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.
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Sharing Interpretation
Here 8 ÷ 2 to mean 8 divided or shared evenly, as between 2 people. 
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Measurement Interpretation
In this interpretation of 8 ÷ 2, we determine how many groups of 2 there are in 8. 
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Our Concept Video Page contains an 8 minute video that discusses how to think conceptually about division of fractions.
Here are more resources for this topic:
A nifty video about partitive and quotitive division from PBS Learning.
This succinct paper from Donna Curry (TERC) explains many of the ideas in this video: Two Ways of Thinking About Division
A video from Graham Fletcher that traces the progression of division from Grade 3 to Grade 5.
A nifty video about partitive and quotitive division from PBS Learning.
This succinct paper from Donna Curry (TERC) explains many of the ideas in this video: Two Ways of Thinking About Division
A video from Graham Fletcher that traces the progression of division from Grade 3 to Grade 5.
