‘A number of’ followed by singular verb or plural?

When a sentence begins with “A number of,” should the verb that follows be singular, or plural?

Compare the following sentences:

1. A number of students were late for class.
2. The number of students in this class is ten.

In sentence 1, ‘A number of’ is an expression of quantity meaning ‘a lot of.’ The subject (noun) is ‘students’, which is plural, so it takes a plural verb, ‘were.’

In sentence 2, ‘The number’ is the subject. ‘Number’ is a singular noun. Hence it is followed by a singular verb ‘is.’

So the trick for getting this right is to identify the essential information (thereby identify the subject). In sentence 1, the intended focus is on the students. But in sentence 2, the point of the sentence is quantity; hence ‘number’ is the key word.