AUSTIN -- Public schools gutted of $5.4 billion two years ago would recapture about half that funding under spending plans offered by the Texas House on Thursday, and top Republicans have not ruled out restoring even more before Gov. Rick Perry signs the state budget.Teachers groups that have marched on the Capitol, demanding that the Legislature reverse classroom cuts that resulted in thousands of layoffs, applauded an additional $1 billion for schools in a 2014-15 budget plan approved by the House Appropriations Committee.School districts hope that lawmakers cough up even more money after the unexpected moves by the House, which came after education emerged as the dominant issue of the 140-day session.Rep. Jim Pitts, R-Waxahachie, the chief budget writer in the House, didn't dampen that optimism."It's not over till it's over," Pitts said.Yet there are realities. Although the $1 billion approved Thursday is on top of the $1.5 billion that the House and Senate previously committed toward undoing the 2011 cuts, Pitts said it's unlikely that all funding slashed two years ago will be restored in the final two months of the session.Democrats and teachers say the money is there -- notably in the projected $12 billion socked away in the state's rainy-day fund. But Perry and Republicans who control the Legislature oppose tapping that piggy bank for anything other than one-time expenses, and school funding is a recurring tab.The Senate did not include the extra $1 billion in the budget plan it overwhelmingly passed this week."The more I talk to the House members, it is the priority of the Texas House, both Republicans and Democrats, to restore public education," Pitts said. "We challenge the Senate to have that same priority."Pitts also revealed Thursday that he wants to slip schools an additional $500 million immediately with a supplemental spending bill. That would let the House boast of putting a total of $3 billion extra toward education, though the money in the supplemental bill wouldn't repeat in the 2014-15 budget."We like what Chairman Pitts is talking about. It's certainly a step in the right direction," said Clay Robinson, spokesman for the Texas State Teachers Association.The House budget spends a total of $93.5 billion in general revenue, which comes largely from sales taxes and is the pot of money that lawmakers wield most control over. The Senate budget spends about $600 million more than the House version.Included in the House budget is $10.9 million for the Texas Department of Public Safety to begin whittling a backlog of sexual assault kits in criminal investigations. Pitts called the spending the "first significant funding of this program by the state."Sharp exchangesAlthough the House budget unanimously passed through committee, the hearing was not without tense exchanges over how the state is spending money.Rep. Sylvester Turner, D-Houston, the vice chairman, asked colleagues why the $161 million tab for fighting wildfires in 2011 -- clearly a one-time expense -- would not come from the rainy-day fund.Turner argued that the money could otherwise go toward education.But using the rainy-day fund is a tough sell in the Legislature, and Rep. Donna Howard, D-Austin, said areas hit hard by wildfires need money by April."I think I know how to add and I think I know how to read and I think I understand what the rules are," Turner said."But what is the rainy-day fund for? Are we just building a reserve just to say we have a nice reserve? Are we never going to touch it?"Pitts expects the full House to take up the budget in early April.