I. Fundamentals
A fraction represents a part of a whole which is expressed in the form a/b.The number on the bottom of the fraction is called the denominator. It represents how many equal parts the whole is divided into. The number on the top of the fraction is called the numerator. It represents how many of the parts we have.
For example, in the fraction 3/4, the numerator tells us the fraction represents 3 equal parts, and the denominator tells us that 4 parts make up a whole.
From this we can see that whenever the numerator and denominator are the same number, the fraction is equal to 1. For instance, 4/4 = 1.
Whole numbers can also be written in fraction form. To do so, place the the whole number over 1. For example, the whole number 5 written as a fraction is 5/1.
II. Conceptual Understanding
Number Line
We can use the number line to help conceptually make sense of fractions. Again using 3/4 as our example, the number line below divides the number one into 4 parts (as represented in the denominator). Starting at zero, we move 3 fractional units (as represented in the numerator) to the right ending at 3/4.
Area Model
Fractions can also be graphically represented through the use of area models. The area models below show a whole can be divided into various fractional parts.
III. Decompose Fractions
IV. Proper Fractions, Improper Fractions & Mixed Numbers
1. Definitions
1. Definitions
A proper fraction is one where the numerator is smaller than the denominator. Examples would include 2/5, 3/4 and 1/6.
A improper fraction is one where the numerator is greater than or equal to the denominator. Examples include 5/2, 7/3 and 6/6.
A mixed number is one that has both a whole number and a fraction. Examples include 3 3/4, 15 2/5 and 17 1/4.
2. Converting Improper Fractions to Mixed Numbers
To convert an improper fraction into a mixed number, simply divide the numerator by the denominator. The answer becomes the whole number and the remainder becomes the numerator of the new fraction. The denominator of the new fraction is the same as the denominator of the old fraction.
VI. Reducing Fractions to Lowest Terms (Simplifying Fractions)
Reducing fractions to their lowest terms involves finding the lowest equivalent fraction. This means that there is no number except 1 that can be divided evenly into both the numerator and the denominator.
To do this, divide both the numerator and denominator by a number that divides evenly into both (common factors). Repeat the process until the fraction is in it's lowest terms.
To do this in one step, divide both the numerator and denominator by the largest number that divides evenly into both (greatest common factor).
Example: Reduce 4/8
The number 2 divides evenly into both 4 and 8.
4/8 / 2/2 = 2/4
Step 1: Write the factors of numerator and denominator.
Step 2: Determine the highest common factor of numerator and denominator.
Step 3: Divide the numerator and denominator by their highest common factor (HCF). The fraction so obtained is in the simplest form.
VII. Common Fraction
A common fraction is a fraction where both the numerator and denominator are both integers (Technically where the denominator is a non-zero integer). For instance, 1/5 is a common fraction but 1.5/2 is not a common fraction since the numerator contains a decimal.
To write fractions that contain a decimal number in the numerator or denominator, we first have to make both numbers into integers.
For example with the fraction 1.5/2 we can see that moving the decimal in the numerator in 1.5 to the right would make it an integer. To do this we multiple both the numerator and denominator by 10.
1.5/2 x 10/10 = 15/20
We then simplify as normal
15/20 = 3/4
VIII. Converting Fractions to Decimals & Converting Decimals to Fractions
See Decimals
IX. Converting Fractions to Percents & Converting Percents to Fractions
See Percentages
X. Multiple Meanings of a Fraction
See Five Meanings of a fraction
Practice:
IXL: Convert between improper fractions and mixed numbers
A improper fraction is one where the numerator is greater than or equal to the denominator. Examples include 5/2, 7/3 and 6/6.
A mixed number is one that has both a whole number and a fraction. Examples include 3 3/4, 15 2/5 and 17 1/4.
2. Converting Improper Fractions to Mixed Numbers
To convert an improper fraction into a mixed number, simply divide the numerator by the denominator. The answer becomes the whole number and the remainder becomes the numerator of the new fraction. The denominator of the new fraction is the same as the denominator of the old fraction.
Example: Convert 9/4 to mixed number
9/4 = 9 ÷ 4 = 2 R1
Write the whole number part 2. Then take the remainder and write it over the original denominator
2 1/4
3. Converting a Mixed Number to an Improper Fraction
Method 1
3. Converting a Mixed Number to an Improper Fraction
Method 1
To convert a mixed number into an improper fraction, multiply the whole number by the denominator and add it to the numerator. This becomes the numerator of the improper fraction; the denominator of the new fraction is the same as the original denominator.
Example: Convert 2 3/4 to an improper fraction
Multiply whole number by denominator
2 x 4 = 8
Then add the answer to the numerator
8 + 3 = 11
Now write the sum on top of the original denominator
11/4
Method 2
This method is really the same as the above method, just worked out. First, convert the whole number into a fraction. Then add the fractions.
Example Convert 6 1/2 to an improper fraction.
Convert the whole number to a fraction
6 = 6/1
Then add the fractions
6/1 + 1/2 = 6/1 x 2/2 + 1/2 = 12/2 + 1/2 = 13/2
V. Equivalent Fractions
Equivalent fractions are fractions that have the same value. They are different ways of representing the same fractional part of a whole. Though they look different each fraction represents the same number. For instance, 1/2 = 2/4 = 3/6 = 4/8.
And here are some of the same equivalent fractions represented on number lines.
Equivalent fractions are fractions that have the same value. They are different ways of representing the same fractional part of a whole. Though they look different each fraction represents the same number. For instance, 1/2 = 2/4 = 3/6 = 4/8.
VI. Reducing Fractions to Lowest Terms (Simplifying Fractions)
Reducing fractions to their lowest terms involves finding the lowest equivalent fraction. This means that there is no number except 1 that can be divided evenly into both the numerator and the denominator.
To do this, divide both the numerator and denominator by a number that divides evenly into both (common factors). Repeat the process until the fraction is in it's lowest terms.
To do this in one step, divide both the numerator and denominator by the largest number that divides evenly into both (greatest common factor).
Example: Reduce 4/8
The number 2 divides evenly into both 4 and 8.
4/8 / 2/2 = 2/4
The number 2 divides evenly into both 2 and 4.
2/4 / 2/2 = 1/2
Alternatively you could do this in one step by using the greatest common factor, which in this case is 4.
4/8 / 4/4 = 1/2
A more precise algorithm for simplifying fractions would be:
Alternatively you could do this in one step by using the greatest common factor, which in this case is 4.
4/8 / 4/4 = 1/2
A more precise algorithm for simplifying fractions would be:
Step 2: Determine the highest common factor of numerator and denominator.
Step 3: Divide the numerator and denominator by their highest common factor (HCF). The fraction so obtained is in the simplest form.
VII. Common Fraction
A common fraction is a fraction where both the numerator and denominator are both integers (Technically where the denominator is a non-zero integer). For instance, 1/5 is a common fraction but 1.5/2 is not a common fraction since the numerator contains a decimal.
To write fractions that contain a decimal number in the numerator or denominator, we first have to make both numbers into integers.
For example with the fraction 1.5/2 we can see that moving the decimal in the numerator in 1.5 to the right would make it an integer. To do this we multiple both the numerator and denominator by 10.
1.5/2 x 10/10 = 15/20
We then simplify as normal
15/20 = 3/4
VIII. Converting Fractions to Decimals & Converting Decimals to Fractions
See Decimals
IX. Converting Fractions to Percents & Converting Percents to Fractions
See Percentages
X. Multiple Meanings of a Fraction
See Five Meanings of a fraction
Practice:
IXL: Convert between improper fractions and mixed numbers
K5 Learning: Convert improper fraction to mixed number
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