The new plan would give priority for 25 percent of the seats at all the district’s middle schools to students who score below grade level on the state tests. Because test scores closely track socioeconomic status and race, the plan would likely increase the number of poor and minority students at schools that are now out of reach for many disadvantaged families.

The video in Mr. Carranza’s tweet captured a meeting about the plan that was held last week at Public School 199, a high-achieving elementary school, where most of the students are white or Asian. In it, some parents erupted at the idea that their children might be shut out of the most popular schools.

“You’re talking about telling an 11-year-old, ‘You worked your butt off and you didn’t get that, what you needed or wanted,’” a woman yells. “You’re telling them, ‘You’re going to go to a school that’s not going to educate you in the same way you’ve been educated. Life sucks!’”

But Irene Butler, who has four grandchildren in the district’s public schools, welcomed the idea. Her grandson is in sixth grade at West Prep, but she said she would have considered other schools if the plan had been in place for him.

“A lot of kids are struggling to get through their classes and need help, but are not getting the help they need,” Ms. Butler, who is black, said. Having an opportunity to go to some of the higher-performing schools, she added, “will also help children from getting frustrated and dropping out.”