GA flight instructor reported a NMAC after turning right rather than left as cleared during departure. The reporter became aware of the issue and corrected the heading.

Narrative

Was cleared for a left turn out and instead turned right at 500 feet AGL. Stayed there for a moment before realizing my mistake and turning the opposite direction; back to the left where I was supposed to be; which put me in the path of departing traffic; although said traffic was not really a factor because at that time I was much higher and further than they would be when I passed as they were also a single engine piston. Did not realize until later I shouldn't have turned but rather advised ATC and let them tell me what to do. They do not have radar so they didn't notice. I did get close to other traffic enough for my warning system to advise me of their position but it was a helicopter flying to the airport from the hospital and was very low. I got sight of them quickly and knew they weren't a factor either. The problem arose due to me not writing down and remembering my clearance properly. Will be avoided in the future by doing this. It was discovered by me realizing my mistake. Contributing factors are distraction and being a new instructor; easily excited and distracted by students. It was on a discovery flight.