The annual fair kicks off the city’s high school application season in September, and Jayda Walker, 13, arrived with a plan.

An eager young woman with an easy smile, Jayda wants to be a divorce lawyer, and at the fair, held at Brooklyn Technical High School, she planned to focus on schools with a legal theme, located in Manhattan. She had already looked through the high school directory, an intimidating tome the size of an old-fashioned phone book, and thought Manhattan offered more variety. Besides, she said, she wanted to get out of the Bronx.

She and her classmates arrived early and were at the front of the line, with hundreds of people behind them eager to get inside.

But for many of the students from Pelham Gardens, and others like them, it was already too late. The sorting of students to top schools — by race, by class, by opportunity — begins years earlier, and these children were planted at the back of the line.

Under a system created during Mayor Michael R. Bloomberg’s administration, eighth graders can apply anywhere in the city, in theory unshackling themselves from failing, segregated neighborhood schools. Students select up to 12 schools and get matched to one by a special algorithm. This process was part of a package of Bloomberg-era reforms intended to improve education in the city and diminish entrenched inequities.

There is no doubt that the changes yielded meaningful improvements. The high school graduation rate is up more than 20 points since 2005, as the administration of Mayor Bill de Blasio has built on Mr. Bloomberg’s gains. The graduation gap between white and black or Hispanic students, while still significant and troubling, has narrowed.

But school choice has not delivered on a central promise: to give every student a real chance to attend a good school.

Fourteen years into the system, black and Hispanic students are just as isolated in segregated high schools as they are in elementary schools — a situation that school choice was supposed to ease.

Within the system, there is a hierarchy of schools, each with different admissions requirements — a one-day high-stakes test, auditions, open houses. And getting into the best schools, where almost all students graduate and are ready to attend college, often requires top scores on the state’s annual math and English tests and a high grade point average.

Those admitted to these most successful schools remain disproportionately middle class and white or Asian, according to an in-depth analysis of acceptance data and graduation rates conducted for The New York Times by Measure of America, an arm of the Social Science Research Council. At the same time, low-income black or Hispanic children like the ones at Pelham Gardens are routinely shunted into schools with graduation rates 20 or more percentage points lower.

While top middle schools in a handful of districts groom children for competitive high schools that send graduates to the Ivy League, most middle schools, especially in the Bronx, funnel children to high schools that do not prepare them for college.

The roots of these divisions are tangled and complex. Students in competitive middle schools and gifted programs carry advantages into the application season, with better academic preparation and stronger test scores. Living in certain areas still comes with access to sought-after schools. And children across the city compete directly against one another regardless of their circumstances, without controls for factors like socioeconomic status.

Ultimately, there just are not enough good schools to go around. And so it is a system in which some children win and others lose because of factors beyond their control — like where they live and how much money their families have.

Measure of America examined the enormous sorting exercise of high school placement by looking at the graduating class of 2015. When they were applying to high school in 2011, about 30 percent of eighth graders were white or Asian.

The Times spent months following the high school application process at Pelham Gardens, where families do not have the advantages that routinely open doors to the city’s best schools. Many families are new to the country, and most are poor.

The school is new, and just its third graduating class will start high school in the fall. But its guidance counselor, Ayana Bryant, knows the application process as well as anybody. She has worked in middle school guidance for 16 years.

On the morning of the fair, she took Jayda and the other students to Brooklyn Tech on a yellow school bus in the hope that they would connect with the right schools. But that hope was tempered by her experience.

“They say it’s choice, but is it really?” she said of the city’s system. “Some schools really aren’t for everybody.”

‘The Beast’

Founded in 2012, Pelham Gardens Middle School occupies a slice of the northeastern Bronx around the corner from a used-car dealership, and not far from the Co-op City housing development. The building, which looks like a giant cinder block painted the color of butter, contains two other schools and a rooftop field of solar panels. About 95 percent of the middle school’s students are black or Hispanic, many of them the children of Jamaican immigrants or immigrants themselves. Most of them come from poor families.

The closest subway line, the No. 5, is about a half-mile away, and traveling to other boroughs requires a significant investment of time. The No. 5 train can make more than a dozen stops before it reaches northern Manhattan.

