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!

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    $\begingroup$ @ChrisC Please just have a look at the other calculator's related questions at this site to see how it is a mission impossible. Moreover, when according to university policy mobile phones are free in the class, automatically, are calculators. $\endgroup$ – Amir Asghari Feb 3 '15 at 16:41
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    $\begingroup$ @ChrisC Moreover again, we are always looking for creating learning opportunities. If calculators can "safely" create some, why to avoid them. $\endgroup$ – Amir Asghari Feb 3 '15 at 16:44
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    $\begingroup$ @JoelFan Not everyone has a 'fancy' phone as they're quite more expensive than a calculator. In fact, I know some math grad students without a phone at all. The utilitarian use of the smart phone with allowing students to use it as a calculator might also be passively allowing cheating (googling the question, etc). $\endgroup$ – Chris C Feb 3 '15 at 20:29
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    $\begingroup$ I am not sure what calculators they have access to, but there are some false counterexamples to Fermat's last theorem that can be mis-evaluated on calculators. E.g., $1782^{12} + 1841^{12} = 1922^{12}$ was tough to detect as an error in the past. But the two expressions are clearly not equal (since, e.g., the left hand side is odd and the right hand side is even). Perhaps showing why some of these equalities cannot hold would be a nice exercise? $\endgroup$ – Benjamin Dickman Feb 3 '15 at 21:52
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    $\begingroup$ @JoelFan I'm with you on this, much rather use a phone or tablet than a clunky calculator, BUT exams specify only a small range of calculators that are acceptable in exams ie. No CAS allowed. Top of the line graphics calculators have insane markups and cost as much as iPad minis, but purposefully limited to conform to exam specs. $\endgroup$ – Richard Feb 3 '15 at 22:03

Reminds me of a story a friend (who is a math teacher) told me. Student comes to him and says his calculator is broken - it shows (-5)^2 = -25.

[C] [-] [5] [x^2] [=]                  shows -25

Q: So, how do you know that's wrong?

A: Because my mobile says 25:

[C] [-] [5] [*] [=]                    shows 25

Q; And how do you know the 25 is correct, and the -25 isn't ?

A: I checked with my friend's mobile, it says 25 as well ....

My friend turned this lesson into one focusing on how to use calculators correctly, and how to check whether or not the result makes sense.

Especially with those cheap calculator apps that come pre-installed with mobiles, and that don't handle operator precedence, you can get a lot of wrong/conflicting results.

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    $\begingroup$ That's an interesting historical fact that $-25$ was one vote away from being equal to $(-5)^2$. Sounds like it must have been a hotly contested election! :) $\endgroup$ – Michael Joyce Feb 6 '15 at 14:10

Calculators can enrich the learning of students if properly used. A simple example; I like to get students to work $2+3\times10$ and then get them to check with a calculator. They are forced into confronting their misconception that the answer is 50 and in the process are more likely to remember the order of operation.

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    $\begingroup$ Helping some 5th graders this morning. They would only use the calculators when the question tells them to. The calculators they had seemed to know the order of operations. ... Except it does $x \wedge y \wedge z$ left-to-right as $(x^y)^z$. $\endgroup$ – Gerald Edgar Feb 3 '15 at 20:06
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    $\begingroup$ Even the cheap calculators sold here now use equation input that evaluates with correct operator precedence. They even show nested fraction, exponents and roots graphically ... Not your grandpas calculator! $\endgroup$ – Richard Feb 3 '15 at 22:08
  • $\begingroup$ I once had a calculator that didn't know about operator precedence (if you typed 2+3x10 it actually did give 50; you needed to explicitly use parentheses or re-order the calculation to get 2+(3x10).). This was no problem if you understood the issue, of course, and I happily used it for a time. If your class is somewhere where students may have a variety of calculators, such a possibility should be kept in mind. The presence of one can be a useful launching point for discussion (eg "how would you make it give the correct answer in this case?" or "should we just trust what the calculator says?") $\endgroup$ – Glen_b Feb 4 '15 at 0:30
  • $\begingroup$ @Glen_b You had one of the new fangled calculators with parentheses! $\endgroup$ – Richard Feb 4 '15 at 5:46
  • $\begingroup$ @Richard Actually, I'm not 100% sure, it was a very cheap calculator, long ago. It either had parentheses, or a memory; whichever it was, it made it (mostly) simple to deal with the lack of operator precedence ... there was just an extra step of thinking required. $\endgroup$ – Glen_b Feb 4 '15 at 6:02

I'm a fan of questions like:

a) Plot $y = x^3 - 19x^2 + 96x - 144$ on your calculator in the standard window.

b) Find an end behavior diagram for $y = x^3 - 19x^2 + 96x - 144$.

c) These answers conflict with each other. What went wrong?

The trick here is that in the standard window, the function looks like a parabola (it has roots at 3, 4, and 12, with 12 outside the window). But this will conflict with their end behavior diagram, and hopefully show them to be skeptical of their calculator.

  • $\begingroup$ It reminded me of a similar experience when graphing two variable functions, say z=x^2+y^2. Most graphing tools draw them in a box, as a result of which, the "border" seems quite different from what is expected. $\endgroup$ – Amir Asghari Feb 3 '15 at 18:45
  • $\begingroup$ Also reminded me of another one that I, again accidentally, found in Calculus 1. To compare X^3 and e^x, we used a graphing tool which only drew the two a little further than their "first" meeting point. As a result, it seemed that from that point on X^3 is bigger than e^x. But, the problem was to justify the other way round. $\endgroup$ – Amir Asghari Feb 3 '15 at 19:04

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