CHICAGO — One by one, teachers at public elementary schools here have been voting to buck their own union and take Mayor Rahm Emanuel up on an unusual offer: to accept bonus pay in exchange for waiving union contract provisions and keeping children at some schools longer each day.

By Friday, nine schools were on board, and City Hall said more might be on the way, even as union leaders objected to the tactic and furiously pledged to take Mr. Emanuel and his school system before a labor relations board.

For Mr. Emanuel, who has portrayed himself as a calmer, more reserved leader in his year of transformation from White House chief of staff to mayor, the uncompromising and blunt approach is both a keeping of a campaign promise and a reminder, as one political analyst here put it, that “Rahm is still Rahm.”

Mr. Emanuel and his top aides say that when it comes to hours of class instruction, Chicago’s 482 public elementary schools compare unfavorably with schools in other cities, including Houston, Los Angeles and New York. They say they merely want children to spend more time learning at school — a notion supported by plenty of parents.