# MA.5.FR.2.4

Extend previous understanding of division to explore the division of a unit fraction by a whole number and a whole number by a unit fraction.

### Clarifications

Clarification 1: Instruction includes the use of manipulatives, drawings or the properties of operations.

Clarification 2: Refer to Situations Involving Operations with Numbers (Appendix A).

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Fractions
Status: State Board Approved

## Benchmark Instructional Guide

• NA

### Vertical Alignment

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### Purpose and Instructional Strategies

The purpose of this benchmark is for students to experience division with whole number divisors and unit fraction dividends (fractions with a numerator of 1) and with unit fraction divisors and whole number dividends. This work prepares for division of fractions in grade 6 (MA.6.NSO.2.2) in the same way that in grade 4 (MA.4.FR.2.4) students were prepared for multiplication of fractions.
• Instruction should include the use of manipulatives, area models, number lines, and emphasizing the properties of operations (e.g., through fact families) for students to see the relationship between multiplication and division (MTR.2.1).
• Throughout instruction, students should have practice with both types of division: a unit fraction that is divided by a non-zero whole number and a whole number that is divided by a unit fraction.
• Students should be exposed to all situation types for division (refer to Situations Involving Operations with Numbers (Appendix A).
• The expectation of this benchmark is not for students to use an algorithm (e.g., multiplicative inverse) to divide by a fraction.
• Instruction includes students using equivalent fractions to simplify answers; however, putting answers in simplest form is not a priority.

### Common Misconceptions or Errors

• Students may believe that division always results in a smaller number, which is true when dividing a fraction by a whole number, but not when dividing a whole number by a fraction. Using models will help students develop the understanding needed for computation with fractions.

### Strategies to Support Tiered Instruction

• Instruction includes making the connection to models and tools previously used to understand division as equal groups or sharing. The teacher uses models to develop the understanding needed for computation with fractions.
• For example, 8 ÷ $\frac{\text{1}}{\text{4}}$ can be shown using a model of 8 wholes divided into parts of size $\frac{\text{1}}{\text{4}}$. The quotient would be the total number of $\frac{\text{1}}{\text{4}}$ pieces. The model below would show that 8 ÷ $\frac{\text{1}}{\text{4}}$ = 32.

• For example, $\frac{\text{1}}{\text{4}}$ divided into $\frac{\text{1}}{\text{4}}$ ÷ 8 can be represented using the below model. One-fourth is divided into 8 equal parts, each part is $\frac{\text{1}}{\text{32}}$ of the whole.

• Instruction includes real-world situations to interact with the content. The teacher provides students with a division expression with a real-world context and provides items to represent the situation to allow connections to be made.
• For example, the teacher provides students with the following situation: “The teacher brought in 8 brownies to split between the class. She cut the brownies into pieces of size $\frac{\text{1}}{\text{4}}$ so there would be enough for the whole class. How many $\frac{\text{1}}{\text{4}}$ pieces will there be?" The teacher provides students with images of eight brownies (or models to represent them) and has them divide or cut them into $\frac{\text{1}}{\text{4}}$ pieces to determine how many pieces they will have (32 pieces).
• For example, the teacher provides students with the following situation: “The teacher baked a pan of brownies. All but $\frac{\text{1}}{\text{4}}$ of the pan was eaten. She brought in the remaining $\frac{\text{1}}{\text{4}}$ and divided it into 8 equal pieces for her co-teachers. What 4 fraction of the whole pan will each person get?” The teacher provides students with an image of a pan of brownies with $\frac{\text{1}}{\text{4}}$ left (or model to represent it). The students divide the $\frac{\text{1}}{\text{4}}$ portion into 8 equal pieces. The teacher then connects the 4 remaining part of the brownies to the whole pan so that students can make the connection to the total number of the smaller pieces representing $\frac{\text{1}}{\text{32}}$ of the whole.

Part A. Emily has 2 feet of ribbon to make friendship bracelets. Use models and equations to answer the questions below.
• a. How many friendship bracelets can she make if each bracelet uses 2 feet of ribbon?
• b. How many friendship bracelets can she make if each bracelet uses 1 foot of ribbon?
• c. How many friendship bracelets can she make if each bracelet uses 1 half foot of ribbon?
• d. How many friendship bracelets can she make if each bracelet uses 1 third foot of ribbon?
• e. How many friendship bracelets can she make if each bracelet uses 1 fifth foot of ribbon?
Part B. Do you see any patterns in the models and equations you have written? Explain.

