Why grade? Are late assignments acceptable or a failed grade?

I did my student teaching in a fourth grade classroom. The mentor teacher gave the students the lesson and let them work on it for a short time in class. If the students did not finish, the assignment was turned into homework. Homework was collected each morning and placed in the appropriately labeled basket. I liked this method as the students that were able to finish in class did not need to finish the assignment at home. The students that were slower were given extra time to finish. The problem I encountered was that the same students did not turn in their homework due to a soccer game or their family went out to eat. I circled the area that did not have a grade in my grade book (it looked like a zero but it was an indication to me that it was not yet turned in) and would tell those students they owed me work by posting their name and assignment on the board. These students would be worked with during a down time in class so I could help them one on one. I don't necessarily agree that extra work needs to be given but I do believe that lessons need to be completed to understand the objective the student is given by the teacher. If a student continues not to try, then they need to have a special meeting to set up an individual educational plan so the student will be able to meet a differentiated set of lessons and objectives.