In this section, students develop their ability to reason about and generate equivalent fractions. They begin by using number lines as a tool for finding equivalent fractions and verifying equivalence of two fractions.

Through repeated reasoning, students notice regularity in the visual representations and begin to make sense of a numerical way to determine equivalence and generate equivalent fractions (MP8). Students generalize that fraction  is equivalent to fraction . 

Note that students do not need to describe this generalization in algebraic notation. Given their understanding of the size of fractions and relationships between fractions, however, students should be able to explain the concept with fractions that have denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.  

As they identify and generate equivalent fractions numerically, students apply their knowledge of factors and multiples from an earlier unit.

In this section, students revisit ideas and representations of fractions from grade 3, working with denominators that now include 5, 10, and 12. Students use physical fraction strips, diagrams of fraction strips, tape diagrams, and number lines to make sense of the size of fractions and fractional relationships.

Students reason about the relationship between fractions where one denominator is a multiple of the other denominator (such as   and  , or and  ). They consider different ways to represent these relationships. Students also compare fractions to benchmarks, such as and 1.

The work in this section prepares students to reason about equivalence and comparison of fractions in the subsequent lessons.

By the time they reach this section, students have an expanded set of understandings and strategies for reasoning about the size of fractions. Here, they further develop these skills and work to compare fractions with different numerators and different denominators.

To make comparisons, students may use visual representations, equivalent fractions, and their understanding of the size of fractions (for instance, relative to benchmarks, such as and 1). Students may rely on the meaning of the numerator and denominator, and choose a way to compare based on the numbers at hand. Students record the results of comparisons with symbols  ,  , or .

At the end of the section, students learn to write equivalent fractions with a particular denominator as a way to compare any fractions, which provides another opportunity to apply the idea of factors and multiples. Even though students have a numerical strategy, they are still encouraged to use flexible methods to reason about the relative size of fractions.