# MA.5.AR.1.3

Solve real-world problems involving division of a unit fraction by a whole number and a whole number by a unit fraction.

### Examples

Example: A property has a total of acre and needs to be divided equally among 3 sisters. Each sister will receive of an acre.

Example: Kiki has 10 candy bars and plans to give of a candy bar to her classmates at school. How many classmates will receive a piece of a candy bar?

### Clarifications

Clarification 1: Instruction includes the use of visual models and equations to represent the problem.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• NA

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

The purpose of this benchmark is to connect division of fraction concepts to real-world scenarios (MTR.7.1). This work builds on the multiplication of fractions by whole numbers in grade 4 (MA.4.AR.1.3), and prepares them for grade 6 (MA.6.NSO.2.3) where they will solve real-world fraction problems using all four operations with fractions (MTR.7.1).
• During instruction, it is important for students to have opportunities to extend their understanding of the meaning of fractions, how many unit fractions are in a whole, and their understanding of division of fractions as involving equal groups or shares and the number of objects in each.
• Students should use visual fraction models and reasoning to solve word problems involving division of fractions.
• For example, to assist students with solving the problem, “The elephant eats 4 lbs of peanuts a day. His trainer gives him $\frac{\text{1}}{\text{5}}$ of a pound at a time. How many times a day does the elephant eat peanuts?” use the following diagram to show how 4 ÷ $\frac{\text{1}}{\text{5}}$ can be visualized to assist students with solving.
• The expectation of this benchmark is not for students to use an algorithm (e.g., multiplicative inverse) to divide fractions.
• 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 opportunities to engage in teacher-directed practice using visual representations to solve real-world problems involving division of a unit fraction by a whole number or a whole number by a unit fraction. The teacher directs students on how to use models or equations based on real-world situations. Through questioning, the teacher guides students to explain what each fractional portion represents in the problems used during instruction and practice.
• For example, the teacher displays and reads aloud the following problem: “Julio has 6 packages of cookies. He is making gift bags for people at school. Each bag will contain $\frac{\text{1}}{\text{3}}$ of a package of cookies. How many gift bags can he make?” Using models, the teacher solves the problem with guided questioning having students explain how to use models to solve this question. The teacher guides students to create an equation to represent the problem. This is repeated with multiple real-world examples that involve division of a unit fraction by a whole number or a whole number by a unit fraction.
• Teacher provides opportunities to use hands-on models and manipulatives to solve real- world problems involving division of a unit fraction by a whole number or a whole number by a unit fraction. Students explain how each model represents the real-world situation. The teacher directs students how to use models or equations based on real- world examples and through questioning guide students to explain what each fractional portion represents in the problems used during instruction and practice.
• For example, the teacher displays and reads aloud the following problem: “Shelton made some lemonade. The pitcher of lemonade holds 8 cups. If each of the glasses that he uses can hold $\frac{\text{1}}{\text{2}}$ cup, how many servings of lemonade can he share?” Using fraction bars or fraction strips, the teacher models solving the problem with explicit instruction and guided questioning. Students explain how to use fraction bars or fraction strips as a model to solve this question and use an equation to represent the problem. This is repeated with multiple real-world problems that involve multiplication of a whole number by a fraction or a fraction by a whole number.

Sonya has $\frac{\text{1}}{\text{2}}$ gallon of chocolate chip ice cream. She wants to share her ice cream with 6 friends. How much ice cream will each friend get?

### Instructional Items

Instructional Items 1

Betty has 12 sheets of tissue paper to add to her holiday gift bags. Each gift bag needs $\frac{\text{1}}{\text{3}}$ sheet of tissue paper. How many holiday gift bags can Betty fill?

*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.AR.1.AP.3: Solve one-step real-world problems involving division of a whole number by a unit fraction.

## Related Resources

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

## 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.

Type: Lesson Plan

## Original Student Tutorials

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

Chocolate Shop Challenge Part 1: Dividing Unit Fractions and Whole Numbers Using Fraction Bar Models:

Divide unit fractions by whole numbers and divide whole numbers by unit fractions in this chocolate-themed, interactive tutorial.

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

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

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.

The purpose of this task is to have students add fractions with unlike denominators and divide a unit fraction by a whole number. This accessible real-life context provides students with an opportunity to apply their understanding of addition as joining two separate quantities.

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.

## Original Student Tutorials Mathematics - Grades K-5

Chocolate Shop Challenge Part 1: Dividing Unit Fractions and Whole Numbers Using Fraction Bar Models:

Divide unit fractions by whole numbers and divide whole numbers by unit fractions in this chocolate-themed, interactive tutorial.

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

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.

## Student Resources

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

## Original Student Tutorials

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

Chocolate Shop Challenge Part 1: Dividing Unit Fractions and Whole Numbers Using Fraction Bar Models:

Divide unit fractions by whole numbers and divide whole numbers by unit fractions in this chocolate-themed, interactive tutorial.

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

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

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.

The purpose of this task is to have students add fractions with unlike denominators and divide a unit fraction by a whole number. This accessible real-life context provides students with an opportunity to apply their understanding of addition as joining two separate quantities.

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.

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.

The purpose of this task is to have students add fractions with unlike denominators and divide a unit fraction by a whole number. This accessible real-life context provides students with an opportunity to apply their understanding of addition as joining two separate quantities.

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.