She wakes to the sound of breathing. The smaller children lie tangled beside her, their chests rising and falling under winter coats and wool blankets. A few feet away, their mother and father sleep near the mop bucket they use as a toilet. Two other children share a mattress by the rotting wall where the mice live, opposite the baby, whose crib is warmed by a hair dryer perched on a milk crate.

Slipping out from her covers, the oldest girl sits at the window. On mornings like this, she can see all the way across Brooklyn to the Empire State Building, the first New York skyscraper to reach 100 floors. Her gaze always stops at that iconic temple of stone, its tip pointed celestially, its facade lit with promise.

“It makes me feel like there’s something going on out there,” says the 11-year-old girl, never one for patience. This child of New York is always running before she walks. She likes being first — the first to be born, the first to go to school, the first to make the honor roll.

Even her name, Dasani, speaks of a certain reach. The bottled water had come to Brooklyn’s bodegas just before she was born, catching the fancy of her mother, who could not afford such indulgences. It hinted at a different, upwardly mobile clientele, a set of newcomers who over the next decade would transform the borough.

Dasani’s own neighborhood, Fort Greene, is now one of gentrification’s gems. Her family lives in the Auburn Family Residence, a decrepit city-run shelter for the homeless. It is a place where mold creeps up walls and roaches swarm, where feces and vomit plug communal toilets, where sexual predators have roamed and small children stand guard for their single mothers outside filthy showers.

It is no place for children. Yet Dasani is among 280 children at the shelter. Beyond its walls, she belongs to a vast and invisible tribe of more than 22,000 homeless children in New York, the highest number since the Great Depression, in the most unequal metropolis in America.

Nearly a quarter of Dasani’s childhood has unfolded at Auburn, where she shares a 520-square-foot room with her parents and seven siblings. As they begin to stir on this frigid January day, Dasani sets about her chores.

Her mornings begin with Baby Lele, whom she changes, dresses and feeds, checking that the formula distributed by the shelter is not, once again, expired. She then wipes down the family’s small refrigerator, stuffed with lukewarm milk, Tropicana grape juice and containers of leftover Chinese. After tidying the dresser drawers she shares with a sister, Dasani rushes her younger siblings onto the school bus.

“I have a lot on my plate,” she says, taking inventory: The fork and spoon are her parents and the macaroni, her siblings — except for Baby Lele, who is a plump chicken breast.

“So that’s a lot on my plate — with some corn bread,” she says. “That’s a lot on my plate.”

Dasani guards her feelings closely, dispensing with anger through humor. Beneath it all is a child whose existence is defined by her siblings. Her small scrub-worn hands are always tying shoelaces or doling out peanut butter sandwiches, taking the ends of the loaf for herself. The bond is inescapable. In the presence of her brothers and sisters, Dasani has no peace. Without them, she is incomplete.

Today, Dasani rides the creaky elevator to the lobby and walks past the guards, the metal detector and the tall, iron fence that envelops what she calls “the jail.” She steps into the light, and is met by the worn brick facade of the Walt Whitman projects across the street.

She heads east along Myrtle Avenue and, three blocks later, has crossed into another New York: the shaded, graceful abode of Fort Greene’s brownstones, which fetch millions of dollars.

“Black is beautiful, black is me,” she sings under her breath as her mother trails behind.

Dasani suddenly stops, puzzling at the pavement. Its condition, she notes, is clearly superior on this side of Myrtle.

“Worlds change real fast, don’t it?” her mother says.

In the short span of Dasani’s life, her city has been reborn. The skyline soars with luxury towers, beacons of a new gilded age. More than 200 miles of fresh bike lanes connect commuters to high-tech jobs, passing through upgraded parks and avant-garde projects like the High Line and Jane’s Carousel. Posh retail has spread from its Manhattan roots to the city’s other boroughs. These are the crown jewels of Mayor Michael R. Bloomberg’s long reign, which began just seven months after Dasani was born.

In the shadows of this renewal, it is Dasani’s population who have been left behind. The ranks of the poor have risen, with almost half of New Yorkers living near or below the poverty line. Their traditional anchors — affordable housing and jobs that pay a living wage — have weakened as the city reorders itself around the whims of the wealthy.

Long before Mayor-elect Bill de Blasio rose to power by denouncing the city’s inequality, children like Dasani were being pushed further into the margins, and not just in New York. Cities across the nation have become flash points of polarization, as one population has bounced back from the recession while another continues to struggle. One in five American children is now living in poverty, giving the United States the highest child poverty rate of any developed nation except for Romania.

This bodes poorly for the future. Decades of research have shown the staggering societal costs of children in poverty. They grow up with less education and lower earning power. They are more likely to have drug addiction, psychological trauma and disease, or wind up in prison.

