Multiplication and Division of Fractions

Students deepen their understanding of fraction multiplication and begin to explore to fraction division (and fractions as division), as well as apply this new understanding to the context of line plots.

Unit Summary

In Grade 5 Unit 5, students continue their exploration with fraction operations, deepening their understanding of fraction multiplication from Grade 4 and introducing them to fraction division.

Students began learning about fractions very early, as described in the Unit 4 Unit Summary. However, students’ exposure to fraction multiplication only began in Grade 4, when they learned to multiply a fraction by a whole number, interpreting this as repeated addition. For example, $${4\times {2\over3}}$$ is thought of as 4 copies of 2 thirds. This understanding is reliant on an understanding of multiplication as equal groups (3.OA.1). In Grade 4, however, students also developed an understanding of multiplicative comparison (4.OA.1), which will be of particular importance to the new ways in which students will interpret fraction multiplication in this unit.

The unit begins with students developing a new understanding of fractions as division. In the past, they’ve thought of fractions as equal-sized partitions of wholes, but here they develop an understanding of a fraction as an operation itself and represent division problems as fractions (5.NF.3). Students now see that remainders can be interpreted in yet another way, namely divided by the divisor to result in a mixed-number quotient. Then, students develop a new understanding of fraction multiplication as fractional parts of a set of a certain size (5.NF.4), which is a new interpretation of multiplicative comparison. Students use this understanding to develop general methods to multiply fractions by whole numbers and fractions, including mixed numbers. Throughout this work, students develop an understanding of multiplication as scaling (5.NF.5), “an important opportunity for students to reason abstractly” (MP.2) as the Progressions notes (Progressions for the Common Core State Standards in Mathematics, Number and Operations - Fractions, 3-5, p. 14). Then, students explore division of a unit fraction by a whole number and a whole number by a unit fraction (5.NF.7), preparing students to divide with fractions in all cases in Grade 6 (6.NS.1). Then, students also solve myriad word problems, seeing the strategies they used to solve word problems with whole numbers still apply but that special attention should be paid to the whole being discussed (5.NF.6, MP.4), as well as write and solve expressions involving fractions as a way to support the major work (5.OA.1, 5.OA.2). Finally, students make line plots to display a data set of measurements in fractions of a unit and solve problems involving information presented in line plots (5.MD.2), a supporting cluster standard that supports the major work of this and the past unit of using all four operations with fractions (5.NF).

In Unit 6, students will learn to multiply and divide decimals, relying on their understanding of these operations with fractions developed in this unit to do so. In Grade 6, students encounter the remaining cases of fraction division (6.NS.1). “Work with fractions and multiplication is a building block for work with ratios. In Grades 6 and 7, students use their understanding of wholes and parts to reason about ratios of two quantities, making and analyzing tables of equivalent ratios, and graphing pairs from these tables in the coordinate plane. These tables and graphs represent proportional relationships, which students see as functions in Grade 8” (NF Progression, p. 20). Students will further rely on this operational fluency throughout the remainder of their mathematical careers, from fractional coefficients in functions to the connection between irrational numbers and non-repeating decimals.

Pacing: 27 instructional days (24 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 5th Grade Scope and Sequence Recommended Adjustments.

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Assessment

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

Unit Prep

Intellectual Prep

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

While students learned the two meanings of division in Grade 3, they are still important when working with 5.NF.7. To read about the two interpretations of fraction division, read the following blog post by Bill McCallum and Kristin Umland, Fraction division part 2: Two interpretations of division. This is part 2 of 4 of blogposts on fraction division, and the other parts are optional.
Read the following table that includes models used in this unit.

Set model	Example: Use a set model to solve $$\frac{1}{4}\times12$$.
Area model	Example: Use an area model to solve $$\frac{2}{5}\times\frac{3}{4}$$.
Tape diagram	Example: Use a tape diagram to solve $$\frac{1}{2}\div3$$.
Number line	Example: Use a number line to solve $$2\div\frac{1}{4}$$.
Line plot

Essential Understandings

Properties of operations still hold even when the terms are fractions, and they can be used to simplify a computation.
Fractions can be interpreted as division. As a result, a remainder in a division problem can be divided by the divisor and represented as a fraction.
Multiplication of a fraction and a whole number can be thought of as repeated addition (e.g., $${4\times {{{2\over3}}}}$$ can be seen as 4 groups of $${{2\over3}}$$) or can be thought of as a partition of a certain size of set (e.g., $$4\times{{{2\over3}}}$$ can be seen as 2 parts of a partition of 4 into 3 equal parts).
Multiplication can result in a smaller or equivalent product. Similarly, division can result in a larger or equivalent quotient.

Unit Materials, Representations and Tools

Scissors
Two-sided counters
Set models
Area models
Tape diagrams
Number lines
Line plots
Inch Grid Paper
Fraction Cards

Additional Unit Practice

Unit Practice

With Fishtank Plus you can access our Daily Word Problem Practice and our content-aligned Fluency Activities created to help students strengthen their application and fluency skills.

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Lesson Map

Topic A: Fractions as Division

1

5.NF.B.3

Model fractions as division using area models and solve word problems involving division of whole numbers with answers in the form of fractions or mixed numbers.

2

5.NF.B.3

Model fractions as division using tape diagrams and solve word problems involving division of whole numbers with answers in the form of fractions or mixed numbers.