### Instructional Items

Instructional Item 1

• What is the quotient of $\frac{\text{1}}{\text{3}}$ ÷ 5?
•  a. $\frac{\text{1}}{\text{15}}$
• b. 15
• c. $\frac{\text{5}}{\text{3}}$
• d. $\frac{\text{3}}{\text{5}}$

Instructional Item 2

How many fourths are in 8 wholes?
• a. 4
• b. 8
• c. 16
• d. 32

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
5012070: Grade Five Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7712060: Access Mathematics Grade 5 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012065: Grade 4 Accelerated Mathematics (Specifically in versions: 2019 - 2022, 2022 and beyond (current))
5012015: Foundational Skills in Mathematics 3-5 (Specifically in versions: 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.5.FR.2.AP.4: Explore the division of a one-digit whole number by a unit fraction. Denominators are limited to 2, 3 or 4.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Educational Game

Ice Ice Maybe: An Operations Estimation Game:

This fun and interactive game helps practice estimation skills, using various operations of choice, including addition, subtraction, multiplication, division, using decimals, fractions, and percents.

Various levels of difficulty make this game appropriate for multiple age and ability levels.

Multiplication/Division: The multiplication and addition of whole numbers.

Percentages: Identify the percentage of a whole number.

Fractions: Multiply and divide a whole number by a fraction, as well as apply properties of operations.

Type: Educational Game

## Formative Assessments

Fractions Divided by Whole Numbers:

Students are given a division expression and asked to write a story context to match the expression and use a visual fraction model to solve the problem.

Type: Formative Assessment

Bags of Fudge:

Students are asked to solve a word problem involving division of a whole number by a fraction.

Type: Formative Assessment

Relay Race:

Students are asked to solve a word problem involving division of a fraction by a whole number.

Type: Formative Assessment

Whole Numbers Divided by Fractions:

Students are given a division expression and asked to write a story context to match the expression and use a visual fraction model to solve the problem.

Type: Formative Assessment

## Image/Photograph

Clipart ETC Fractions:

Illustrations that can be used for teaching and demonstrating fractions. Fractional representations are modeled in wedges of circles ("pieces of pie") and parts of polygons. There are also clipart images of numerical fractions, both proper and improper, from halves to twelfths. Fraction charts and fraction strips found in this collection can be used as manipulatives and are ready to print for classroom use.

Type: Image/Photograph

## Lesson Plans

Natural Disaster Dividing Fractions:

In this lesson, students will extend learning of dividing unit fractions and whole numbers within the context of governmental response to an emergency situation.

Type: Lesson Plan

It's My Party and I'll Make Dividing by Fractions Easier if I Want to!:

During this lesson students will relate their understanding of whole number division situations to help them interpret situations involving dividing by unit fractions. They will then develop models and strategies for representing the division of a whole number by a unit fraction.Â

Type: Lesson Plan

## Original Student Tutorials

Carnival Craziness!:

Learn to divide whole numbers by unit fractions as you help Allie and Cameron create equal shares of candy and prizes for guests at a carnival in this interactive tutorial.

Type: Original Student Tutorial

Chocolate Shop Challenge Part 2: Dividing Unit Fractions and Whole Numbers Using Number Lines:

Solve real-world word problems involving dividing a unit fraction by a whole number and dividing a whole number by a unit fraction using number lines in this chocolate-themed, interactive tutorial.

This is part 2 of a 2-part series. Click HERE to open "Chocolate Shop Challenge Part 1: Dividing Unit Fractions and Whole Numbers Using Fraction Bar Models"

Click HERE to open the related tutorial, "David Divides Desserts: Divide a Unit Fraction by a Whole Number"

Type: Original Student Tutorial

David Divides Desserts: Divide a Unit Fraction by a Whole Number:

Learn to solve word problems involving division of a unit fraction by a whole number by using models, expressions, equations, and strategic thinking in this interactive, dessert-themed tutorial.

Type: Original Student Tutorial

Share and Share Alike:

Learn how to divide a unit fraction by a whole number to share yummy picnic goodies equally in this interactive tutorial.

Type: Original Student Tutorial

How many servings of oatmeal?:

This task provides a context for performing division of a whole number by a unit fraction. This problem is a "How many groups?'' example of division: the "groups'' in this case are the servings of oatmeal and the question is asking how many servings (or groups) there are in the package.

Painting a room:

The purpose of this task is to provide students with a situation in which it is natural for them to divide a unit fraction by a non-zero whole number. Determining the amount of paint that Kulani needs for each wall illustrates an understanding of the meaning of dividing a unit fraction by a non-zero whole number.

