Important: If you have not synced your course roster yet, check out the Gradescope Roster Sync Guide before you try to sync your grades to Moodle.
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To push any grades from Gradescope to Moodle, there has to be an assignment in Moodle to associate the grades with. You can find detailed instructions in our article, How do I create an Assignment in Moodle? Just below is the quick version of those instructions:
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Go to your course in Moodle, make sure editing is turned on, and click on the "Add an activity or resource" link at the bottom of the topic area where you want to create the assignment.
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Click on the Assignment link to begin creating the assignment in Moodle.
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Give your assignment a name that makes sense to your students. It should probably match the name of the assignment in Gradescope, but it does not have to.
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By default, Moodle sets the Due date to be one week after you create the assignment. If you like, you can modify that Due date to reflect the actual date/time that the assignment is due. (Please remember that Moodle uses military time, so 5:00 PM, for example, would be 17:00.)
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You'll want to expand the "Submission types" section and make sure that "Online text" and "File submissions" are both de-selected. In other words, you'll want to be sure to clear the check mark in the "File submissions" box, so that the "Online text" and "File submissions" check boxes are both empty. Students will not be submitting their assignment via Moodle. The assignment is just a placeholder for the grades that you will download from Gradescope.
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The other default settings should be fine, and Gradescope will set the maximum grade value when you sync, so you can just scroll down to the bottom of the page and click the "Save and return to course" button.
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Return to your Gradescope window, click on the Assignments link, and select the homework assignment that you would like to sync to your Moodle gradebook.
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In the menu on the left, go to "Review Grades."
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At the bottom of the screen, you should see an option to Post Grades to Moodle. (Don't click on the button yet.) 
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Before posting the grades to Moodle, you need to make sure that your students are all properly synced. To do so, check and ensure they have a link under the MOODLE column. If the link is not there, you may need to run through the roster sync process again.
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If your students are all synced, click the Post Grades to Moodle button.
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You will see an information box with details about your sync with Moodle. Under the heading "Linked Moodle Assignment," click edit and select the Moodle assignment for the grades to attach to.
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Select the Post Grades button.
Congratulations! Your Gradescope assignment score is now synced with your Moodle gradebook! You will need to repeat this process for any other assignment grades that you wish to sync with Moodle.
If you’d like to learn about Gradescope, send us an email at dps@bucknell.edu or check out the help resources at gradescope.com.