Topic B: Multiplying a Fraction by a Whole Number

3

5.NF.B.4.A

5.NF.B.6

Multiply a fraction by a whole number using set models.

4

5.NF.B.4.A

5.NF.B.6

Multiply a fraction by a whole number using tape diagrams and number lines.

5

5.NF.B.4.A

5.NF.B.5

5.NF.B.5.A

5.NF.B.5.B

Relate multiplication of a fraction by a whole number to multiplication of a whole number by a fraction and use this to develop a general method to multiply any fraction by any whole number (or vice versa).

6

5.NF.B.6

Solve real-world problems involving multiplication with mixed numbers and create real-world contexts for expressions involving multiplication with mixed numbers.

17

5.NF.B.5

5.NF.B.5.A

5.NF.B.5.B

Interpret multiplication as scaling.

Topic E: Dividing with Fractions

18

5.NF.B.7.A

5.NF.B.7.C

Divide a unit fraction by a whole number.

19

5.NF.B.7.B

5.NF.B.7.C

Divide a whole number by a unit fraction.

20

5.NF.B.7.C

Solve real-world problems involving division with fractions and create real-world contexts for expressions involving division with fractions.

Topic F: Fraction Expressions and Word Problems

21

5.NF.B.3

5.NF.B.6

5.NF.B.7

Solve real-world problems involving multiplication and division with fractions.

22

5.OA.A.1

5.OA.A.2

5.NF.B.3

5.NF.B.4

5.NF.B.5

5.NF.B.6

5.NF.B.7

Write and evaluate numerical expressions involving operations with fractions.

Topic G: Line Plots

23

5.MD.B.2

Create line plots.

24

5.MD.B.2

Solve problems involving information presented in a line plot (dot plot).

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

Measurement and Data

5.MD.B.2 — Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

Measurement and Data

5.MD.B.2 — Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

Number and Operations—Fractions

5.NF.B.3 — Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Number and Operations—Fractions

5.NF.B.3 — Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
5.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Number and Operations—Fractions

5.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.4.A — Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

Number and Operations—Fractions

5.NF.B.4.A — Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
5.NF.B.4.B — Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

Number and Operations—Fractions

5.NF.B.4.B — Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
5.NF.B.5 — Interpret multiplication as scaling (resizing), by:

Number and Operations—Fractions

5.NF.B.5 — Interpret multiplication as scaling (resizing), by:
5.NF.B.5.A — Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

Number and Operations—Fractions

5.NF.B.5.A — Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
5.NF.B.5.B — Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

Number and Operations—Fractions

5.NF.B.5.B — Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
5.NF.B.6 — Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Number and Operations—Fractions

5.NF.B.6 — Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
5.NF.B.7 — Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade

Number and Operations—Fractions

5.NF.B.7 — Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade
5.NF.B.7.A — Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

Number and Operations—Fractions

5.NF.B.7.A — Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
5.NF.B.7.B — Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

Number and Operations—Fractions

5.NF.B.7.B — Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
5.NF.B.7.C — Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Number and Operations—Fractions

5.NF.B.7.C — Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Operations and Algebraic Thinking

5.OA.A.1 — Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

Operations and Algebraic Thinking

5.OA.A.1 — Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
5.OA.A.2 — Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Operations and Algebraic Thinking

5.OA.A.2 — Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Foundational Standards

Measurement and Data

3.MD.C.7

Measurement and Data

3.MD.C.7 — Relate area to the operations of multiplication and addition.
4.MD.A.2

Measurement and Data

4.MD.A.2 — Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
4.MD.B.4

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

3.NF.A.1

Number and Operations—Fractions

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.
4.NF.A.1

Number and Operations—Fractions

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.B.3

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.4

Number and Operations—Fractions

4.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
5.NF.A.1

Number and Operations—Fractions

5.NF.A.1 — Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
5.NF.A.2

Number and Operations—Fractions

5.NF.A.2 — Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Operations and Algebraic Thinking

3.OA.A.1

Operations and Algebraic Thinking

3.OA.A.1 — Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
3.OA.A.2

Operations and Algebraic Thinking

3.OA.A.2 — Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
3.OA.B.6

Operations and Algebraic Thinking

3.OA.B.6 — Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
3.OA.D.8

Operations and Algebraic Thinking

3.OA.D.8 — Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
4.OA.A.1

Operations and Algebraic Thinking

4.OA.A.1 — Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
4.OA.A.2

Operations and Algebraic Thinking

4.OA.A.2 — Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.
4.OA.A.3

Operations and Algebraic Thinking

4.OA.A.3 — Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Future Standards

Number and Operations in Base Ten

5.NBT.B.7

Number and Operations in Base Ten

5.NBT.B.7 — Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Ratios and Proportional Relationships

6.RP.A.1

Ratios and Proportional Relationships

6.RP.A.1 — Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."
6.RP.A.2

Ratios and Proportional Relationships

6.RP.A.2 — Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Expectations for unit rates in this grade are limited to non-complex fractions. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."

Statistics and Probability

6.SP.A.1

Statistics and Probability

6.SP.A.1 — Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages.
6.SP.A.2

Statistics and Probability

6.SP.A.2 — Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
6.SP.A.3

Statistics and Probability

6.SP.A.3 — Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
6.SP.B.4

Statistics and Probability

6.SP.B.4 — Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

The Number System

6.NS.A.1

The Number System

6.NS.A.1 — Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
6.NS.B.4

The Number System

6.NS.B.4 — Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1—100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
7.NS.A.2

The Number System

7.NS.A.2 — Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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.