# Finding Greatest Common Factors (GCF) Using Prime Factorization

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## Objective

SWBAT determine the GCF of two numbers less than 100 using prime factorization

#### Big Idea

Every positive integer (except the number 1) can be represented as a product of one or more prime numbers.

7 minutes

Students work in pairs to solve the Think About It problem.

After a few minutes of work time, I ask students what they tried to do, to solve this problem.  Students generally create t-charts to factor the numbers, as they learned to do in the previous lesson. I ask students what was tricky about this problem.  They may say that it was time consuming to come up with all of the factor pairs of 63.  They also may say that they're unsure that they found all of the factor pairs of 63.

I frame the lesson by telling students that in this lesson, we are going to learn another strategy for determining the GCF of two numbers, which will serve us especially when we have larger numbers.  We are going to use the prime factorization of each number and determine the prime factors that both numbers have in common.  This method is more efficient than listing factors, and eliminates the possibility that we will miss a factor pair.  It’s really useful, too, when you aren’t sure you know all of the factor pairs for a number.

## Intro to New Material

10 minutes

I model how to create a factor tree, using the Think About it Problem.  I ask students to share with me any way they think of to multiply to 63, with the exception of 63 x 1.  Students help me break down each factor into further factor pairs, until we have only prime numbers left.  We circle the prime factors to show that we are done.

Prime factorization is more accurate than listing factors to find the GCF because you eliminate the chance of a missing factor.  This is especially helpful when the numbers we're working with are large.

I fill out the notes for students:

To determine the GCF of two numbers using prime factorization, we can first determine the prime factorization of each number. Then we determine the common prime factors.  The product of those common prime factors is the GCF of the two numbers.

Prime factorization is writing a number as the product of its prime factors.

We can use a factor tree to help us determine the prime factors of a number.

As we work through example two, I am asking questions to anticipate common misconceptions.I ask students if we multiply all of the common factors from both prime factorization lists together.  I also ask both numbers share a common factor that is repeated (i.e. both have two 2’s) do we write one 2 or two 2’s?  I want to get a sense from the class if these misconceptions exist, so that I can address them before I release students to practice.

## Partner Practice

15 minutes

Students work in pairs on the Partner Practice problem set.  As students work, I circulate around the room and check in with each group.  I'm looking for:

• Are students explaining their thinking to their partner?
• Are students writing their work in the work space?
• Are students identifying the prime factors in their factor tree by circling them?
• Are students expressing the prime factors as a multiplication sentence?
• Are students determining the common prime factors to find the GCF?

• How did you determine the GCF of _____?
• Once you listed the prime factors, how did you determine the GCF?

For students who need extra support, I'll have them write a list of the prime factors they'll see most often in the margin of their work space (2, 3, 5, 7).  It's important to stress to students that these are not the only prime factors that they'll encounter, but this list will help my struggling students to know when to stop factoring.

## Independent Practice

15 minutes

Students work on the Independent Practice problem set. As they are working, I circulate around the room and check in with each student.  I'm looking for common errors.

Students may stop factoring when they still have composite factors.  If I see this, I will ask students how they know a number is prime.  I'll also ask them to name ways to multiply to the composite number that they've stopped with.

Students may think there is one particular way they have to start to factor a number.  If I suspect this is an issue, I'll have a student create two factor trees for the same number, starting each one off with a different factor pair.  I want students to determine that they can use any factor pair to get the factor trees started.

Students may think they should multiply all of the common prime factors of the two numbers together.  Alternatively, students may choose only one instance of a common prime factor if that factor is repeated (i.e. if both numbers have three 2’s).  If I see either of these mistakes, I'll ask questions about the concept of greatest common factor and help students to make sense of and recontextualize the answer they've come to.

## Closing and Exit Ticket

8 minutes

After independent work time, I have students compare with their partners the work for Problem 4. I then display completed factor trees on the document camera, and ask for a student to explain how to go from trees to pieces of cheese.

We then look at Problem 9.  I ask students which strategy for determining the GCF they prefer to use.  Students turn and tell their partner about their preferences, and I have 2-3 students share out.  I want my students talking about math and problem solving strategies often.

Students complete the Exit Ticket independently to close the lesson.  My key shows what student work can look like.