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

Lesson 6: Area with Fractional Side Lengths (< 1)

In this Grade 5 lesson from Illustrative Mathematics Chapter 2, students calculate the area of rectangles with fractional side lengths less than 1 by applying fraction multiplication. Students connect the area formula (length × width) to multiplying unit fractions and other fractions less than 1, building on their understanding of fractions as quotients. This lesson deepens conceptual understanding of how multiplying two fractions less than 1 produces a product smaller than either factor.

Section 1

Finding a Unit Fraction of a Set

Property

Finding a unit fraction of a set is equivalent to dividing the total number in the set ( $N$ ) by the denominator ( $b$ ). This calculation finds the size of one equal part.

\frac{1}{b} \text{ of } N = N \div b

Examples

Section 2

Multiplying a Whole Number by a Unit Fraction

Property

Multiplying a whole number, $n$ , by a unit fraction, $\frac{1}{d}$ , is equivalent to repeated addition.
This relationship is generalized by the rule:

n \times \frac{1}{d} = \underbrace{\frac{1}{d} + \frac{1}{d} + \dots + \frac{1}{d}}_{n \text{ times}} = \frac{n}{d}

Examples

Section 3

Procedure: Multiplying a Whole Number by a Fraction

Property

To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.

\frac{a}{b} \times c = \frac{a \times c}{b}

Examples

Section 4

Equivalence of '1/b of c' and 'c × 1/b'

Property

Finding a unit fraction of a whole number is equivalent to multiplying the whole number by the unit fraction. The word "of" in this context implies multiplication.

\frac{1}{b} \text{ of } c = c \div b = c \times \frac{1}{b}

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Finding a Unit Fraction of a Set

Property

Finding a unit fraction of a set is equivalent to dividing the total number in the set ( $N$ ) by the denominator ( $b$ ). This calculation finds the size of one equal part.

\frac{1}{b} \text{ of } N = N \div b

Examples

Section 2

Multiplying a Whole Number by a Unit Fraction

Property

Multiplying a whole number, $n$ , by a unit fraction, $\frac{1}{d}$ , is equivalent to repeated addition.
This relationship is generalized by the rule:

n \times \frac{1}{d} = \underbrace{\frac{1}{d} + \frac{1}{d} + \dots + \frac{1}{d}}_{n \text{ times}} = \frac{n}{d}

Examples

Section 3

Procedure: Multiplying a Whole Number by a Fraction

Property

To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.

\frac{a}{b} \times c = \frac{a \times c}{b}

Examples

Section 4

Equivalence of '1/b of c' and 'c × 1/b'

Property

Finding a unit fraction of a whole number is equivalent to multiplying the whole number by the unit fraction. The word "of" in this context implies multiplication.

\frac{1}{b} \text{ of } c = c \div b = c \times \frac{1}{b}

Overview

Lesson overview

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

Overview

Section 1

Finding a Unit Fraction of a Set

Property

Finding a unit fraction of a set is equivalent to dividing the total number in the set ( $N$ ) by the denominator ( $b$ ). This calculation finds the size of one equal part.

\frac{1}{b} \text{ of } N = N \div b

Examples

Section 2

Multiplying a Whole Number by a Unit Fraction

Property

Multiplying a whole number, $n$ , by a unit fraction, $\frac{1}{d}$ , is equivalent to repeated addition.
This relationship is generalized by the rule:

n \times \frac{1}{d} = \underbrace{\frac{1}{d} + \frac{1}{d} + \dots + \frac{1}{d}}_{n \text{ times}} = \frac{n}{d}

Examples

Section 3

Procedure: Multiplying a Whole Number by a Fraction

Property

To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.

\frac{a}{b} \times c = \frac{a \times c}{b}

Examples

Section 4

Equivalence of '1/b of c' and 'c × 1/b'

Property

Finding a unit fraction of a whole number is equivalent to multiplying the whole number by the unit fraction. The word "of" in this context implies multiplication.

\frac{1}{b} \text{ of } c = c \div b = c \times \frac{1}{b}