Generate fractions between any two quantities
Understanding that between any two fractions there are an infinite number of fractions (fraction density) is important. Students should use a variety of strategies for identifying such fractions. This helps students to compare fractions and assess fractions for equivalence.
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:
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.
Johanning, D. I. (2011). Estimation's role in calculations with fractions.Mathematics Teaching In the Middle School, 17(2), 96-102.