## Lesson: Factors and Fractions Introducing the Concept

Your students have had an opportunity to review prime numbers, prime factorization, and divisibility rules. These will prove helpful to them as they develop an understanding of the greatest common factor and least common multiple. An understanding of these two concepts is essential when working with fractions.

Materials: paper and pencil

Preparation: none

Prerequisite Skills and Concepts: Students should know how to find the prime factorization of a number. They should also be familiar with the meaning of the terms factor and multiple.

• Say: Take out a sheet of paper and a pencil, and find the prime factorization of 36.
Have a volunteer come to the board and explain how to find the prime factorization of 36. Have another volunteer show how he or she found it, using a different method.
• Ask: Who can tell me what the factors of the number 12 are?
(1, 2, 3, 4, 6, and 12) Now, who can tell me what some multiples of 12 are? (12, 24, 36, and so on)
• Ask: Who can tell me what a factor is? (A factor is one of two or more numbers multiplied together to get a product.) Who can tell me what a multiple of a number is?
It is a number that is the product of the given number and some other number.
• Say: Today you are going to learn two new terms, The first term is greatest common factor, or GCF. The word common suggests that we should look for a number that is a common factor for the two or more numbers involved, which means the number must divide evenly into those numbers. Since we are looking for the greatest common factor, we are looking for the greatest number that evenly divides all the numbers. Let's look at a specific example.
• Say: There are many ways to find the GCF of 18 and 30. One way is to write the prime factorization of the two numbers.
On the board, write:18 = 2 x 3 x 3 and30 = 2 x 3 x 5.
• Say: One factor that is common to both numbers is 2. (Circle the 2 in both numbers.) Are there any other factors that are common to both? (3) Right. Are there any others? (no) Thus, our greatest common factor is 2 x 3, or 6.
• Say: Now try to find the GCF for 20 and 28 at your desk.
After the class has had time to find the GCF, have a volunteer come to the board to show what he or she did.
• Say: Let's try three more examples. Find the GCF for (a) 20 and 60, (b) 15 and 28, and (c) 24, 32, and 40.
Write the three problems on the board for students to copy.
The first example above was chosen so that students could realize that the GCF could be one of the numbers involved, since the GCF of 20 and 60 is 20. The second example was chosen to show them that the GCF of two numbers might only be 1. The third example was chosen to indicate that you can find the GCF for more than two numbers at a time.
• Say: The next term we will discuss is least common multiple. Since we are looking for a common multiple, we should be looking for a number that is divisible by all the numbers, and we want it to be the least number possible. Let's work through an example together. The least common multiple is abbreviated LCM.
• Say: In order to find the LCM of 30 and 12, we will first find the prime factorization for the two numbers.
On the board, write the following:30 = 2 x 3 x 5 and12 = 2 x 2 x 3.
• Say: Since both numbers have to be factors of the LCM, we can take all the factors of one of the numbers, 30: 2 x 3 x 5. (Write2 x 3 x 5 on the board.) Now we look at the next number, 12 in this case. Since only one factor of 2 appears so far, we need to add another 2 to our list, because 2 x 2 has to divide the LCM. However, the other factor, 3, already appears in the LCM being created. Therefore, the LCM for 30 and 12 is 2 x 2 x 3 x 5, or 60.
• Ask: Who can summarize for us what we need to do?
Write the numbers in their prime factorization form and then select the highest power that any factor appears in any number and multiply those numbers.
• Say: Now try this one at your desk: Find the LCM for 18 and 20.
Have a student volunteer to show his or her work on the board and explain the work.
• Say: Let's try three more.
On the board, write Find the LCM of (a) 20 and 60, (b) 8 and 15, and (c) 6, 10, and 14.
The first exercise will illustrate that the LCM can be one of the two numbers, since the LCM of 20 and 60 is 60. The second example indicates that sometimes the LCM is the product of the two numbers. This is the case when the two numbers have no common factor. The third example shows that you can find the LCM for more than two numbers. Have students come to the board to show and explain their work.