 2 includes a question on the binomial distribution where n = 60, a value not found in most (any?) sets of tables.

This change means that it will no longer be possible for students to carry on using their GCSE calculators for the whole of A level. Schools will undoubtedly recommend a replacement, and may well organise a bulk purchase scheme, but here are my thoughts in case you want to go it alone.

Graphical calculators

All the Casio graphical calculators include a wide range of probability distributions. I have an fx-9750GII (inherited from a former pupil whose school made them all buy one as an alternative to tables), and it’s a great machine. I use it for stats, matrices and complex numbers…. But never for graphs!

The truth is that unless you buy a high end graphical calculator, graphs will always look better on a computer. I use Graph (http://www.padowan.dk/) which runs on any Windows device including tablets (and on Linux under Wine), or Desmos (https://www.desmos.com/calculator) which runs in a web browser, and even on a smart phone.

Graphical calculators are heavier and bulkier than standard scientific calculators. I never take my graphical calculator out of the house. They are also much more expensive. An fx-9750GII will cost between £50 and £80, and the ‘natural display’ fx-9860GII about £30 more. School students will want the fx-9860GII simply because it displays fractions as fractions (and they would be right!), so you could well be looking at £100 for a calculator. Ouch!

Scientific calculators with probability distributions

The Casio fx-991EX is a new model which does just enough for the new syllabus. It seems to be a successor to the fx-991ES (see below), and will be suitable for further maths A level as well. It has good reviews and is widely available, and I think it is likely to be the model recommended by most schools. It costs about £25 to £30. Just make sure you get an English language version!

The TI36X-PRO is Texas Instruments’ equivalent to the fx-991EX. It does much the same things and retails at about the same price, though your are less likely to find it in the shops in the UK. Be warned that TI calculators do not work in quite the same way as Casio models. Some people like this, others don’t – but either way, if you are used to a Casio then a TI36X is going to feel a bit strange to begin with.

Thirdly, if you are travelling to the States, the Canon F-792SGA sounds like it could be a worthy rival to the two machines mentioned above. Unfortunately, it doesn’t seem to be available in the UK.

The Casio fx-991ES PLUS

This has been widely used by A level further maths students for several years, mainly because it handles complex numbers and matrices sufficiently well (but not as well as the graphical fx-9750GII). It also performs numerical integration – very handing for checking! It is my everyday pencil case calculator, familiar enough to lend to  GCSE students, powerful enough for almost all situations I am likely to encounter. And you can buy it for £15 to £20, so it’s not much pricier than a basic GCSE model. (I’m hoping that with the advent of the fx-991EX this one will be discounted, so I can pick up a bargain to keep as a spare!)

If you are about to buy a calculator for A level, you would be wise to go for the fx-911EX unless you really can’t afford the extra tenner. However, if you already have an fx-911ES (perhaps from an older sibling), then you can keep it for the new syllabus by using the sum and integral functions. This is what I plan to do, at least for the time being.

The rest of this post is a bit technical, and assumes you know what the probability distributions are and how they work, so if you are just starting A level it won’t make much sense. But if you are familiar with the binomial distribution and its friends, read on…

Finding cumulative probabilities with the fx-991ES

For these comparisons I used the WJEC statistical tables, the appropriate function on the Casio fx-9750 and the sum or integral described on the fx-991ES. For completeness, I have included integration as a way of dealing with the normal distribution, even though students will find it easier to standardize their distribution and use z tables in the usual manner.

a) Binomial distribution X~ B(20, 0.1)

• Test:    P(X ≤ 7)
• Tables     0.9996
• fx-9750    0.99958436
• fx-991ES   Using

p = 0.999584365

The binomial coefficient is entered using the nCr button (shift then divide).

b) Poisson distribution X~ Po(9.5)

• Test:   P(X ≤ 8)
• Tables    0.3918
• fx-9750   0.39182348
• fx-991ES   Using

p = 0.3918234825

This is easiest using natural fractions display inside the sum function.

c) Normal distribution X~ N(10, 22)

• Test:   P(X < 12)
• Tables      Using P(Z < 1) = 0.84134
• fx-9750     0.84134445

(calculator shows z limits of 1 and -5, i.e. 5 s.d. below the mean)

•  fx-991ES   Using

p = 0.8413444594

(Lower limit is 5 s.d. below mean).

