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Scenic Routes to RootsAuthor(s): Alan GrahamSource: Mathematics in School, Vol. 19, No. 5 (Nov., 1990), pp. 44-45Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214734 .
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'Routes
'Routes
ITo
IRoots by Alan Graham, Centre for Mathematics Education, Open University
How often have you wanted to calculate a cube root, only to find that you didn't have a cube root key on your calculator? Well, probably not very often! But just in case you ever do get into this predicament, how might you tackle it, armed with a standard scientific calculator?
There are actually a number of possible approaches, each of which illustrates how the calculator can help shed a little light on various interesting mathematical ideas.
I've outlined three of these approaches below, and, to keep the explanations simple, have assumed that I am trying to find the cube root of 8.
Method (a) Using the 'Index' Key Provided you know that = 81/3, it isn't hard to come up with the following five-stroke key sequence:
873 3-
This sequence is explained below.
Key Press Explanation
8 Enter the number whose cube root you wish to find
-Yx Raised to the power of ..
3 A quick way of entering
E- Complete the calculation
Method (b) Using the 'Logarithm' Key log
One way of raising a number to a power is to take the logarithm of the number, multiply by the power and then take the antilogarithm. This three-step procedure is sum- marised in the diagram below:
Raising to a power via the logarithm
How do I raise A to the power n?
A A"
Step 1 Step 3
log A Step 2 (log A) x n
This approach gives rise to the following key sequence for finding 813
8 IloglO S~IINVI /Logj
Key Press Explanation
8 Enter the number whose cube root you wish to find
log Step 1 in the diagram
S3 - Step 2 in the diagram
-e Step 3 in the diagram
44 Mathematics in School, November 1990
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Method (c) Iteration A third method is to use iteration. First I need to find a suitable iterative formula, so here goes:
x3=8
x4 = 8x
4 = x=,e8 This gives the iterative formula
Xn+1
So, taking some arbitrary starting value, say, x, = 2.5 the key sequence required is:
2.5 F-
repeat loop However, the whole process can be speeded up by using
the calculator's constant to perform the " x 8" part, thus:
7x8 2.5 [ O F
repeat loop.
Key Press Explanation
8 Set up the calculator constant x 8
2.5 Enter the value of x,
Multiply by 8 (using the constant)
[-e e Take the fourth root
Continue the iterative process and check whether xn settles down
So What? Well, what general educational issues, if any, can be drawn from these three approaches to calculating a cube root? To start with a negative reaction, you may have felt that a fairly high level of mathematical understanding and competence was necessary in order to be aware that these alternative key sequences might produce the same result (knowing that /8=81'3 or finding a suitable iterative formula, for example). A more positive response might be to say that solving the same problem in several ways can help students to see connections between different (and hitherto seemingly unconnected) areas of math- ematics.
With luck they may be encouraged to explore further connections for themselves, asking questions such as:
" What other equations could be solved iteratively? " What happens if I try to find e/ and so on.
Exercises where pupils have to work with a restricted set of keys can, I believe, help to release creative energy in many pupils and they can begin to explore the intercon- nectedness of mathematical functions. For example, being denied the use of the 7y key may lead them to realise that this key really corresponds to repeated multiplication. (At a lower level pupils should discover that multiplication can be thought of as repeated addition and division repeated subtraction.)
Pupils are also having to think "algorithmically" in this sort of investigation. This means having to think through
the underlying generality of the calculation and express it concisely (i.e. in as few key presses as possible) or, if not concisely, then at least elegantly!
As a follow-up exercise, try going through all the function and operation keys on your scientific calculator and see how many you can perform using other keys. But, don't forget: be as concise and elegant as possible!
Finally, here are two pupil worksheets which I prepared for a seven-year-old and a fifteen-year-old, both based on this same principle of "restricted keys".
Restricted Keys 1 You can only use the following keys on your calculator
32000 What keys would you press to make the following
Make Keys Pressed
4
7
0
33
30
65 '+
54 3 a- 3 , 300 3
230
254
Name
Make up some more of your own!
Restricted Keys 2 Use your calculator to do each of the following calculations WITHOUT using the "banned key".
Calculation Banned Key Key Sequence
1) 24/6
2) 5x7 -
O
3)15 \ C
4) log 100 0 log oo -
O oe a 5) log 130 (to d.p.) lo
,e .
I5 l3 \ --c \I=
6) tan 52' E
ta n c
7) (-3) x (-5) 3-6 ,,4 -4q=
xMR = \S
8) sin 59 xsin cos x S<14 , .,
O e
Make up some more of your own!
Name
Mathematics in School, November 1990 45
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