Introduce children to improper fractions and mixed numbers. Teach children how to convert between the two.
For more help, check out our Introduction to Fractions Lessons Page
Credit goes to EasyTeaching
What Are Improper Fractions?
Improper fractions are a type of fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). In other words, the fraction represents a value that is equal to or greater than one whole. Improper fractions are called “improper” because they don’t fit the traditional notion of fractions, where the numerator is usually smaller than the denominator.
Here’s a detailed explanation of improper fractions:
Components of an Improper Fraction
Numerator – The numerator is the top number in the fraction. It represents the number of equal parts you have or are considering.
Denominator – The denominator is the bottom number in the fraction. It represents the total number of equal parts that make up one whole.
Characteristics of Improper Fractions
Numerator Greater Than or Equal to Denominator – In an improper fraction, the numerator is equal to or larger than the denominator. For example, 5/4, 7/7, and 12/9 are all improper fractions.
Value Equal to or Greater Than One Whole – Improper fractions represent values that are equal to or greater than one whole. For example, 5/4 represents 1 whole and 1/4 more, 7/7 represents exactly 1 whole, and 12/9 represents 1 whole and 3/9 more, which can be simplified to 1 whole and 1/3.
Can Be Mixed Numbers – Improper fractions can be converted into mixed numbers, which consist of a whole number part and a proper fraction part. For example, the improper fraction 5/4 can be written as the mixed number 1 1/4.
Examples of Improper Fractions
5/3 – This improper fraction represents a value greater than one whole. It is equivalent to 1 whole and 2/3.
8/8 – While this fraction may seem unusual, it is also an improper fraction. It represents exactly one whole, as 8 equal parts make up a whole, and all 8 parts are present.
11/10 – Another improper fraction, it represents 1 whole and 1/10 more.
When Do Improper Fractions Arise?
Improper fractions commonly arise in various mathematical contexts, such as:
When dividing one whole into more than one part.
When dealing with situations where the numerator is greater than or equal to the denominator.
In advanced mathematics, when working with equations or mathematical operations that yield such fractions.
In real-life scenarios involving measurements and calculations where values can exceed one whole.
Improper fractions are a valid and useful representation of quantities, even though they might initially seem unconventional when compared to the more familiar proper fractions where the numerator is smaller than the denominator.
What Are Mixed Numbers?
Mixed numbers are a combination of a whole number and a proper fraction. They are a way to represent values that are greater than one whole while also indicating the fractional part of that value. Mixed numbers are useful in everyday life and mathematics for representing quantities that involve both whole units and fractions. Here’s a detailed explanation of mixed numbers:
Components of a Mixed Number
Whole Number Part – The whole number part represents the whole units or the complete objects in a quantity. It is typically written before the fraction part.
Fraction Part – The fraction part represents a portion or part of a whole unit. It consists of a numerator (the top number) and a denominator (the bottom number).
Characteristics of Mixed Numbers
Whole Number Part – The whole number part of a mixed number can be any integer (positive, negative, or zero) or can be omitted entirely if the value is exactly one whole. For example, 3, -2, and 0 are all valid whole number parts.
Fraction Part – The fraction part is always a proper fraction, meaning the numerator is smaller than the denominator. For example, 1/2, 3/4, and 5/6 are proper fractions.
Relationship Between Parts – The whole number and fraction parts together represent a single value. For example, the mixed number 2 1/4 represents two whole units and one-fourth of another whole unit.
Examples of Mixed Numbers
2 1/2 – This mixed number represents 2 whole units and 1/2 (one-half) of another whole unit.
-3 3/4 – This mixed number represents -3 whole units and 3/4 (three-fourths) of another whole unit. The negative sign indicates a value less than zero.
1 0/8 – This mixed number represents 1 whole unit with no fractional part. The fraction part is 0/8, which simplifies to 0.
Conversion Between Improper Fractions and Mixed Numbers
Mixed numbers can be converted to improper fractions and vice versa. Here’s how:
Divide the numerator by the denominator.
The result becomes the whole number part.
The remainder (if any) becomes the numerator of the fraction part, and the denominator remains the same.
For example, to convert the improper fraction 7/3 to a mixed number:
7 ÷ 3 = 2 with a remainder of 1.
The whole number part is 2.
The fraction part is 1/3.
So, 7/3 as a mixed number is 2 1/3.
From Mixed Number to Improper Fraction
Multiply the whole number by the denominator of the fraction part.
