Partitive and Quotitive Division of Fractions

What is 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 between partitive and quotitive division is summarized in this diagram.


8 dots are divided into 2 groups of 4 dots each and shown as 4 dots plus 4 dots.
Sharing Interpretation, or Partitive Division. Here 8 ÷ 2 to mean 8 divided or shared evenly, as between 2 people.


8 dots are divided into four circles of 2 dots each, showing 2+2+2+2
Measurement Interpretation, or Quotitive Division. ​In this interpretation of 8 ÷ 2, we determine how many groups of 2 there are in 8.

More resources describing partitive and quotitive division of fractions

​Our Concept Video Page contains an 8-minute video that discusses how to think conceptually about the 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.

This page was last updated on June 21, 2023