Members of Miami-Dade’s teachers union decided Monday to sign onto the district’s efforts to compete for up to $32 million in federal money.

The support came just in time and amid contract negotiations.

The district needed the United Teachers of Dade to endorse its application for another round of the Obama Administration’s education competition, Race to the Top.

Union stewards had previously voted against it, because of concerns that Miami-Dade’s proposal was too narrow, focusing on a Cambridge curriculum, new labs and middle school math teachers.

The application deadline was Tuesday, though federal officials said it would be extended because of Hurricane Sandy, which threatened the East Coast.

Broward County Public Schools also intends to compete for the grant.

Unlike an earlier round of the competition, in which states submitted proposals, this time schools districts will apply directly and compete. It’s called Race to the Top D, which stands for district.

Fifteen to 25 districts around the country are expected to win a piece of the nearly $400 million being awarded. Districts can create a proposal for all or a part of its schools, for specific grades or particular subjects, but to win they must show they will “personalize education for all students in their schools,” according to the U.S. Department of Education. Winners will be announced in December.

Miami-Dade’s union stewards reversed their position amid contract negotiations.

UTD President Karen Aronowitz said she called an emergency meeting at Miami Jackson Senior High School after the issue came up during a bargaining session Monday morning. Stewards -- teachers and education support professionals elected to represent their schools -- left work early to discuss and vote. Some saw the grant as leverage in ongoing negotiations, in which the union is pushing for raises after three years without, and feared rejecting potential money would send the wrong message about the need for more education resources.

Aronowitz said the vote, taken by voice, was overwhelmingly in favor: “They said, ‘No, Go for it.’ ”

Said Superintendent Alberto Carvalho in a statement: “We are pleased that common sense and collaboration continue to forge educational opportunities for Miami-Dade County Public Schools students.”

With the grant, Broward hopes to create “personalized learning environments” at 12 middle schools over the course of four years, said Nadine Drew, spokeswoman for the district.

“We will focus on math and science in the first year, then reading/language arts and social studies, then elective and special subjects, and in the fourth year, we will focus on re-engineering the school to prepare students for college and career success,” Drew said in an email.

Miami-Dade’s spokesman, John Schuster, said the district wants to replicate the iPrep model, which blends traditional teaching with virtual classes, for middle school math programs. “There will be an iPrep math lab, and instruction will follow four distinct models. This will be an opt-in program: Schools will vote to participate and two volunteer teachers will participate in each school,” Schuster said.

Dade and Broward will have a lot of competition.

In August, nearly 900 districts around the country told federal officials they intended to apply. Some have since decided to skip it, including Palm Beach County, where the teachers union and the district couldn’t agree on a proposal.

In an earlier round of Race to the Top, Florida won hundreds of millions. Miami-Dade has slated most of its portion of that -- $17 million a year for four years -- on bonuses for high-performing teachers.