Darn those calculators, which I refer to as devil boxes!
There are subtleties to be learned by doing things the old fashioned way--any master craftsman will agree with this statement. Note that I didn't say that we should ignore technological innovations; I'm no Luddite. I said there are subtleties to be learned.
This web site lists a few algorithms but the site itself is pretty dry. This site is much better. My favorite, though, is a mnemonic device in the form of a poem, found here:
From "The Poetical Geography" by George Van Waters, 1857
Square rootDivide into periods of two figures each,
The number you know, as the pedagogues teach,
- In the left hand period find the greatest square,
Which from it subtract, and to what remains there
Bring the next period down for a Dividend (fair):
Place the root of the square at the right hand of all,
And two times the root a Divisor we call.
Then try the Divisor, see how many times
The Dividend holds it (by prose or by rhymes).
Of its right hand figure exclusive, you know,
And write in the root the number't will go,
Then to the Divisor the same figure tie,
And by the same figure the whole multiply;
The product then take from the Dividend (penned),
And of that which remains, make a new dividend;
By bringing the period that's next, along side,-
And for a Divisor that's new and untried,
Just double the figures that stand in the root,
And work as before, till the answer is got.
Isn't that wild? Multidisciplinary education--poetry and math :-)