C172 flight instructor reported an NMAC when they took off from a non-towered airport in front of an aircraft on short final.

Narrative

The event in question is a near miss on my takeoff roll with a Cessna 310 on final. The event began when my student and I were holding short of Runway XX at ZZZ. The plan was to practice normal take-offs and landings in the pattern. We had just finished the run-up; and were monitoring ZZZ CTAF as we taxied to the hold short line. We were aware of an aircraft on downwind and an aircraft on final. The downwind traffic was yielding to traffic on final. The traffic on final was a Cessna 310. Both my student and I made visual contact with the traffic-it was on a three mile final approximately six hundred ft. above the ground. We agreed that we had adequate spacing; and elected to take-off. We also made the decision to expedite the takeoff to give the 310 extra space. This entire decision making process took ten seconds. My student made his departure call; and we took the runway. As he made his departure call I made an additional call stating that we would expedite for the 310 on final just to give him extra room. Immediately after we received a call stating 'no need to expedite-you have plenty of space.'At this point; in the interest of safety; I told my student that there was no need to expedite his takeoff; and that he could make a normal takeoff roll. I assumed that the caller was the 310; and it did not occur to me that anyone else would make that callout. I thought it was polite for the person in question to make the call; and I came back over frequency to say thank you. Upon review of the audio files; unbeknownst to both myself and my student; in making this call I stepped on the 310 saying he was on a one mile final. The next point of contact with the 310 came in the form of visual contact with the traffic approximately 100 ft. above me. My student and I were already wheels up past rotate speed. I took the controls from my student and kept us below the traffic until he was clear. Ideally I would have liked to abort the takeoff; but I was uncomfortable doing so at our speed and our position on the runway. The 310 made a go around and made a left hand turn. Once he was well clear; we joined right traffic for XX. Upon conclusion of the flight I reported the incident immediately to my lead instructors. The 310 was on a three mile final at the time my student and I decided to make our takeoff. I believed that we had adequate spacing. I decided to expedite just to be safe; and when my student and I heard the 'plenty of room' callout we discontinued the expedited take-off; and resumed a normal. The decision to takeoff was ultimately mine. While I am intimately familiar with runway incursion avoidance and related safety practices; I hold both myself and my students to the highest standards of safety possible. I intend to complete the ALC-48: Hold Short for Runway Safety FAA Webinar immediately; and review runway incursion avoidance with every single one of my students. My biggest takeaway is the necessity to give adequate spacing for high performance/multi engine aircraft. In the future I will not take the runway if a high performance/multi engine aircraft is within a four mile final. I will also ensure that any decision regarding traffic spacing is verified visually; and will use more caution in regard to traffic callouts made by unverified aircraft.