Let me start by sharing what happened in my class today. The subject was complex number and I started with the historical problem of "finding two numbers whose sum is equal to 10 and whose product is equal to 40". Okay, as you know from experience or say from this question, nowadays there is no way to avoid student's over-reliance on calculator. Thus soon after students formalized the problem and came to $x^2-10x+40=0$, the room was filled with the sound of click, click, click of inputting the equation into the calculator to find its roots. But, the calculator for the reason that you know gave students an Error message, and then naturally, they wanted to know why. Indeed, I hadn't planned for that, but suddenly it occurred to me what a good use of calculator: create a situation for misusing it purposefully.
Somehow my story is a possible answer to this question: Appropriate ways/sayings to discourage undergraduate students' over-reliance on calculators that in itself includes some other calculator's related questions. But, Here is my own "question" that is asking for some other such misuse of calculator that if used purposefully could create a rewarding learning situation.
There are some , say "positive", uses of calculator. For example, the problems about that broken calculator that hasn't got a numbers of keys and we want to use other keys to get a certain result. Though nice, these uses are not what the question is about. Our calculator works perfectly!