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Most mistakes with apostrophes occur with pronouns, especially possessive pronouns. For example, the word for belonging to it is its, notit's. Also, the word for belonging to us is ours, notour's.
Apostrophes are not normally used to make plurals, and and you should never use an apostrophe to pluralize a word. For example, the plural of apple is apples, notapple's. The plural of orange is oranges, notorange's.
How do I start to write about the apostrophe? and how might it present problems to learners?
I would start with the main uses of apostrophes:
1. To show possession
2. To mark missing letters in a word, as in contractions.
3. To show plurals of lower case letters, numbers, and some acronyms/initialisms.
As Ron said, the main problems occur with mixing up contractions and possessives and using them in possessive pronouns such as its and theirs.
the pattern analysis of the apostrophe 
Linear Mode
