6.3: Multiplication and Division

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Fraction multiplication and division are among the most conceptually challenging areas of elementary mathematics—and yet they are often taught through memorized rules that students don’t truly understand (Feike et al., 2018). Research and practice show that conceptual understanding must come first. Children need to grasp what they’re doing and why it makes sense before they are expected to perform symbolic procedures fluently. This means centering instruction around models, representations, and real-world contexts. Area models, number lines, diagrams, and hands-on tools help children visualize fraction operations and construct their own meaning (NCTM, 2020).

Ultimately, procedural fluency is important—but only when built on a strong foundation of understanding. When students can see and explain what it means to multiply or divide fractions, they become confident and competent mathematicians, not just rule-followers.

Wonder, Play, Grow

Your group will be assigned one of the problems below. Before collaborating, take a few minutes to work independently on your problem. In order to solve the problem, think of producing a solution by drawing a picture or using a number line. Try NOT to use procedures at this point. You may want to use the virtual fraction bars to help you see the solutions. https://polypad.amplify.com/p#fraction-bars

Task 1 - Alisa had a ½ liter of juice in a bottle. She drank ¾ of the juice that was in the bottle. How many liters of juice did she drink?

Task 2 - The distance between Rosa’s house and her school is ¾ mile. She ran ⅓ of the way to school. How many miles did she run?

The tasks can be found at Illustrative Math. https://tasks.illustrativemathematic...t-standards/NF

Multiplication

Trying to extend the repeated-addition approach of whole-number multiplication to fraction multiplication is an interesting challenge. Consider the following cases (Musser, 2014).

Case 1: A Whole Number times a Fraction

$3 \times \dfrac{1}{4} = \dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{4} = \dfrac{3}{4}$

$8 \times \dfrac{1}{2} = \dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{2} + \dfrac{1}{2} = 4$

The repeated-addition approach works well since the first factor is a whole number.

Case 2: A Fraction Times a Whole Number

$\dfrac{1}{2} \times 8$

We cannot apply the repeated-addition approach, since that would literally say to add 8 one-half times. But if multiplication is to be commutative, then 1/2 x 8 would have to be equal to 8 x 1/2=4, as in case 1. We can then interpret 1/2 x 8 as taking "one-half" of 8, which is 4.

Case 3: A Fraction time a Fraction

Next, it is impossible to use the repeated-addition approach, so we apply the technique of case 2.

$\dfrac{1}{3} \times \dfrac{2}{5}$ means $\dfrac{1}{3}$ of $\dfrac{2}{5}$

First picture 2/5, subdividing a rectangular region vertically into 5 equal parts. Then horizontally subdividing a region into three equal parts (1/3). The intersecting shaded region (small rectangles = 1/21) consists of 2 of the 15 smallest rectangle. Therefore, $\frac{1}{3} \times \frac{2}{5}$ = $\frac{2}{15}$, see Figure 6.3.1.

The first rectangle is divided vertically into 5 equal parts, with 2 parts shaded blue, representing 2 / 5 . The second rectangle overlays a 3-row horizontal grid onto the same vertical fifths. One horizontal row (or 1/ 3 ) is shaded, intersecting with the 2 /5 vertical section. The third image highlights the overlapping area (2 out of 15 total smaller rectangles) in green, showing that 1 /3 × 2/ 5 = 2 /15 . — Figure $\PageIndex{1}$: Fraction Times a Fraction

Example $\PageIndex{1}$

Use a diagram (case 3) to model $\dfrac{1}{2} \cdot \dfrac{3}{4}$.

Answer

First shade in $\dfrac{3}{4}$ of the rectangle.

$A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded.$

We will take $\dfrac{1}{2}$ of this $\dfrac{3}{4}$, so we heavily shade $\dfrac{1}{2}$ of the shaded region.

$A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded. The rectangle is divided by a horizontal line, creating eight equal pieces. Three of the eight pieces are darkly shaded.$

Notice that 3 out of the 8 pieces are heavily shaded. This means that $\dfrac{3}{8}$ of the rectangle is heavily shaded. Therefore, $\dfrac{1}{2}$ of $\dfrac{3}{4}$ is $\dfrac{3}{4}$, or $\dfrac{1}{2} \cdot \dfrac{3}{4} = \dfrac{3}{8}$.

