Understanding that between any two fractions there are an infinite number of fractions (fractiondensity) is important. Students should use a variety of strategies for identifying such fractions. This helps students to compare fractions and assess fractions for equivalence.
BACKGROUND
Comparing and ordering fractions allows students to develop a sense of fraction as quantity, as well as a sense of the size of a fraction, both necessary prior knowledge components for understanding fraction operations (Johanning, 2011).
Comparing and ordering fractions with different fractional units (or denominators) leads students to identify the need for equivalent fractions. When students determine an equivalent fraction they are changing the unit of measure by either splitting or merging the partitions of the original fraction. The following illustration demonstrates these concepts using an area model:
Splitting to determine an equivalent fraction for 2/3
Merging to determine an equivalent fraction for 6/8
The exploration of equivalence allows students to develop an understanding of equivalent fractions as simply being a different way of naming the same quantity; it also supports them in viewing the fraction as a numeric value. A solid understanding of equivalence helps students with fractions operations, especially addition and subtraction.
References:
Johanning, D. I. (2011). Estimation's role in calculations with fractions. Mathematics Teaching In the Middle School, 17(2), 96102.

TASKS
Generating Fractions
between Two Numbers (Fraction Density)
This is a set of progressive prompts that will elicit the use of various strategies used to find fractions between two numbers. The prompts, when used in sequence, support students in choosing a strategy based upon the fractions being considered. Encourage students to build models/representations and create contexts to aid in visualization. This series of prompts can be used for a variety of grades and purposes. The time required will vary depending on the grade level and student readiness.