Add the result to the numerator of the fraction part.
The denominator remains the same.
For example, to convert the mixed number 4 5/6 to an improper fraction:
4 (whole number) × 6 (denominator) = 24.
24 (result from step 1) + 5 (numerator of the fraction part) = 29.
The denominator is 6.
So, 4 5/6 as an improper fraction is 29/6.
Mixed numbers are commonly used in everyday life, especially in situations involving measurements and quantities that involve both whole units and fractions, such as recipes, distances, and time. They provide a clear and intuitive way to represent such values.
How to Convert Improper Fractions to Mixed Numbers
Converting improper fractions to mixed numbers is a straightforward process. An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). A mixed number, on the other hand, consists of a whole number part and a proper fraction part. Here’s a step-by-step explanation of how to convert improper fractions to mixed numbers:
Step 1 – Divide the numerator by the denominator.
Divide the numerator of the improper fraction by the denominator. This division will give you the whole number part of the mixed number.
Step 2 – Find the remainder.
If the division in step 1 results in a remainder, take note of it.
Step 3 – Write the mixed number.
Write down the whole number part obtained in step 1 as the whole number part of the mixed number.
Write the remainder (if any) as the numerator of the proper fraction part.
Keep the denominator the same as in the original improper fraction.
Step 4 – Simplify (if necessary).
If the improper fraction can be simplified further, simplify it after converting it to a mixed number.
Let’s illustrate these steps with an example:
Example – Convert the improper fraction 13/4 to a mixed number.
Step 1 – Divide the numerator by the denominator.
13 ÷ 4 = 3 with a remainder of 1.
Step 2 – Find the remainder.
The remainder is 1.
Step 3 – Write the mixed number.
Write down the whole number part (3) as the whole number part of the mixed number.
Write the remainder (1) as the numerator of the proper fraction part.
Keep the denominator (4) the same as in the original improper fraction.
So, 13/4 as a mixed number is 3 1/4.
Simplified Version (Optional)
The fraction 1/4 can be simplified further because the numerator (1) and denominator (4) have a common factor of 1. Divide both the numerator and denominator by this common factor to simplify it.
1 ÷ 1 = 1 and 4 ÷ 1 = 4.
So, the simplified mixed number is 3 1/4, and there’s no further simplification needed.
In summary, converting an improper fraction to a mixed number involves dividing the numerator by the denominator, finding the remainder, and writing the whole number part along with the proper fraction part. The mixed number represents the same value as the original improper fraction but in a different format.
How to Convert Mixed Numbers to Improper Fractions
Converting mixed numbers to improper fractions is a straightforward process. A mixed number consists of a whole number part and a proper fraction part, while an improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). Here are the steps to convert mixed numbers to improper fractions:
Step 1 – Multiply the whole number part by the denominator of the fraction part.
Multiply the whole number part of the mixed number by the denominator of the proper fraction part. This result will become the new numerator of the improper fraction.
Step 2 – Add the result from step 1 to the numerator of the fraction part.
Add the result obtained in step 1 to the numerator of the proper fraction part. The denominator remains the same as in the original proper fraction.
Step 3 – Write the improper fraction.
The numerator obtained in step 2 becomes the new numerator of the improper fraction, and the denominator remains the same as in the original proper fraction.
Step 4 – Simplify (if necessary).
If the improper fraction can be simplified further, simplify it after converting it from the mixed number.
Let’s illustrate these steps with an example:
Example – Convert the mixed number 2 3/5 to an improper fraction.
Step 1 – Multiply the whole number part by the denominator.
2 (whole number part) × 5 (denominator of the fraction part) = 10.
Step 2 – Add the result from step 1 to the numerator of the fraction part.
10 (result from step 1) + 3 (numerator of the fraction part) = 13.
Step 3 – Write the improper fraction.
The improper fraction has a numerator of 13 and a denominator of 5 (which remains the same as in the original fraction).
So, 2 3/5 as an improper fraction is 13/5.
Simplified Version (Optional)
The fraction 13/5 can be simplified further because the numerator (13) and denominator (5) have no common factors other than 1. Therefore, it is already in its simplest form, and no further simplification is needed.
In summary, converting a mixed number to an improper fraction involves multiplying the whole number part by the denominator of the fraction part, adding the result to the numerator of the fraction part, and keeping the denominator the same. This conversion allows you to represent the same value as the original mixed number but in a different format.