**Launching GCF and LCM with a Real World Context**

Teachers often want to know how they can teach greatest common factor and the least common multiple in a way that is more concrete or constructivist. Using Least Common Multiple and Greatest Common Factor Worksheets is fine but we can do better. They’ve been teaching factor trees for years with the same mixed bag level of understanding. I’ve put together a list of my favorite strategies for teaching GCF and LCM. These strategies will help your students develop a deeper conceptual understanding of GCF and LCM, all while finding the GCF and LCM more efficiently.

Math teachers are all too familiar with the “When will we use this?” or “Why do we need to know this?” questions. Just like adults, it’s natural for students to want to know the practical applications of what they are learning. I recommend launching your greatest common factor and least common multiple unit with a context that students can connect to. I have found the most success from starting with the concrete and then moving towards the abstract. For example, GCF and LCM story problems helps students see why this skill is useful. Story problems can be inspired from your class, community, or popular books and movies.

**GCF Lesson Exploration**

One way to do this is to create a greatest common factor worksheet with a problem similar to the one I created below:

You are adding candy toppings to some cakes. You have 24 Hershey’s Kisses and 32 M&Ms. You want each cake to have the same number of Hershey’s Kisses and the same number of M&Ms. You also don’t want any left over Kisses or M&Ms. What is the greatest number of cakes you can toppings to if you want each cake to have the exact same number of Kisses and the exact same number of M&Ms?

Feel free to tweak the wording to better fit the interests/tastes of your class or to create a GCF worksheet of your own. I also like to bring in real Hershey’s Kisses and M&Ms as fun manipulatives or just to have them as treats! You can always use two colored manipulatives as well.

After you’ve gotten several creative responses from your class ask students if there’s another mathematical way to find their answer. How could we make this process more efficient?

(We give you some Least Common Multiple and Greatest Common Factor Worksheets to practice with your students at the end of this lesson.)

**LCM Lesson Exploration**

You can use this same method when you introduce least common multiple. My students know I’m a huge Harry Potter fan so I like to use the problem below, but feel free to create your own LCM worksheet based on your school’s sports teams or another context that you think will best resonate with your class.

Professor McGonagall and Professor Snape are trying to coordinate Quidditch practice. To avoid any high-jinxes before the big match, they want to be on the field when Gryffindor and Slytherin next have practice on the same day. The Gryffindor team plays every 5 days and the Slytherin team plays every 3 days. How soon will both teams have games on the same day again?

Provide the class with two color manipulatives to aid their understanding of this LCM worksheet. Ask students if there’s another mathematical way to find their answer. How could we make this process more efficient?

(At the bottom of this lesson we give you Least Common Multiple and Greatest Common Factor Worksheets to use with your students.)

## Vocabulary

Before going into formal instruction into GCF and LCM I find it useful to go over some key vocabulary words: factor, multiple, prime and composite. It’s always helpful to do a quick vocabulary check with units such as GCF and LCM because part of the battle is knowing what is being asked. This review can be done as a whole class warm-up or as a GCF and LCM station. You can use many vocabulary building/reviewing strategies; the objective here is to make sure the class has the language foundation necessary for working in this unit.

## Rods

When we get to the upper grades, too often we find that few visual or kinesthetic models exist for teaching math. A great concept building GCF and LCM activity is finding GCF or LCM using rods.

With all the push for automaticity it’s easy to slip into teaching memorization of procedures and skip conceptual understanding. This greatly limits the amount of students who are able to make the leap to mathematical abstraction. For this reason when teaching something like GCF and LCM, I like to start with the concrete and then move to the abstract, seeking efficiency each step of the way.

This is where rods come in handy. You can use rods to find the GCF and LCM! You can make a GCF station or an LCM station or later, a station with both GCF and LCM using rods!

If your students have never used rods, model how to use this tool. For example, you can model finding the greatest common factor for 8 and 12. Go through the process of finding the different factors for 8 with the rods, starting with the 1 rods, then lining up the 2 rods underneath and so on. This is a great place to teach into how to organize your factors into a list as you discover them with the rods. Then do the same for 12 and model looking for the GCF in each list.

After sufficient practice, you can remove the rods scaffolding. Help students generalize that just like they used this list to find common factors, this list can be used to help them find the GCF for larger numbers. You can then teach prime factorization and the factor tree.

To use rods in an LCM station, start by asking students to make two trains of the same size using rods for the numbers you’ve chosen. For example, in the Quidditch problem above I would model making one train out of 3’s rod and the other train out of 5’s rods.

