Last weekend my father-in-law taught Kate how to solve square roots longhand. He writes out 4225 and shows her - "divide it into pairs of digits, so start with 42. What's the biggest integer no greater than the square root of 42?" "Six." "Right, so that's the first digit of the square root. You write it up here, square it, subtract from 42, get six. Do you see?" Kate asks a couple clarifying questions, then nods. "Now, bring down the next two digits, just like with long division except two at a time, and you have 625. Take the solution you have so far - 6 - double it to 12 - always double it - write that to the left, and add a digit to the right - that digit is going to multiply by the entire number, to be no greater than 625. So - if it were 1, you'd have 121 x 1; if it were 2, you'd have 122 x 2. So what digit should we write?" Kate thinks. "Five!" "Exactly! So that's the next digit, and now you have 125 x 5, which is exactly 625, so you have your answer, 65."
Kate loved learning it. (Of course! Getting this kind of lesson from your infinitely cool grandpa? Win.)
Robert walks into the room. "What's going on?" "I'm teaching Kate to take square roots. We just learned the square root of 4225." Three second pause. "Oh, you mean sixty-five."
Dad later showed Robert the same method. Robert: "Why do you double the number you bring down?" Me: "Because of x^2 + 2xy + y^2." Robert: (thinks) "Oh, right."