Flight Instructor with student reported taking off while an airplane was on short final approach to the same runway. The landing aircraft conducted a go around.

Narrative

While holding short of runway at a non-towered airport; I checked my EFB for traffic. My student and I confirmed traffic by looking outside. There was an aircraft on downwind who looked like they would be doing a regular downwind pattern to base so I told my student they may proceed onto the runway after making their call to enter the runway and departure. As our plane passed the hold short lines; the aircraft started to turn base for a short approach. Perhaps I missed their call on downwind for their short approach? We continued a no delay takeoff and departed with no issue. Aircraft on short approach initiated a go around and overtook our slower plane to the right and left ample separation. They continued to fly an extended upwind to a crosswind for another lap in the traffic pattern. We followed behind slowly. Us listening to the radio calls more carefully could have avoided a go around for the other aircraft. Also; noticing the speed of the other aircraft and holding short until they landed and departed on a touch and go would have been the best action; instead of assuming they were doing a regular pattern.