Lesson 4.1: Visualize Fractions

New Concept

Fractions represent parts of a whole. This lesson shows how to visualize different types of fractions, convert between them, find equivalent forms, and place them on a number line to compare their values.

What’s next

Next, you’ll master these skills through interactive examples and practice cards that bring fractions to life.

Property

A fraction is written $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. In a fraction, $a$ is called the numerator and $b$ is called the denominator. A fraction is a way to represent parts of a whole. The denominator $b$ represents the number of equal parts the whole has been divided into, and the numerator $a$ represents how many parts are included. The denominator, $b$, cannot equal zero because division by zero is undefined.

Examples

• A chocolate bar is split into 12 equal squares. If you eat 5 squares, you have eaten $\frac{5}{12}$ of the bar.
• A class has 25 students. If 14 are girls, then $\frac{14}{25}$ of the class consists of girls.
• An hour has 60 minutes. If you spend 20 minutes on homework, you have used $\frac{20}{60}$ of the hour.

Explanation

Think of a pizza! The denominator is how many equal slices you cut it into, and the numerator is how many of those slices you get to eat. It's all about representing parts of a complete item or group.

Property

The fraction $\frac{a}{b}$ is a proper fraction if $a < b$ and an improper fraction if $a \geq b$.

Examples

• Proper Fraction: You finished $\frac{3}{4}$ of your homework. This is a proper fraction because $3 < 4$.
• Improper Fraction: A group of friends eats $\frac{9}{8}$ of a pizza. This is an improper fraction because $9 \geq 8$.
• Property of One: If you have $\frac{5}{5}$ of a cake, you have the whole cake, which is 1. Any number (except zero) divided by itself is one.

Explanation

A proper fraction represents less than one whole, like $\frac{1}{2}$ of a cookie. An improper fraction represents one whole or more, like $\frac{3}{2}$ cookies, because the numerator is bigger than or equal to the denominator.

Property

A mixed number consists of a whole number $a$ and a fraction $\frac{b}{c}$ where $c \neq 0$. It is written as follows.

Examples

• A baker uses $3 \frac{1}{2}$ cups of flour for a recipe. This means three full cups and one half cup.
• A movie is $2 \frac{1}{4}$ hours long. This means it lasts for two full hours and an additional quarter of an hour.
• A child is $4 \frac{1}{3}$ feet tall. This represents four full feet plus one-third of a foot.

Explanation

A mixed number is a simpler way to write an improper fraction. It combines a whole number with a proper fraction, making it easier to understand how many wholes and parts you have, like $2 \frac{1}{2}$ pizzas.

Property

To convert an improper fraction to a mixed number:

1. Divide the denominator into the numerator.
2. Identify the quotient, remainder, and divisor.
3. Write the mixed number as $\text{quotient} \frac{\text{remainder}}{\text{divisor}}$.

Examples

• To convert $\frac{13}{5}$: Divide 13 by 5 to get a quotient of 2 and a remainder of 3. The mixed number is $2 \frac{3}{5}$.
• To convert $\frac{9}{2}$: Divide 9 by 2 to get a quotient of 4 and a remainder of 1. The mixed number is $4 \frac{1}{2}$.
• To convert $\frac{22}{7}$: Divide 22 by 7 to get a quotient of 3 and a remainder of 1. The mixed number is $3 \frac{1}{7}$.

Explanation

Think of this as making full groups. If you have $\frac{7}{3}$ (seven-thirds), you can make two full groups of three-thirds, with one-third left over. This gives you the mixed number $2 \frac{1}{3}$.

Property

To convert a mixed number to an improper fraction:

1. Multiply the whole number by the denominator.
2. Add the numerator to the product found in Step 1.
3. Write the final sum over the original denominator.

Examples

• To convert $3 \frac{1}{4}$: First, multiply $3 \times 4 = 12$. Then, add the numerator, $12 + 1 = 13$. The improper fraction is $\frac{13}{4}$.
• To convert $5 \frac{2}{3}$: First, multiply $5 \times 3 = 15$. Then, add the numerator, $15 + 2 = 17$. The improper fraction is $\frac{17}{3}$.
• To convert $7 \frac{3}{5}$: First, multiply $7 \times 5 = 35$. Then, add the numerator, $35 + 3 = 38$. The improper fraction is $\frac{38}{5}$.

Explanation

To change $2 \frac{1}{4}$ into just fourths, you break down the wholes. Two wholes are eight-fourths ($2 \times 4$). Add the extra one-fourth, and you have nine-fourths in total, which is written as $\frac{9}{4}$.

Property

Equivalent fractions are fractions that have the same value. If $a$, $b$, and $c$ are numbers where $b \neq 0$ and $c \neq 0$, then

Examples

• To find a fraction equivalent to $\frac{2}{5}$, multiply the numerator and denominator by 3: $\frac{2 \cdot 3}{5 \cdot 3} = \frac{6}{15}$. So, $\frac{2}{5}$ is equivalent to $\frac{6}{15}$.
• Find a fraction equivalent to $\frac{1}{6}$ that has a denominator of 24. Since $6 \times 4 = 24$, you multiply the numerator by 4: $\frac{1 \cdot 4}{6 \cdot 4} = \frac{4}{24}$.
• The fractions $\frac{1}{3}$, $\frac{2}{6}$, and $\frac{4}{12}$ are all equivalent because they represent the same value, just with the whole divided into different numbers of parts.

Explanation

Equivalent fractions look different but represent the exact same amount, like $\frac{1}{2}$ and $\frac{2}{4}$ of a pizza. You create them by multiplying the numerator and denominator by the same non-zero number, which is like multiplying by 1.