Dasani does not need the proof of abstract research. All of these plights run through her family. Her future is further threatened by the fact of her homelessness, which has been shown, even in short spells, to bring disastrous consequences.

Dasani’s circumstances are largely the outcome of parental dysfunction. While nearly one-third of New York’s homeless children are supported by a working adult, her mother and father are unemployed, have a history of arrests and are battling drug addiction.

Yet Dasani’s trials are not solely of her parents’ making. They are also the result of decisions made a world away, in the marble confines of City Hall. With the economy growing in 2004, the Bloomberg administration adopted sweeping new policies intended to push the homeless to become more self-reliant. They would no longer get priority access to public housing and other programs, but would receive short-term help with rent. Poor people would be empowered, the mayor argued, and homelessness would decline.

But the opposite happened. As rents steadily rose and low-income wages stagnated, chronically poor families like Dasani’s found themselves stuck in a shelter system with fewer exits. Families are now languishing there longer than ever — a development that Mr. Bloomberg explained by saying shelters offered “a much more pleasurable experience than they ever had before.”

Just three days before the mayor made that comment at a news conference in August 2012, an inspector at Auburn stopped by Dasani’s crowded room, noting that a mouse was “running around and going into the walls,” which had “many holes.”

“Please assist,” the inspector added. “There is infant in room.”

Dasani was about to start sixth grade at a promising new school. This would be a pivotal year of her childhood — one already marked by more longing and loss than most adults ever see.

A tangle of three dramas had yet to unspool.

There was the question of whether Dasani’s family would remain intact. Her mother had just been reunited with the children on the condition that she and her husband stay off drugs. The city’s Administration for Children’s Services was watching closely. Any slips, and the siblings could wind up in foster care, losing their parents and, most likely, one another.

The family’s need for a home was also growing desperate. The longer they stayed in that one room, the more they seemed to fall apart. Yet rents were impossibly high in the city, and a quarter-million people were waiting for the rare vacancy in public housing. Families like Dasani’s had been leaving the state. This was the year, then, that her parents made a promise: to save enough money to go somewhere else, maybe as far as the Pocono Mountains, in Pennsylvania.

Dasani could close her eyes and see it. “It’s quiet and it’s a lot of grass.”

In the absence of this long-awaited home, there was only school. But it remained to be seen whether Dasani’s new middle school, straining under budget cuts, could do enough to fill the voids of her life.

For children like Dasani, school is not just a place to cultivate a hungry mind. It is a refuge. The right school can provide routine, nourishment and the guiding hand of responsible adults.

But school also had its perils. Dasani was hitting the age when girls prove their worth through fighting. And she was her mother’s daughter, a fearless fighter.

She was also on the cusp of becoming something more, something she could feel but not yet see, if only the right things happened and the right people came along.

Dasani is a short, wiry girl whose proud posture overwhelms her 4-foot-8 frame. She has a delicate, oval face and luminous brown eyes that watch everything, owl-like. Her expression veers from wonder to mischief. Strangers often remark on her beauty — her high cheekbones and smooth skin — but the comments never seem to register.

What she knows is that she has been blessed with perfect teeth. In a family where braces are the stuff of fantasy, having good teeth is a lottery win.

On the subway, Dasani can blend in with children who are better off. It is an ironic fact of being poor in a rich city that the donated garments Dasani and her siblings wear lend them the veneer of affluence, at least from a distance. Used purple Uggs and Patagonia fleeces cover thinning socks and fraying jeans. A Phil & Teds rain cover, fished from a garbage bin, protects Baby Lele’s rickety stroller.

Dasani tells herself that brand names don’t matter. She knows such yearnings will go unanswered, so better not to have them. But once in a while, when by some miracle her mother produces a new pair of Michael Jordan sneakers, Dasani finds herself succumbing to the same exercise: She wears them sparingly, and only indoors, hoping to keep them spotless. It never works.

Best to try to blend in, she tells herself, while not caring when you don’t.

She likes being small because “I can slip through things.” In the blur of her city’s crowded streets, she is just another face. What people do not see is a homeless girl whose mother succumbed to crack more than once, whose father went to prison for selling drugs, and whose cousins and aunts have become the anonymous casualties of gang shootings, AIDS and domestic violence.

“That’s not gonna be me,” she says. “Nuh-uh. Nope.”

Dasani speaks with certainty. She often begins a sentence with “Mommy say” before reciting, verbatim, some new bit of learned wisdom, such as “camomile tea cures a bad stomach” or “that lady is a dope fiend.” She likes facts. She rarely wavers, or hints at doubt, even as her life is consumed by it.

When strangers are near, Dasani refers to Auburn as “that place.” It is separate from her, and distant. But in the company of her siblings, she calls it “the house,” transforming a crowded room into an imaginary home.

