Fraction Equivalence and Ordering

Students are exposed to general methods and strategies to recognize and generate equivalent fractions, and learn to compare fractions with different numerators and different denominators.

Unit Summary

In this 13-day unit, students develop general methods and strategies to recognize and generate equivalent fractions as well as to compare and order fractions. 

Students began their study of fractions in Grades 1 and 2, where students learned to partition rectangles and circles into halves, thirds, and fourths. In Grade 3, students developed an understanding of fractions as numbers rather than simply equal parts of shapes. Students work with number lines, which help to “reinforce the analogy between fractions and whole numbers” (Progressions for the Common Core State Standards in Math, p. 4). Students also begin their work with recognizing and generating equivalent fractions in simple cases, using a visual fraction model to support that reasoning. This also involves the special case of whole numbers and various fractions, e.g., $${1={2\over2}={3\over3}={4\over4}...}$$.  Lastly, students begin to compare fractions in cases where the two fractions have a common numerator or common denominator. 

Thus, students begin this unit where they left off in Grade 3, extending their understanding of and strategies to recognize and generate equivalent fractions. Students use area models, tape diagrams, and number lines to understand and justify why two fractions $${{{a\over b}}}$$ and $${{{(n\times a)}\over{(n\times b)}}}$$ are equivalent, and they use those representations as well as multiplication and division to recognize and generate equivalent fractions. Next, they compare fractions with different numerators and different denominators. They may do this by finding common numerators or common denominators. They may also compare fractions using benchmarks, such as “see[ing] that $${{7\over 8} < {{13\over12}}}$$ because $${{7\over8}}$$ is less than $$1$$ (and is therefore to the left of $$1$$) but $${13\over12}$$ is greater than $$1$$ (and is therefore to the right of $$1$$)” (Progressions for the Common Core State Standards in Math, pp. 6–7).

Students engage with the practice standards in a variety of ways in this unit. For example, students construct viable arguments and critique the reasoning of others (MP.3) when they explain why a fraction $${{{a\over b}}}$$ is equivalent to a fraction $${{{(n\times a)}\over{(n\times b)}}}$$. Students use appropriate tools strategically (MP.5) when they choose from various models to solve problems. Lastly, students look for and make use of structure (MP.7) when considering how the number and sizes of parts of two equivalent fractions may differ even though the two fractions themselves are the same size.  

Students will only work with fractions of the form $${{{a\over b}}}$$, including fractions greater than $$1$$. Students will develop an understanding of mixed numbers in Unit 6, where they will use fraction addition to see the equivalence of fractions greater than $$1$$ and mixed numbers. Beyond that special case, students will encounter all cases of addition and subtraction of fractions with like denominators, as well as multiplication of a whole number by a fraction. Then, in Unit 7, students will work with decimal fractions, understanding decimal notation for fractions and comparing decimal fractions, including adding decimal fractions with respective denominators $$10$$ and $$100$$. Students continue their work with fraction and decimal computation in Grades 5 and 6. Thus, the property that “multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction” forms the basis for much of their upcoming work in Grade 4, as well as Grades 5 and 6.

Pacing: 13 instructional days (11 lessons, 1 flex day, 1 assessment day)

Assessment

This assessment accompanies Unit 5 and should be given on the suggested assessment day or after completing the unit.

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Unit Prep

Essential Understandings

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  • “The numerical process of multiplying the numerator and denominator of a fraction by the same number, $$n$$, corresponds physically to partitioning each unit fraction piece into $$n$$ smaller equal pieces. The whole is then partitioned into $$n$$ times as many pieces, and there are $$n$$ times as many smaller unit fraction pieces as in the original fraction” (Progressions for the Common Core State Standards in Math, p. 6). 
  • “It is possible to over-emphasize the importance of simplifying fractions. There is no mathematical reason why fractions must be written in simplified form, although it may be convenient to do so in some cases” (Progressions for the Common Core State Standards in Math, p. 6).

Vocabulary

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benchmark fraction

common denominator

common numerator

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Unit Materials, Representations and Tools

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  • Area model
  • Fraction Cards - one set per pair of students
  • Fraction strip/tape diagram
  • Number line

Intellectual Prep

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Intellectual Prep for All Units

  • Read and annotate “Unit Summary” and “Essential Understandings” portion of the unit plan. 
  • Do all the Target Tasks and annotate them with the “Unit Summary” and “Essential Understandings” in mind. 
  • Take the unit assessment.  

Unit-Specific Intellectual Prep

  • When referred to fractions throughout Units 5 and 6, use unit language as opposed to “out of” language (e.g., $$\frac{3}{4}$$ should be described as “3 fourths” rather than “3 out of 4”). To understand why, read the blog post, Say What You Mean and Mean What You Say by Illustrative Mathematics.
  • Read the blog post Fractions: Units and Equivalence from one of the lead writers of the Common Core State Standards, Bill McCallum.

Lesson Map

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Number and Operations—Fractions
  • 4.NF.A — Extend understanding of fraction equivalence and ordering.

  • 4.NF.A.1 — Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

  • 4.NF.A.2 — Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Foundational Standards

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Number and Operations—Fractions
  • 3.NF.A.1

  • 3.NF.A.2

  • 3.NF.A.3

  • 3.NF.A.3.D

Operations and Algebraic Thinking
  • 4.OA.A.1

  • 4.OA.A.2

Future Standards

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Number and Operations—Fractions
  • 4.NF.B.3

  • 4.NF.C.5

  • 5.NF.A.1

  • 5.NF.B.5

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.