To find out which day Slytherin and Gryffindor will have practice together I keep adding rods to each train until both trains match up. I can model several multiples that would work to show students how to make a multiples list. I point out that multiples go on forever so we need to know when to stop. Since I’m looking for the next time the Quidditch team will play, I’m looking for the least common multiple so I don’t need to go very far. As soon as I find the LCM, my search is over.

**Prime Factorization and the Ladder Method**

Once students have a solid conceptual understanding of LCM and GCF, I like to push for more efficient strategies. I have class discussions on the level of efficiency and usefulness of each strategy given the circumstances.

This is usually about the time I would teach the good ol’ standby: prime factorization and the factor tree. I still teach into prime factorization, but recently, I’ve discovered a new method that I’m a much bigger fan of since it’s an even more efficient strategy! It’s called the Ladder Method or the Cake Method. I love that this method allows students to find GCF and LCM in one model!

### To Find GCF Using the Ladder Method:

### For example from the Cake problem above let’s make a new example…

1.) **Write your two numbers on one line: **24 and 32

2.) **Then Draw an L Shape:** (this can also look like an upside down division symbol). Place the two numbers inside the L.

3.) **Drive out Prime Numbers Starting with the Smallest:** Next find the smallest prime number that goes into both numbers. This is a good time to reference your prime numbers list if your students don’t already have the first few sets of prime numbers memorized.

4.) **LCM Makes and L:** The smallest prime number that goes into 24 and 32 is 2. Put the 2 on the outside of the L. Determine how many times 2 goes into 24 and write 12 underneath the L. Then do the same for how many times 2 goes into 32 and write 16 next to the number 12. Repeat this step until no more prime factors will evenly go into both sets of numbers. You’ll be left with 3 and 4.

5.) **GCF is Down the Left Side:** To find the GCF I put a big G around the prime numbers on the outside of the ladder. These are the numbers I will multiply to get my GCF: 2x2x2. That gives me 8 as the GCF.

6.) **Simplified Fraction:** Now to find my LCM I need to multiply all of the factors. For the LCM I put an L around all the factors. That is 2x2x2x3x4. When I multiply this out I get 96.

Here is another sample using the Cake Method to find the GCF and the LCM of the numbers 24 and 36.

Like I said, I love this method because it’s an efficient way to find GCF and LCM in one model. There are many different strategies and techniques for teaching your students how to find the greatest common factor and the least common multiple. These are some of my favorites because they provide real and strong conceptual understanding, while also challenging students to develop ever more efficient ways to find the GCF and LCM.

## Prime Factorization Video Lesson

## Least Common Multiple and Greatest Common Factor Worksheets

**Least Common Multiple and Greatest Common Factor 6th Grade Math ( **CCSS.MATH.CONTENT.6.NS.B.4 **)**

- 2-7 Assignment SE – Least Common Multiple and Greatest Common Factor
- Homework Assignment – Teacher Edition ( Members Only )
- 2-7 Bell Work SE – Least Common Multiple and Greatest Common Factor
- Bell Work – Teacher Edition ( Members Only )
- 2-7 Exit Quiz SE – Least Common Multiple and Greatest Common Factor
- Exit Quiz – Teacher Edition ( Members Only )
- 2-7 Guide Notes SE – Least Common Multiple and Greatest Common Factor
- Least Common Multiple and Greatest Common Factor Guided Notes – Teacher Edition ( Members Only )
- 2-7 Interactive Notebook – Least Common Multiple and Greatest Common Factor
- LCM and GCF Interactive Notebook – Teacher Edition ( Members Only )
- 2-7 Slide Show – Least Common Multiple and Greatest Common Factor
- LCM and GCF PowerPoint – Teacher Edition ( Members Only )

**Greatest Common Factor 7th Grade Pre-Algebra ( **CCSS.MATH.CONTENT.6.NS.B.4**)**

- 4-3 Assignment Student Edition – Prime Factorization and Greatest Common Factor
- Homework Assignment – Teacher Edition – ( Members Only )
- 4-3 Bell Work SE – Prime Factorization and Greatest Common Factor
- Bell Work – Teacher Edition – ( Members Only )
- 4-3 Exit Quiz SE – Prime Factorization and Greatest Common Factor
- Exit Quiz – Teacher Edition – ( Members Only )
- 4-3 Guided Notes SE – Prime Factorization and Greatest Common Factor
- Guided Notes – Teacher Edition – ( Members Only )
- Lesson Plan – ( Members Only )
- Online Activities – ( Members Only )
- 4-3 Slide Show – Prime Factorization and Greatest Common Factor
- Greatest Common Factor PowerPoint ( Members Only )

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