Teachers have had to take on more and more students, relaxing rigor and achievement along the way, according to interviews. While teachers do not have the burden of a full day of classes, they field questions from families, monitor students’ progress and review and grade schoolwork. Complaints about low pay and high class loads — with some high school teachers managing more than 250 students — have prompted a unionization battle at Agora, which has offices in Wayne, Pa.

A look at a forthcoming study by researchers at Western Michigan University and the National Education Policy Center shows that only a third of K12’s schools achieved adequate yearly progress, the measurement mandated by federal No Child Left Behind legislation.

Some teachers at K12 schools said they felt pressured to pass students who did little work. Teachers have also questioned why some students who did no class work were allowed to remain on school rosters, potentially allowing the company to continue receiving public money for them. State auditors found that the K12-run Colorado Virtual Academy counted about 120 students for state reimbursement whose enrollment could not be verified or who did not meet Colorado residency requirements. Some had never logged in.

“What we’re talking about here is the financialization of public education,” said Alex Molnar, a research professor at the University of Colorado Boulder School of Education who is affiliated with the education policy center. “These folks are fundamentally trying to do to public education what the banks did with home mortgages.”

The online companies can tailor their programs by reducing curriculum and teachers. During a presentation at the Virginia legislature this year, a representative of Connections explained that its services were available at three price points per student:

Option A: $7,500, a student-teacher ratio of 35-40 to 1, and an average teacher salary of $45,000.

Option B: $6,500, a student-teacher ratio of 50 to 1, with less experienced teachers paid $40,000.

Option C: $4,800 and a student-teacher ratio of 60 to 1, as well as a narrower curriculum.

Despite lower operating costs, the online companies collect nearly as much taxpayer money in some states as brick-and-mortar charter schools. In Pennsylvania, about 30,000 students are enrolled in online schools at an average cost of about $10,000 per student. The state auditor general, Jack Wagner, said that is double or more what it costs the companies to educate those children online.

“It’s extremely unfair for the taxpayer to be paying for additional expenses, such as advertising,” Mr. Wagner said. Much of the public money also goes toward lobbying state officials, an activity that Ronald J. Packard, chief executive of K12, has called a “core competency” of the company.