C172 INSTRUCTOR WITH STUDENT REPORT CRITICAL CONFLICT WITH PIPER TWIN ON FINAL.

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MY PVT STUDENT AND I WERE TRYING TO DO SLOW FLT; STALLS; OUT IN THE PRACTICE AREA. AFTER WE TOOK OFF I NOTICED THE CEILING STARTING TO COME DOWN MUCH LOWER THAN THE METARS HAD SPECIFIED PRIOR TO TAKING OFF. WE STAYED BELOW THE CLOUD LAYER AND REMAINED VFR BUT IT WAS APPARENT WE WERE UNABLE TO DO AIRWORK. I DECIDED THAT BECAUSE THE WX WAS VFR OVER ZZZ ARPT WE WOULD PRACTICE TFC PATTERNS; TKOFS; AND LNDGS. WE WERE THE ONLY ONES IN CLOSED TFC WITH INTERMITTENT TFC DEPARTING THE FIELD. ON EITHER THE 4TH OR 5TH TFC PATTERN WE WERE CLRED FOR TOUCH AND GO ON OUR XWIND LEG JUST AS WE HAD BEEN ON THE PREVIOUS PATTERNS. ON DOWNWIND I HEARD THE CTLR TALKING TO A CHEROKEE WHO WAS COMING IN FOR AN APCH. WE TURNED BASE AROUND 900 FT AND STARTED OUR DSCNT. TURNING FINAL MY STUDENT SAW THE CHEROKEE OUT HIS L WINDOW AND TOLD ME THERE WAS AN AIRPLANE DIRECTLY BEHIND US AT OUR ALT. I TOOK CTLS AND CLBED OUT STRAIGHT OVER THE RWY. THE CHEROKEE WENT DOWN AND TO THE R AND CONTINUED LNDG. IMMEDIATELY AFTER ATC CAME ON THE RADIO AND TOLD US THAT HE HAD SEQUENCED US BEHIND THE CHEROKEE. I APOLOGIZED AND TOLD HIM WE'D BE MAKING ANOTHER PATTERN. I DID NOT HEAR ANY XMISSION ABOUT US BEING #2 FOR LNDG AND AS SUCH NO CALL BACK WAS MADE TO CONFIRM CLRED TO LAND. MY STUDENT ALSO CONFIRMED THAT NO RADIO CONTACT WAS MADE CHANGING OUR LNDG CLRNC. SUPPLEMENTAL INFO FROM ACN 787856: THE APCH SPD OF THE TWIN ENG PIPER WAS CONSIDERABLY FASTER THAN THE C172 WHICH MOST LIKELY WAS AN ATTRIBUTE THAT LED TO THIS INCIDENT.