Half doesn't or half don't?

Solution 1:

You could make a logical case for either variant, but the only one that native speakers of English would say is

Half [of] the students don't bother to show up.

This is a case in which the outcome is dictated by synesis (see http://en.wikipedia.org/wiki/Synesis ).

Solution 2:

English has a handful of premodifiers which, despite formally appearing to be a prepositional phrase, are not such for determination of the subject or for agreement with the verb. Other are a number of and a lot of, and in fact, half of the.

These premodifying phrases act as adjectives, and the grammatical subject remains the word following, which is either singular or plural, and thus so too much be the verb.

• A lot of the people are scared.
• A lot of the sky is blue.
• Half of the people are scared.
• Half of the sky is blue.

In other word, the premodifier does not affect grammatical number for subject–verb concordance.

Note that this is a special case, not the normal one. Therefore, other prepositional phrases would work differently, and indeed shift the subject and thus affect concordance:

• One of your best traits is your humor.
• Several of our membership have gone to their glory.

And sometimes you just cannot test, or require contextual clues.

• The best of us at our ring-tossing competition is made Mayor for the Day.
• The best of us at our ring-tossing competition are allowed to keep their rings.