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Fractions as division

Interpret fractions as division

Here you will learn about interpreting fractions as division, including understanding a fraction as a division equation, understanding a division equation as a fraction, and solving word problems involving understanding a fraction as division.

Students will first learn about interpreting fractions as division as part of number and operations–fractions in $5$ th grade.

What is interpreting fractions as division?

Interpreting fractions as division is when you understand that a fraction represents a division operation between its numerator and denominator. In other words, when you have a fraction $\cfrac{a}{b}$ , you can interpret it as $"a$ divided by $b"$ or $a \div b.$

To do this, you can divide the numerator by the denominator to find the quotient. The numerator of the fraction would become the dividend and the denominator would become the divisor in a division equation; you would write the numerator, then a division sign, then the denominator.

For example,

$\cfrac{3}{4}$ can be interpreted as $"3$ divided by $4,"$ and written as $3 \div 4.$

$\cfrac{6}{2}$ can be interpreted as $"6$ divided by $2,"$ and written as $6 \div 2,$ which equals $3.$

Alternatively, you can also write a division equation as a fraction.

To do this, you would write the dividend of the equation as the numerator of a fraction and the divisor of the equation as the denominator of the fraction.

For example,

3 \div 4 = \cfrac{3}{4}

6 \div 2 = \cfrac{6}{2} = 3

Interpreting fractions as division can help us understand word problems involving the division of whole numbers leading to answers in the form of fractions.

For example,

$4$ friends are sharing $3$ sandwiches equally. What fraction of a sandwich will each friend get?

The $3$ pizzas are each divided by $4.$

The fourths are equally distributed among the $3$ friends.

So each friend gets $3$ fourths, or $\cfrac{3}{4}.$

What is interpreting fractions as division?

$What is interpreting fractions as division?$

Common Core State Standards

How does this relate to $5$ th grade math?

Number and Operations—Fractions (5.NF.B.3)
Interpret a fraction as division of the numerator by the denominator $(\cfrac{a}{b} = a \div b).$ Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, example, by using visual fraction models or equations to represent the problem.

For example, interpret $\cfrac{3}{4}$ as the result of dividing $3$ by $4,$ noting that $\cfrac{3}{4}$ multiplied by $4$ equals $3,$ and that when $3$ wholes are shared equally among $4$ people, each person has a share of size $\cfrac{3}{4}.$

If $9$ people want to share a $50$ -pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

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How to interpret fractions as division

In order to interpret fractions as division:

Identify the numerator and denominator of the fraction.
Write a division equation using the numerator as the dividend and the denominator as the divisor.
Perform the division. (Note that the quotient may be the fraction you started with.)

Interpret fractions as division examples

Example 1: fraction as division equation

Write $\cfrac{2}{3}$ as a division equation. Then solve.

Identify the numerator and denominator of the fraction.

In the fraction $\cfrac{2}{3}, 2$ is the numerator and $3$ is the denominator.

2Write a division equation using the numerator as the dividend and the denominator as the divisor.

2 \div 3=

3Perform the division. (Note that the quotient may be the fraction you started with.)

2 \div 3=\cfrac{2}{3}

Example 2: improper fraction as division

Write $\cfrac{8}{4}$ as a division equation. Then solve.

Identify the numerator and denominator of the fraction.

In the fraction $\cfrac{8}{4}, 8$ is the numerator and $4$ is the denominator.

Write a division equation using the numerator as the dividend and the denominator as the divisor.

8 \div 4=

Perform the division. (Note that the quotient may be the fraction you started with.)

8 \div 4=2

Example 3: mixed number as division

Write $3 \cfrac{1}{2}$ as a division equation. Then solve.

Identify the numerator and denominator of the fraction.

Since this is a mixed number, we first need to convert it to an improper fraction.

$3 \cfrac{1}{2}=\cfrac{7}{2}$

In the fraction $\cfrac{7}{2}, 7$ is the numerator and $2$ is the denominator.

Write a division equation using the numerator as the dividend and the denominator as the divisor.

7 \div 2=

Perform the division. (Note that the quotient may be the fraction you started with.)

7 \div 2=3 \cfrac{1}{2}

How to write a division equation as a fraction

In order to write a division equation as a fraction:

Identify the dividend and divisor.
Write a fraction using the dividend as the numerator and the divisor as the denominator.

Example 4: division equation as a fraction

Write $2 \div 5$ as a fraction.

Identify the dividend and divisor.

In the equation $2 \div 5, 2$ is the dividend and $5$ is the divisor.

Write a fraction using the dividend as the numerator and the divisor as the denominator.

2 \div 5=\cfrac{2}{5}

Example 5: division equation as a fraction

Write $9 \div 3$ as a fraction.

Identify the dividend and divisor.

In the equation $9 \div 3, 9$ is the dividend and $3$ is the divisor.

Write a fraction using the dividend as the numerator and the divisor as the denominator.