Each year, about 160 children from Pelham Gardens join the flood of 80,000 eighth graders applying for the city’s public high schools. The field on which they compete is enormous: They have to choose from 439 schools that are further broken up into 775 programs. One program may admit students based on where they live, while another program at the same school may admit only those with strong grades.

The sheer number of choices offers up great possibilities, but it can also make the system maddeningly complex, with so many requirements, open houses, deadlines and portfolios to keep track of. Yaslin Turbides helps middle schoolers apply through the Cypress Hills Local Development Corporation, a nonprofit organization in Brooklyn. She said that she and her colleagues called the application system “the beast.”

The process can become like a second job for some parents as they arm themselves with folders, spreadsheets and consultants who earn hundreds of dollars an hour to guide them. But most families in the public school system have neither the flexibility nor the resources to match that arsenal.

At a school like Pelham Gardens, much navigation falls to the guidance counselor, and Ms. Bryant is tireless in trying to help the children make the best decisions. She pulls groups of students into her office to explain the basics, trying to get each of them in front of her at least once. She takes them on visits to high schools. She writes tip sheets over the summer, and puts together curriculums so seventh-grade teachers can acquaint their students with the process. She warns students away from the lowest-performing schools.

Ms. Bryant grew up in the Soundview Houses, a public housing development in the eastern Bronx, an area where she still lives. In addition to being a guidance counselor, she has worked for years at a program that helps low-income students prepare for the specialized high school exam, which admits students to a group of top-performing schools like Stuyvesant High School and the Bronx High School of Science. Her daughter went to Beacon School, on the West Side of Manhattan, one of the highest-performing schools in the city.

In January, Ms. Bryant pulled some students into her office so they could practice for their interviews at University Heights Secondary School, a top school in the Bronx where there are 20 applicants for every spot. Two students, Darnell Donaldson and Samantha Suriel, were seated in front of her while the rest of the group observed. Ms. Bryant asked them to tell her about themselves.

“I like to stand out in class and participate a lot,” Darnell said.

“I like to study sometimes, or read books,” Samantha said quietly, her face pointed toward the floor.

“Why should I pick you over him?” Ms. Bryant asked Samantha. Flummoxed, she stammered for a moment before saying she had been on the honor roll since sixth grade.

Ms. Bryant asked if anyone had a comment or critique.

“When you walk in, you should introduce yourself, out of politeness,” a girl named Ivyana Rivas offered.

Ms. Bryant then asked the children whom they would accept.

A boy called out, “No one!”

After all the students had practiced interviewing, Ms. Bryant stood up from her chair. “Some of you, when you shook my hand, you acted like you didn’t want to touch me,” she said.

“Now I know some of us are germophobes, and Ms. Bryant is one,” she grinned, and her audience chuckled. “I will wash my hands later. But you’ve got to make sure your hand is in their palm,” she said, reaching out to shake a boy’s hand.

“Your hand is in their palm. Your hand is in their palm,” she repeated as she reached for them one by one.

Still, most of Ms. Bryant’s time is taken up by the mandated student meetings and accompanying paperwork for children in special education. She is also on the attendance and college committees, and acts as the child-abuse liaison. How much time can she squeeze out of the day to help children find the right high school?

Sharp Differences

The citywide graduation rate for all kinds of high schools is 72.6 percent, according to the Education Department. But that average masks sharp variations between schools based on their admissions methods. When Measure of America analyzed the rate for each method, it found that selectivity and graduation rates declined essentially in lock step, and that as graduation rates fell, the students were more likely to be poor and black or Hispanic.

Schools use a variety of methods to select the students they admit.

But graduation remains a meaningful measure of a school, and of the opportunities it provides. If parents felt they had another option, how many would be happy to send their children to a school where more than a quarter of students do not graduate?

For its analysis, Measure of America looked at how individual students were admitted to high school for the 2011-12 academic year, breaking data down by the children’s race, the borough in which they lived and whether they were poor enough to qualify for services like free lunch. Then it examined graduation rates for the different admissions methods four years later. The Education Department said the analysis contained “major methodological issues,” but it did not dispute the findings.

