That divide highlights a harsh truth about the sources of school segregation in New York City.

Many people, including Mayor de Blasio, point to segregated neighborhoods as the cause of separate schools. In fact, many of the city’s school zones and districts encompass a mix of families. And by opening up every school to any family in a district, “school choice” systems like the one in District 15 offer a golden opportunity to override divided neighborhoods and make schools integrated.

Instead, district parents, schools, and officials have made choices that reinforce segregation.

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In the whiter, wealthier northern half of District 15, competition is fierce for a seat at a Big Three school.

“You have to battle for your so-called ‘choice,’” said Antonia Martinelli, a Gowanus parent and blogger who put M.S. 51 and 447 at the top of her son’s application. Otherwise, “there’s a fear that your child won’t get into a good enough high school.”

Some parents pay a private consultant $400 for a two-hour consultation about the district’s admission process. Others rely on their social circles. Because parents believe that attending one of the school tours will increase their odds of admission, many wait at their computers for the exact moment when online registration begins. The spots are usually snatched up within hours.

“It’s almost a full-time job,” said Rhonda Keyser, whose child attends M.S. 51.

But ultimately, the competition is within a narrow group of parents.

Eight of the district’s 25 elementary schools send half or more of their students to one of the Big Three, according to city data. Those elementary schools are on average 64 percent white and just 17 percent low-income. Districtwide, 31 percent of students are white and 65 percent are considered poor.

Monica Kipiniak’s son attends the School for International Studies, one of several district and charter schools where families who did not make it into the Big Three are starting to venture. She said the fight for Big Three seats favors wealthier parents with the time and ability to navigate the process and to ensure their students are strong academically.

“There’s no question,” she said, “that for many reasons, kids who come from more affluent families end up going to the more desirable schools.”

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Just a few subway stops away, the southern end of District 15 can seem worlds apart from that frenzy.

In Sunset Park, an immigrant-filled neighborhood home to many Hispanic families and its own Chinatown, many parents are daunted by the application process and opt to only apply to local middle schools they know, said Julie Stein Brockway, co-director of the Center for Family Life in Sunset Park.

Still, she said many would consider applying to northern-end schools if they were encouraged to. But even though her social-service agency works with hundreds of local families, she said only charter schools have asked her for help recruiting Sunset Park students—never one of the Big Three.