Can I create an open form to save and continue later?

This would be a form where I could post questions to students we teach and then if they didn't finish the questions, they could come back to it the next week. Is this possible?

As far as I understand, you need an option to "Save and Continue" for your form. 

We do have such option available at JotForm. 

Please visit this link for detailes: 

How-to-Save-Forms-and-Continue-Later

However, the proccess is much more easier with Card Layout. You need just to enable  "Continue Forms Later" option at your Form's Settings tab, so the "Save & Continue" becomes active.

Your respondents will see this option like bellow:

We will gladly assist you if you need more help.