Kristen Lewis, one of the directors of Measure of America, said the data revealed, in essence, two separate public school systems operating in the city. There are some great options for the families best equipped to navigate the application process. But there are not enough good choices for everyone, so every year thousands of children, including some very good students, end up in mediocre high schools, or worse.

“The average kid has to be able to get a good education, because most people are average,” Ms. Lewis said. “It’s great that the highfliers are succeeding, and they deserve the chance to succeed. But so do the average kids.”

Feeders

A common mantra among teachers, administrators and politicians is that a child’s ZIP code should not dictate the quality of his or her education. If the schools in a child’s neighborhood are failing, the system is supposed to let him or her attend a better school elsewhere.

But in practice, children who grow up in neighborhoods with low-performing elementary schools tend to go to low-performing middle schools, then on to high schools with low graduation rates and even lower college-readiness rates.

Middle school choice generally revolves around the city’s community school districts. In some districts, children tend to go to their assigned neighborhood schools, while in others there is competition to get into middle schools that require good grades or entrance exams. There are a few programs open to high achievers from all over the city, or to children from a particular borough. Some districts have a greater number of top-performing middle schools than others.

An analysis by the Center for New York City Affairs at the New School found that half of all students who got top scores on state tests came from just 45 middle schools out of more than 500. And 60 percent of students who went to specialized high schools came from those same 45 schools. None of those middle schools are in the Bronx.

These feeder schools, like NYC Lab Middle School in the Chelsea section of Manhattan or the Marie Curie Middle School in Bayside, Queens, have several advantages over a place like Pelham Gardens. For one thing, many have competitive admissions themselves, so students come in with strong academic records, and the school’s job is to keep them advancing. At schools like Pelham, most new students arrive behind grade level. Teachers there have the more difficult task of helping students catch up.

Last year, 146 seventh graders at Pelham Gardens took the state tests. On the English exam, 29 passed, which requires a score of at least 3 out of 4. Fifteen did that well in math. Only seven scored at least a 3 on both tests.

This means that a majority of the children had no real chance of getting into the most selective schools, like Manhattan/Hunter Science High School or Townsend Harris High School in Queens, where students must have a 3 or higher on the tests. The high school directory lists 29 programs in the city that did not accept anyone with a score lower than 3 on the math exam.

Advanced courses taught at many feeder schools also create an advantage. Laura Zingmond is a senior editor at the InsideSchools website, a project of the Center for New York City Affairs. Ms. Zingmond said that students from some middle schools had simply done more interesting work than their peers elsewhere.

“It’s not even that their writing skills are better, but what they’re writing about is better,” Ms. Zingmond said. When it is time to apply to high school and create a portfolio, which is required in some cases, they “will just have a lot more they can pull from.”

By contrast, Ms. Turbides, who helps middle schoolers apply to high school in eastern Brooklyn, said the opposite was often true.

“We have teachers who have no idea they need to be holding on to students’ work, and they throw it out because they have no place to put it,” Ms. Turbides said. “So for a lot of these students, when it comes time to submit a portfolio, they have nothing to submit.”

The Fair

In the imposing Brooklyn Tech building, each of the city’s five boroughs had been given a floor or two, and the high schools had set up displays at slender tables. Students rushed from one to the next asking questions and printing their names on clipboards. Many tables were sheathed in school colors and littered with swag. There were branded pens, mints and lip balm. George Washington Carver High School, a school in Queens with a veterinary program, brought along a guinea pig called Mugsy, a turtle named Frankie and a rabbit who went by Marshmallow.

“Ms. Bryant, this hallway is giving me a heart attack!” a Pelham Gardens student named Tabitha Gonzalez called out on one of the floors devoted to Manhattan schools. “There are so many schools I just don’t even know.”

On the floor for Bronx schools, the halls were quieter, with fewer students exploring what they had to offer. Bronx schools find themselves on the wrong side of many important measures. They had the lowest graduation rate in the city last year, at just 65 percent. Of the students who should have graduated, more than one in 10 had dropped out.

By noon, the hallways were stuffy, and two dozen Pelham Gardens students sat against the walls of an empty corridor, resting their legs.

