Partitive and Quotitive Division
An important distinction in division is between situations that call for a partitive (also called fair share or sharing) model of division, and those that call for a quotitive (also called subtraction or measurement) model of division. This distinction is critical for the division of fractions work that students begin in 5th grade. Students should have exposure to both types of division starting in 3rd grade.
The distinction is summarized in this diagram.
Our video page contains an 8 minute video that discusses how to think conceptually about division of fractions.