Twenty hands shot into the air after Ashley Tam asked a question of her third graders during a math lesson on Tuesday morning. One boy threw his arm up with such force that his desk jumped off the ground with him. A nonprofit, nonpartisan news organization providing local coverage of Chicago and the surrounding area for The New York Times. It was the first week of the much-anticipated longer school day, as well as the second day back from winter break at Genevieve Melody Elementary School, not a time when one would expect a high level of student enthusiasm. “I think the kids have adapted faster than we have,” said Tiffany Tillman, the assistant principal at Melody. The principal, Nancy Hanks, agreed, saying she had picked the wrong week to try to kick her coffee habit. Melody is one of 13 Chicago public schools at which teachers voted to waive portions of their union contract and lengthen the school day to seven-and-a-half hours. In exchange, those teachers receive a stipend of roughly $800 to $1,200 and schools receive $75,000 to $150,000 each in discretionary funds. Mayor Rahm Emanuel’s push for a longer school day has been criticized by the Chicago Teachers Union, which accused school officials of unlawfully coercing and bribing teachers at individual schools in an attempt to “make the union irrelevant.” At Melody, 16 teachers voted for the longer day, with 4 against it, but teachers and administrators say they are less concerned with the politics than in making a longer day work. “It’s a bigger deal to everybody else than everybody over here at Melody,” Ms. Hanks said. “At the end of the day, we have to be here together. You just don’t need that kind of tension coming to work.” Melody and the 12 other “pioneer schools” that are starting longer days this year are being closely watched because next year all of Chicago’s public schools will move to a seven-and-a-half-hour day — making Chicago the first major city in recent years to add substantial school time district-wide. The longer day includes 90 minutes of additional instructional time and more time for breaks. A sweeping education-reform law passed last year gives school district administrators the power to unilaterally lengthen the school day and year, but the details will have to be resolved in bargaining with the teachers’ union. How well the longer day works at Melody and the other schools will influence their decisions, officials said. While most parents, teachers and administrators support lengthening the current 5 hour and 45 minute school day, many questions remain: What will schools do with the additional time? Will teachers be compensated? Can young children handle the longer school day? How can the district, already financially stretched, afford to add instruction? In many ways, Melody is a microcosm of the city’s system — failing by most accounts, but pushing aggressively to improve. The school, on the West Side, has been on probation for five years, falls 500 students below optimum enrollment standards and, according to federal guidelines, has been failing for years. But with an ambitious, Harvard-educated principal and a sizable experienced staff, the school is registering promising growth. “Last year, we made good movement for the first time in 10 years and we’re hoping to move more,” said Tyrone Covington, who has been a physical education teacher at Melody for 20 years. “We have no other way to go but up.” Although the extra time was spent differently according to grade, all Melody classes added instruction in math and reading. In Ms. Tam’s class, the longer day also meant more time for science and social studies. But she said her students were tired by the end of the day on Monday. “It’s a long day for 8-year-olds,” she said. “They’re adjusting well, but by 3 they wanted to nap.” Some parents on the North Side, where the public schools are performing better, have raised concerns in meetings and surveys that the seven-and-a-half-hour day may be too long for their children and could cut into extracurricular activities. Ms. Hanks said she understands, but added, “I need my kids to be able to read. Can you give a little bit of your piano time so that my kids over here on this side of the city can learn how to read?” What works at Melody may not work at another school, Ms. Hanks said. The district announced on Wednesday that 30 schools with the most innovative plans for using the extra time next year will be awarded $100,000 each. All schools have been asked to devise a new schedule and submit it by the end of February. Officials say the best ideas may be applied across the district. For Ms. Hanks, the main concern is not the final details worked out by the city and the teachers’ union, but rather the adjustment that teachers and parents will have to make to the stark change in their daily routine. “You can pay attention to the nuts and the bolts of the schedule all you want, but if you don’t mentally prepare and take the adults through the transition, it’s not going to be successful,” Ms. Hanks said.