Reba and Interpretrator are both correct about "its" and "it's," so I feel a bit foolish about going on and on (as that's what makes student's eyes glaze over).
Sounds defensive, but remember that English teachers and interpreters are pretty much in the same boat. We don't make the rules. The people who speak the language have made them by usage. We just try to explain them.
Teaching apostrophe use is difficult, because (as you've said) the darned mark is used in this case for contractions and possessives. Here are some short memory jogs I've found helpful:
The key is no possessive pronoun ever uses an apostrophe. So its is like his. You wouldn't write "hi's," would you? Likewise its, his, hers, yours, ours, theirs are all in the same class.
The contraction is usually for "it is," and a handy recall tool is saying the apostrophe is like the "i" so "it's" still uses four letters like "it is."
You always put the apostrophe in place of the missing letter(s) or number(s). So another it is is 'tis, as in "'Tis the season to be jolly." Only in this case the first "i" is replaced.
The exception you were looking for is "it has," as in "It's come to my attention that languages can be a puzzle at times."