Before you view the video, see if you can model $\dfrac{1}{4} \cdot \dfrac{3}{5}$ and $\dfrac{2}{3} \cdot \dfrac{4}{7}$.

This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.

Definition: Fraction Multiplication

If $a, b, c,$ and $d$ are numbers where $b ≠ 0$ and $d ≠ 0$, then

\[\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}\]

Example $\PageIndex{2}$

Multiply, and write the answer in simplified form: $\dfrac{3}{4} \cdot \dfrac{1}{5}$.

Answer

	$\dfrac{3}{4} \cdot \dfrac{1}{5}$
Multiply the numerators, and multiply the denominators.	$\dfrac{3 \cdot 1}{4 \cdot 5}$
Simplify.	$\dfrac{3}{20}$

There are no common factors, so the fraction is simplified.

Example $\PageIndex{3}$

Multiply $2 \cdot \frac{3}{7}$.

Answer: to multiply $2 \cdot \frac{3}{7}$, we think of “2” as $\frac{2}{1}$, and compute this way \[2 \cdot \frac{3}{7} = \frac{2}{1} \cdot \frac{3}{7} = \frac{2 \cdot 3}{1 \cdot 7} = \frac{6}{7} \ldotp \nonumber \]

Wonder, Play, Grow

Work together to find each product and then check your work with the Desmos Scientific Calculator.

a) $\dfrac{3}{4} \cdot \dfrac{28}{15}$

b) $\dfrac{2}{3} \cdot \dfrac{5}{13}$

c) $2\dfrac{1}{3} \cdot \dfrac{2}{5}$

d) $\dfrac{18}{13} \cdot \dfrac{39}{72}$

e) $\dfrac{50}{15} \cdot \dfrac{39}{55}$

f) $3\dfrac{4}{7} \cdot 2\dfrac{1}{2}$

GIVE ONE TAKE ONE: Make up a multiplication problem, find the product, then share your problem with a peer. They will then share the problem they created. You can then check each other's work.

Properties of Fraction Multiplication

Let $\dfrac{a}{b}$ be any fraction.

Multiplicative Identity Property for Fraction Multiplication: $\dfrac{a}{b} \cdot 1$ = $\dfrac{a}{b}$ = $1 \cdot \dfrac{a}{b}$

Multiplicative Inverse Property for Fraction Multiplication: For every nonzero fraction $\dfrac{a}{b}$, there is a unique fraction $\dfrac{b}{a}$ such that $\dfrac{a}{b} \cdot \dfrac{b}{a}$= 1.

IMPORTANT NOTE: $\dfrac{b}{a}$ is called the multiplicative inverse or reciprocal of $\dfrac{a}{b}$.

Understanding reciprocals is essential because they are the key to dividing with fractions. Dividing by a fraction is conceptually and procedurally more complex than multiplying, but the reciprocal simplifies the process: dividing by a fraction is the same as multiplying by its reciprocal. This “invert and multiply” rule transforms a tricky division into a familiar multiplication (Finan, 2006). When students get to algebra, the reciprocal is also critical in solving equations, where it allows us to isolate variables by “undoing” a fraction multiplication (Musser et al., 2014). Simply put, a reciprocal reverses the numerator and denominator, and when multiplied by the original fraction, the result is always 1—making it a powerful tool for both conceptual understanding and procedural fluency.

Example $\PageIndex{4}$

Find the reciprocal of each number. Then check that the product of each number and its reciprocal is $1$.

$\dfrac{4}{9}$
$7$

Answer

The reciprocal of $\dfrac{4}{9}$ is $\dfrac{9}{4}$.

$\dfrac{4}{9} \cdot \dfrac{9}{4}$

$\dfrac{36}{36} $

$1 \; \checkmark $

$\dfrac{7}{1}$

$\dfrac{1}{7} $

$7 \cdot \left(\dfrac{1}{7}\right) = 1 \; \checkmark $

Division

Division of fractions is a difficult concept for many children (and adults). This difficulty is partly due to the lack of simple concrete models (Musser et al. , 2014). While the foundation for multiplication and division begins as early as second grade by developing an understanding of place value and a part/whole conception of numbers the formal study of division of fractions typically occurs later (Feike et al., 2018). At this stage, a strong emphasis on conceptual understanding using models, real-world contexts, and the connection to the meaning of division is crucial, rather than solely relying on rote procedures.

