A Question of Logic: Three Things You Should Know about Your CalculatorAuthor(s): Alan GrahamSource: Mathematics in School, Vol. 13, No. 2 (Mar., 1984), pp. 28-29Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30216203 .

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A

QUESTION OF

LOGIC

SThree thingsyou thingsyou should know

bou fyourcalculafor

by Alan Graham, Centre for Mathematics Education, The Open University

Have a look at the calculator key sequences below and write down your prediction as to what answer the calculator would give. Once you've predicted, ("guessed"), then "press" the sequences to see how, you did.

GUESS PRESS

(a) 2 3

(b) 2i- 3 --4 this may be

E (c) 4

i- -3

1 on

your calculator (d) 3 ]]2~]

(e) 2-3 ON/C14~

(f) 3-2EJ _

Well, did you get any surprises? On my calculator (a Texas TI 30) the answers were as follows:

(a) 10 (b) 14 (c) 1 (d) 1 (c) 6

You may have found that your calculator gave different answers to mine. So what is going on here? Well, not only do calculators have a variety of different keys available, but in some cases even keys with the same label will do different things. Since most calculators that appear in the classroom are provided courtesy of F. Christmas Esq., they will come in all colours, shapes, sizes and, most important, logics. One of the first things a teacher needs to do is to discover how the various machines available differ. This will help to antici- pate and resolve any apparent inconsistencies that might crop up when the children are using their own machines.

Here, then, are one or two simple tips to help you to know what to look for.

Type of Logic - algebraic or arithmetic This is perhaps the most basic difference between cal- culators. At the cheap end of the calculator market, most machines are programmed to carry out the operations arithmetically - i.e. in the order in which they are fed into the machine. Thus, for sequence (b)

2 [13Wx41=

a calculator with arithmetic logic will perform the addition of 2 and 3 before multiplying the result by 4 (giving the answer 20). A machine with algebraic logic, however, obeys the conventions of precedence in algebra, where multipli- cation and division must be carried out before addition and subtraction. (Note for the over 40's - you might have

learnt this as B.O.D.M.A.S. at school!). An "algebraic" machine will defer adding the 2 until it has multiplied the 3 by 4. Hence the alternative answer of 14. If your calculator is programmed with algebraic logic you might like to find ways of getting it to produce the answer 20 for this basic sequence (two possible ways are given in the footnote).

Key stroke errors - and how to correct them Have you ever used a calculator to add up a long column of figures and then pressed the wrong key halfway through? Most people sigh sorrowfully, swear silently, switch off and start again. Providing you have a half-decent calculator, however, there is of course an easier solution. Sequences (c) and (d) may have helped you to decide what to do if you press the wrong operation by mistake. On many machines (including mine), all but the last of a sequence of operations will be ignored. So if you accidentally press [- instead of

-+, this is corrected by pressing a next. This does not work for some calculators. Pressing 6

--] may confuse some machines into displaying 12, while 6 -- may give 0. Under these circumstances it is probably advisable to switch off and start again!

A more common error is to press the wrong number. Sequence (e) should confirm how your calculator copes with this without having to start again from the beginning. Incidentally, you might like to explore this next sequence on your machine:

2 ON 3 ON- 41 is for constant

Almost all calculators have a constant facility. What this means is that you can set the machine up to perform the same "sum" repeatedly at the touch of the O key. The teaching possibilities of this calculator feature are enormous as I hope the two examples (A and B) below will indicate.

First, however, you will need to know how to set up the constant function. If the machine has a key marked K-, then this can be done directly. On my model, for example, if I want to set it up to add 5, I press:

51E ll% Now I can add 5 to any number I choose by pressing the

number and then Thus:

4 = gives the answer 1

10 E 7'givess the answer 15 and so on.

28 Mathematics in School, March 1984

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If the machine does not have a [ key, the chances are that its constant facility comes on automatically after you key in a suitable calculation. For example, key in this simple ad- dition sum:

Now press 3 --4 rl 1 3

10l 100 ~

With many calculators the addition sum ([j 4) is carried over as a constant addition and so 4 will be added to 1, 10, 100 or whatever, giving 5, 14, 104 ... and so on. With other models this sequence will give answers of 4, 13, 103 ... Can you see why?

