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

Lesson 4: Relate Division and Multiplication

In this Grade 5 lesson from Illustrative Mathematics Chapter 2, students explore the relationship between division and multiplication, learning that dividing by a number produces the same result as multiplying by its reciprocal. Students apply this understanding within the context of fractions as quotients, connecting expressions like a ÷ b to fraction multiplication. This foundational skill prepares students to fluently multiply and divide fractions throughout the chapter.

Section 1

Writing Quotients as Mixed Numbers

Property

To convert an improper fraction quotient $\frac{a}{b}$ to a mixed number, find the whole number quotient $q$ and remainder $r$ from the division $a \div b$ . The mixed number is written as $q\frac{r}{b}$ .

Examples

Section 2

Quotients Relative to 1

Property

By comparing the dividend ( $a$ ) and the divisor ( $b$ ), we can predict if the quotient will be greater than, less than, or equal to 1.

If the dividend is larger than the divisor ( $a > b$ ), the quotient is greater than 1: $\frac{a}{b} > 1$
If the dividend is smaller than the divisor ( $a < b$ ), the quotient is less than 1: $\frac{a}{b} < 1$
If the dividend is equal to the divisor ( $a = b$ ), the quotient is exactly 1: $\frac{a}{b} = 1$

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Writing Quotients as Mixed Numbers

Property

Examples

Section 2

Quotients Relative to 1

Property

By comparing the dividend ( $a$ ) and the divisor ( $b$ ), we can predict if the quotient will be greater than, less than, or equal to 1.

If the dividend is larger than the divisor ( $a > b$ ), the quotient is greater than 1: $\frac{a}{b} > 1$
If the dividend is smaller than the divisor ( $a < b$ ), the quotient is less than 1: $\frac{a}{b} < 1$
If the dividend is equal to the divisor ( $a = b$ ), the quotient is exactly 1: $\frac{a}{b} = 1$

Overview

Lesson overview

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

Overview

Section 1

Writing Quotients as Mixed Numbers

Property

Examples

Section 2

Quotients Relative to 1

Property

By comparing the dividend ( $a$ ) and the divisor ( $b$ ), we can predict if the quotient will be greater than, less than, or equal to 1.

If the dividend is larger than the divisor ( $a > b$ ), the quotient is greater than 1: $\frac{a}{b} > 1$
If the dividend is smaller than the divisor ( $a < b$ ), the quotient is less than 1: $\frac{a}{b} < 1$
If the dividend is equal to the divisor ( $a = b$ ), the quotient is exactly 1: $\frac{a}{b} = 1$