When Coach Natalie Randolph and the Coolidge High School Colts take the field Thursday for the D.C. Public Schools football championship, her team will be, in many ways, a rarity in DCPS. She will have 45 players invested in a program that’s as much about academics and character-building as it is about football, as much about getting kids off the streets and into classrooms as it is about playing in the Turkey Bowl, the city’s marquee public school event.

Across the field from Coolidge will be Dunbar, a perennial powerhouse with more than its share of National Football League alumni (including three recent first-round draft picks). Dunbar, in many ways, has struggled with the elements that have been plaguing DCPS high school athletics: It is one of three teams to forfeit games this season for on-field fights, and its coach, Willie Bennett, is working on an interim basis after the previous coach was removed for using ineligible players barely a month into the season.

Randolph’s players are the new kids to the league’s 42nd annual Thanksgiving Day party at Eastern High — this is just their second appearance in the game since 1987; Dunbar has played in 12 of the past 13 games. And, of course, there’s the fact that Randolph will be the first woman ever to coach in this game and is believed to be the only woman currently coaching a high school football team in the country.

Her arrival here is hardly an accident. It’s pretty close to the script as DCPS would have written when she was hired amid national headlines in March 2010.

All she has done is put in the work.

“I don’t think there is another coach in DCIAA history that has done what Natalie has done,” Coolidge Principal Thelma Jarrett said. “We’re talking about boys in the ninth grade that were walking the halls constantly. You’re talking about boys that gave up constantly, not gangs, but neighborhood crews. All of these schools have these challenges. But we have Natalie and people said she couldn’t do it and couldn’t go in the locker room and couldn’t relate to men. But she has exceeded those expectations.”

It’s not uncommon for coaches, particularly in DCPS, to go beyond their job descriptions. But Coolidge players say that in Randolph they see a uniquely genuine person, someone who is tough when needed (“no study hall, no practice”) but will cheer louder than anyone when they earn an A.

Randolph, 31, invites players over and orders a dozen pizzas or puts food on the grill for them. This week, she traded in her two-door coupe for a larger, four-door sports utility vehicle mainly so she can give more players rides home, to football camps over the summer or on college visits.

“I have a dog and these guys,” she said, smiling. “That’s it.”

In the 20 months since she was hired, Randolph has poured her life into the job. She has juggled teaching three environmental science classes and coaching her football team on three hours of sleep a night.

“I don’t know how she is still upright and functioning with the million ways she has spread herself,” said Eileen Pascucci, who teaches science and, along with teacher Pam Cogas, helps run the study hall for the football team. “She seriously does everything.”

Randolph has spent late nights alternating between drawing up lesson plans and planning the team’s defense, a byproduct of her increased role this season due to a smaller coaching staff. She has helped her players overcome their personal challenges, from poor grades to behavioral struggles.

“I don’t have any other choice,” Randolph said. “I have to succeed. I don’t have a choice.

To her players, she’s an inspiring and motivating figure.

“We trust her,” said senior defensive end-tight end Chuck Gaines, who met Randolph when he took her environmental science class as a ninth grader and considers having her as a coach a life-changing experience. “Not that we didn’t trust coaches before. But we actually trust Ms. Randolph. We are her life, basically. She sacrifices her life.”

Coaching high school football at a public school in the District is unlike coaching anywhere else in the area. Not only are resources scarce for schools and the distractions of the city plentiful for kids, but the DCPS out-of-boundary policy that allows students to change schools for any reason makes assembling — and keeping — a talented roster an enormous challenge.

“Right now you’ve got kids that will go to one of four high schools because those are the schools that win,” notes Mayor Vincent C. Gray (D), a Dunbar alum who has long been a supporter of Randolph’s. “It’s a cycle that is not going to change until we have a more evenly-distributed talent pool.”

Randolph’s roster of 45 at the school located at Fifth and Tuckerman streets in Northwest is one of the biggest in the DCIAA, and it’s not by coincidence. Her players want to play for her.

“It makes me feel great that we can do this for her and do this for ourselves, to take the first lady to the Turkey Bowl in her second year,” senior wide receiver-defensive back Calvin Brown said. “That’s a great accomplishment. It pays off, the hard work that we put in pays off to the Turkey Bowl.”

At first, Randolph, a District native, never envisioned herself teaching after college. The 1998 Sidwell Friends graduate studied environmental science at Virginia, where she was a scholarship track athlete before earning a master’s degree in education. Her first job out of college was doing environmental outreach for the D.C. Department of Public Works, speaking to schools about the importance of recycling.

“I was going to be fine in aquatic ecology, doing research on a boat somewhere,” she said.

But football had a long hold on her. “She wanted to play on Sidwell’s team and she was actually faster than anyone on the team,” her father, Nathaniel Randolph, told The Post in 2006. “But then I told her, ‘You know, some of them may try to hurt you because you’re a girl.’ ” (Nathaniel Randolph died in a 2008 car accident.)

From 2004 to 2008, she got her chance to play, putting on pads as a wide receiver for the D.C. Divas professional team. In 2008, she joined the Coolidge faculty but didn’t coach until 2010, when a fellow teacher urged her to apply for the school’s football opening.

This year, under far less attention, Randolph’s imprint on the team has grown. The Colts are 8-2, a four-win improvement over last season, behind a talented crop of seniors. She has shepherded an improvement in the team’s overall grade-point average from 2.75 last fall to 3.0 this season. The players, who take their cue from a stricter Randolph, are more disciplined and focused.

Before he began playing for Randolph, junior Hasan Jenkins often found trouble, dealing with family problems, roaming the school’s hallways, arriving late to class and paying little attention to his grades. The football coach has guided his maturation the past two seasons — his grade-point average improved from 2.0 to 3.5 and he has become a starting linebacker.

“She motivated me to become a better person,” Jenkins said. “She helped me out and told me there are better things than being one of them hoodlums on the street.”

Said principal Jarrett: “Now he is one of the most respected and respectful boys. Who would have thought?”

Jenkins is one example of the way Randolph’s methods have taken hold off the field. On it, Randolph can also see a difference.

“It’s easier the second year,” Randolph said. “They know the plays. They know the system.”

The players know she has taken care of them. Now they’re ready to take care of her.

“We are going to leave our hearts out on this field,” Gaines said. “We’re going to ball out for Ms. Randolph, we’re going to ball out for Coolidge.”