Second-grader Jacob Stone and fourth-grader Thomas Daniel trick-or-treated in Harlem with Wanda Fisher.

As it became clear on Wednesday that city schools would not be able to reopen this week because of damage to the city’s infrastructure, concern deepened at the Department of Education.

The department has ramped up a push to toughen academic standards this year, and a week off eight weeks into the semester — even if the days are likely to be made up later — could set back those efforts.

So department officials started compiling worksheets, suggested study schedules, test preparation guides, and lists of television shows with educational merit for students in each grade. On Wednesday night, they emailed principals to ask them to send a message alerting families to the new resources.

“We know that you and your students are eager to get back to school, and we are working hard to reopen schools as soon as possible,” Deputy Chancellor Shael Polakow-Suransky wrote in the message to families, which schools without power could have difficulty distributing. “In the meantime, we are encouraging students and their parents to continue learning at home during this time away from school.”

He suggested that families look to a silver lining in this week’s storm clouds: “Extra time at home is an opportunity to begin or continue planning for your future after graduation,” he wrote.

It’s an approach that some families and schools have taken since early in the week, when Hurricane Sandy’s danger passed for the many New Yorkers living out of the flood zone and in areas that retained electrical power.

When her nephews finished the homework they brought home on Friday, Wanda Fisher said her husband started quizzing them on mental math problems.

“You did math, multiplication, subtraction, division,” Fisher reminded the boys, second-grader Jacob Stone and fourth-grader Thomas Daniel, as they trick-or-treated in Harlem on Wednesday afternoon.

“And then we did plus/minus, and addition,” Stone added. A second-grader at P.S. 200, Stone said he had spent the days since the storm “reading and watching TV and seeing the hurricane.”

Other parents said they also had pushed their children to go above and beyond the homework assignments they received last week.

“I want them to be safe, but I want them to keep up with their education, too,” said Rosy Lopez, whose son Joseph is in first grade in Harlem’s P.S. 46. She said she had made sure Joseph had tackled all of the work that his teachers had sent home, which included social studies, and math. Then he read books about pirates and the movie “Toy Story.”

Some schools were able to give families additional assignments to keep students busy and engaged with academics.

Ralph Martinez, the principal of P.S. 89 in the Bronx, said his teachers had posted new homework assignments and practice materials online using the program Jupiter Grade. But he said not every student has internet access at home.

Martinez, who said he was not concerned about his Bronxwood school building because of its elevated location, had driven into Manhattan on Wednesday with his two sons, who attend Catholic school, to buy ice and stock up on other supplies. They live in New Jersey and have been without power since Tuesday.

“Many of our teachers live in Rockland County. Fortunately they’re okay, but without light, like I am,” Martinez said. “We have been emailing each other back and forth, almost every day.”

Not every school had taken that approach as of Wednesday, and some that were most affected by the storm are not likely to be able to. Department officials said 200 school buildings were “not operational” because of the storm, including 86 that did not have power.

“Because our building is closed and because many staff members are dealing with power outages at home, there will be no online assignments emailed to students, as some parents have inquired,” Millennium High School told families on Wednesday. “We recommend that students take the opportunity to do review work and read until school reopens.”

Millennium is housed in a Lower Manhattan office building whose basement was flooded. The building’s management company informed the school that no one would be able to enter until Monday, according to an email sent to parents on Wednesday from the school’s parent coordinator on behalf of its interim principal, who she said did not have power.

For high school seniors, the days off come at an opportune time: Most colleges have their first application deadline this week. (Many colleges have extended the deadline for students affected by Sandy.)

Adelya Baimukhamedova, a senior at the High School of Environmental Studies, said she had taken the time so far to catch up on assignments and put the finishing touches on college applications. No teachers assigned new homework since Friday, she said, but “they sent emails to reschedule exams. And I’m caught up with my homework now.”

Dyani Lebron, a fifth-grader at Manhattan’s P.S. 116, also said she had not heard from her teachers this week. She finished her weekend homework on Saturday under the assumption that this week would be a regular school week, so for now, she has been reading “The Mysterious Benedict Society.”

Lebron was walking on the Upper West Side Wednesday afternoon with Jocelyn Alvarez, a senior at Norman Thomas High School, which is in the process of phasing out and now has only an 11th and 12th grade this year. Alvarez said she is worried the storm could deepen the school’s difficulties.

“It’s already very disorganized,” Alvarez said. “Classes, schedules, students mixed in with the wrong grade. There are students who are behind and need to catch up, and this hurricane has made it worse.”

At more thriving schools, some teachers found innovative ways to trouble-shoot the situation. At Stuyvesant High School, which is located next to the West Side Highway and currently does not have power, longtime computer science teacher Mike Zamansky resumed classes for some of his students on Wednesday by online Google chat.

About 40 students watched the class live and others watched a recorded video afterwards, said Zamansky, who documented the experience on his blog.

“People keep talking about recorded lectures … but if anything, today’s experience just confirms to me that there’s nothing like an in-class teacher, particularly with a small group of students,” Zamansky wrote. “That said, I think this was a good experience and my students seem to agree. We spent part of an otherwise unproductive day in a productive manner and we’re planning on doing it again tomorrow.”