Exploring Equivalency and Comparing Fractions

Primary Standards:

MAFS.3.NF.1.3: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

  1. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
  2. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
  3. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
  4. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Connecting Standards:

MAFS.3.NF.1.1: Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

Content Knowledge:

In this Unit, as students build fraction models, they should notice that the same fraction can be named in different ways.   all represent the same fractional amount or point on a number line, they are equivalent.  A critical understanding about equivalent fractions is that for the fractions to be equivalent the wholes must be the same.

Although it is not the goal of the standard to learn the “procedure” for making equivalent fractions until fourth grade, students’ experiences and observations with fraction models allow them to generate some simple equivalent fractions in 3rd grade.  In 3rd grade, their experiences are primarily focused on understanding the meaning of equivalence and recognizing simple equivalent fractions.  In addition to understanding that one fraction can be equivalent to another, students should be able to recognize that fractions can be equivalent to a whole (or wholes).


GCG 1 – Learning Goal: As a mathematician, I will be able to use strategies to fluently multiply by 9  

  • Step 1: Explore x9 facts with x10 facts to build fluency (e.g., 9×6 is the same as 10 groups of 6, less one group)
  • Step 2: Decide when to apply the commutative/ distributive property or use other known facts for efficiency (e.g., students already know 9×2, they wouldn’t need to use a distributive strategy or x10 minus 1 group)

GCG 2 – Learning Goal: As a mathematician, I will be able to identify equivalent fractions using area models

  • Step 1: Identify equivalent fractions given visual area fraction models
  • Step 2: Use area models to generate and justify equivalent fractions
  • Step 3: Use area models to generate fractions equivalent to whole numbers (including with 1 as a denominator)

GCG 3 – Learning Goal: As a mathematician, I will be able to identify equivalent fractions using linear models

  • Step 1: Identify and explain equivalency in fractions using equal distances (on linear models)
  • Step 2: Use linear models to generate and justify equivalent fractions
  • Step 3: Identify whole numbers as equivalent fractions on a number line

GCG 4 – Learning Goal: As a mathematician, I will be able to compare fractions with the same denominator or same numerator

  • Step 1: Use models (area & linear) to compare two fractions with the same denominator
  • Step 2: Use models (area & linear) to compare two fractions with the same numerator
  • Step 3: Use the <, >, or = symbols to compare two fractions with the same denominator or same numerator
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