A 1969 report from the NAACP Legal Defense and Educational Fund Inc. found examples of exactly that. As a result, Congress amended the law in 1970 to thwart such maneuvers. It required districts to spend Title I dollars on additional education for poor children, above and beyond what they already received from other sources — that is, to supplement, not to supplant.

But from the start, the rule proved hard to enforce, since few districts maintain separate bank accounts and accounting ledgers for each school. Under the previous version of the education act, passed in 2001, the Education Department enforced a complicated set of policies that essentially required districts and auditors to posit a counterfactual: How much money would poor students have received without Title I? If state law requires students to be taught for six hours a day, for example, a district couldn’t use Title I funds to teach poor children for the sixth hour, because that would leave them no better off than before.

This requirement proved cumbersome for schools and regulators alike. So lawmakers modified it when they passed the latest version of the law in 2015. They kept the “supplement, not supplant” provision. But districts would no longer have to identify particular services as “supplemental.” And the department was prohibited from requiring a “specific methodology” for distributing state and local funds.

The department’s proposed rule — the one that prompted the lobbyists’ letter — affirmed that districts could decide for themselves how to distribute their money among schools, whether based on head counts, student-to-teacher ratios, a percentage increase over the previous year’s school budget or anything else. However, the result of the methodology had to be that the district spent at least as much state and local money in each Title I school as in non-Title I schools. That had to be the nonnegotiable starting point — otherwise, how could Title I money, by any reasonable definition of the word, be considered “extra”?

The department did not mention teacher salaries, but they make up the bulk of school budgets and would thus be included in calculations of spending.

Normally, teachers unions and school advocates support Democratic politicians and are the mortal enemies of conservative Republicans. Yet this time, they found an enthusiastic supporter in Senator Lamar Alexander, Republican of Tennessee and chairman of the Senate Education Committee. He’s generally opposed to federal regulations of education on principle, regardless of the issue. In this way, left-leaning interest groups like local teachers unions are ideologically aligned with Tea Party Republicans.