Factors and Fractions

Number theory has interested and motivated mathematicians for centuries. It is a branch of mathematics that deals with topics such as even and odd numbers, prime and composite numbers, divisibility, greatest common factor (GCF) and least common multiple (LCM), and terminating and repeating decimals. These topics lay the foundation for the development of fraction concepts.

Your students will have worked with prime and composite numbers and prime factorization in Grade 5. However, it is a good idea to review the definitions of prime numbers and composite numbers. Prime numbers are numbers greater than 1 that have as factors only themselves and 1. Two is the first prime number and it is the only even prime number, since all other even numbers are divisible by two. In a similar way, 5 is the only prime number ending in 5, since all whole numbers that end in 5 are divisible by 5. Thus, all the other prime numbers must end in 1, 3, 7, or 9. However, a whole number is not a prime number just because it ends in 3, 7, or 9. Some examples of numbers that are not prime and end in 1, 3, 7, or 9 are 21, 33, 27, and 9. The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Composite numbers are numbers greater than 1 that have more than 2 factors. This means they are divisible by numbers other than themselves and one. For example, the composite number 12 can be divided by 2, 3, 4, and 6 as well as by 1 and 12. Geometrically this means that if given a set of 12 blocks, you could construct a rectangle with dimensions 1 by 12, 2 by 6, or 3 by 4. You can construct only one rectangle with a prime number of blocks. Some composite numbers have only one other factor, such as 4 (1, 2, and 4), 9 (1, 3, and 9), and 25 (1, 5, and 25). When we find the prime factorization of a composite number, the result is always the same, no matter how we started to factor the number. For example, look at the three factor trees for 24 below. The prime factorizations are all 2³ 3.

Equivalent fractions are different representations of the same number. For example, is the same as , or . Similarly, the number 1 can be represented as , or . Since multiplying by 1 does not change the value of a fraction, we can change any fraction into an equivalent fraction by multiplying by 1. This is done by multiplying both the numerator and the denominator by the same number. For example, = , or = . In a similar fashion, we can divide a number by 1 and not change the value of the number. This is done by dividing both the numerator and the denominator by the same number. For example, = , or = . In a similar fashion, we can divide a number by 1 and not change the value of the number. For example,   = ÷ ,

Two very useful concepts to understand when working with fractions are the greatest common factor (GCF) and the least common multiple (LCM). The GCF for two or more numbers is the greatest number that divides evenly into those numbers. For example, the GCF for 72 and 54 is 18, since 18 is the greatest number that divides evenly into both numbers without a remainder. There are several ways to find the GCF of two numbers. One way is to write the prime factorization of the numbers and circle the common primes as shown below.

Thus, the GCF of 54 and 72 is 2 3 3, or 18. Another way to find the GCF for two numbers is to divide both numbers by their common factors successively until there are no more common factors. The product of all the common factors you divided by is the GCF. See the example below.

Since 3 and 4 have no common factor other than 1, the GCF for 72 and 54 is 2 3 3, or 18. One of the most common uses for the GCF is in the simplification of fractions by dividing the numerator and denominator by their GCF. Let's look at the fraction . We found the GCF to be 18. Dividing both the numerator 54 and the denominator 72 by 18 gives us , when simplified.

The LCM of two or more numbers is the least nonzero number that those numbers will divide. For example, the LCM of 18 and 24 is 72, since that is the least nonzero number into which 18 and 24 will divide evenly. There are several ways to find the LCM of two or more numbers. One way is to write the prime factorizations of the numbers and multiply the greatest powers of the prime numbers that appear in any of the numbers. Let's begin finding the LCM for 18 and 24, by writing the prime factorizations of the two numbers.

18 = 2 3² and 24 = 3 2³ The greatest power of 2 that appears is 2³ and the greatest power of 3 is 3². Thus the LCM of 18 and 24 is 2³ 3², or 72.

Another way to find the LCM is to list the multiples of each number until you find the first multiple that appears in each list. Applying this method to find the LCM of 18 and 24, we see that the multiples of 18 are 18, 36, 54, 72, 90, and so on, and the multiples of 24 are 24, 48, 72, 96, and so on, so the LCM of 18 and 24 is 72.

One of the primary uses for the LCM is in adding and subtracting fractions with unlike denominators. One way to add or subtract those fractions is to find the least common denominator (LCD), which is the LCM of the denominators.

Exponents are mathematical symbols that facilitate working with expressions. Students need to know what these symbols mean. A whole number exponent is a symbol for the repeated multiplication of a number by itself. In the expression 3⁵, the exponent (5) tells how many times the base (3) is used as a factor. Thus, 3⁵ means 3 3 3 3 3. Just as students learn how to order and compare whole numbers, they need to learn how to compare and order fractions, decimals, and mixed numbers. One method of comparing fractions is to find equivalent expressions for the fractions by finding a common denominator for the fractions. The common denominator need not be the least common denominator, which is the LCM for the denominators involved. For example, to order the fractions 5/6, 3/4, and 7/9, we first need to find a common denominator. The product of 6 9 4, the number 216, is a common denominator. Thus, changing each of the above fractions to equivalent fractions using 216 as the denominator we get

Since 167 < 168 < 180, it follows that < < . We could also have done this by finding the LCM (36) and using it to find the equivalent fractions as shown below.

Again, since 27 < 28 < 30, it follows that < < . When comparing or ordering numbers including fractions and decimals, we can first change them into the same form—write them all as decimals or as fractions—and it is then easy to compare or order them. For example, to order the numbers , 0.72, and from least to greatest, we can change them all into fractions with a denominator of 100.

Thus, < < . Since = 0.75 and = 0.76, we can also write them as decimals: 0.72 < 0.75 < 0.76. When ordering mixed numbers, first order them based on the whole-number parts. If the whole-number parts are the same, then order them based on their fractional parts or decimals. For example, to determine which is greater, 1 or 1.37, you could rewrite 1 = = and 1.37 = 1 = . Since 140 > 137,1 > 1.37.

If, when one whole number is divided by another nonzero whole number, the division is carried out far enough, the division process will either terminate or a pattern of numbers will begin to repeat. For example, the fractions , , and can be written as terminating decimals. = 0.625      = 0.52      = 0.45

Numbers such as , , and will eventually repeat when written as decimals.

The bar in the expressions above is used to show the digits that repeat. In the case of , only the 5 repeats. In the decimal for , the digits 428571 repeat; for , only 36 repeats.

A terminating decimal can be written as a fraction simply by writing it the way you say it—3.75 = three and seventy-five hundredths = 3—and then combining if necessary to produce a fraction: + = . So any terminating decimal can be written as a fraction.