When I import a Google classroom, it imports the assignments. What happens after that when I add a new assignment via Google Classroom? (In other words: Can I easily import all of the new assignments since the initial import? or, do I need to import at the very end to have all assignments import?)
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Yes, you will be able to import all new assignments that were created in the Google Classroom using the same import button. Also, if you delete an assignment from the Classroom then after next import that assignment will disappear. In other words, you can consider "import" as an option to sync with the specified Google Classroom. Same will happen with students.