In reality, Auburn is neither. The forbidding, 10-story brick building, which dates back almost a century, was formerly Cumberland Hospital, one of seven public hospitals that closed because of the city’s 1970s fiscal crisis.

In 1985, the city repurposed the former hospital into a shelter for families. This was the dawn of the period known as “modern homelessness,” driven by wage stagnation, Reagan-era cutbacks and the rising cost of homes. By the time Mayor Bloomberg took office in 2002, New York’s homeless population had reached 31,063 — a record for the city, which is legally obligated to provide shelter.

Among the city’s 152 family shelters, Auburn became known as a place of last resort, a dreaded destination for the chronically homeless.

City and state inspectors have repeatedly cited the shelter for deplorable conditions, including sexual misconduct by staff members, spoiled food, asbestos exposure, lead paint and vermin. Auburn has no certificate of occupancy, as required by law, and lacks an operational plan that meets state regulations. Most of the shelter’s smoke detectors and alarms have been found to be inoperable.

There are few signs that children live at Auburn. Locked gates prevent them from setting foot on the front lawn. In a city that has invested millions of dollars in new “green spaces,” Auburn’s is often overrun with weeds.

Inside, prepackaged meals are served in a cafeteria where Dasani and her siblings wait in one line for their food before heading to another line to heat it in one of two microwaves that hundreds of residents share. Tempers fly and fights explode. The routine can last more than an hour before the children take their first bite.

Hand-washed clothes line the guards on the windows, which are shaded by gray wool blankets strung from the ceiling. A sticky fly catcher dangles overhead, dotted with dead insects.

There is no desk or chair in the room — just a maze of mattresses and dressers. A flat-screen television rests on two orange milk crates.

To eat, the children sit on the cracked linoleum floor, which never feels clean no matter how much they mop. Homework is a challenge. The shelter’s one recreation room can hardly accommodate Auburn’s hundreds of children, leaving Dasani and her siblings to study, hunched over, on their mattresses.

Sometimes it feels like too many bodies sharing the same air. “There’s no space to breathe ’cause they breathe up all the oxygen,” Dasani says.

She carves out small, sacred spaces: a portion of the floor at mealtime, an upturned crate by the window, a bathroom stall.

The children spend hours at the playgrounds of the surrounding housing projects, where a subtle hierarchy is at work. If they are seen enough times emerging from Auburn, they are pegged as the neighborhood’s outliers, the so-called shelter boogies.

Nothing gnaws at Dasani more.

A mucus-stained nose suggests a certain degradation, not just the absence of tissues, but of a parent willing to wipe or a home so unclean that a runny nose makes no difference. Dasani and her siblings can get hungry enough to lose their concentration in school, but they are forever wiping one another’s noses.

When Dasani hears “shelter boogies,” all she can think to say is what her mother always tells her — that Auburn is “just a pit stop.”

“But you will live in the projects forever, as will your kids’ kids, and your kids’ kids’ kids.”

She knows the battle is asymmetrical.

The projects may represent all kinds of inertia. But to live at Auburn is to admit the ultimate failure: the inability of one’s parents to meet that most basic need.

Dasani ticks through their faces, the girls from the projects who might turn up at this new school. Some are kind enough not to gossip about where she lives. ■ The others might be distracted by the sheer noise of this first day — the start of sixth grade, the new uniform, the new faces. She will hopefully slip by those girls unseen.

She approaches the school’s steps on a clear September morning. Fresh braids fall to one side of her face, clipped by bright yellow bows. Her required polo and khakis have been pressed with a hair straightener, since Auburn forbids irons.

Her heart is pounding. She will be sure to take a circuitous route home. She will focus in class and mind her manners in the schoolyard. She only has to climb those steps.

Minutes pass.

“Come on, there’s nothing to be scared about,” her 34-year-old mother, Chanel, finally says, nudging Dasani up the stairs.

She passes through the metal detector, joining 507 other middle and high school students at the Susan S. McKinney Secondary School of the Arts.

Housed in a faded brick building two blocks from Auburn, McKinney is a poor-kids’ version of LaGuardia Arts, the elite Manhattan public school that inspired the television series “Fame.” Threadbare curtains adorn its theater. Stage props are salvaged from a nearby trash bin. Dance class is so crowded that students practice in intervals.

An air of possibility permeates the school, named after the first African-American woman to become a physician in New York State.

There is Officer Jamion Andrews, the security guard who moonlights as a rap lyricist, and Zakiya Harris, the dance teacher who runs a studio on the side. And there is Faith Hester, the comedic, eyelash-batting humanities teacher who wrote a self-help book titled “Create a Life You Love Living” and fancies her own reality show.

The children also strive. Among them is a voice that periodically lifts the school with a “Madama Butterfly” aria. When the students hear it, they know that Jasmine, a sublimely gifted junior, is singing in the office of the principal, Paula Holmes.

