Mercedes Schneider says it best: “First, a word on what a foundation is: A foundation is a non-governmental entity that is established as a nonprofit corporation or a charitable trust, with a principal purpose of making grants to unrelated organizations, institutions, or individuals for scientific, educational, cultural, religious, or other charitable purposes.
This broad definition encompasses two foundation types: private foundations and grantmaking public charities.
Gates is funding Common Core State Standards via grants throught his Bill and Melinda Gates Foundation. Thus, it is possible for one foundation to give money to another foundation. Organizations create associated foundations [501(c)3] for the tax breaks available to nonprofits.
For example, both AFT and NEA have foundations; in donating to AFT and NEA, Gates donated to the AFT and NEA foundations. Next, for the definition of institute:
An institute is an organization or association designed to study or promote something.
This institutes supported with Gates CCSS money promote CSS. The foundations receiving Gates CCSS money support organizations that promote CCSS.