In the 1950s and 1960s schools taught a method of calculating square roots by hand, that resembled long division. A number of demonstrations of the method, via Youtube clips, can be found on the web. My favourite is this one, by Anita Govilkar. Not only is the exposition clear, but Ms Govilkar’s carefully enunciated Indian English is a delight to listen to.
Explanations of just why the method works are less frequently encountered. In this pdf, I give my take on the workings of the method.