The number of students is rising :
the number is the subject of the sentence and hence the verb is in the singular too.
students here is an attribute and as Xpert said is a matter of subject - verb agreement.
A number of students are rising:
A number of - attribute
students -is the subject of the sentence and, as a consequence, the verb should be in the plural too
'Is' is correct in the above example because when we precede number by ‘The’ it denotes a unit and is SINGULAR. When It’s preceded by ‘a’ it means ‘several’ or ‘many’ and is ‘PLURAL’
A number of pupils are absent today.(not A number of pupils is absent today.)
The number of pupils is increasing.(not The number of pupils are increasing.)
It is ONE ENTITY of things, people. IS is logical like in other languages.
People tend to apply the LAW OF PROXIMITY, the closest noun in plural, verb in plural