Also, the students for whom this thing is intended (those who would rather just get the answer) will probably study the calculator, learning how it works and probably learning ways to beat it. Then we can return to business as usual: students who "get it" and reason about the problem will continue to do so; students who just grab for answers will continue to grab for answers; and teachers will be complacent--everyone's getting their results by typing an accurate-enough estimate into the calculator, right?--until people realize what's happening, and this will become one more failed educational experiment.

Here's the first approach that occurred to me. How quickly can you type in estimates, and how accurate do they have to be? (The first thing I would do with one of these things is find that out, simply for curiosity and control over my tools.) If, say, it's "same order of magnitude of the correct answer", then you just have to guess 1, 10, 100, and so on (maybe .1, .01, etc. if the numbers are small). If it's "within a factor of 2", then you could guess 1, 4, 16, 64, and so on (or something that's easier to type; maybe guess 1 10 100 1000, 4 40 400 4000).

It may take a while to do, but if the problem is difficult and frightening, it is easier and safer to apply some brute force strategy like this--even if you had to type "1 2 3 4 5 6 7 8 9 10 20 30 40 50 ...". I wouldn't be surprised if the answer-grabbers got really fast at it (1e5 2e5 3e5 ... even handles many digits in more or less constant time). It would still take longer than a normal calculator, but that would be the case for people who estimated it "the right way" too.

Maybe you could put in some kind of penalty for repeated wrong guesses, but I'm skeptical. First, even the students who understand what they're doing will make some wrong guesses, and you don't want to punish them too hard; second, I suspect it would actually be hard to do that: the student can turn the calculator off and on at any time, and I think giving it some non-volatile storage of timestamps of wrong estimates would make the calculator somewhat more complex and expensive than it looks like it's supposed to be.

(Actually, I think the "problem" of "misusing" the calculator is not limited to pure answer-grabbers. I could see myself respecting the challenge and doing my estimates, then being off by a bit and making a correct estimate that was right next to it, then realizing I could just be lazy about my estimates--truncate everything to [single digit] * 10^n, or whatever--and just make two or three guesses if necessary to cover the relatively wide range. I might possibly still enjoy the experience or the challenge of making estimates and not considering myself done until I can justify my estimate--just like I enjoy doing this stuff http://setgame.com/set/puzzle_frame.htm even though I can write a program to solve those puzzles, or not solve them at all--but if I was bored with the assignment and just wanted to finish it, or annoyed with the people who thought such a calculator was a good idea, then there would be nothing stopping me from using perverse strategies.)

Edit: Turns out it is somewhat more complicated. The degree of tolerance depends on the perceived difficulty of the mental calculation (in particular, transcendental functions give a large room for error). http://qamacalculator.com/qama/complicated.jsp

That is kind of an impressive thing, actually. The way error bounds are determined might be somewhat complicated to work out, and maybe even a determined answer-grabber would have to do a bit of estimating work... I dunno. Maybe you could figure out that "as long as there's a sine in the expression, then the error bound is at least this much".

OH MY GOD you would simply take any expression and put "times sin(89°)" or something in it, something close to 1. Get your answer, then use that as an estimate for the original thing. Maybe put in a ton of little transcendental-but-you-know-it's-about-1 expressions in there; maybe you can get your error bounds so wide that you need only guess once. Now maybe the programming will notice things like that, will give small error bounds for something very close to a known thing like sin 90. But then put in A * (your expression) * B, where A and B are transcendental expressions whose ratio is very close to 1, but which individually are definitely not 1, and which don't obviously cancel out. Like e and 2^-1.44.

I am, let's say, 95% confident that kids would figure out something that would defeat this.