PA28 INSTRUCTOR PLT AND STUDENT TURNED ONTO THE WRONG TXWY AT NIGHT AT RIC.

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AFTER LNDG ON RWY 20; WE EXITED AT TXWY C. THE TWR CTLR INSTRUCTED US TO 'MAKE NEXT R ON TXWY A; L ON TXWY R TO COMPANY CTR.' THE STUDENT READ BACK THE CLRNC AS WE CLEANED UP THE AIRPLANE. WE FOLLOWED THE TXWY LINE TO THE R AND AS WE PASSED THE NEXT INTXN; I REALIZED WE HAD MISTAKENLY TURNED ONTO TXWY U INSTEAD OF TXWY A. BY THAT TIME; I COULD SEE TXWY A AND WHERE WE WERE SUPPOSED TO TURN L ON TXWY R. I POINTED OUT TO THE STUDENT THAT WE WRE ON TXWY U; AND THAT IF WE TURNED L AT THE END; WE WOULD BE ON TXWY R AS INSTRUCTED. WE TURNED ONTO TXWY R AND TAXIED TO THE COMPANY CTR PARKING AREA WITHOUT HEARING ANYTHING FROM THE CTLR. ONE CONTRIBUTING FACTOR WAS THE BRIGHT LIGHTS OF THE TERMINAL; WHICH TENDED TO MAKE IT DIFFICULT TO READ THE TXWY SIGNS. A SECOND CONTRIBUTING FACTOR WAS THE TAXI INSTRUCTIONS TO 'MAKE NEXT R ON TXWY...;' BECAUSE OUR NEXT R WAS IN FACT TXWY U. TXWY A WOULD HAVE BEEN OUR SECOND R; AND WOULD HAVE FIRST REQUIRED A SLIGHT L TURN WHILE FOLLOWING TXWY C. A THIRD CONTRIBUTING FACTOR WAS FATIGUE. I HAD INSTRUCTED FOR ALMOST 4 HRS SO FAR THAT DAY AND WAS STARTING TO FEEL TIRED. THERE WAS ALSO AN ELEMENT OF GET THEREITIS; SINCE WE HAD REACHED OUR DEST; AND I NEEDED TO USE THE RESTROOM. THIS WAS THE SECOND TIME THAT I HAD FLOWN TO RIC AS AN INSTRUCTOR AND I THOUGHT THAT I HAD A HANDLE ON THE TXWY LAYOUT. THE STUDENT AND I WERE REFERRING TO THE CURRENT ARPT DIAGRAM AT THE TIME. IN RETROSPECT; I SHOULD HAVE HAD THE STUDENT INDICATE THAT WE WERE UNFAMILIAR WITH THE FIELD; AND ASK FOR PROGRESSIVE TAXI INSTRUCTIONS. WHEN I NOTICED THAT WE WERE ON THE WRONG TXWY; I SHOULD HAVE HAD THE STUDENT STOP THE AIRPLANE; ADMIT OUR MISTAKE AND THE SOURCE OF THE CONFUSION; AND ASK THE CTLR FOR FURTHER INSTRUCTIONS. HAD THE ARPT BEEN BUSY; WE COULD HAVE TAXIED INTO THE PATH OF ANOTHER TAXIING AIRPLANE. IN THE FUTURE; I WILL BE MORE CAREFUL ABOUT ALLOWING STUDENTS TO BEGIN TAXIING AT UNFAMILIAR ARPTS IF THERE IS ANY QUESTION AS TO OUR ASSIGNED TAXI RTE. I WILL ALSO BE SURE TO DOUBLECHK OUR POS ON THE ARPT AND MAKE SURE I GET CLARIFICATION FROM THE CTLR IF THE TAXI RTE ASSIGNED IS UNCLR; EVEN IF THE STUDENT THINKS HE KNOWS WHERE HE'S GOING. I WILL ALSO PLACE MORE INSTRUCTIONAL EMPHASIS ON LOOKING FOR AND INTERPING RWY/TXWY SIGNAGE; RATHER THAN FOLLOWING THE CTLR'S DIRECTIONS 'MAKE NEXT R...' WITHOUT VISUALLY VERIFYING THE PROPER RWY/TXWY.