Learn on PengiIllustrative Mathematics, Grade 5Chapter 2: Fractions as Quotients and Fraction Multiplication

Lesson 1: Sharing Situations and Division as Fractions

In this Grade 5 lesson from Illustrative Mathematics Chapter 2, students explore the relationship between division and fractions by working through equal-sharing situations involving sandwiches. Students build on their prior knowledge of whole-number division to understand that dividing, such as 3 sandwiches shared among 4 people, produces a fractional quotient like three-fourths. This lesson lays the foundation for the standard 5.NF.B.3, helping students see that a fraction represents the value of a division expression.

Section 1

Modeling Partitive Division: Finding the Group Size

Property

To find the amount for a single share in an equal sharing situation (Partitive Division), divide the total amount by the number of shares. The result can be expressed as a fraction or a mixed number.

\text{Size of one share} = \frac{\text{Total amount}}{\text{Number of shares}}

Examples

To model $3 \div 4$ , draw a tape diagram representing the whole number 3, and divide it into 4 equal parts. The value of one part is $\frac{3}{4}$ .
To model $5 \div 2$ , draw 5 tape diagrams, each representing 1 whole. To share them into 2 equal groups, each group receives 2 whole tapes and $\frac{1}{2}$ of the last tape, showing that $5 \div 2 = 2\frac{1}{2}$ .

Section 2

Writing Division as a Fraction

Property

A division problem can be written as a fraction, where the dividend is the numerator and the divisor is the denominator.

a \div b = \frac{a}{b}

Examples

Section 3

Modeling Sharing Problems with Tape Diagrams

Property

A division problem $a \div b$ can be modeled with a tape diagram.
The dividend, $a$ , represents the total amount being shared.
The divisor, $b$ , is the number of equal groups to divide the total into.
The size of one group is the quotient, which is the fraction $\frac{a}{b}$ .

Examples

To model $3 \div 4$ , draw a tape diagram representing the whole number 3, and divide it into 4 equal parts. The value of one part is $\frac{3}{4}$ .
To model $5 \div 2$ , draw 5 tape diagrams, each representing 1 whole. To share them into 2 equal groups, each group receives 2 whole tapes and $\frac{1}{2}$ of the last tape, showing that $5 \div 2 = 2\frac{1}{2}$ .

Explanation

A tape diagram is a visual tool used to represent division. The total quantity being divided (the dividend) is drawn as a tape or a series of tapes. This total is then partitioned into a number of equal sections corresponding to the divisor. The size or value of one of these sections represents the quotient, showing how a division problem is equivalent to a fraction.

Section 4

Modeling Sharing with Shares Greater Than One

Property

When sharing $a$ items among $b$ people where $a > b$ , each person's share will be greater than 1. The total share can be found by first distributing whole items, and then partitioning the remaining items equally.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Modeling Partitive Division: Finding the Group Size

Property

\text{Size of one share} = \frac{\text{Total amount}}{\text{Number of shares}}

Examples

To model $3 \div 4$ , draw a tape diagram representing the whole number 3, and divide it into 4 equal parts. The value of one part is $\frac{3}{4}$ .
To model $5 \div 2$ , draw 5 tape diagrams, each representing 1 whole. To share them into 2 equal groups, each group receives 2 whole tapes and $\frac{1}{2}$ of the last tape, showing that $5 \div 2 = 2\frac{1}{2}$ .

Section 2

Writing Division as a Fraction

Property

A division problem can be written as a fraction, where the dividend is the numerator and the divisor is the denominator.

a \div b = \frac{a}{b}

Examples

Section 3

Modeling Sharing Problems with Tape Diagrams

Property

Examples

To model $3 \div 4$ , draw a tape diagram representing the whole number 3, and divide it into 4 equal parts. The value of one part is $\frac{3}{4}$ .
To model $5 \div 2$ , draw 5 tape diagrams, each representing 1 whole. To share them into 2 equal groups, each group receives 2 whole tapes and $\frac{1}{2}$ of the last tape, showing that $5 \div 2 = 2\frac{1}{2}$ .

Explanation

Section 4

Modeling Sharing with Shares Greater Than One

Property

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Modeling Partitive Division: Finding the Group Size

Property

\text{Size of one share} = \frac{\text{Total amount}}{\text{Number of shares}}

Examples

To model $3 \div 4$ , draw a tape diagram representing the whole number 3, and divide it into 4 equal parts. The value of one part is $\frac{3}{4}$ .
To model $5 \div 2$ , draw 5 tape diagrams, each representing 1 whole. To share them into 2 equal groups, each group receives 2 whole tapes and $\frac{1}{2}$ of the last tape, showing that $5 \div 2 = 2\frac{1}{2}$ .

Section 2

Writing Division as a Fraction

Property

A division problem can be written as a fraction, where the dividend is the numerator and the divisor is the denominator.

a \div b = \frac{a}{b}

Examples

Section 3

Modeling Sharing Problems with Tape Diagrams

Property

Examples

To model $3 \div 4$ , draw a tape diagram representing the whole number 3, and divide it into 4 equal parts. The value of one part is $\frac{3}{4}$ .
To model $5 \div 2$ , draw 5 tape diagrams, each representing 1 whole. To share them into 2 equal groups, each group receives 2 whole tapes and $\frac{1}{2}$ of the last tape, showing that $5 \div 2 = 2\frac{1}{2}$ .