The correct option is D The teachers' attendance book recorded teachers' attendance through the year's classes.
Regular nouns are nouns that form their plurals by adding either the letter s or es (in the given sentence - teachers). According to general rule, an apostrophe after the s is used to indicate the plural possession of such regular nouns.
However, the possession of singular nouns require an apostrophe followed by s (here, year).
Thus, according to rules, the correct answer is Option D. The remaining options are incorrect.