## Add and Subtract Fractions

Fractions
Up to this point, students have been taught about a fraction of a whole and a fraction of a group. The fraction when applied to a whole pizza or a rectangular area means to divide the object or area into 4 equal parts and consider 3 of these equal parts. The fraction when applied to a group of 16 soccer players means to divide the number of players into 4 equal groups (4 each) and consider 3 of the 4 equal groups (12 players). In fact, the idea of a fraction of a group is really the same as that of a fraction of a whole if the 16 soccer players are considered as one whole soccer team.

Another way to help students with fraction concepts is to relate fractions to a number line, in particular, fractions of an inch on a ruler. A unit fraction is defined as a fraction with a numerator of 1 (for example, , , , ). In a unit fraction, , one whole unit is divided into n equal parts. One of these smaller parts is the amount represented by the unit fraction.

On the number line, the unit fraction represents the length of a segment when a unit interval on the number line is divided into n equal segments. The point to the right of 0 on the number line at a distance from 0 will be . The fraction can represent the quotient of m and n, or m × n. If the fraction is defined in terms of the unit fraction , the fraction means m unit fractions of . In terms of distance along the number line, the fraction means the length of m abutting segments each of length . The point is located to the right of 0 a distance m × from 0. The numerator of the fraction tells how many segments. The denominator tells the size of each segment.

A straightforward way to show that fractions represent a solution to a division problem is by using equivalent fractions. What is 35 ÷ 7? It is 5 because 35 equals 7 × 5. What is 5 ÷ 7? This is more difficult because 5 is not a multiple of 7. However, 5 = = 5 × × 7 = , and equals 35 unit fractions of . Just as 35 divided by 7 is 5, 35 unit fractions of divided by 7 is 5 unit fractions of . So 5 ÷ 7 = .

Equivalent Fractions
From the definition of fraction, students should understand that for each counting number m, m = and = 1. So there are many different names for the same point on the number line.

Equivalent fractions are fractions that represent the same point on the number line. As shown above for the number 1, there are many fractions that name the same amount. The number can represent the quantity of pizza given by 3 parts of a pizza cut into 4 equal parts. If each of the 4 parts is in turn cut into 3 equal parts, the pizza has been cut into 4 × 3 = 12 equal parts. The same quantity of pizza can now be written as 3××3 =. This principle is summed up in the cancellation law for fractions that states m × × r = .

The cancellation law for fractions does not require any knowledge about multiplication of fractions. Once multiplication by fractions is understood, the cancellation law for fractions can be expressed as m × × r = × = × 1 = .

Simplest Form
A fraction is in simplest form if the numerators and denominators are as small as possible. A more formal way of stating this is to say that the numerator and denominator have no common factors other than 1. The traditional way of simplifying fractions is to find the GCF of the numerator and denominator and then divide both the numerator and denominator by that GCF. Other ways include canceling all factors common to the numerator and denominator, dividing the numerator and denominator by the same number as many times as possible, or taking the prime factorizations of the numerator and denominator and canceling common factors.

Example:

Simplify

Comparison of Fractions
Which is greater, or ? When comparing fractions with like denominators, it is enough to compare the numerators. Since 4 unit fractions of is greater than 3 unit fractions of , is greater than . Which is greater, or ? When comparing fractions with unlike denominators, it is usually necessary to find a common denominator. In the example, any common multiple of 12 and 8 will do. For simplicity, choose 12 × 8 = 96.

Now, = 7 × × 8 = and = 5 × × 12 = . Since 60 > 56, > .

Should students want to use only the least common denominator, suggest they try it both ways and decide for themselves which method is quicker or easier.

Addition and Subtraction of Fractions
To add or subtract fractions with a common denominator, find the sum or difference of the numerators and keep the common denominator. For example, in + , three unit fractions of plus two unit fractions of equals five unit fractions of . So the sum is .

When adding and subtracting fractions with unlike denominators, a common denominator must be found. One way to find a common denominator is to use the product of the denominators.

Examples:

Add where b ≠ 0, d ≠ 0 (because division by 0 is undefined)

Subtract.

Using the least common multiple of the denominators to get the least common denominator is another way of adding or subtracting fractions.

Example:

Add + . Write then answer in simplest form. Add by using the product of the denominators as the common denominator.

Proper and Improper Fractions
A fraction like is said to be proper because the numerator is less than the denominator. A fraction like is said to be improper because the numerator is not less than the denominator. Whole numbers can also be written as improper fractions, such as 1 = and 2 = . The improper fraction can be written as the mixed number 1.

Addition and Subtraction of Mixed Numbers
To add mixed numbers, first add the fractions and then the whole numbers. The result is written in simplest form. A mixed number is in simplest form when the fractional part of the mixed number is less than 1 and is in simplest form.

Example:

Notice that if the LCD of 24 were used, the result would be 6, or 7

When subtracting mixed numbers, first compare the fractional parts of the numbers to decide whether the mixed number needs to be renamed in order to complete the subtraction.

Example:

Notice that if the LCD of 24 is used, the answer 1 is obtained in simplest form. Remind students that addition can be used to check subtraction results.

Teaching Model 10.3: Add Fractions With Unlike Denominators