# Whole Number Division

Student View (Opens in new window) ## Instructor Overview

Students explore methods for dividing a whole number by a fraction.

# Key Concepts

In earlier grades, students learned to think of a whole number division problem, such as 8 ÷ 4, in terms of two types of equal groups.

Divisor as the Number of Groups

Divide 8 into 4 equal groups and find the size of each group.

Divisor as the Group Size

Divide 8 into groups of 4 and find the number of groups.

To divide a fraction by a whole number in Lesson 2, students used the first interpretation. For example, to find $\frac{8}{9}$ ÷ 4, they divided 8 ninths into 4 equal groups and found that there were 2 ninths in each group.

To divide a whole number by a fraction, the second interpretation is most helpful. For example, to find 3 ÷ $\frac{3}{4}$, we find the number of groups of 3 fourths in 3 wholes. The diagram in the Opening shows that there are 4 groups, so 3 ÷ $\frac{3}{4}$ = 4.

Just as with whole number division, the quotient when a whole number is divided by a fraction is not always a whole number. Below is a model for 2 ÷ $\frac{3}{5}$. The model shows that there are 3 groups of 3 fifths in 2 wholes plus $\frac{1}{3}$ of another group ($\frac{1}{3}$ of a group of 3 fifths is 1 fifth). Therefore, 2 ÷ $\frac{3}{5}$ = 3$\frac{1}{3}$.

Notice that once we have divided the 2 wholes into fifths, we are finding the number of groups of 3 fifths in 10 fifths. This is simply 10 ÷ 3.

These models can help explain that the “multiply by the reciprocal” method of dividing a whole number by a fraction works. To find 2 ÷ 35, we can multiply 2 by 5 to find the total number of fifths in 2 and then divide the result (10) by 3 to find the number of groups of 3 of these fifths in 2. So, $2÷\frac{3}{5}=\frac{2×5}{3}=2×\frac{5}{3}$.

ELL: Encourage students to verbalize their explanations. To help students gain confidence and increase their understanding, allow those that share the same language of origin to speak in small groups using their prefered language.

# Goals and Learning Objectives

• Use models and other methods to divide a whole number by a fraction.