Learn on PengiIllustrative Mathematics, Grade 6Unit 4 Dividing Fractions

Lesson 3: Algorithm for Fraction Division

In this Grade 6 lesson from Illustrative Mathematics Unit 4, students explore the algorithm for dividing fractions by identifying patterns when dividing by unit fractions and non-unit fractions. Using tape diagrams, students discover that dividing by a fraction is equivalent to multiplying by its reciprocal, working through expressions such as 6 ÷ 1/3 and 6 ÷ 2/3 to generalize the rule as a ÷ b/c = a × c/b. The lesson builds conceptual understanding of fraction division before formalizing the procedure.

Section 1

Connecting Division by a Unit Fraction to Multiplication

Property

We noticed in the previous visual models that dividing by a unit fraction (1b\frac{1}{b}) produces the same result as multiplying by the denominator (bb).
To solve a÷1ba \div \frac{1}{b}, you can ask: "aa is 1b\frac{1}{b} of what number?". The shortcut is to multiply the whole number by the denominator:

a÷1b=a×ba \div \frac{1}{b} = a \times b

Examples

Section 2

Procedure: Dividing a Whole Number by a Non-Unit Fraction

Property

To divide a whole number by a non-unit fraction, we can use a two-step process or simply multiply by the reciprocal.
The reciprocal of a fraction bc\frac{b}{c} is cb\frac{c}{b} (flipping the numerator and denominator).

a÷bc=a×cba \div \frac{b}{c} = a \times \frac{c}{b}

Examples

Section 3

Dividing a Fraction by a Whole Number

Property

To divide a fraction by a whole number, you apply the same rule: multiply the fraction by the reciprocal of the whole number.
Since the reciprocal of a whole number cc is 1c\frac{1}{c}, this operation makes the fractional parts smaller.

ab÷c=ab×1c=ab×c\frac{a}{b} \div c = \frac{a}{b} \times \frac{1}{c} = \frac{a}{b \times c}

Examples

Section 4

Algorithm: Dividing Any Fraction by Multiplying by the Reciprocal

Property

The general algorithm for dividing any two numbers (integers or fractions) is to multiply the dividend by the reciprocal of the divisor.
Division is the inverse operation of multiplication.

(a/b)÷(c/d)=ab×dc(a/b) \div (c/d) = \frac{a}{b} \times \frac{d}{c}

Examples

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Connecting Division by a Unit Fraction to Multiplication

Property

We noticed in the previous visual models that dividing by a unit fraction (1b\frac{1}{b}) produces the same result as multiplying by the denominator (bb).
To solve a÷1ba \div \frac{1}{b}, you can ask: "aa is 1b\frac{1}{b} of what number?". The shortcut is to multiply the whole number by the denominator:

a÷1b=a×ba \div \frac{1}{b} = a \times b

Examples

Section 2

Procedure: Dividing a Whole Number by a Non-Unit Fraction

Property

To divide a whole number by a non-unit fraction, we can use a two-step process or simply multiply by the reciprocal.
The reciprocal of a fraction bc\frac{b}{c} is cb\frac{c}{b} (flipping the numerator and denominator).

a÷bc=a×cba \div \frac{b}{c} = a \times \frac{c}{b}

Examples

Section 3

Dividing a Fraction by a Whole Number

Property

To divide a fraction by a whole number, you apply the same rule: multiply the fraction by the reciprocal of the whole number.
Since the reciprocal of a whole number cc is 1c\frac{1}{c}, this operation makes the fractional parts smaller.

ab÷c=ab×1c=ab×c\frac{a}{b} \div c = \frac{a}{b} \times \frac{1}{c} = \frac{a}{b \times c}

Examples

Section 4

Algorithm: Dividing Any Fraction by Multiplying by the Reciprocal

Property

The general algorithm for dividing any two numbers (integers or fractions) is to multiply the dividend by the reciprocal of the divisor.
Division is the inverse operation of multiplication.

(a/b)÷(c/d)=ab×dc(a/b) \div (c/d) = \frac{a}{b} \times \frac{d}{c}

Examples