Skyhawk 172/Cutlass 172
Flight Crew Flight Instructor; Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Commercial
Flight Crew Last 90 Days 190; Flight Crew Total 877
Workload
Conflict NMAC
General None Reported / Taken
Environment - Non Weather Related; Human Factors
Me and my student were on an IFR flight plan from ZZZ to ZZZ1 and were communicating with Approach. We were maintaining 4000 ft as assigned. At around XA:05pm approach advised us that there was a traffic on opposite direction; indicated 3500 ft; a Cherokee. At that time there were several other traffics in the area. We saw the traffic passing below us on the opposite direction with a safe separation; so we advise ATC that we have traffic insight. I also confirm with the ads-b traffic on MFD that they were at least 500 ft below us and would not create any collision hazard. Approach then hand us off [to another frequency]. While switching frequency on the MFD; I saw another ads-b traffic on MFD approaching us from the left hand side. The traffic seemed to be maneuvering; climbing rapidly; and converging with us at the same altitude. I immediately look up and see an aircraft passing underneath us from left to right. The separation would have been less than 200 ft. I then make an initial call to approach and advise them that we just had a traffic crossing beneath us almost the same altitude. The controller acknowledge but did not give us any other information. Approach did not advise us of that traffic either prior to handing us off. The converging identification was [callsign] as shown on the ads-b traffic.What I would have done different would be to keep my eye outside more; especially near the area with high volume of flight training activities. Multiple traffics in the area made it harder to judge which one is closer or heading toward us. I was busy setting up frequency and reading approach plate; where I loose sight of the outside traffic for a moment. I also wouldn't rely too much on ATC to provide traffic information in busy airspace even on IFR flight plan.
C-172 Instructor reported an NMAC event during a training flight. The Instructor reported the conflict to ATC who acknowledged the call.