Types of Fractions, Mixed Numbers and Equivalent Fractions for Primary 4

Fractions

Performance Objectives

By the end of the lesson, students should be able to:

1. differentiate between proper and improper fractions;
2. change improper fractions into mixed numbers and vice versa;
3. convert fractions into decimal and vice versa;
4. form equivalent fractions;
5. solve Quantitative Reasoning involving fractions

What Are Fractions?

A fraction represents a part of a whole. When a whole object or group is divided into equal parts, a fraction shows how many of those parts are being considered.

Fractions have two main parts: a numerator and a denominator. The numerator is the top number, and it tells how many parts are taken or being counted. The denominator is the bottom number, and it tells the total number of equal parts the whole is divided into.

For example, if a cake is cut into 8 equal slices, and 3 slices are eaten, the fraction of the cake eaten is $\frac{3}{8}$. Here, 3 is the numerator and 8 is the denominator.

Proper and Improper Fractions

Fractions can be grouped into different types based on the relationship between their numerator and denominator.

Proper Fractions

A proper fraction is a fraction where the numerator is smaller than the denominator. This means that a proper fraction represents a value less than one whole.

Examples of proper fractions include:

1. $\frac{1}{2}$
2. $\frac{3}{4}$
3. $\frac{5}{8}$
4. $\frac{7}{10}$

In real-life situations, proper fractions are used to describe parts of a single item. For instance, if a pizza is cut into 6 slices and 2 slices are left, the fraction of pizza left is $\frac{2}{6}$. This is a proper fraction because 2 is smaller than 6.

Improper Fractions

An improper fraction is a fraction where the numerator is equal to or larger than the denominator. This means that an improper fraction represents a value equal to or greater than one whole.

Examples of improper fractions include:

1. $\frac{5}{4}$
2. $\frac{7}{3}$
3. $\frac{10}{10}$
4. $\frac{12}{5}$

In real-life, improper fractions can describe situations where there is more than one whole. For example, if a class needs 5 halves of an apple for a snack, this can be written as $\frac{5}{2}$. This is an improper fraction because 5 is larger than 2.

Mixed Numbers

A mixed number is a number that combines a whole number and a proper fraction. Mixed numbers are another way to represent values greater than one whole.

Examples of mixed numbers include:

1. $1\frac{1}{2}$
2. $2\frac{3}{4}$
3. $3\frac{1}{5}$

A mixed number like $1\frac{1}{2}$ means one whole and one-half. This is the same as $\frac{3}{2}$ as an improper fraction.

Changing Improper Fractions to Mixed Numbers

To change an improper fraction into a mixed number, divide the numerator by the denominator. The quotient (the whole number result of the division) becomes the whole number part of the mixed number. The remainder becomes the new numerator, and the denominator stays the same.

The procedure involves the following steps:

1. Divide the numerator by the denominator.
2. The whole number result is the whole number part of the mixed number.
3. The remainder is the new numerator.
4. The denominator stays the same.

Example 1: Change $\frac{7}{3}$ to a mixed number.

Divide 7 by 3:

$$
\begin{array}{r}
2 \\
3 \overline{)7} \\
-6 \\
\hline
1
\end{array}
$$

The whole number is 2. The remainder is 1. The denominator is 3.

So, $\frac{7}{3} = 2\frac{1}{3}$.

Example 2: Change $\frac{11}{4}$ to a mixed number.

Divide 11 by 4:

$$
\begin{array}{r}
2 \\
4 \overline{)11} \\
-8 \\
\hline
3
\end{array}
$$

The whole number is 2. The remainder is 3. The denominator is 4.

So, $\frac{11}{4} = 2\frac{3}{4}$.

Changing Mixed Numbers to Improper Fractions

To change a mixed number into an improper fraction, multiply the whole number by the denominator, then add the numerator. This result becomes the new numerator, and the denominator stays the same.

The procedure involves the following steps:

1. Multiply the whole number by the denominator.
2. Add the numerator to the result from step 1. This is the new numerator.
3. Keep the original denominator.

Example 1: Change $2\frac{1}{3}$ to an improper fraction.

Multiply the whole number (2) by the denominator (3): $2 \times 3 = 6$.

Add the numerator (1) to the result: $6 + 1 = 7$.

The new numerator is 7. The denominator is 3.

So, $2\frac{1}{3} = \frac{7}{3}$.

Example 2: Change $3\frac{2}{5}$ to an improper fraction.

Multiply the whole number (3) by the denominator (5): $3 \times 5 = 15$.

Add the numerator (2) to the result: $15 + 2 = 17$.

The new numerator is 17. The denominator is 5.

So, $3\frac{2}{5} = \frac{17}{5}$.

Decimal Fractions

Decimal fractions, often called decimals, are another way to write fractions where the denominator is a power of 10 (like 10, 100, 1000, etc.). They use a decimal point to separate the whole number part from the fractional part.

For example, $\frac{1}{10}$ can be written as $0.1$, and $\frac{25}{100}$ can be written as $0.25$.

Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator.

The procedure involves the following steps:

1. Divide the numerator by the denominator.
2. Place a decimal point in the quotient and add zeros to the numerator as needed.

Example 1: Convert $\frac{1}{2}$ to a decimal.

Divide 1 by 2:

$$
\begin{array}{r}
0.5 \\
2 \overline{)1.0} \\
-1.0 \\
\hline
0
\end{array}
$$

So, $\frac{1}{2} = 0.5$.

Example 2: Convert $\frac{3}{4}$ to a decimal.

Divide 3 by 4:

$$
\begin{array}{r}
0.75 \\
4 \overline{)3.00} \\
-2.8 \downarrow \\
\hline
20 \\
-20 \\
\hline
0
\end{array}
$$

So, $\frac{3}{4} = 0.75$.

Example 3: Convert $\frac{7}{10}$ to a decimal.

Divide 7 by 10:

$$
\begin{array}{r}
0.7 \\
10 \overline{)7.0} \\
-7.0 \\
\hline
0
\end{array}
$$

So, $\frac{7}{10} = 0.7$.

Converting Decimals to Fractions

To convert a decimal to a fraction, write the decimal as a fraction with a denominator that is a power of 10. The number of decimal places tells you the power of 10 to use.

The procedure involves the following steps:

1. Write the decimal number without the decimal point as the numerator.
2. For the denominator, write 1 followed by as many zeros as there are digits after the decimal point.
3. Simplify the fraction if possible.

Example 1: Convert $0.5$ to a fraction.

The digit 5 is in the tenths place. So, the denominator is 10.

$0.5 = \frac{5}{10}$.

Simplify the fraction by dividing both numerator and denominator by 5:

$\frac{5 \div 5}{10 \div 5} = \frac{1}{2}$.

So, $0.5 = \frac{1}{2}$.

Example 2: Convert $0.25$ to a fraction.

The digits 25 are in the hundredths place. So, the denominator is 100.

$0.25 = \frac{25}{100}$.

Simplify the fraction by dividing both numerator and denominator by 25:

$\frac{25 \div 25}{100 \div 25} = \frac{1}{4}$.

So, $0.25 = \frac{1}{4}$.

Equivalent Fractions

Equivalent fractions are fractions that look different but represent the same value or the same part of a whole. They are equal in value.

For example, $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{3}{6}$ are equivalent fractions because they all represent one-half of a whole.

Forming Equivalent Fractions

To form equivalent fractions, multiply or divide both the numerator and the denominator by the same non-zero number. This does not change the value of the fraction.

The procedure involves the following steps:

1. Choose a non-zero whole number.
2. Multiply the numerator by that number.
3. Multiply the denominator by the same number.

Example 1: Form an equivalent fraction for $\frac{1}{2}$.

Multiply both the numerator and denominator by 2:

$\frac{1 \times 2}{2 \times 2} = \frac{2}{4}$.

So, $\frac{1}{2}$ is equivalent to $\frac{2}{4}$.

Example 2: Form an equivalent fraction for $\frac{3}{4}$.

Multiply both the numerator and denominator by 3:

$\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$.

So, $\frac{3}{4}$ is equivalent to $\frac{9}{12}$.

Example 3: Form an equivalent fraction for $\frac{6}{10}$.

Divide both the numerator and denominator by 2:

$\frac{6 \div 2}{10 \div 2} = \frac{3}{5}$.

So, $\frac{6}{10}$ is equivalent to $\frac{3}{5}$.

Quantitative Reasoning Involving Fractions

Quantitative reasoning involves solving problems using numbers and mathematical concepts. When solving quantitative reasoning problems with fractions, it is important to understand what the fractions represent and how they relate to each other.

Example 1: If a basket contains 20 fruits, and $\frac{1}{4}$ of them are oranges, how many oranges are there?

To find the number of oranges, multiply the total number of fruits by the fraction:

Number of oranges = $\frac{1}{4} \times 20$

Number of oranges = $\frac{20}{4}$

Number of oranges = $5$ oranges.

There are 5 oranges in the basket.

Example 2: A tailor used $\frac{1}{3}$ of a roll of fabric on Monday and $\frac{1}{3}$ on Tuesday. What fraction of the fabric roll was used in total?

To find the total fraction used, add the fractions:

Total fraction used = $\frac{1}{3} + \frac{1}{3}$

Total fraction used = $\frac{1+1}{3}$

Total fraction used = $\frac{2}{3}$.

The tailor used $\frac{2}{3}$ of the fabric roll in total.

Example 3: A cake was cut into 8 equal slices. If 5 slices were eaten, what fraction of the cake was left?

Total slices = 8

Slices eaten = 5

Slices left = Total slices – Slices eaten = $8 – 5 = 3$ slices.

Fraction of cake left = $\frac{\text{Slices left}}{\text{Total slices}} = \frac{3}{8}$.

So, $\frac{3}{8}$ of the cake was left.