Carnegie Learning will operate as a subsidiary of Apollo Group, whose holdings include the online University of Phoenix.

Apollo Group Inc., the nation’s largest for-profit college chain and owner of the online University of Phoenix, said Aug. 2 that it has agreed to acquire Carnegie Learning Inc., a publisher of research-based math curricula, for $75 million.

In a separate transaction, Apollo said it will acquire related technology from Carnegie Mellon University for $21.5 million, payable over a 10-year period.

The acquisitions allow Apollo to accelerate its efforts to incorporate adaptive learning into its academic platform and to provide tools to help raise student achievement in mathematics, which supports improved retention and graduation rates, the company said.

Apollo’s holdings include the University of Phoenix, the nation’s largest online college. Carnegie Learning will operate as a subsidiary of Apollo Group.

Apollo anticipates the acquisitions will add value over the long term, but they’ll result in charges of 7 cents per share to 9 cents per share to earnings in fiscal 2012. The acquisitions are expected to be completed during the first quarter of fiscal 2012.

Carnegie Learning’s Cognitive Tutor software, used by districts such as Chicago Public Schools, follows students’ actions within the software to determine each student’s strategy in solving a problem. Each action the student takes is associated with one or more skills, which are references to knowledge components within the cognitive model. The performance of individual students in these skills is tracked over time (and displayed to students in the software’s “skillometer”).

Then, the software uses each student’s skill profile to pick problems that emphasize the skills on which the student is weakest. When the program determines that all the skills in a section of the curriculum have been mastered sufficiently, the student can move on to the next section of curriculum, which introduces new skills.