The school matriarch closes her eyes as she listens. It may be her only tranquil moment.

Miss Holmes is a towering woman, by turns steely and soft. She wears a Bluetooth like a permanent earring and tends toward power suits. She has been at McKinney’s helm for 15 years and runs the school like a naval ship, peering down its gleaming hallways as if searching the seas for enemy vessels.

Students stammer in her presence. She leaves her office door permanently open, like a giant, unblinking eye. A poster across the hall depicts a black man in sagging jeans standing before the White House, opposite President Obama. “To live in this crib,” the poster reads, “you have to look the part.”

Miss Holmes has no tolerance for sagging — sartorial, attitudinal or otherwise.

McKinney’s roots run deep. Dasani’s own grandmother studied there as a girl. Most of the middle school students are black, live in the surrounding projects and qualify for free or reduced meals. They eat in shifts in the school’s basement cafeteria, watched over by the avuncular Frank Heyward, who blasts oldies from a boombox, telling students, “I got shoes older than you.”

For all of McKinney’s pluck, its burdens are great. In the last six years, the city has cut the school’s budget by a quarter as its population declined. Fewer teachers share a greater load. After-school resources have thinned, but not the needs of students whose families are torn apart by gun violence and drug use. McKinney’s staff psychologist shuttles between three schools like a firefighter.

And now, a charter school is angling to move in. If successful, it will eventually claim McKinney’s treasured top floor, home to its theater class, dance studio and art lab. Teachers and parents are bracing for battle, announced by fliers warning against the “apartheid” effects of a charter co-location.

Dasani knows about charter schools. Her former school, P.S. 67, shared space with one. She never spoke to those children, whose classrooms were stocked with new computers. Dasani’s own school was failing by the time she left.

At McKinney, Dasani quickly draws the notice of the older students, and not because she is short, though the nickname “Shorty” sticks. It is her electricity. When they dote on her, she giggles. But say the wrong thing and she turns fierce, letting the four-letter words fly.

It is still September when Dasani’s temper lands her in the principal’s office.

“Please don’t call my mother,” Dasani whispers.

Miss Holmes is seated in a rolling pleather chair held together by duct tape. She stares at the anguished girl. She has been at McKinney long enough to know when a child’s transgressions at school might bring a beating at home.

The principal slowly scoots her chair up to Dasani and leans within inches of her face.

“O.K.,” she says softly. “Let’s make a deal.”

From that day forward, Dasani will be on her best behavior. In turn, Miss Holmes will keep what happens at school in school.

With that, she waves Dasani off, fighting the urge to smile. She can’t help but like this feisty little girl.

Dasani closes her eyes and tilts her head toward the ceiling of her classroom. She has missed breakfast again. ■ She tries to drift. She sees Florida. For a child who has never been to the beach, television ads are transporting. She is walking in the sand. She crashes into the waves.

“Dasaaaaaani!” her teacher sings out.

She opens her eyes.

There is Miss Hester, batting those lashes.

Both she and another teacher, Kenya Mabry, were raised in the projects. They dress and talk with a polish that impresses Dasani, who studies them.

Miss Hester is also watching Dasani. She does not yet know where Dasani lives, or how hungry she gets. But Miss Hester finds two things striking: how late she arrives some mornings and how capable this girl is in spite of it. Without even trying, she keeps up.

Dasani possesses what adults at McKinney consider an intuitive approach to learning, the kind that comes when rare smarts combine with extreme life circumstances. Her intelligence is “uncanny” and “far surpasses peers her age,” one counselor writes. “Student is continuously using critical analysis to reflect upon situations and interactions.”

Principal Holmes is also taking note. She can already see in this “precocious little button” the kind of girl who could be anything — even a Supreme Court justice — if only she harnesses her gifts early enough. “Dasani has something that hasn’t even been unleashed yet,” Miss Holmes says. “It’s still being cultivated.”

For now, Dasani’s most honed skill might be obfuscation. She works hard to hide her struggles, staying quiet as other children brag about their new cellphones or sleepovers with friends.

If there is one place she feels free, it is dance class. When she walks into McKinney’s studio, and the music starts, her body releases whatever she is feeling.

“When I’m happy I dance fast,” she says. “When I’m sad I dance slow. When I’m upset I dance both.”

Dasani has been dancing for as long as she can remember, well before she earned her first dollar a few years ago break-dancing in Times Square. But the study of dance, as something practiced rather than spontaneous, this is new. She is learning to point her toes like a ballerina, and to fall back into a graceful bridge.

Perhaps it is no accident that amid the bedlam of Dasani’s home life — the missed welfare appointments and piles of unwashed clothes — she is drawn to a craft of discipline.