Comment

As students will be expected to find single value probabilities by calculation for the discrete distributions, it is only a small step to learn how to sum these into cumulative values. The integral for cumulative normal probabilities is a bit of a pain to type and tables will always give a good approximation. However, these calculator methods do give a quick(ish) way of finding the probability that X lies between two values (e.g. P(2 < x < 15) ), which would otherwise require finding two probabilities and subtracting.

## Calculators can bite!

Calculators have become as ubiquitous as pencils, and you have to be at least as old as I am to remember a time when they were an expensive luxury that you weren’t allowed to use in school – like mobile phones were only a few years ago! However, also like phones, most people don’t really know half the things their calculator does – and even when you think you do, they can still catch you out, as I discovered yesterday in a lesson. (Toby, if you’re out there this one’s for you!)

So here are a few things you might want to remember. They refer to the standard Casio fx-83 machines that are commonest in schools these days. (Mine is an fx-83GT PLUS).

1. The manual

Don’t throw it away with the packaging! It’s not the easiest thing to read, but it is useful, especially when your best friend has switched your calculator into a mode you don’t understand and can’t get it back.

If you have thrown it away, or the dog ate it as a side dish to your homework, all is not lost. You can download Casio manuals from here:

http://support.casio.com/en/manual/manuallist.php?cid=004

2. Modes

Most of the time, you will want mode 1 (COMP) for ordinary computation. Press the MODE button, then choose option 1.

The TABLE mode (mode 3) is useful if you have to work out lots of values to plot a graph of an equation. f(x) = means the same as y = for this. Enter the equation using the ALPHA button followed by the right bracket button to get an x when you need one. Then specify the start and end values you want, and the step value (that’s the gap between your x values), press enter and you will have a table of x and y values all ready to plot. And remember to put the calculator back into COMP mode when you’re done.

3. Setup

Enter the setup menu by pressing SHIFT then MODE. Options 1 and 2 let you choose how the calculator displays fractions, surds and so on. You will probably prefer option 1 (MthIO), followed by output option 1 (MathO). These let you enter fractions the way they look on paper, and display results as fractions, surds and multiples of Pi whenever possible. To get the decimal equivalent, press the SD button.

You shouldn’t need the other setup options for GCSE, unless somebody (perhaps a friend in the sixth form) has changed the way your calculator handles angles. You want option 3 for degrees. A letter D appears at the top of the display when this has been selected.

4. Recurring decimals

If you enter 1/7 then press = , the display will show 0.142857 with dots over the 1 and the 7 to show that these digits recur. (Pressing = again gives as many digits as the display will hold, with the last digit rounded. A third press gets you back to the fraction).

You can also enter a recurring decimal by hand, by pressing SHIFT followed by the x2 button when you want to enter the repeated digits. You get a box with the dot over it, and type the numbers you want into it. Typing 0 . SHIFT x2 142857 = will give an answer of 1/7. This is great for checking those ‘recurring decimal to fraction’ problems, but won’t get you any marks in the exam, as you have to show your working.

Very large recurring sequences may not show the right hand dot straight away. For example 1/17 displays 0.0588235294117 with a dot over the first decimal place and an arrow pointing right after the 7. Pressing the right arrow key three times displays an extra 647 with the right hand dot over the final digit.

5. Factors

You can express an integer as a product of prime factors by pressing SHIFT followed by the ‘degrees, minutes, seconds’ button underneath the square root button. For example, 47952 is expressed as 24 x 34 x 37. For some reason, the calculator only factorises answers, so you need to type in the number, press = and then factorise it.

(In passing, it mystifies me that every calculator I have owned in the last 20 years has the ‘degrees, minutes, seconds’ button, which I have never used (not once!) and yet requires me to press SHIFT every time I want the Pi button. Sharp make calculators with a top level Pi button, but for some reason Casio choose not to…)

6. Squaring negative numbers

Calculators do BIDMAS! That is, when you enter something they look at brackets first, then indices, then division, multiplication, addition and subtraction.

