Montgomery County’s failure rate for the June final exam in Algebra 1 was so steep — 82 percent for high school students — that district officials say they decided to add 15 percentage points to all test grades, recalculate scores and reprint thousands of report cards.

Report cards across the system were mailed Friday, three days late, and many parents said they received them Saturday.

The last-minute scramble to adjust for the poor results came as perhaps the most vivid example so far of fallout from widespread exam failures in the high-performing school system.

High rates of exam failure go back years in Montgomery but came to public attention last year, when district data showed that a majority of 30,000 students taking key high school math exams failed the biggest test of the semester.

A work group was created to study the issue, and two weeks ago Montgomery Schools Superintendent Joshua P. Starr offered a five-point plan to address widespread math failure. But about the same time, teachers and principals began to realize that the most recent exams, taken by nearly 11,600 students, had hit a striking low mark, officials said.

More than eight of 10 high school students failed the June exam, while middle school students — who tend to be higher performers — failed at a rate of 23 percent. A semester earlier, 61 percent of high school students and 12 percent of middle school students failed.

Erick Lang, Montgomery’s associate superintendent for curriculum and instructional programs, said the main cause of the failure spike appears to have been a loss of instructional time in the spring semester, as teachers prepared students for state exams required for graduation. Time also was lost to weather-related school closings.

“We believe that’s a huge factor,” he said of the state exam preparation time. For some schools, it meant two weeks lost, and for some it meant three weeks, he said. “That’s a big chunk out of instructional time.”

The missing instruction came as Montgomery’s Algebra 1 course and exam were aligned for the first time this year with Common Core state standards, which are rolling out across the country. Lang noted that the state test — the High School Assessment, or HSA — includes algebra and other topics, such as data analysis.

“We have a new curriculum we’re implementing, and that curriculum is not completely aligned,” Lang said.

Other math classes did not show a similar uptick in failure rates, Starr wrote in a memo to the board.

The addition of 15 percentage points to exam scores shifted the failure rate to 68 percent for high schoolers and 12 percent for middle schoolers, resembling previous June exam results. This meant 623 high school students and 758 middle school students who would have failed instead passed the Algebra 1 exam.

Under state requirements, middle schoolers must pass the final to earn course credit.

“I don’t know what other choice we had than to curve those grades,” said School Board Vice President Patricia O’Neill. She and others said they did not want Montgomery students penalized for problems not of their making. “We can’t hold children accountable for adult problems.”

Most Montgomery students who fail exams do not fail the course. Exams are worth 25 percent of a course grade.

In Starr’s memo to the board, he said an analysis showed that wrong answers most frequently were related to the final unit of the year and that the problems were not significantly more difficult, suggesting that students did not complete the final course topics.

School board member Michael Durso said the algebra problems underscore broader issues with exam failure and raise questions about whether Montgomery has gotten to the bottom of it. “I guess for me the questions still outnumber the answers,” he said.

Others also questioned the 15-point add-on.

“I think it definitely puts a question to the integrity of your assessment when you have to put a curve to every child,” said Merry Eisner-Heidorn, a parent advocate who served on the math work group that Starr created.

In his memo to the board, Starr said officials had considered other options, such as eliminating some exam items from their grading, but decided it might hurt some students as it helped others. He said the major concern was “not penalizing students for things beyond their control.”

The district is offering summer reteaching and retesting at no cost, officials said, and all fees will be waived for students taking ­second-semester Algebra 1 in summer school.