RWY INCURSION OCCURS WHEN A PA28 INSTRUCTOR PLT ALLOWS HIS STUDENT PLT TO TAXI ACROSS AN ACTIVE RWY AT FRG; NY.

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AT FARMINGDALE'S REPUBLIC ARPT; I INADVERTENTLY TAXIED TO THE WRONG RWY AND IN SO DOING; I CROSSED OVER THE ACTIVE RWY BY MISTAKE. I DID NOT IMMEDIATELY REALIZE MY MISTAKE BECAUSE I WAS ADJUSTING MY PAX'S (STUDENT) SHOULDER HARNESS WHILE I ALLOWED HIM TO TAXI THE ACFT. ONCE I CROSSED OVER THE RWY; WE CONTINUED TO THE RUN-UP AREA WHEN I FINALLY REALIZED THAT I WAS NOT POSITIONED CORRECTLY AND THAT I CROSSED OVER THE ACTIVE RWY. I ADVISED THE TWR THAT I WAS HOLDING SHORT OF THE WRONG RWY FOR TKOF; AND THEY QUESTIONED MY ACTIONS. AFTER A FEW MINS; I WAS ALLOWED TO DEPART ON THE INACTIVE RWY WITHOUT INCIDENT. I REALIZE THE GRAVITY OF MY MISTAKE AND TAKE FULL RESPONSIBILITY FOR BECOMING DISTR WHILE TAXIING AND NOT SUPERVISING MY STUDENT FULLY. DESPITE HAVING THE CURRENT ATIS INFO; A TAXI CLRNC; AND BEING THOROUGHLY FAMILIAR WITH REPUBLIC ARPT; MY LACK OF ATTN CREATED A POTENTIALLY DANGEROUS SIT. I AM GRATEFUL THAT MY ERROR DID NOT CAUSE AN ACCIDENT; AND TO MY KNOWLEDGE; NO OTHER PLTS WERE AFFECTED BY MY MISTAKE (ABORTED TKOF; GAR; ETC). I GREATLY APPRECIATE THE PROFESSIONAL MANNER WITH WHICH THE REPUBLIC AIR TFC CTLRS HANDLED THIS MATTER AT THE TIME OF ITS OCCURRENCE. I RPTED THIS INCIDENT TO MY EMPLOYER WHO ADMONISHED ME FOR MY LACK OF STUDENT SUPERVISION AND ATTN WHILE TAXIING.