My department wants me to keep track of my hours worked. Is that appropriate?

Yes.  Syracuse University has a responsibility under the FLSA to maintain precise records of the hours worked by non-exempt employees, and also retains the right under the FLSA to track the hours worked by exempt employees.  There are various methods for tracking time, including time sheets, sign in/out sheets, and time clocks. A common method is for staff members to record their hours worked each day on a form, and then submit that form to the payroll person in their department at the end of the payroll week.