The District will begin winter with more than 700 homeless families in shelters and overflow motel rooms — a figure 250 percent higher than in any previous year — and it’s partly by design.

To try to tamp down on a perennial surge in families that seek rooms when the temperature drops below freezing, Mayor Muriel E. Bowser’s administration has gone on the offensive. With shelters still full from last winter, it has placed more than 400 families in motel rooms before cold weather triggers a city law requiring it to do so.

Whether the mayor’s plan will prevent a rush for more placements, however, remains to be seen, advocates say. Demand has already been higher than expected and the number now in shelters is 20 percent higher than the administration estimated it would be just 60 days ago.

How the city manages the inevitable and perhaps record-setting — winter influx marks the first in a series of new tests for Bowser (D), who has promised to end chronic family homelessness in the nation’s capital within two years and to end all chronic homelessness in the District within five years.

Kurt Runge of Miriam's Kitchen, right, delivered a bucket of toy soldiers to D.C. Council budget director Jennifer Budoff last month. Advocates return to council members' offices every few weeks to update the number of homeless vets. The District is trying to house all homeless veterans by the end of the year. (Aaron C. Davis/The Washington Post)

D.C. officials expect 800 to 1,000 more families to require shelter this winter. To house all of them, the city will need to accelerate efforts to transition families into temporary apartments and other settings.

But Laura Green Zeilinger, who oversees the agency coordinating city homeless services, says Bowser’s administration is not worried about numbers as much as how it is remaking city services for the homeless.

“The way we judge our success in hypothermia season is not ‘did we somehow predict the right number that would need shelter,’ but whether we were successful at meeting the needs of families that sought shelter,” Zeilinger said.

Managing a growing population in shelter and overflow motel rooms, which could cost more than $12 million this year, will be one of many challenges the administration will face simultaneously.

Bowser’s team is also remaking the process for how it decides which families are placed on a path to long-term housing assistance. It is also launching programs to provide short-term rentals and other assistance in an attempt to keep some families from falling into homelessness.

At the same time, the District is racing to find apartments for more than 200 homeless veterans to meet a year-end goal to house all homeless veterans before 2016.

Bowser’s team is also at odds with a growing number of advocates for the homeless about the best way to proceed with closing the city’s dilapidated family homeless shelter at the site of the former D.C. General Hospital campus.

Bowser has proposed building a series of dormitory-style shelters in neighborhoods across the city. Families in those shelters would use shared bathrooms located on each floor. The arrangement would save on construction costs and prevent families from becoming too established in the temporary shelters, the mayor’s office has said.

Advocates and D.C. Council Chairman Phil Mendelson (D) have expressed deep skepticism about that the $40 million plan, saying it would be only marginally better than the shared bathrooms now in use at D.C. General. Homeless families have complained of unsanitary conditions in the shared restrooms, awkward situations of whether to send young boys into men’s bathrooms by themselves or with their mothers into women’s rooms, and even instances of finding people having sex in shared bathrooms at D.C. General.

Bowser wrote to Mendelson late last week saying a majority of homeless advocates support her plan. If he does not as well, she said, Mendelson risks forcing the city to keep D.C. General open longer. That, she argued, would undermine her campaign pledge to close the shelter.

Mendelson said the issue boils down to one of cost vs. “general dignity” for the families in the city’s care. He has not said whether he will support the mayor’s proposal for dormitory-style shelters, and said he would announce his intentions before Tuesday, when the D.C. Council is set to take up that issue.

Late last week, an attorney with the Washington Legal Clinic for the Homeless wrote to every council member, saying Bowser had“mischaracterized” their support for her plan and that she had played down what advocates said was a critical need for private bathrooms for homeless families.

“It has become clear that the Administration believes as a matter of policy that children and their parents will be safe and healthy with shared bathrooms. We strongly disagree,” wrote Amber Harding, an attorney with the clinic.

The city will first have to deal with a growing population in shelters this winter.

The District is one of a handful of jurisdictions in the country that make shelter a right.

Last year, an estimated 850 families were projected to need shelter, a 16 percent increase from the year before. It turned out to be more than that: 1,942 families applied for shelter and 1,007 were offered housing, city documents show.

The city finished the winter with 879 families in shelters.

Bowser’s administration has reduced the backlog in recent months, moving more than 400 families into subsidized apartments and other transitional housing, but as of early last week, 715 families remained.