Why is $12 ÷ 3 = 4$? We previously modeled this with counters. How many groups of $3$ counters can be made from a group of $12$ counters?

$Four red ovals are shown. Inside each oval are three grey circles.$

Figure $\PageIndex{2}$

There are $4$ groups of $3$ counters. In other words, there are four $3$s in $12$. So, $12 ÷ 3 = 4$.

What about dividing fractions? Suppose we want to find the quotient: $\dfrac{1}{2} \div \dfrac{1}{6}$. We need to figure out how many $\dfrac{1}{6}$s there are in $\dfrac{1}{2}$. We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in Figure $\PageIndex{3}$. Notice, there are three $\dfrac{1}{6}$ tiles in $\dfrac{1}{2}$, so $\dfrac{1}{2} \div \dfrac{1}{6} = 3$.

$A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.$

Figure $\PageIndex{3}$

Example $\PageIndex{5}$

Model: $\dfrac{1}{4} \div \dfrac{1}{8}$.

Answer: We want to determine how many $\dfrac{1}{8}$s are in $\dfrac{1}{4}$. Start with one $\dfrac{1}{4}$ tile. Line up $\dfrac{1}{8}$ tiles underneath the $\dfrac{1}{4}$ tile.

Example $\PageIndex{6}$

Model: $2 ÷ \dfrac{1}{4}$.

Answer

We are trying to determine how many $\dfrac{1}{4}$s there are in $2$. We can model this as shown.

$Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.$

Because there are eight $\dfrac{1}{4}$s in $2$, $2 ÷ \dfrac{1}{4} = 8$.

Using fraction tiles, we showed that $\dfrac{1}{2} \div \dfrac{1}{6} = 3$. Notice that $\dfrac{1}{2} \cdot \dfrac{6}{1} = 3$ also. How are $\dfrac{1}{6}$ and $\dfrac{6}{1}$ related? They are reciprocals. This leads us to the procedure for fraction division.

Definition: Fraction Division

If $a, b, c,$ and $d$ are numbers where $b ≠ 0$, $c ≠ 0$, and $d ≠ 0$, then

\[\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c} \]

To divide fractions, multiply the first fraction by the reciprocal of the second.

We need to say $b ≠ 0$, $c ≠ 0$ and $d ≠ 0$ to be sure we don’t divide by zero.

Example $\PageIndex{7}$

Divide, and write the answer in simplified form: $\dfrac{7}{18} \div \dfrac{14}{27}$.

Answer

Add texts here. Do not delete this text first.

Multiply the first fraction by the reciprocal of the second. $\dfrac{7}{18} \cdot \dfrac{27}{14} $
Multiply. $\dfrac{7 \cdot 27}{18 \cdot 14} $
Rewrite showing common factors. $\dfrac{\cancel{\textcolor{red}{7}} \cdot \cancel{\textcolor{red}{9}} \cdot 3}{\cancel{\textcolor{red}{9}} \cdot \cancel{\textcolor{red}{7}} \cdot 2}$
Remove common factors. $\dfrac{3}{2 \cdot 2} $
Simplify. $\dfrac{3}{4} $

Now that we just learned the "invert and multiply" rule. Here are some other ways you may not have seen to divide fractions.

Common Denominator Method

Think About and Discuss

Solve each of the following fraction division problems until you see a pattern.

$\frac{6}{4} \div \frac{3}{4} \qquad \frac{6}{10} \div \frac{3}{10} \qquad \frac{8}{9} \div \frac{4}{9} \qquad \frac{6}{33} \div \frac{2}{33} \qquad \frac{5}{4} \div \frac{2}{4} \qquad \frac{5}{2} \div \frac{2}{2} \qquad \frac{5}{10} \div \frac{2}{10}$

What do you notice?

