<p>Like some on the list, I always play with numbers in my head both for fun and to get a “ballpark” answer to crosscheck against the formal answer. However, I’m not so much against someone not knowing the mechanics of long division. To me the “why does it work” and “what else can we do using this technique” is more fun to teach a kid than to get the actual answer.</p>
<p>To take it a step further - I suspect many engineering students of today don’t know how to do square roots by hand, long-division style, but by putting commas and taking down 2 digits at a time, etc. And in this day and age, the ability to do so is not important. But it has a really cool use. Whenever I’ve shown the technique to middle/high schoolers when I used to help out in math/chess clubs, there was always one or two that I could really get involved. After the 5 minutes spent on learning this, we would spend 30 minutes discussing why it worked, and talk about how this kind of technique could be used elsewhere and what else can be done.</p>