This means that when you enter -3 then press x2 = you get an answer of -9.

You did know that’s wrong, didn’t you? (A minus times a minus makes a plus?)

It only does this when you actually type in the negative number. (It ‘knows’ the sign of the answer to something like 4 – 7 and will square it correctly.) So to make sure you aren’t embarrassed by a negative square:

a) Type in the number, press =, then square it.

b) Put brackets round the number, then square.

c) Don’t put the minus sign in at all!! After all, you know the answer should be

positive. Save keystrokes, save energy.

7. Mixed fractions

The best way to enter these is to press the shift button followed by the fractions button (in MthIO mode), then type the numbers into the boxes.

e.g for 2 ½ press shift, fraction, 2, right arrow, 1, down arrow, 2

This gives you the correct decimal value of 2.5 when you press = and SD.

The way NOT to do it is to type the fraction, then backspace and enter the whole number to the left of it.

e.g fraction, 1, down arrow, 2

then either right arrow twice or left arrow 4 times to get the cursor on the left side,

then 2 for the whole number

If you now press =, the answer is 1.

The calculator has interpreted what you did as 2 x ½ !!

I have no idea why this happens, but we had a fun time in yesterday’s lesson figuring out exactly what was going on….

Don’t do it!

## Surds are pointless, right?

September: season of mists and mellow whatsisname… And surds revision for students beginning A level maths.

GCSE pupils hate surds, and are horrified to discover that once you are in the sixth form you are expected to use the blasted things all the time. How ridiculous is that? I mean, who cares that 12 = 23 ? Who would ever use something that stupid in real life? They’re worse than fractions!

It’s a fair argument. We have phones that work as calculators these days, and tablets that work as calculators, and computers…. oh, and calculators as well. If I need the value of 12, a few keystrokes give me 3.464101615, which is more than enough precision for any practical purpose. So what’s the point of surds?

The point is, of course, that there is no point. No decimal point. 12 is an irrational number, which means that it can’t be written as a fraction and its decimal expansion goes on forever without repeating. So 3.464101615 is only an approximate value for 12, good enough for government work (to quote the great uncle of a friend who surveyed railroads in the Amercan west) but WRONG! If you square it on your calculator, the answer will be 12 (it is on all mine), but this is because the calculator is rounding 11.999999999045608225 to 10 significant figures.

(If you want to try full precision multiplication like this, a good resource is at https://www.mathsisfun.com/calculator-precision.html Or you could try it with a pencil and paper…)

Using surds saves us from unnecessary rounding errors. Suppose we approximate 12 as 3.464 to save space.

Then 312 x 712 = 3( 3.464) x 7( 3.464) = 251.985216

But 12 x 12 = 12, so the correct answer is 3 x 7 x 12 = 252….

AND it’s actually easier to work this out than to get the approximate answer with a calculator. And less error prone – the more buttons you press, the more mistakes you make.

KEY FACT

12 is the number which when squared gives us 12.

It doesn’t matter that we don’t know exactly what the value is, because whenever we square it the surd vanishes and we get a proper number, with NO errors.

We also get better answers by rationalising the denominator of a surd fraction.

For instance:  is nasty to work out as it stands.

We have a square root which we can only evaluate approximately and, worse, we have to divide by it. That’s two potential sources of error.

By rationalising the denominator to obtain

we still have the surd on top but now we only have to divide by 4, which gives us as the answer. Easier and safer.

(If you don’t know how to rationalise the denominator, there are lots of good videos on You Tube, such as https://www.youtube.com/watch?v=T_cZqeLNCHM to show you how it works.)

The same principle applies to ‘exact values’ calculations involving π. If you ask most children what π is, they are likely to give the approximate value 3.14. In fact:

π is the number you get when you divide the circumference of a circle by its diameter.

If we leave the symbol for π in our calculations, and we get lucky, it will cancel out. At worst, we carry out one computation with 3.14 (or whatever) at the end. Either way, we simplify the calculation hugely. Again, fewer keystrokes = fewer errors.

So learn to love surds, and be right more often.

(PS  I’m new to WordPress.  I’m sure there has to be a better way of incorporating fractions that bringing them over from Libre Office as images.  I’ll look into it…)