If you need to move a student from one class to another (ex: Advanced Algebra to regular Algebra) and need to move the graded assignments as well,  first go to the main, hosted database and to  Scheduling=>Assign Students to Classes by Class and add the student to the new class; or go to Assign Students to Classes by Student and add the class to the student. DON'T remove the student from the old class yet! 

After changing the student's schedule, sync the portal.

To move the student's assignments to the new class: go to Gradebook=>Academic=>Gradebook=>By Assignment. Look up the teacher and new class. Click the “Move Student” button on the top toolbar. From the drop down, choose the student, the “Move From Teacher” and “Move From Class”. For each assignment on the list, use the drop down to indicate whether the assignment should be merged with an existing assignment, added to the teacher’s gradebook, or ignored. When done, click the “Import Selected Assignments” button at the top right corner. Let the teacher know that the assignments, with the new student in it, will show up in her gradebook. When calculating grades, assignments from other classes will only affect the student that is in it.

After you’ve moved the assignments for the student, you can go back to Schedule Classes (by Class or by Student) and remove the student from the old classes.