Jayda was among them, and though she had started the day aiming for Manhattan schools, she said that she had changed her mind. She preferred the schools on the fourth floor, the ones in the Bronx.

“Manhattan schools didn’t explain themselves,” she said. “They would say their school was better, but never said how it was better.”

She said her favorite school she saw that day was DeWitt Clinton, a large Bronx high school that was represented at the fair by a passionate principal. But DeWitt Clinton is in trouble. It is part of Renewal Schools, a city program for struggling schools, and only 48 percent of its students graduated last year.

Making Choices

By Ms. Bryant’s estimate, there are only about 15 high schools in the Bronx doing a consistently solid job. The rest have disappointing records, with too few students graduating and even those who earn diplomas not possessing the skills they need for college.

But families at Pelham Gardens often choose schools in their home borough anyway. Getting to a school in Upper Manhattan, like the Manhattan Center for Science and Mathematics, a successful school in East Harlem, can take an hour. Plenty of parents do not want to sign up their 14-year-olds to leave the house before sunrise and spend three hours a day on the train.

So they choose more local options, like Herbert H. Lehman High School, in the Westchester Square neighborhood of the Bronx, just two miles away from Pelham Gardens. But there, only 52 percent of students graduated last year. Ms. Bryant has warned Pelham Gardens students about the school, but 15 students have matched there.

“The options in the Bronx are very limited,” said Nathaniel Query, whose daughter, Emma, is an eighth grader at Pelham Gardens. She is an honor roll student with straight A’s, he said.

“In Manhattan there are so many good choices, even if not specialized schools, just good schools,” Mr. Query said. “And Brooklyn and Queens are too far.”

Sukanya Thompson was one of Pelham Gardens’ strongest students. A tall girl with colorful braces — green in the fall, red in the spring — she was born in Jamaica and came to the United States when she was 3. She hopes one day to be an obstetrician, and she had the grades and scores to apply to screened schools, like the Manhattan Center, which has a 96 percent graduation rate. But her mother is a nurse in the northern Bronx, and she wants to be able to attend parent-teacher conferences and other events at her daughter’s school.

“I thought going to Manhattan would be cool, but she said no,” Sukanya said. “She said to put good Bronx schools at the top of the list and good Manhattan schools at the bottom.”

Despite her strong report cards and a confidence that she would do well, Sukanya still described the process as a stressful puzzle. She said she changed her list multiple times, cycling through her sixth version by early November.

“When I move one thing on my list, the whole thing changes,” she said. “It’s like if you take out the middle section of a building, the whole thing will fall down.”

In the spring, Sukanya was matched with her first choice, Eximius College Preparatory Academy in the Bronx, a limited unscreened school, which means all it requires is that applicants visit or sign in at a high school fair. When Sukanya was applying in the fall, the most recent graduation rate for the school was 86 percent. Those figures have since been updated, and the rate at Eximius has dipped to 81 percent.

“I’m excited, it was my first choice,” she said. “I wish it could be higher,” she said of the graduation rate, “but it’s still O.K.”

Experts on the process caution that a school’s admissions method is not necessarily an indicator of quality. There are limited unscreened schools that perform well, just as there are screened schools they would not recommend.

Indeed, there is an enormous range among programs, even if they share an admissions model.

Take screened schools. Some screened programs set their threshold for English or math grades at an 85 or higher, or require a 2.9 average on state tests. These most selective programs are in successful schools that are mostly white and Asian.

Another class of high schools also contributes to the sorting of students by race and class, schools that give preference to students living nearby.

The best of these zoned programs are in white and Asian neighborhoods, and are effectively out of reach for black and Hispanic students. Zoned programs in black and Hispanic neighborhoods have low graduation rates.

Over the course of three years, none of the children at Pelham Gardens have gotten into one of the city’s specialized high schools, even though Ms. Bryant encourages lots of students to take the test. Still, some students have gotten into very good programs. Ten students have matched at the Manhattan Center, and one has gone to the highly regarded Beacon.

The average graduation rate for the schools to which students at Pelham Gardens were assigned was 72.3 percent, about the same as the city rate. Even so, almost 60 percent of the students matched at schools with below-average graduation rates, including schools where less than half of students graduate.

