C172 INSTRUCTIONAL FLT INVOLVED IN NMAC WITH ACFT IN TFC PATTERN.

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ON R DOWNWIND TO RWY 31L DUE TO RWY 31R CLOSED FOR ACFT WITH BLOWN TIRE ON RWY. STUDENT HAD CLBED ABOVE TPA ON DOWNWIND TO 1300 FT. I HAD JUST POINTED ALT ERROR OUT TO HIM AND HE WAS IN A DSCNT BACK TO 1100 FT. I HEARD A XMISSION FROM TWR DIRECTING AN INBOUND ACFT TO MAKE R TFC (DID NOT DISCERN ACFT N-NUMBER). AS WE CAME ABEAM RWY 31 APCH END I REMINDED STUDENT TO REDUCE PWR; APPLY 10 DEG FLAPS AND BEGIN DSCNT. AT THAT POINT I RECEIVED A CALL FROM TWR THAT TFC WAS 12 O'CLOCK POS 1 MI; IMMEDIATELY FOLLOWED BY A CALL 'PIPER (THE OTHER ACFT) YOU'RE GOING THE WRONG WAY.' I FIRST SAW THE PIPER +/-50-100 FT HIGH AND 100 FT AT MY 11 O'CLOCK POS. TWR KEPT CALLING 'PIPER YOU'RE GOING THE WRONG WAY.' I DID NOT INTERRUPT TWR'S CALL BECAUSE THE PIPER HAD ALREADY PASSED US. THE PIPER FINALLY RESPONDED THAT HE WOULD DO A 360 AND REENTER DOWNWIND. WE LANDED AND CONTINUED IN L PATTERN FOR RWY 31L FOR 1 MORE LNDG; THEN TERMINATED. MY ERROR WAS IN BEING INVOLVED IN MY STUDENT'S ACTIONS AND NOT VISUALLY DETECTING THE APCHING HEAD ON TFC BEFORE THE CALL FROM TWR. I DO NOT KNOW IF THE PIPER EVER SAW US; CLASSICAL LOW WING HIGHER THAN HIGH WING BLIND SPOT.