9 \div 3=\cfrac{9}{3}

This fraction can be simplified to $\cfrac{3}{1}$ or $3.$

Example 6: word problem

$6$ friends are sharing $5$ candy bars. What fraction of a candy bar will each friend get?

Identify the dividend and divisor.

First, you need to write an equation to represent the word problem.

$5$ candy bars are equally divided among $6$ people. So your equation will be $5 \div 6$ ( $5$ candy bars $\div \, 6$ people)

In the equation $5 \div 6, 5$ is the dividend and $6$ is the divisor.

Write a fraction using the dividend as the numerator and the divisor as the denominator.

5 \div 6=\cfrac{5}{6}

This means that each person will get $\cfrac{5}{6}$ of a candy bar.

Teaching tips for interpreting fractions as division

Add fraction word problems to your lesson plans and worksheets that involve real-life scenarios where division is evident. For instance, divide a pizza or a cake into slices to represent fractions, showing how each slice is a part of the whole. Also provide division problems where the answer is a fraction or decimal.

If needed, show students step-by-step how to use long division to show how the division equation and fraction are equivalent. Finding relevant math videos can allow students to watch the process as many times as needed.

Involve the concept of equivalent fractions and show how multiplying or dividing the numerator and denominator by the same number results in an equivalent fraction. For example, $\cfrac{1}{2}$ is equivalent to $\cfrac{2}{4},$ and both represent $1 \div 2$ and $2 \div 4,$ etc.

Mixed fractions can be included as well; students can convert to an improper fraction, then a division equation.

Easy mistakes to make

Misunderstanding the relationship
Students may confuse the relationship between the numerator and denominator, mistakenly thinking that the numerator represents the number of parts rather than the dividend.

Misapplying the concept
Students may incorrectly apply the concept of fractions as division in situations where they need to multiply fractions or subtract fractions, etc. instead, leading to computational errors.

Not simplifying the fraction
Students may forget about simplifying fractions before interpreting them as division, leading to more complex division problems and potentially incorrect solutions.

Practice interpret fractions as division questions

4 \div 1

2 \div 4

1 \div 4

3 \div 4

To change a fraction into a division equation, the numerator becomes the dividend and the denominator becomes the divisor.

That means $\cfrac{1}{4}=1 \div 4.$

5 \div 9

4 \div 9

1 \cfrac{4}{5}

9 \div 5

To change a fraction into a division equation, the numerator becomes the dividend and the denominator becomes the divisor.

That means $\cfrac{9}{5}=9 \div 5.$

\cfrac{2}{3}

\cfrac{3}{2}

\cfrac{1}{3}

\cfrac{1}{2}

To change a division equation into a fraction, the dividend becomes the numerator and the divisor becomes the denominator.

That means $2 \div 3=\cfrac{2}{3}.$

\cfrac{7}{8}

\cfrac{8}{7}

1 \cfrac{7}{8}

1 \cfrac{8}{7}

To change a division equation into a fraction, the dividend becomes the numerator and the divisor becomes the denominator.

That means $8 \div 7=\cfrac{8}{7}$ or $1 \cfrac{1}{7}.$

\cfrac{4}{5}

\cfrac{5}{4}

4 \div 5

5 \div 4

Mary is dividing $4$ cakes by the $5$ platters, or $4 \div 5.$

Since the problem asks what fraction of each cake will go on each platter, we need to use the division equation to write the fraction.

$4 \div 5= \cfrac{4}{5},$ so $\cfrac{4}{5}$ of each cake will go on each platter.

$7 \div 8$ cups

$8 \div 7$ cups

$\frac{8}{7}$ or $1 \cfrac{1}{7}$ cups

$\cfrac{7}{8}$ cups

You are dividing the $7$ cups of juice by $8$ people, or $7 \div 8.$

Since the problem asks how many cups of juice each person will get, the answer will not be an equation, but we need to use the division equation to write the fraction.

$7 \div 8 = \cfrac{7}{8},$ so each person will get $\cfrac{7}{8}$ cups of juice.

Interpret fractions as division FAQs

How do you interpret fractions as division?

Fractions can be interpreted as division by dividing the numerator by the denominator. For example, $\cfrac{3}{4}$ means $3 \div 4$ or dividing $3$ into $4$ equal parts or equal groups. Similarly, $\cfrac{5}{2}$ means $5 \div 2,$ or dividing $5$ into $2$ equal parts.

What is the difference between interpreting fractions as division and dividing fractions?

Interpreting fractions as division is understanding a fraction as representing a division operation. Dividing fractions is performing the operation of division of fractions, where one fraction is divided by another.

How does interpreting fractions as division help us understand fractions?

Interpreting fractions as division helps us see fractions as pieces of a whole. By thinking of fractions as dividing a number into equal parts, we can better understand their meaning and use them more effectively in math.

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