Origami Stars:

The purpose of this task is to present students with a situation in which they need to divide a whole number by a unit fraction in order to find a solution. Calculating the number of origami stars that Avery and Megan can make illustrates student understanding of the process of dividing a whole number by a unit fraction.

How many marbles?:

This task is intended to complement "How many servings of oatmeal?" and "Molly's run.'' All three tasks address the division problem 4÷1/3 but from different points of view. This task provides a how many in each group version of 4÷1/3. This task should be done together with the "How many servings of oatmeal" task with specific attention paid to the very different pictures representing the two situations.

Dividing by One-Half:

This task requires students to recognize both "number of groups unknown" (part (a)) and "group size unknown" (part (d)) division problems in the context of a whole number divided by a unit fraction. It also addresses a common misconception that students have where they confuse dividing by 2 or multiplying by 1/2 with dividing by 1/2.

Banana Pudding:

The purpose of this task is to provide students with a concrete situation they can model by dividing a whole number by a unit fraction. For students who are just beginning to think about the meaning of division by a unit fraction (or students who have never cooked), the teacher can bring in a 1/4 cup measuring cup so that students can act it out. If students can reason through parts (a) and (b) successfully, they will be well-situated to think about part (c) which could yield different solution methods.

## Professional Development

Fractions, Percents, and Ratios, Part A: Models for Multiplication and Division of Fractions:

This professional development module shows teachers how to use area models to understand multiplication and division of fractions.

Type: Professional Development

## Teaching Idea

Divide Fractions:

This interactive resource provides three activities which model the concept of dividing fractions, as well as mixed numbers, by using number lines or circle graphs.  It includes the equation showing the standard algorithm.

Type: Teaching Idea

## Tutorial

How Do You Divide Fractions?:

This five-minute video answers the question "Must one always invert and multiply?" when dividing fractions. An alternative algorithm is presented which works well in certain cases. The video focuses on sense-making in using either method, and on judging the reasonableness of answers.

Type: Tutorial

## MFAS Formative Assessments

Bags of Fudge:

Students are asked to solve a word problem involving division of a whole number by a fraction.

Fractions Divided by Whole Numbers:

Students are given a division expression and asked to write a story context to match the expression and use a visual fraction model to solve the problem.

Relay Race:

Students are asked to solve a word problem involving division of a fraction by a whole number.

Whole Numbers Divided by Fractions:

Students are given a division expression and asked to write a story context to match the expression and use a visual fraction model to solve the problem.

## Original Student Tutorials Mathematics - Grades K-5

Carnival Craziness!:

Learn to divide whole numbers by unit fractions as you help Allie and Cameron create equal shares of candy and prizes for guests at a carnival in this interactive tutorial.

Chocolate Shop Challenge Part 2: Dividing Unit Fractions and Whole Numbers Using Number Lines:

Solve real-world word problems involving dividing a unit fraction by a whole number and dividing a whole number by a unit fraction using number lines in this chocolate-themed, interactive tutorial.

This is part 2 of a 2-part series. Click HERE to open "Chocolate Shop Challenge Part 1: Dividing Unit Fractions and Whole Numbers Using Fraction Bar Models"

Click HERE to open the related tutorial, "David Divides Desserts: Divide a Unit Fraction by a Whole Number"

David Divides Desserts: Divide a Unit Fraction by a Whole Number:

Learn to solve word problems involving division of a unit fraction by a whole number by using models, expressions, equations, and strategic thinking in this interactive, dessert-themed tutorial.

Share and Share Alike:

Learn how to divide a unit fraction by a whole number to share yummy picnic goodies equally in this interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorials

Carnival Craziness!:

Learn to divide whole numbers by unit fractions as you help Allie and Cameron create equal shares of candy and prizes for guests at a carnival in this interactive tutorial.

Type: Original Student Tutorial

Chocolate Shop Challenge Part 2: Dividing Unit Fractions and Whole Numbers Using Number Lines:

Solve real-world word problems involving dividing a unit fraction by a whole number and dividing a whole number by a unit fraction using number lines in this chocolate-themed, interactive tutorial.

This is part 2 of a 2-part series. Click HERE to open "Chocolate Shop Challenge Part 1: Dividing Unit Fractions and Whole Numbers Using Fraction Bar Models"

Click HERE to open the related tutorial, "David Divides Desserts: Divide a Unit Fraction by a Whole Number"

Type: Original Student Tutorial

David Divides Desserts: Divide a Unit Fraction by a Whole Number:

Learn to solve word problems involving division of a unit fraction by a whole number by using models, expressions, equations, and strategic thinking in this interactive, dessert-themed tutorial.

