PA-28 Cherokee/Archer/Dakota/Pillan/Warrior
Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Commercial
Flight Crew Last 90 Days 226; Flight Crew Total 685
Communication Breakdown; Situational Awareness
ATC Issue All Types; Conflict NMAC
Air Traffic Control Issued Advisory / Alert; Flight Crew Took Evasive Action
Human Factors
My Student and I were doing Pattern work at ZZZ. Tower cleared us for takeoff on Runway XX and told us to fly straight out and she would call our crosswind. Shortly after we took off; Tower cleared another aircraft for takeoff (which appeared to be another GA aircraft). In the upwind I noticed on our traffic page on our avionics that the traffic behind us seemed like it was starting to catch us. Around 800 ft AGL Tower told us to turn a right crosswind. When we turned I could see the traffic heading straight for us and Tower frantically said 'Traffic'. I cannot recall how far away the traffic was at the time but I do remember that the traffic did appear to be slightly above us. The other traffic and I both told Tower that we had each other in sight and he kept climbing while I pulled power and pitched down to around 550 ft AGL to give extra spacing between the two aircraft. On our traffic page it said that the traffic passed over us by 400-500 ft.
Flight Instructor flying PA-28 aircraft reported NMAC during departure climb.