This leads to our first alternative fraction division method:

Common denominator method

If two fractions have the same denominator, then when you divide them, you can just divide the numerators. In symbols,

\[\frac{a}{d} \div \frac{b}{d} = \frac{a}{b} \ldotp \nonumber \]

Missing Factor Method

We know that we can always turn a division problem into a “missing factor” multiplication problem. Can that help us compute fraction division? Sometimes!

Example $\PageIndex{8}$

For each division problem, rewrite it as a missing factor multiplication question. Then find the quotient using what you know about multiplying fractions.

$\dfrac{9}{10} \div \dfrac{3}{5}$
$\dfrac{25}{12} \div \dfrac{5}{2}$

Answer

a) $\dfrac{3}{5} \cdot \dfrac{?}{?}$= $\dfrac{9}{10}$

$\dfrac{?}{?}$=$\dfrac{3}{2}$, so $\dfrac{9}{10} \div \dfrac{3}{5}$ = $\dfrac{3}{2}$

b) $\dfrac{5}{2} \cdot \dfrac{?}{?}$= $\dfrac{25}{12}$

$\dfrac{?}{?}$=$\dfrac{5}{6}$, so $\dfrac{25}{12} \div \dfrac{5}{2}$ = $\dfrac{5}{6}$

Unfortunately, the missing factor method doesn’t always work out so nicely. For example,

\[\frac{3}{4} \div \frac{1}{3} = \_\_ \nonumber \]

can be rewritten as

\[\frac{1}{3} \cdot \_\_ = \frac{3}{4} \ldotp \nonumber \]

There isn’t a nice ratio of whole numbers that obviously fills in the blank, but we’ll come back to this idea and resolve it soon.

Dividing Across

Let's let Dr. Howie Hua explain this process before we try it for ourselves.

Example $\PageIndex{9}$

Divide.

$\dfrac{8}{15} \div \dfrac{2}{3}$
$\dfrac{4}{7} \div \dfrac{3}{2}$

Answer

$\dfrac{8}{15} \div \dfrac{2}{3}$ = $\dfrac{8÷2}{15÷3}$ = $\dfrac{4}{5}$ - WOW Right?
$\dfrac{4}{7} \div \dfrac{3}{2}$ = $\dfrac{8}{14} \div \dfrac{21}{14}$ = $\dfrac{8÷21}{14÷14}$ = $\dfrac{8/21}{1}$ = $\dfrac{8}{21}$

Wonder, Play, Grow

Find the following quotients using the most convenient division method. Then share with your peers how you completed the problems.

a) $\dfrac{17}{4} \div \dfrac{4}{11}$

b) $\dfrac{3}{4} \div \dfrac{5}{7}$

c) $\dfrac{6}{25} \div \dfrac{2}{5}$

d) $\dfrac{5}{19} \div \dfrac{13}{11}$

Fraction Division Retrieval Practice

Complex Fractions

Consider the following situation. You have 3/4 of a liter of juice left. Each small cup holds 1/3 of a liter. Question: How many cups can you fill?

To solve this, you would compute $\dfrac{\frac{3}{4}}{\frac{1}{3}}$. This is an example of a complex fraction. Although fractions are made of whole numbers in elementary school, numbers other than whole numbers are used in numerators and denominators. Complex fractions are used to divide fractions.

\[
\dfrac{\dfrac{3}{4} \cdot \dfrac{3}{1}}{\dfrac{1}{3} \cdot \dfrac{3}{1}}
= \dfrac{\dfrac{3}{4} \cdot \dfrac{3}{1}}{1}
= \dfrac{9}{4}
\nonumber\]

Notice that the multiplicative inverse of the denominator 1/3 was used to form a complex fraction form of one.

Example $\PageIndex{10}$

Find $\dfrac{1}{2} \div \dfrac{3}{5}$

Answer: \[
\dfrac{\dfrac{1}{2} \cdot \dfrac{5}{3}}{\dfrac{3}{5} \cdot \dfrac{5}{3}}
= \dfrac{\dfrac{1}{2} \cdot \dfrac{5}{3}}{1}
= \dfrac{5}{6}
\nonumber\]

	\(\dfrac{3}{4} \cdot \dfrac{1}{5}\)
Multiply the numerators, and multiply the denominators.	\(\dfrac{3 \cdot 1}{4 \cdot 5}\)
Simplify.	\(\dfrac{3}{20}\)