Type: Original Student Tutorial

Share and Share Alike:

Learn how to divide a unit fraction by a whole number to share yummy picnic goodies equally in this interactive tutorial.

Type: Original Student Tutorial

## Educational Game

Ice Ice Maybe: An Operations Estimation Game:

This fun and interactive game helps practice estimation skills, using various operations of choice, including addition, subtraction, multiplication, division, using decimals, fractions, and percents.

Various levels of difficulty make this game appropriate for multiple age and ability levels.

Multiplication/Division: The multiplication and addition of whole numbers.

Percentages: Identify the percentage of a whole number.

Fractions: Multiply and divide a whole number by a fraction, as well as apply properties of operations.

Type: Educational Game

How many servings of oatmeal?:

This task provides a context for performing division of a whole number by a unit fraction. This problem is a "How many groups?'' example of division: the "groups'' in this case are the servings of oatmeal and the question is asking how many servings (or groups) there are in the package.

Painting a room:

The purpose of this task is to provide students with a situation in which it is natural for them to divide a unit fraction by a non-zero whole number. Determining the amount of paint that Kulani needs for each wall illustrates an understanding of the meaning of dividing a unit fraction by a non-zero whole number.

Origami Stars:

The purpose of this task is to present students with a situation in which they need to divide a whole number by a unit fraction in order to find a solution. Calculating the number of origami stars that Avery and Megan can make illustrates student understanding of the process of dividing a whole number by a unit fraction.

How many marbles?:

This task is intended to complement "How many servings of oatmeal?" and "Molly's run.'' All three tasks address the division problem 4÷1/3 but from different points of view. This task provides a how many in each group version of 4÷1/3. This task should be done together with the "How many servings of oatmeal" task with specific attention paid to the very different pictures representing the two situations.

Dividing by One-Half:

This task requires students to recognize both "number of groups unknown" (part (a)) and "group size unknown" (part (d)) division problems in the context of a whole number divided by a unit fraction. It also addresses a common misconception that students have where they confuse dividing by 2 or multiplying by 1/2 with dividing by 1/2.

Banana Pudding:

The purpose of this task is to provide students with a concrete situation they can model by dividing a whole number by a unit fraction. For students who are just beginning to think about the meaning of division by a unit fraction (or students who have never cooked), the teacher can bring in a 1/4 cup measuring cup so that students can act it out. If students can reason through parts (a) and (b) successfully, they will be well-situated to think about part (c) which could yield different solution methods.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

## Image/Photograph

Clipart ETC Fractions:

Illustrations that can be used for teaching and demonstrating fractions. Fractional representations are modeled in wedges of circles ("pieces of pie") and parts of polygons. There are also clipart images of numerical fractions, both proper and improper, from halves to twelfths. Fraction charts and fraction strips found in this collection can be used as manipulatives and are ready to print for classroom use.

Type: Image/Photograph

How many servings of oatmeal?:

This task provides a context for performing division of a whole number by a unit fraction. This problem is a "How many groups?'' example of division: the "groups'' in this case are the servings of oatmeal and the question is asking how many servings (or groups) there are in the package.

Painting a room:

The purpose of this task is to provide students with a situation in which it is natural for them to divide a unit fraction by a non-zero whole number. Determining the amount of paint that Kulani needs for each wall illustrates an understanding of the meaning of dividing a unit fraction by a non-zero whole number.

Origami Stars:

The purpose of this task is to present students with a situation in which they need to divide a whole number by a unit fraction in order to find a solution. Calculating the number of origami stars that Avery and Megan can make illustrates student understanding of the process of dividing a whole number by a unit fraction.

How many marbles?:

This task is intended to complement "How many servings of oatmeal?" and "Molly's run.'' All three tasks address the division problem 4÷1/3 but from different points of view. This task provides a how many in each group version of 4÷1/3. This task should be done together with the "How many servings of oatmeal" task with specific attention paid to the very different pictures representing the two situations.

Dividing by One-Half:

This task requires students to recognize both "number of groups unknown" (part (a)) and "group size unknown" (part (d)) division problems in the context of a whole number divided by a unit fraction. It also addresses a common misconception that students have where they confuse dividing by 2 or multiplying by 1/2 with dividing by 1/2.

Banana Pudding:

The purpose of this task is to provide students with a concrete situation they can model by dividing a whole number by a unit fraction. For students who are just beginning to think about the meaning of division by a unit fraction (or students who have never cooked), the teacher can bring in a 1/4 cup measuring cup so that students can act it out. If students can reason through parts (a) and (b) successfully, they will be well-situated to think about part (c) which could yield different solution methods.