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When I want to point out to my students that getting the right result is not enough, I like to show them the example:

$$\frac{16}{64} = \frac{1\hskip-.1cm- \hskip-.4cm{6}}{-\hskip-.2cm{6}\,4} = \frac14.$$

This example works well in some of my classes where the error is just the right level of "stupid": The students feel absolutely safe from making this error, and they also do not feel that this is a "kindergarten" example too far removed from their level of mathematics.

But I would like to have some more examples like this for students with less and with more knowledge about mathematics.

What are some good simple preferrably amusing examples that getting the right result is not enough?

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    $\begingroup$ I have posted a feature request for MathJax on meta. $\endgroup$
    – Jim Belk
    Mar 13, 2014 at 21:32
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    $\begingroup$ Please take a moment to refine your question based on this advice. I think the better question here is to ask not for examples, but concrete techniques to use in this situation. $\endgroup$
    – Jon Ericson
    Mar 13, 2014 at 22:27
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    $\begingroup$ I notice that the close votes are not for "duplicate", but for "opinion-based" and "off-topic". Since this is exactly the type of question I have committed to this site for, I think that it would be a very good idea to discuss this on meta. $\endgroup$
    – user11235
    Mar 14, 2014 at 9:16
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    $\begingroup$ @JonEricson If the suggested example given by the OP is not concrete then what can be? Lists of concrete examples for mathematics teachers are not shopping lists. They are very useful resources and the more the merrier. Teachers have repeat lessons, new examples help them stay interested in their work. I can imagine looking up this question at some future date after it's been given a chance to grow and getting some very good ideas to use in lessons. And, a correct answer coincidentally given by false mathematics, is not an opinion based matter. $\endgroup$
    – Geoff Pointer
    Mar 14, 2014 at 22:53
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    $\begingroup$ @user11235 I do agree that the question could be better framed. For example: What ways are there to emphasise to students the importance of understanding processes as opposed to just getting the right answer? The difficulty of course is you're not supposed to ask what is the best way or what are good ways because of the opinion based bogey man, but every answer to this question will be an opinion about how to go about this and so what? That's what this site is supposed to be about, Mathematics Education. The content may be concrete, but the delivery is very opinion based. $\endgroup$
    – Geoff Pointer
    Mar 14, 2014 at 23:11

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