How do I add an assignment and also have it listed under a respective module?
If you go to the module in question and click the "+", then select "Add Assignment [New Assignment]", that will create the assignment in the Assignments area and link it from the module.
There are more details on creating an assignment atHow do I create an assignment?
You can add assignment by clicking +Assignment in the module.
If you are what to move the assignment between the module, you can either drag and drop or click the three horizontal dots and select the move option from the dropdown.
Hope this helps.
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