# Divide a Fraction by a Fraction

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Students explore methods of dividing a fraction by a unit fraction.

# Key Concepts

In this lesson and in Lesson 5, students explore dividing a fraction by a fraction.

In this lesson, we focus on the case in which the divisor is a unit fraction. Understanding this case makes it easier to see why we can divide by a fraction by multiplying by its reciprocal. For example, finding $\frac{3}{4}÷\frac{1}{5}$ means finding the number of fifths in $\frac{3}{4}$. In this lesson, students will see that this is $\frac{3}{4}$ × 5.

Students learn and apply several methods for dividing a fraction by a unit fraction, such as $\frac{2}{3}÷\frac{1}{4}$.

• Model $\frac{2}{3}$. Change the model and the fractions in the problem to twelfths: $\frac{8}{12}÷\frac{3}{12}$. Then find the number of groups of 3 twelfths in 8 twelfths. This is the same as finding 8 ÷ 3.
• Reason that since there are 4 fourths in 1, there must be $\frac{2}{3}$ × 4 fourths in $\frac{2}{3}$. This is the same as using the multiplicative inverse.
• Rewrite both fractions so they have a common denominator: $\frac{2}{3}÷\frac{1}{4}=\frac{8}{12}÷\frac{3}{12}$. The answer is the quotient of the numerators. This is the numerical analog to modeling.

# Goals and Learning Objectives

• Use models and other methods to divide fractions by unit fractions