1. The number of social problems is rising sharply during financial downturn.
2. A number of social problems are caused by financial downturn.
In sentence 1, the subject is 'number' specified by the determiner 'the', so 'is' is the verb-to-be to keep the agreement. In sentence 2, the subject is 'social problems', and the phrase 'a number of' is a quantifier to modify the subject 'social problem’. Therefore, the verb-to-be is 'are'.
Again, it shows how the article 'the' and 'a' have the specific and general reference respectively.
Again, it shows how the article 'the' and 'a' have the specific and general reference respectively.