Recently, I read a paper about student's understanding of fraction. Most students mentioned in the paper had given $5/8$ as their answer to the following question:
What fraction is shaded?
For reasons that only an "educated eye" might see, the author of the paper was expecting $5/4$ as the answer. Surprised of the conviction of our educated eyes, I put a voting question (on social media) where "normal" people could vote for $5/8$, $5/4$, or $other$. The result, here and here, is telling. To make the voters more curious, I used different quantifications of the shaded area to "prove" here that $1=2$. For a similar argument, see also Benjamin's answer below.
I am now in the position of explaining the problem to more than 400 voters and the aim of this question is to get help to do so. Obviously, the question is about what fraction of what is shaded. More generally, the question is about unit conversion in disguise. So far so good. The problem started when I was going to write about the meaning of the equal sign in unit conversion in comparison with its meaning in arithmetic. Then I realized that I don't know (and this is my question) how these meanings are similar to and/or different from each other. Going into more details, I even don't know, I should think of $5/8$ and $5/4$, or of $5/8$ or $5/4$ as possible answers. Similarly, in writing $1m=100cm$ for a length, is the length quantified by $1$ and $100$, or, $1$ or $100$?
Just in case that you are wondering why I have asked this question here, I shall add that because we are the only community that rightly cares about different meanings of equal signs and the difficulties that they cause for our students, and of course, sometimes for us :)
Added. It's worth mentioning why I perceive this post related (at least partly) to unit conversion. I hope the following equalities tell it all:
$1 REC(tangle) = 2 SQU(are)$
Thus, $5/8 REC = 5/8 * 2 SQU = 5/4 SQU$
Of course, you might rightly question what is measured by these fictional units. For this, we might create some fictional story as well, say measuring pizza in Pizzaland where asking for 2 RECs pizza or equivalently 4 SQUs pizza is quite meaningful.
PS. I hope it can be seen now why $1=2$ in the original title, and in the body of the post, is not a contradiction if understood in the context of unit conversion.