There is no half year claiming of a dependent.
The age test for a child is:
Under 19 at the end of the year OR
A student under age 24 at the end of the year OR
Permanently and totally disabled at any time during the year, regardless of age.
The child can earn income and as long as you provide over half of child's support and the child resides with you over half the year, you can claim the child.