This little 15-minute TedTalk, Arthur Benjamin races calculators from the audience doing the calculations in his head. He is very entertaining and I think students would enjoy the presentation, since they are often under the impression that nobody has every done calculations without a calculator. He squares a random 4-digit number from the audience in his head just as fast as the calculators.
For those of you that teach probablity, he does a neat trick guessing the missing digits of 7-digit numbers (this is somewhere around the 7-minute mark).
For elementary ed, there is a nice “number” on guessing the day of the week when someone was born based on the birth date (8-9 minute mark).
For you algebra teachers, he does the square of a 5-digit number, going through the thought process out loud (starting around the 11-minute mark). Cleverly, he squares the number by breaking it into a binomial:
(57683)2 = (57000+683)2 = 570002 + 57000*683*2 + 6832
I’ll let you work on that one in your head…