Here it is the 3 [a part of the original sum which the machine adopts as its constant function. So each time the [E key is pressed the machine will add 3 (rather than 4) to any input. A few models may trigger the constant function when an operation is pressed twice in succession (try 5 ]-I 2 7 10 -). Typically, the constant function is killed as soon as any key other than a number or ] is pressed.

Calculator activities based on *

Example A Guess and Press

(i) 2 ...... (ii) 1E 10 E; E ; .... (iii) o.1 a 0 o ; [a ; -; ..... (iv) 2 1 I; ; ..... (iv) I=.

(v) 10 7a 1 J ; [; ; ..... (vi) 1 i1000000 ; O ; E;.....

Example B Guess the Number This is a game for 2 people designed for a calculator which has a constant function. It has been written assuming a key. If your calculator's constant is set up in a different way then adapt the instructions accordingly.

Instructions Player A chooses any number between 1 and 100 (say 40) and, unseen by B, presses

40 [I~IKIO (note: the final 0 is pressed in order to wipe the 40 from the display)

Player B has then to guess which number (denominator) A has chosen by trying different numbers (numerators) and pressing -. B's aim is to guess A's number as quickly as possible. B then chooses a number for A to guess.

Sample Play: B's attempts to discover the denominator of 40 are as follows:

B presses Display Comments

1. 5 ] 0.125 5 is too small 2. 24 j 0.6 24 is too small 3. 50 O 1.25 50 is too big 4. 40 E 1 40 is the denominator

B's score= 4

Scores are recorded as the number of "guesses" for each hidden number. The player with the smallest total score after, say, 10 rounds is the winner.

*Footnote: these activities assume a a key. However, they can be easily adapted to accommodate machines with an automatic constant.

Although there may be other respects in which cal- culators differ, these three seem to me to be the most important for the teacher. Here they are again:

Slogic - arithmetic or algebraic Smethods of correcting keying-in errors Soperation of the constant function - K or automatic.

It is a worthwhile exercise to take stock of all the various calculators which your pupils use and check out each model according to these three features. The chart below shows how this can be done fairly quickly.

(a) 20 - arithmetic

Check the Press logic machine's logic 2 3 4 14-algebraic

logic

(b) Check how the machine corrects keying-in errors 2+34

Press

(c) Check If it has Press whether the a [ key 4

+ [~I ; = machine has Does it

a constant Try pressing add 3

facility 4 ; E l or4?

Ifithas 3 l4E;E no [a key

Do I hear you asking, "Why can't calculators be standard- ised?". Why not indeed? The fact remains that at present they aren't. Calculators are manufactured to be effective calculating aids, not teaching/learning aids. As a result they may possess a number of valuable features which aren't even disclosed in the operating instructions. Should the day come when a calculator is manufactured according to pedagogic criteria ... well, when that day comes what would you bid for? How about a fixed decimal point in the centre of the display, or a sign change key (+ / -)? Would you, perhaps, banish the percentage key (%o)? Alternatively, if each child has her/his own hand-held micro-computer by the end of the decade, what pedagogic considerations should influence its design. I don't have clear answers to any of these questions. Like many other teachers I'm simply experimenting and exploring possibilities.

Some of these issues are addressed in the Open Univer- sity Pack for teachers "Calculators in the Primary School" (PM537). Further information about this course can be obtained by writing to Liz Dawtry, Centre for Continuing Education, The Open University, Milton Keynes. Currently in preparation is the secondary equivalent "Calculators in the Secondary School". If you want to hear more about how this is developing, please write to me.

Footnote Here are two ways of coaxing your "algebraic" calculator to calculate "arithmetically". (i) 2 [1+ 3 a- [x] 4 W-

(ii) 2 2+3 [1 [7 4 j (for calculators with "bracket" keys.)

Mathematics in School, March 1984 29

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