Factors and Fractions

Students will remember concepts more readily when they are connected to other concepts. The GCF and LCM are the underlying concepts for finding equivalent fractions and adding and subtracting fractions, which students will do in Chapter 4.

Materials: paper and pencil

Preparation: none

Prerequisite Skills and Background: Students should know what prime and composite numbers are and how to find the prime factorization of a number. Students should also have a fundamental understanding of fractions.

• Ask: In the previous lesson we worked with two new concepts: the greatest common factor and the least common multiple. Who can explain to us what the greatest common factor of two numbers is?
  Students should tell you that the GCF is the greatest number that divides evenly into both numbers.

• Ask: Now, what is the least common multiple of two numbers?
  Students should say that the LCM is the least number that is a multiple of both numbers.

• Say: Today we are going to look at other ways of finding the GCF and LCM and some applications of these ideas. Let's start by having you find the GCF of 28 and 42.
  Give students a few minutes to find the GCF at their desks and then have a student volunteer show it on the board for the class to see.

• Say: Another way of doing this would be to look at the two numbers and ask yourself "What number divides both those numbers?" Since the numbers are even, they are both divisible by 2. So we could start by dividing both numbers by 2.
  Show the following division exercises on the board.
  

  We see that 2 is a factor of both numbers.

• Ask: Can you think of any number that divides both 14 and 21? (Yes, 7 does.)
  On the board, divide 7 into each of these numbers as shown below.
  

  Now we see that 2 and 7 are factors of both numbers.

• Say: Since no other number besides 1 divides both 2 and 3, then 2 7, or 14, is the GCF for 28 and 42. Now use this method to find the GCF for 24 and 42 at your desks.
  Have a volunteer come to the front of the class and show how to find the GCF using the method above.

• Ask: Which method do you like better?
  Discuss with the class the two methods and the advantages of one method over the other. Some students may say they like dividing out common factors, because it makes the process simpler. Others may like the prime factorization method because it seems clearer to them. Explain that both methods are good methods, but that students should practice both methods to be sure they can do them.

• Say: We are going to find the LCM using another method today. For example, to find the LCM of 28 and 42, we could start by writing out the multiples of each until we find the first one that is common.
  Write the following on the board:

  
  The multiples of 28 are 28, 56, 84, 112, 140, . . .
  The multiples of 42 are 42, 84, 126, . . .

• Say: We can see that the LCM of 28 and 42 is 84, because it is the first multiple of both numbers. Let's use this method to find the LCM of 24 and 42.
  After the students have had time to find the LCM, have a volunteer come up front and explain how he or she found the LCM.

• Ask: Which method do you like better, the prime factorization method or repeating the multiples of each number?
  Some students may like the repeating-multiples method better because it makes more sense to them. Others may prefer the prime-factorization method in cases where you might have to write out 10 or 12 multiples of each number. Explain that either method is fine, but that the latter method could be very time consuming and may lead to mistakes.

• Say: The GCF and LCM of two numbers are important ideas when working with fractions. Equivalent fractions are fractions that represent the same amount, like and . We can find equivalent fractions for any fraction by multiplying or dividing the fraction by 1. For example, the fractions , , , and are all representations of 1. We can change the fraction into an equivalent fraction by multiplying by .
  Thus, , so is a fraction equivalent to . Notice that we multiplied both the numerator and denominator by the same number. Use this technique to change into an equivalent fraction by multiplying by 1, written as .

• Ask: What did you get for an answer? () Good. Now change the following fractions into equivalent fractions with the indicated denominators.
  Write the following on the board:
  Have a volunteer explain how he or she solved each problem.

• Say: We say a fraction is in simplest form if the GCF of the numerator and denominator is 1. To simplify a fraction, we divide the numerator and denominator by the GCF of the two numbers. One way to simplify the fraction is to find the GCF of 20 and 36. How might we do this?
  Find the GCF on the board with the class.

  
  The multiples of 20 are 1, 2, 4, 5, 10, and 20.
  The multiples of 36 are 1, 2, 3, 4, 6, 9, 18, and 36.

• Say: Now, if we divide both 20 and 36 by 4 we get the following.
  Write the following on the board for the class to see.
  

• Say: Now try this problem at your desks: Simplify and I'll walk around to see how you are doing.
  Have a volunteer explain how he or she solved the problem. Show students that they could have written the fractions in factored form and divided out the common factors.
  

20 = 2² 5.

Similarly, to deepen your students' understanding of the GCF, create an open-ended request, such as "Find three pairs of numbers whose GCF is 12." Once again, there are many solutions to this problem, including the following: 12 and 36, 36 and 24, and 36 and 60. Students should realize that all the number pairs are multiples of the GCF (12) but have no other common factor.