The additional instructional time is proposed to begin in the 2012-13 school year.
The grants also come with a research component that will allow the foundation to share with other districts the lessons that it learns from the six grantees.
Arlene Ackerman, the superintendent since 2008, is leaving amid discontent in some corners with her leadership.
The district started screening applicants for food service jobs for tobacco use, and is considering expanding that restriction to all new hires.
A Missouri middle school has been transformed in part, the principal says, because the school changed the way the students feel about themselves and how they treat each other.
New Jersey and Connecticut join several other districts and states that are investigating allegations of teacher cheating on state tests, including Philadelphia, the District of Columbia and Georgia.
The district's NAEP scores are believed to be due to real improvements and not cheating, education officials say.
The board that oversees the district released a terse statement saying it remains committed to working with Superintendent Arlene Ackerman.
The school board threatened to delay school until the city made an overdue tax revenue payment to the 105,000-student district
This corruption allegation is one of several in the district over the past five years.