C172 flight instructor reported a near midair collision with a DA40 that was not transmitting intent...

Narrative:

I was flying with student in a C172. We headed toward [home airport]; I made the top of stack request 10 miles north. 3500 was taken; 4000 was open and there was a GPS hold at 4500. I made my call to take 4000 ft in 7 minutes. I was following the published procedures.as we got closer 3500 feet became open. I made the announcement to take 3500. Crossed the VOR; tear drop entry; made outbound call; turned inbound. Student told me he was turning inbound. I made the procedure turn inbound call for the ILS. As we got closer to the VOR; another plane came up from beneath us and almost caused a midair. The DA40 passed inches beneath us. I was able to increase the aoa just enough to get us out the way. As the wake passed I had a moment of doubt as [to] whether or not there was contact on the main gear - I believe that although we did not make contact we were inches from contact. I was able to see (albeit) the two pilots in the cockpit. I made an announcement on frequency that there was a plane in the stack that was not talking on frequency; they replied saying they had made intention to take 3500 in 1 minute. That however is not the procedure and I did not hear this call. The student was able to verify that he heard nothing. The other plane then made his procedure call over the VOR. This is when I was able to take down the tail number.

Original NASA ASRS Text

Title: C172 flight instructor reported a NMAC with a DA40 that was not transmitting intentions or following proper procedure.

Narrative: I was flying with student in a C172. We headed toward [home airport]; I made the top of stack request 10 miles north. 3500 was taken; 4000 was open and there was a GPS hold at 4500. I made my call to take 4000 ft in 7 minutes. I was following the published procedures.As we got closer 3500 feet became open. I made the announcement to take 3500. Crossed the VOR; tear drop entry; made outbound call; turned inbound. Student told me he was turning inbound. I made the procedure turn inbound call for the ILS. As we got closer to the VOR; another plane came up from beneath us and almost caused a midair. The DA40 passed inches beneath us. I was able to increase the AOA just enough to get us out the way. As the wake passed I had a moment of doubt as [to] whether or not there was contact on the main gear - I believe that although we did not make contact we were inches from contact. I was able to see (albeit) the two pilots in the cockpit. I made an announcement on frequency that there was a plane in the stack that was not talking on frequency; they replied saying they had made intention to take 3500 in 1 minute. That however is not the procedure and I did not hear this call. The student was able to verify that he heard nothing. The other plane then made his procedure call over the VOR. This is when I was able to take down the tail number.

Data retrieved from NASA's ASRS site and automatically converted to unabbreviated mixed upper/lowercase text. This report is for informational purposes with no guarantee of accuracy. See NASA's ASRS site for official report.

Attributes
ACN	1512955
Time
Date	201801
Local Time Of Day	1201-1800
Place
Locale Reference	ZZZ.Airport
State Reference	US
Environment
Flight Conditions	VMC
Light	Daylight
Aircraft 1
Make Model Name	Skyhawk 172/Cutlass 172
Operating Under FAR Part	Part 91
Flight Phase	Initial Approach
Flight Plan	None
Aircraft 2
Make Model Name	DA40 Diamond Star
Operating Under FAR Part	Part 91
Flight Phase	Initial Approach
Person 1
Function	Instructor Pilot Not Flying
Qualification	Flight Crew Multiengine Flight Crew Commercial Flight Crew Flight Instructor Flight Crew Instrument
Experience	Flight Crew Last 90 Days 120 Flight Crew Total 492 Flight Crew Type 350
Events
Anomaly	Conflict NMAC Deviation - Procedural Published Material / Policy

C172 flight instructor reported a near midair collision with a DA40 that was not transmitting intentions or following proper procedure.