# Fractions

Students deepen their understanding of halves, thirds, and fourths to understand fractions as equal partitions of a whole, and are exposed to additional fractional units such as fifths, sixths, eighths, ninths, and tenths.

## Unit Summary

In Unit 6, students extend and deepen Grade 1 work with understanding halves and fourths/quarters (1.G.3) as well as Grade 2 practice with equal shares of halves, thirds, and fourths (2.G.3) to understanding fractions as equal partitions of a whole. Their knowledge becomes more formal as they work with area models and the number line. Throughout the module, students have multiple experiences working with the Grade 3 specified fractional units of halves, thirds, fourths, sixths, and eighths. To build flexible thinking about fractions, students are exposed to additional fractional units such as fifths, ninths, and tenths.

Students begin the unit by partitioning different models of wholes into equal parts (e.g., concrete fraction strips and pictorial area models) (3.G.2), allowing this supporting cluster content to enhance the major work of Grade 3 with fractions. They identify and count equal parts as 1 half, 1 fourth, 1 third, 1 sixth, and 1 eighth in unit form before introduction to the unit fraction $\frac{1}{b}$ (3.NF.1). Then, they make copies of unit fractions to build non-unit fractions, understanding unit fractions as the basic building blocks that compose other fractions (3.NF.1). Next, students transfer their work to the number line. They begin by using the interval from 0 to 1 as the whole and then extend to mark fractions beyond a whole. Noticing that some fractions with different units are placed at the exact same point on the number line, they come to understand equivalent fractions (3.NF.3a). Students recognize that whole numbers can be written as fractions, including writing 1 as $\frac{1}{1}, \frac{2}{2}, \frac{3}{3}$, etc., as well as writing whole numbers as fractions with a denominator of 1, e.g., 2 as $\frac{2}{1}$, 3 as $\frac{3}{1}$, etc. Lastly, students use their understanding of the number of units and the size of each unit to compare fractions in simple cases, such as when dealing with common numerators or common denominators by reasoning about their size (3.NF.3d). Lastly, students measure lengths with fractional units and use data generated by measuring multiple objects to create line plots (3.MD.4). Students “use their developing knowledge of fractions and number lines to extend their work from the previous grade by working with measurement data involving fractional measurement values” (MD Progression, p. 10), thus using this supporting cluster work to enhance the major work of fractions.

This unit affords ample opportunity for students to engage with the Standards for Mathematical Practice. Students will develop an extensive toolbox of ways to model fractions, including area models, tape diagrams, and number lines (MP.5), choosing one model over another to represent a problem based on its inherent advantages and disadvantages. Students construct viable arguments and critique the reasoning of others as they explain why fractions are equivalent and justify their conclusions of a comparison with a visual fraction model (MP.3). They attend to precision as they come to more deeply understand what is meant by equal parts, and being sure to specify the whole when discussing equivalence and comparison (MP.6). Lastly, in the context of line plots, “measuring and recording data require attention to precision (MP.6)” (MD Progression, p. 3).

Unfortunately, “the topic of fractions is where students often give up trying to understand mathematics and instead resort to rules” (Van de Walle, p. 203). Thus, this unit places a strong emphasis on developing conceptual understanding of fractions, using the number line to represent fractions and to aid in students' understanding of fractions as numbers. With this strong foundation, students will operate on fractions in Grades 4 and 5 (4.NF.3—4, 5.NF.1—7) and apply this understanding in a variety of contexts, such as proportional reasoning in middle school and interpreting functions in high school, among many others.

Pacing: 31 instructional days (28 lessons, 2 flex days, 1 assessment day)

For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 3rd Grade Scope and Sequence Recommended Adjustments. • Expanded Assessment Package
• Problem Sets for Each Lesson
• Student Handout Editor
• Vocabulary Package

## Unit Prep

### Essential Understandings

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• “Unit fractions [are] basic building blocks for fractions, in the same sense that the number 1 is the basic building block of the whole numbers. Just as every whole number can be obtained by combining ones, every fraction can be obtained by combining copies of one unit fraction” (NF Progression, p. 7).
• “The number line reinforces the analogy between fractions and whole numbers. Just as 5 is the point on the number line reached by marking off 5 times the length of the unit interval from 0, so $\frac{5}{3}$ is the point obtained in the same way using a different interval as the unit of measurement, namely the interval from 0 to $\frac{1}{3}$” (NF Progression, p. 8).
• With both equivalence and comparison of fractions, it is important to make sure that each fraction refers to the same whole. For example, it is possible for a fourth of a large pizza to be greater than half of a small pizza.
• One can compare fractions with the same denominator by thinking about the number of units. For example, just as 5 inches is greater than 3 inches because it has a greater measurement in the same size, 5 eighths is greater than 3 eighths because it is made of more unit fractions of the same size.
• One can compare fractions with the same numerator by thinking about the size of the unit. For example, just as 2 inches is greater than 2 centimeters because inches are larger than centimeters, 2 thirds is greater than 2 fifths because thirds are a larger unit than fifths.
• The numerical axis of a line plot is simply a segment of a number line. Further, “the number line diagram in a line plot corresponds to the scale on the measurement tool used to generate the data” (MD Progression, p 3).

### Vocabulary

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denominator

equivalent fraction

fraction

fractional unit

numerator

unit form

unit fraction

unit interval

whole

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### 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.

## Assessment

This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.

## Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

### Core Standards

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##### Geometry
• 3.G.A.2 — Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

##### Measurement and Data
• 3.MD.B.4 — Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.

##### Number and Operations—Fractions
• 3.NF.A — Develop understanding of fractions as numbers.

• 3.NF.A.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.

• 3.NF.A.2 — Understand a fraction as a number on the number line; represent fractions on a number line diagram.

• 3.NF.A.2.A — Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

• 3.NF.A.2.B — Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

• 3.NF.A.3 — Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

• 3.NF.A.3.A — Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

• 3.NF.A.3.B — 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.NF.A.3.C — Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Example: express 3 in the form 3 = 3/1; recognize that 6/1 = 6. Example: locate 4/4 and 1 at the same point of a number line diagram.

• 3.NF.A.3.D — 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.

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• 2.G.A.3

• 2.MD.A.1

• 2.MD.A.2

• 2.MD.B.6

• 2.MD.D.9

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• 4.MD.B.4

• 4.NF.A

• 4.NF.B

• 6.NS.C.6

### 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.