Fraction Operations

Students start to operate on fractions, learning how to add fractions with like denominators and multiply a whole number by any fraction.

Unit Summary

In this unit, students begin their work with operating with fractions by understanding them as a sum of unit fractions or a product of a whole number and a unit fraction. Students will then add fractions with like denominators and multiply a whole number by any fraction. Students will apply this knowledge to word problems and line plots.

In Grade 3, students developed their understanding of the meaning of fractions, especially using the number line to make sense of fractions as numbers themselves. They also did some rudimentary work with equivalent fractions and comparison of fractions. In Grade 4 Unit 5, they deepened this understanding of equivalence and comparison, learning the fundamental 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” (NF Progression, p. 6).

Thus, in this unit, armed with a deep understanding of fractions and their value, students start to operate on them for the first time. The unit is structured so that students build their understanding of fraction operations gradually, first working with the simplest case where the total is a fraction less than 1, then the case where the total is a fraction between 1 and 2 (to understand regrouping when operating in simple cases), and finally the case where the total is a fraction greater than 2. With each of these numerical cases, they first develop an understanding of non-unit fractions as sums and multiples of unit fractions. Next, they learn to add and subtract fractions. And finally, they apply these understandings to complex cases, such as word problems or fraction addition involving fractions where one denominator is a divisor of the other, which helps prepare students for similar work with decimal fractions in Unit 7. After working with all three numerical cases in the context of fraction addition and subtraction, they work with fraction multiplication, learning strategies for multiplying a whole number by a fraction and a mixed number and using those skills in the context of word problems. Finally, students apply this unit’s work to the context of line plots. Students will solve problems by using information presented in line plots, requiring them to use their recently acquired skills of fraction addition, subtraction, and even multiplication, creating a contextual way for this supporting cluster content to support the major work of the grade. The unit provides lots of opportunity for students to reason abstractly and quantitatively (MP.2) and construct viable arguments and critique the reasoning of others (MP.3).

Students’ understanding of fractions is developed further in Unit 7, in which students explore decimal numbers via their relationship to decimal fractions, expressing a given quantity in both fraction and decimal forms (4.NF.5—7). Then, in Grade 5, students extend their understanding and ability with operations with fractions (5.NF.1—7), working on all cases of fraction addition, subtraction, and multiplication and the simple cases of division of a unit fraction by a whole number or vice versa. Students then develop a comprehensive understanding of and ability to compute fraction division problems in all cases in Grade 6 (6.NS.1). Beyond these next few units and years, it is easy to find the application of this learning in nearly any mathematical subject in middle school and high school, from ratios and proportions in the middle grades to functional understanding in algebra.

Pacing: 25 instructional days (22 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 4th Grade Scope and Sequence Recommended Adjustments.

• Expanded Assessment Package
• Problem Sets for Each Lesson
• Student Handout Editor
• Vocabulary Package

Assessment

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

Unit Prep

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

Essential Understandings

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• “The meaning of addition is the same for both fractions and whole numbers, even though algorithms for calculating their sums can be different. Just as the sum of $4$ and $7$ can be seen as the length of the segment obtained by joining together two segments of lengths 4 and 7, so the sum of ${{{{2\over3}}}}$ and ${{{{8\over5}}}}$ can be seen as the length of the segment obtained joining together two segments of length ${{{{2\over3}}}}$ and ${{{{8\over5}}}}$” (Progressions for the Common Core State Standards in Mathematics, 3-5 Numbers and Operations - Fractions, p. 7).
• Quantities cannot be added or subtracted if they do not have like units. Just like one cannot add 4 pencils and 3 bananas to have 7 of anything of meaning (unless one changes the unit of both to “objects”), the same applies for the units of fractions (their denominators). This explains why one must find a common denominator to be able to add fractions with unlike denominators when adding and subtracting fractions. Further, when you add or subtract quantities with like units, their units do not change. Just like one adds 5 bananas and 2 bananas and gets 7 bananas, one adds 5 eighths and 2 eighths and gets 7 eighths.
• “Converting a mixed number to a fraction should not be viewed as a separate technique to be learned by rote, but simply as a case of fraction addition. Similarly, converting an improper fraction to a mixed number is a matter of decomposing the fraction into a sum of a whole number and a number less than 1” (Progressions for the Common Core State Standards in Mathematics, 3-5 Numbers and Operations - Fractions, p. 8).
• “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 Mathematics, 3-5 Numbers and Operations - Fractions, p. 6). Thus, students should not be expected to simplify fractions in all cases where it’s possible to do so.

Vocabulary

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mixed number

fraction greater than one

Unit Materials, Representations and Tools

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• Area model
• Buttons (with various diameters)
• Fraction strips (made from paper, or the plastic kind)
• Line plot
• Number line
• Rulers
• Tape diagram

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Measurement and Data
• 4.MD.B.4 — Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

Number and Operations—Fractions
• 4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

• 4.NF.B.3.A — Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

• 4.NF.B.3.B — Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

• 4.NF.B.3.C — Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

• 4.NF.B.3.D — Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

• 4.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

• 4.NF.B.4.A — Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

• 4.NF.B.4.B — Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

• 4.NF.B.4.C — Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

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• 3.MD.B.4

• 3.NF.A.1

• 3.NF.A.2

• 4.NF.A.1

• 4.NF.A.2

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• 4.MD.A.2

• 5.MD.B.2

• 4.NF.C.5

• 5.NF.A.1

• 5.NF.B.4

• 5.NF.B.7

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.