Sean P. Corcoran, an associate professor of economics and education policy at New York University, has researched the choice process and how students match. He said that the best option is for students to reach for the best possible school for which they are qualified, and indeed, most students get one of their top choices. But in many cases, students reach either too low or too high.

“I went into this many years ago thinking kids were uninformed and were just making bad choices, but it’s a lot more complicated than that,” Dr. Corcoran said. “People are trying to make good choices based on the obstacles that are there.”

For Jayda, the process ended with an offer from Mott Haven Village Preparatory High School in the South Bronx, which was her sixth choice, she said. Last year, 59 percent of Mott Haven students graduated.

“At first, I was really upset,” Jayda said. She thought about reapplying in the second round of admissions, but the only schools available would be those that did not fill up in the first round. “I thought about it and decided I didn’t want leftover schools.”

At least, she said, it is in a different neighborhood.

Better Schools

In the winter, after eighth graders had submitted their final applications, Ms. Bryant arranged for some of them to speak to the seventh grade about their experiences.

“If I could do seventh grade over, I would change my whole mind-set,” said a girl from behind a pair of glasses. “My mind-set was: ‘Oh, this is middle school; it doesn’t mean nothing. My high school will depend on my eighth-grade grades, so I’ll do better then, but I’ll just slack now.’ If I could go back, I would change everything.”

Dennis Walcott, a schools chancellor during the Bloomberg administration who helped devise the choice system, said information allowed families “to be smart consumers,” and let students go to school anywhere in the city, even far from their own neighborhoods.

“That flexibility was built into the system,” Mr. Walcott said, “and to me that’s how you move away from a segregated school environment and you open up the process.”

The changes the Bloomberg administration made to the system have shown significant improvements. The four-year graduation rate has risen by more than 20 points, and the gap in graduation rates between white and black or Hispanic students has narrowed. But there remains a divide of 14 percentage points between whites and blacks, and 15 between whites and Hispanics.

These improvements make real differences in the lives of students, but they leave plenty of room for children to fall through the cracks. And high schools remain exceedingly segregated. The isolation of black and Hispanic students in high schools is nearly as complete as in neighborhood school

Josh Wallack, a deputy chancellor for strategy and policy in the Education Department, said that choice and market forces could not be left to mold the system on their own.

Mr. Wallack said the de Blasio administration had tried to make information about high schools more accessible. In the fall, it started a mobile website called School Finder, which allows families to search the high school directory on their phones. It has also taken a range of steps to strengthen the school system over all, like expanding prekindergarten and offering more computer science and Advanced Placement courses. He described this as a holistic approach to improving all schools so that students were better prepared, no matter where they landed in high school.

“Choice is a really important part of our system, but it’s not enough just to have choices,” Mr. Wallack said.

“You have to have good systems and personal help in figuring out what the best fit is for you,” he continued. “We know different families need different kinds of help, and that is what we aim to provide.”

Denise Williams, the sharp and energetic principal at Pelham Gardens, said it was too soon to judge how well the system had worked for her students, since only three classes had been matched with high schools so far.

“I like the idea of choice, for students to have the opportunity to branch out from where they’re from,” she said. “On the flip side, it is a huge system. Any time you’re serving this many kids, the process itself is going to be challenging.”

Many educators believe that some sort of preference should be created for low-income students, for example. That way, children whose parents do not have the ability to take them to open houses across the city are not competing so directly with those from families that can make the high school quest their mission. The city is experimenting with that idea now, allowing two Manhattan high schools with high graduation rates, Harvest Collegiate High School and Central Park East High School, to test that idea.

In the meantime, Mr. Wallack said that “as a matter of practice,” the city was not adding any screened seats at this point, and that the number of those seats had fallen by about 675 since 2015. The city is not looking to expand limited unscreened seats, either, he said.

But no matter how well the algorithm works or how much information families have, as long as there are low-performing schools, there will always be children assigned to attend them. And who are those children likely to be?

“They do need to have a choice,” Ms. Bryant said from behind her desk. “But are the opportunities these kids have real opportunities?”

She gave a gentle shrug. “They just need better schools.”