Principal Victoria Holmes gives a high five to a student while making the rounds at Dodge Park Elementary School in Landover. (Amanda Voisard/THE WASHINGTON POST)

In her first lesson for her fourth-grade math class Monday, Shoneka Robbins told her students at Dodge Park Elementary School to imagine a store stocked with sugary treats. Sourballs. Fudge toppings. Gummy worms.

There was a world of options but limited funds. The fake money each child received amounted to $3.

“You have to make some decisions,’’ Robbins told the class.

It’s a feeling educators across Prince George’s County, on the first day of school for the 127,000-student system, know all too well. Budget cuts have left one of the county’s most prized schools without some of the tools it most prized.

Class sizes at Dodge Park have increased by about two students. The pre-kindergarten program, which staff members said was key for students to start reading early, was reduced to half-days. The librarian can come in only twice a week. And funding for after-school programs? Gone.

Still, the school of about 500 children prides itself on defying the odds. Seven years ago, 19 percent of the students from this low-income Landover neighborhood were at grade level in reading and math. Now, proficiency rates have skyrocketed: About 86 percent tested at grade level in math last school year, and about 95 percent did so in reading.

Superintendent William R. Hite Jr. attributed the success to a “can-do, will-do” culture.

“They make no excuses for the challenges and do not allow any excuses from their students or their parents,” Hite said.

D.C. schools also opened Monday, and schools in Anne Arundel, Calvert and St. Mary’s counties open this week. Those in Charles, Frederick, Howard, Loudoun and Montgomery counties open next week. Most Northern Virginia schools resume after Labor Day.

Atop a grassy hill and surrounded by pine and oak trees, Dodge Park is perched above a concrete-laden community of apartment complexes where most students live. Almost all students are black or Hispanic. More than 83 percent of families cannot afford the full cost of a school lunch.

Dodge Park’s sky-blue hallways are full of signs of hard work and optimism: Bar graphs show the uptick in test scores. “We Will Move Beyond Proficient” reads one of many signs hanging from ceilings. The new principal, Victoria Holmes, put up a banner of caterpillars turning into butterflies.

Holmes spent the past few years as assistant principal to Judith White, a recently promoted and popular leader who started math and literacy nights for parents and upgraded technology in the 45-year-old building.

On her first day as principal, Holmes left her office to help steer the swells of traffic scaling the hill to the parking lot. As students entered, she smiled and hugged them.

She tried out her Spanish with a parent trying to find her child’s classroom.

“Diez y uno?’’ she said. The parent gave her a confused look. She counted the numbers in Spanish on her fingers.

“Once!” she said. The parent smiled. Room 11.

More than an hour after the 7:45 a.m. start time, foot traffic slowed. Inside the school, a first-grade teacher was instructing students on how to sit in the ready position — hands folded, eyes open. The pre-kindergarten teacher was teaching counting by placing magnets on a white board. All was going well.

“I have big shoes to fill,’’ ­Holmes said, exhaling.

Holmes worked in tandem with White, who crafted the strategy to turn Dodge Park around. They coordinated a prep period during which teachers learned best practices from each other. They emphasized relating lessons to anything — movies, music, candy — that would forge a connection.

“We got better at what we did, which was teach,’’ Holmes recalled. “The staff was young and energetic then, and most of those teachers are still here. We got better and understood data. That was the key.”

With no active parents association, White encouraged teachers to call homes early and often.

“There’s a real family atmosphere in this school,’’ said Marjorie Fennell, 36, a mother of two. “I know that, because when my daughter broke her finger, Dr. White stayed on the phone with me for two hours.”

Fennell’s daughter, 11-year-old Ronnell Caldwell, is in her second year at the school. She said Ronnell’s reading has improved to above grade-level.

“The parents here, they’re low income. . . . The teachers here seem so concerned with breaking the cycle,” Fennell said.

When the day ended, Holmes said she was thrilled that she saw “more parents in school than ever before.” Maybe this year, she said, they’ll be able to start a parents group.

Her staff is also stepping up. When after-school programs were cut, teachers volunteered to run clubs.

Mostly, Holmes encouraged the staff not to sweat the big things they don’t have but to focus on little things that work. Start class on time. Be attentive to students. Be prepared.

“I keep telling my teachers to think of the butterfly effect,’’ ­Holmes said. “The small things can make a big difference in how they learn.”