City educators gave out answers to state test questions, inflated Regents exam scores, and coached students to change incorrect responses dozens of times in recent years, according to reports from a slew of investigations into test improprieties.

Responding to a Freedom of Information Law request by GothamSchools for information about complaints about test security, the Department of Education released 97 reports from investigations that concluded violations had taken place. The reports were completed between 2006 and 2012 by the Department of Education’s Office of Special Investigations and the independent Special Commissioner of Investigation.

Thirty-eight of the reports documented relatively minor violations of administrative protocol. In multiple cases, for example, investigators found that teachers had photocopied exam books when there were too few before getting official permission.

But 59 of the reports substantiated allegations about cheating, some of them serious.

One of the people found to have participated in cheating in a newly released report told GothamSchools today that an administrative trial ultimately concluded that no misconduct had taken place. The department did not immediately provide details about what happened in the cases after the investigations were over.

The number of cases are not a comprehensive accounting of the scale in which cheating occurs in schools. Investigators in some of the reports suggest that suspicious activities could go underreported. But the cases do provide a snapshot of what lengths teachers go to — and the ease in which they can attempt them — to inflate their students’ test scores

Some of the reports have been made public in the past. The report dump includes the report on Ruth Ralston, found to have changed students’ answers and lied about it at the High School of Contemporary Arts, and Joyce Plush-Saly, under whose watch teachers at P.S. 58 in Brooklyn gave students test answers in advance.

But the vast majority of the reports had not previously been released. SCI releases reports only about 5 percent of the time when it concludes that wrongdoing has taken place, and OSI rarely releases any reports at all.

The newly released reports include ones in which students recount the creative strategies their teachers used to alert them to incorrect answers and, in some cases, to the correct ones as well. They also include reports about schools — such as Hillcrest High School in 2007 and New Utrecht High School in 2010 — where large numbers of students were found to have received passing Regents exam scores when they actually failed.

A year ago, the Department of Education moved to crack down on Regents exam grade inflation, introducing a new grading system that went into place citywide this month in which teachers no longer grade tests taken at their schools. Officials attributed a significant decline in the number of just-passing scores on exams taken last June to a pilot of the “distributed grading” system. A state decision to eliminate “regrading” has further reduced opportunities to inflate students’ scores, although cheating is still possible.

Department officials emphasized that the number of substantiated allegations about cheating in the last several years has been very small and that the city’s test security requirements have been stricter than what the state has required.

“We have zero tolerance for cheating and all violators are disciplined,” Connie Pankratz, a spokeswoman, said in a statement. “The department goes above and beyond the state’s test security requirements with unannounced visits by test monitors and stricter protocols such as immediately removing completed test materials from schools. These measures work.”

Since last year, the state has been ramping up its test security practices and now is set to exceed the city’s regulations. In addition, a new test security office at the state level is preparing to launch investigations based not only on allegations but also on test score data that raise red flags. Virtually all of the reports released today stemmed from allegations first made by teachers or principals, but a handful suggested that city and state data analysts had played a role in identifying suspicious score patterns.

Just one substantiated report found evidence of cheating based on allegations made by teachers who found that their students were not prepared for the next grade. Students from multiple schools arrived at M.S. 218 in Brooklyn with high scores but low skills, and investigators ultimately concluded that teachers at the elementary schools had provided test answers to fifth-graders the previous year.

Last year, two Brooklyn schools were placed under investigation for the same reason, but teachers and administrators at other schools that experience large “swing rates” in new students’ test scores sometimes do not alert the Department of Education about their suspicions. Department officials have said they do not launch investigations based on data anomalies alone.