The school day is done, the homework is polished off and all that's left to do for the two boys sitting at a table on a rainy afternoon is to talk sports, crack jokes and banter on about Greek mythology.

The debate is made easier because one of them, a sixth-grader at Jordan Middle School, just completed his paper on Cronus, the powerful progeny of the sky and the earth.

His friend, a fifth-grader at Escondido Elementary School, had spent the previous hour boning up on his math but when it comes to the Greeks, he holds his own. He knows his Pericles (who gave us democracy) from his Hercules (the adventurous strongman). This despite the fact that Greek mythology is clearly not his favorite subject ("So fourth grade," he says).

Even so, the boys agree that Cronus deserves some respect. He is, after all, the father of Zeus.

"And Zeus is legit," the younger boy concedes.

"Yeah, but Uranus was even more legit because that's Cronus' dad," his friend replies.

The discussion is taking place against a backdrop of TV noise and sporadic bursts of laughter coming from the other end of the sprawling room in the Opportunity Services Center, a downtown facility that provides shelter and support services to people making a transition from homelessness. Managed by LifeMoves, a nonprofit whose goal is to "break the cycle of homelessness," the program last year received a $5,000 grant from the Palo Alto Weekly Holiday Fund.

Formerly known as InnVision Shelter Network, the nonprofit recently changed its brand to highlight that the fact that its services go beyond providing shelter, according to staff. The after-school tutoring and counseling program for the 23 children who live at the Encino Avenue facility is a prime example of that.

For its users, who tend to be elementary- and early-middle-school students, the room is a place of rigor, leisure and -- above all -- stability. Most get there at about 3 p.m. for snacks, followed by reading time, quiet time, homework time and, finally, play time. This week, with schools just starting up, the homework load is light so most of the children are whiling away the time with "Toy Story 3."

The well-lighted room is part library and part rec room, with two computers, a flat-screen TV and shelves of books, which some of the younger kids riffle through when the urge strikes. One of the younger kids, an Addison Elementary School fourth-grader, takes a break from the movie and fills out a form about the book he just read, which he will then submit to Johanna Mora, the children's services coordinator, in exchange for a prize from the closet. This can be a toy car, a Pokemon card or a One Direction CD. Mora noted that a few new prizes had recently been added to the mix.

"So now they're excited about reading again," Mora said. "They were excited when I started the whole prize thing, but now that there's new prizes, you can see that even more."

Launched 10 years ago, the program has gone through several recent changes. Initially, case managers were in charge of providing mental-health counseling to resident families. About four years ago, mental-health therapists were brought in and the counseling became more specialized. Today, there are about three mental-health professionals in each LifeMoves facility, including the Opportunity Services Center, said Philip Dah, the facility's senior director. Starting last year, the nonprofit began providing focused mental-health therapy for children.

The goal of these is to ease what could be a jarring transition from the streets to the schools, Dah said.

"When you're coming from a household where mom and dad may have mental-health issues or substance-abuse issues and are just coming out of homelessness and now living here, behavior in school may be significantly different from what any other kid in school has to go through," he said. "So the goal is to make sure that, in school, they can assimilate very well."

At the same time, staff also aims to serve as the bridge between center counselors and school staff. Most of the teachers don't know anything about the children's household situations. The program's counselors aim to change that, Dah said.

"What we are trying to do is connect schools and counselors and say, 'This is what the story is,'" Dah said. "You cannot treat all these kids in the same way."

In some cases, the parents need as much help as the kids. Some aren't accustomed to attending conferences with a teachers, for example. At times, Opportunity Center staff members sit in on such meetings, Dah said.

In addition to the counseling and the tutoring services, there are also art activities, sports, games and field trips. Twice a week, Mora takes the group to the Boys and Girls Club, where there are more computers and play space. And during the summer, there are day trips to California's Great America, the Monterey Bay Aquarium and other regional destinations.

Mora said she sees the after-school program as an anchor for children who live at the Opportunity Services Center -- a place where they can get a snack, find someone to talk to or get extra help with homework. Two high school children who live in the Opportunity Center routinely come in to help out their younger neighbors. Community volunteers from Stanford University and local high schools occasionally pop in. And the after-school program also offers the myriad small pleasures of everyday life: the books, the toys, the conversations with peers and room to roam and explore.

"Upstairs, they may have small apartments, but over here there is a decent amount of space -- as well as computers and books," Mora said.

Information about the Holiday Fund, including how to donate, can be found at PaloAltoOnline.com/holiday_fund.