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

Lesson 3: Relate Division and Fractions

In this Grade 5 Illustrative Mathematics lesson from Chapter 2, students explore why a ÷ b = a/b, learning to interpret fractions as division of the numerator by the denominator. Using real-world sharing scenarios, they analyze situations where a quotient is greater than, equal to, or less than 1, and practice solving equations where the unknown may be the numerator, denominator, or quotient. This lesson directly addresses standard 5.NF.B.3 and helps students build flexible reasoning about the relationship between division and fractions.

Section 1

Understanding Fractions as Division

Property

A fraction represents division. The fraction ab\frac{a}{b} means "aa divided by bb" or "a÷ba \div b". The numerator (top number) is what we are dividing, and the denominator (bottom number) is what we are dividing by.

Examples

Section 2

Identifying Parts of a Fair-Share Division Problem

Property

In a fair-share division problem, the total amount is divided into a known number of equal groups to find the size of each group.

Total÷Number of Groups=Size of Group (Unknown)Total \div \text{Number of Groups} = \text{Size of Group (Unknown)}

Examples

Section 3

Distinguishing Dividend and Divisor in Word Problems

Property

To correctly model an equal sharing situation, the division expression must be set up as:

Total Amount Being Shared÷Number of Shares \text{Total Amount Being Shared} \div \text{Number of Shares}

Examples

Section 4

Creating Division Word Problems

Property

To write a division story for an expression like a÷ba \div b, identify the total amount (the dividend, aa) and what it is being divided by (the divisor, bb). The story should ask a question where the answer is the quotient. The question will either be "How many groups of size bb are in aa?" or "What is the size of each group if aa is shared into bb groups?"

Examples

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Understanding Fractions as Division

Property

A fraction represents division. The fraction ab\frac{a}{b} means "aa divided by bb" or "a÷ba \div b". The numerator (top number) is what we are dividing, and the denominator (bottom number) is what we are dividing by.

Examples

Section 2

Identifying Parts of a Fair-Share Division Problem

Property

In a fair-share division problem, the total amount is divided into a known number of equal groups to find the size of each group.

Total÷Number of Groups=Size of Group (Unknown)Total \div \text{Number of Groups} = \text{Size of Group (Unknown)}

Examples

Section 3

Distinguishing Dividend and Divisor in Word Problems

Property

To correctly model an equal sharing situation, the division expression must be set up as:

Total Amount Being Shared÷Number of Shares \text{Total Amount Being Shared} \div \text{Number of Shares}

Examples

Section 4

Creating Division Word Problems

Property

To write a division story for an expression like a÷ba \div b, identify the total amount (the dividend, aa) and what it is being divided by (the divisor, bb). The story should ask a question where the answer is the quotient. The question will either be "How many groups of size bb are in aa?" or "What is the size of each group if aa is shared into bb groups?"

Examples