PHOENIX — High school seniors who can’t pass the AIMS tests may still be able to graduate. Without discussion, the House on Thursday gave preliminary approval to reinstating a program which allows high schoolers to augment their scores with good grades. Without HB 2008 the bonus points, which have existed for the last two years — since AIMS passage became a graduation requirement — will go away.

 Rep. David Schapira, D-Tempe, said the measure will ensure that youngsters who otherwise are good students are able to get a diploma. He said while the AIMS test is important, they should not become the one thing standing in the way of graduation.

 State School Superintendent Tom Horne acknowledged some seniors who otherwise successfully complete their courses will not graduate solely because they did not get passing grades on the math, reading or writing sections of the test, more formally known as Arizona’s Instrument to Measure Standards.

 But Horne said the test is specifically designed to ensure that students know the subjects the state has determined are necessary to graduate. Nor was he persuaded that good students are failing solely because of text anxiety. “Tests are a part of life,’’ he said. “Get over it.’’

 Shortly after the vote, Horne announced he had found enough additional unexpended cash in his agency’s account to reinstate a free tutoring program for students who have previously failed one or more parts of AIMS. Horne had suspended the program in

 February, saying the money had run out. But this is not a complete reversal.

 The funds will be available solely for math tutoring, the one part of AIMS not yet administered this year. Reading and writing test segments were given in February.

 Horne said there is only enough cash to help seniors; the prior program also aided juniors. And Horne said if lawmakers want tutoring for next year’s crop of seniors they will have to give him the money.

 The bonus points system is available to high schoolers who have taken the AIMS test each time it has been offered and participated in remedial work in the areas of failure. It allows good grades in required courses — electives would not count — to be counted for up to 25 percent of a student’s AIMS score.

 If there are enough bonus points, the student graduates even without getting a passing score on each of the three AIMS segments.

 Horne said about 3,000 seniors in each of the last two years got their diplomas solely because they had enough bonus points to get a passing grade on AIMS. And he said without the bonus points they might not graduate.

 But Horne said that isn’t the end of the world. He said they can retake any failed portion again in July — and over and over again after that. “So as soon as they demonstrate the skills we expect of a high school graduate they get their diploma,’’ he said. “So we haven’t given up on anybody.’’

 But Schapira said that presumes the AIMS test is designed to match what the state Board of Education says are the items that students must learn. He said the reverse is true. “We’re teaching to the test,’’ he said. “This isn’t taking what the teachers are teaching in the classroom and putting it in the form of a test,’’ Schapira said. “That’s called a final exam which students are taking and are required to pass in order to pass their courses.’’

 Schapira said if he had his way, AIMS would be scrapped — at least as a graduation requirement.

 He said the test serves a purpose because it does measure what students have learned. And Schapira said Arizona now has several years of data to track achievement. “But using it as a high-stakes test, I think, is a mistake,’’ Schapira said.