Learn on PengiEureka Math, Grade 5Chapter 21: Multiplication of a Whole Number by a Fraction

Lesson 3: Relate a fraction of a set to the repeated addition interpretation of fraction multiplication.

In this Grade 5 Eureka Math lesson from Chapter 21, students learn to connect the concept of a fraction of a set to the repeated addition interpretation of fraction multiplication, such as understanding that 2/3 × 6 can mean both "2/3 of 6" and "6 copies of 2/3 added together." Students use tape diagrams and the commutative property to explore how whole number multiplication strategies extend to fraction multiplication. The lesson builds on prior work with multiplying a fraction times a whole number and prepares students to fluently solve problems like 3/8 × 56 using multiple representations.

Section 1

Commutative Property of Multiplication

Property

Due to the commutative property, multiplying a fraction by a whole number can be interpreted in two equivalent ways: finding a fraction of a set, or as repeated addition of the fraction.

ab×c ("a/b of c")=c×ab ("c groups of a/b")\frac{a}{b} \times c \text{ ("a/b of c")} = c \times \frac{a}{b} \text{ ("c groups of a/b")}

Examples

Section 2

Multiply a Fraction by a Whole Number: Multiply then Divide

Property

To multiply a fraction by a whole number, multiply the numerator by the whole number and place the product over the original denominator.

ab×c=a×cb\frac{a}{b} \times c = \frac{a \times c}{b}

Examples

Section 3

Simplify Before Multiplying Fractions and Whole Numbers

Property

When multiplying a fraction by a whole number, ab×c\frac{a}{b} \times c, you can simplify the calculation by dividing the denominator (bb) and the whole number (cc) by a common factor before performing the multiplication.
This is an application of the standard algorithm a×cb\frac{a \times c}{b}.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Commutative Property of Multiplication

Property

Due to the commutative property, multiplying a fraction by a whole number can be interpreted in two equivalent ways: finding a fraction of a set, or as repeated addition of the fraction.

ab×c ("a/b of c")=c×ab ("c groups of a/b")\frac{a}{b} \times c \text{ ("a/b of c")} = c \times \frac{a}{b} \text{ ("c groups of a/b")}

Examples

Section 2

Multiply a Fraction by a Whole Number: Multiply then Divide

Property

To multiply a fraction by a whole number, multiply the numerator by the whole number and place the product over the original denominator.

ab×c=a×cb\frac{a}{b} \times c = \frac{a \times c}{b}

Examples

Section 3

Simplify Before Multiplying Fractions and Whole Numbers

Property

When multiplying a fraction by a whole number, ab×c\frac{a}{b} \times c, you can simplify the calculation by dividing the denominator (bb) and the whole number (cc) by a common factor before performing the multiplication.
This is an application of the standard algorithm a×cb\frac{a \times c}{b}.

Examples