Small aircraft instructor pilot reported an near midair collision in the pattern.

Narrative:

The event occurred during traffic pattern operations at ZZZ. There were 3 planes in the pattern. 2 PA-28 planes (1 solo student) and a piper malibu. The 2 archers were in left traffic for [runway] [runway] xx and the malibu was in right traffic for [runway] [runway] yy. I was number one with the solo student behind me as number 2. Tower instructed the malibu; who at the time was at around 600 feet MSL on what looked like a right base to [runway] yy; to enter right downwind for [runway] xx as number 3 because he was faster than anticipated. The malibu read back right downwind for [runway] xx and asked if he should keep it in tight. Tower advised him that he was number 3 for the field. At this point; I was left base turning to final (student was flying at the time) and I looked off to my right and I saw the malibu coming straight at me at the same altitude. I took the controls; full power; left steep turn to avoid hitting the other plane. The other plane turned final and landed without a landing clearance. I advised tower of the turn I made and that the plane just cut us off. Tower advised the malibu to copy a phone number after landing and; thankfully; an accident was avoided. Tower then advised me that he did not know why the plane did that.in hindsight; I believe there was a misunderstanding from the malibu because he thought that because he was faster; tower wanted him in first. When he asked tower if he should keep it in tight; tower said you're number 3 for the field instead of saying 'negative extend downwind' or something more of a command to ensure the pilot knew what was happening. The weather was not a factor. The pilot should have exercised better judgement and should have tried to get a better picture of who was in the pattern when tower told him he was number 3.as a CFI; I am going to use this story for all my students about maintaining situational awareness and to always be vigilant and don't expect other pilots to be perfect. If that solo student was number one instead of me; this could have resulted in a tragedy.

Original NASA ASRS Text

Title: Small aircraft instructor pilot reported an NMAC in the pattern.

Narrative: The event occurred during traffic pattern operations at ZZZ. There were 3 planes in the pattern. 2 PA-28 planes (1 solo student) and a piper Malibu. The 2 Archers were in left traffic for [Runway] [Runway] XX and the Malibu was in right traffic for [Runway] [Runway] YY. I was number one with the solo student behind me as number 2. Tower instructed the Malibu; who at the time was at around 600 feet MSL on what looked like a right base to [Runway] YY; to enter right downwind for [Runway] XX as number 3 because he was faster than anticipated. The Malibu read back right downwind for [Runway] XX and asked if he should keep it in tight. Tower advised him that he was number 3 for the field. At this point; I was left base turning to final (student was flying at the time) and I looked off to my right and I saw the Malibu coming straight at me at the same altitude. I took the controls; full power; left steep turn to avoid hitting the other plane. The other plane turned final and landed without a landing clearance. I advised Tower of the turn I made and that the plane just cut us off. Tower advised the Malibu to copy a phone number after landing and; thankfully; an accident was avoided. Tower then advised me that he did not know why the plane did that.In hindsight; I believe there was a misunderstanding from the Malibu because he thought that because he was faster; Tower wanted him in first. When he asked Tower if he should keep it in tight; Tower said you're number 3 for the field instead of saying 'negative extend downwind' or something more of a command to ensure the pilot knew what was happening. The weather was not a factor. The pilot should have exercised better judgement and should have tried to get a better picture of who was in the pattern when Tower told him he was number 3.As a CFI; I am going to use this story for all my students about maintaining situational awareness and to always be vigilant and don't expect other pilots to be perfect. If that solo student was number one instead of me; this could have resulted in a tragedy.

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	1744835
Time
Date	202006
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	PA-28 Cherokee/Archer/Dakota/Pillan/Warrior
Operating Under FAR Part	Part 91
Flight Phase	Final Approach
Route In Use	Visual Approach
Flight Plan	None
Aircraft 2
Make Model Name	PA-46 Malibu/Malibu Mirage/Malibu Matrix
Operating Under FAR Part	Part 91
Person 1
Function	Instructor Pilot Not Flying
Qualification	Flight Crew Flight Instructor Flight Crew Commercial Flight Crew Instrument Flight Crew Multiengine
Experience	Flight Crew Last 90 Days 250 Flight Crew Total 1400 Flight Crew Type 1300
Events
Anomaly	Conflict NMAC Deviation - Procedural Published Material / Policy Deviation - Procedural Clearance Deviation - Track / Heading All Types
Miss Distance	Horizontal 200 Vertical 0