I don’t like being too dependent on calculators. The future might bring an EMP (electromagnetic pulse) that would fry all such gadgets (and cell phones, cars, computers, etc.), and I want to be ready for a post-calculator world, if that happens.
My overarching strategy for doing math in my head is this: don’t have just one single strategy. Instead, devise one, on the fly, based on the problem you are trying to solve.
I do, however, use a few “go to” strategies for certain things, such as finding 25% of something, or multiplying by eight, or similar problems. This involves looking for, and take advantage of, powers of two, as well as their reciprocals. 25% is 1/4, which is simply halving twice, and multiplying by eight is three doublings, since 8 = 2³. With practice, doubling or halving numbers repeatedly and silently, in one’s head, becomes much faster and easier. If done out loud, it becomes easier still, and on paper, it’s extremely easy.
I intend to do more blog-posts in the future with calculator-free calculation strategies, but not all at once — instead, these techniques will be posted one at a time. However, these postings will stop immediately, in the event of an EMP.
RobertLovesPi.net 
I was actually just discussing something along these lines with one of my tutoring students, yesterday. She’s in 4th Grade, and she has a test today on Factors & Multiples, so I was going over Divisibility Rules with her. Her textbook gave her algorithms to determine whether a number was divisible by 2, 3, 5, 6, or 9, but there was no divisibility rule for 4 or 8 (and they are only factoring up to two-digit numbers).
So, I told her that if a number is even, and half that number is also even, then 4 is a factor. Similarly, if a number is even, and half the number is even, and half of that half is even, then 8 is a factor. It can sometimes be quicker to do this sort of halving in one’s head than to divide the number by 4 or 8 and check for a remainder.
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How about the binomial theorem ?
Leslie
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I’m confused . . . what exactly are you asking about the binomial theorem?
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Are you going to show us how to do it?
Leslie
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It’s an application of the distributive property, and numerous “how to” websites can help with it . . . but it’s not likely to be the topic of a post here. The reason for this is simply that my obsessions in mathematics tend to fall on the side of geometry, rather than algebra.
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No problem. Thanks anyway.
Leslie
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I teach a calculator-free physics class. The scores actually improved when I stopped allowing calculators. (Of course, now they have no choice but to show work and actually earn partial credit…) 🙂
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