Fairfax County leaders began discussing next year’s school budget this week in a sometimes tense meeting that focused on funding the elimination of half-day Mondays.

When it began the academic year Tuesday, the county school system ended its 40-year-old practice of dismissing children in lower grades early on Mondays, a move that was popular with parents who previously had to make day-care arrangements for Monday afternoons. The policy also allowed the system to build in a scheduling cushion in the event of inclement weather, effectively adding 13 snow days to the school calendar.

But paying for the change — which adds 2.5 hours of instructional time each week for elementary students and is estimated to cost $7.6 million — has been controversial, as schools officials pushed for the change after the end of the most recent budget discussions and asked county supervisors to pay for it. The schools ultimately opted to cover the costs for this year using leftover funds from last year’s budget when it became clear that the Board of Supervisors was not likely to add funds after a contentious budget process had already concluded.

School Superintendent Karen Garza and School Board members made another plea for county funding Tuesday, with Garza arguing that the change is “a good investment in our children” that could help improve academic achievement in the county’s lower grades. School Board members echoed that idea, arguing that the county supervisors should fund the initiative in future years.

“We can’t pay for it out of thin air,” said School Board member Elizabeth Schultz (Springfield). “But it had to be done for many, many reasons.”

It did not appear that supervisors were willing to commit significant additional funding for the initiative, but supervisors said at Tuesday’s meeting that they support an offer from Chairman Sharon Bulova (D), who proposed that the supervisors defray the costs in next year’s budget. The supervisors are scheduled to vote on the matter this month.

“I remember when my kids got to elementary school and thinking ‘Why is it like this?’ ” Bulova said of half-day Mondays. “So I think it’s about time someone came along like Superintendent Garza, who said this should be fixed.”

Supervisor Jeff C. McKay (D-Lee) said the change will alleviate a burden on single parents and families with parents who both work full time and struggled to find day-care options.

Garza acknowledged that it was unusual for the school board to approve a new initiative after completing the year’s budget process. But Garza explained that she felt compelled to urge an immediate change after experiencing a winter filled with weather delays and school cancellations. The state required Fairfax to add three extra days to the calendar in June to make up for lost time in the classroom.

“Because of our shortened day at the elementary level, we would be continuing to be behind state compliance,” Garza said. “Potentially, we would find ourselves in this same issue year after year of having to worry about how we make up this additional time.”

Supervisor Michael R. Frey (R-Sully) said he was displeased with the way the school system implemented the change after budget negotiations were over. Frey noted that the county increased the tax rate this year to give the schools additional funding and that he didn’t understand why the funding request was added when “the ink hadn’t even dried on the budget.”

“There was not a word said throughout the entire budget discussion to implement this,” Frey said, adding that it seemed as if the full-day Monday initiative was being “shoved down our throats.”

He noted that, during last year’s budget negotiations, the school board had frequently stated that employee compensation was the top priority and only later proposed extending full-day Mondays as a key funding issue.

“If I’m Joe Taxpayer, why would I believe anything that’s being said?” Frey said. “When you can change your priorities and pivot that quickly between adopting the budget and the start of the fiscal year, your credibility has been lost. Significantly. And it’s going to hurt. . . . This is going to be hard to overcome.”