Greatest Common Factor & Least Common Multiple

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Students will be able to solve problems involving greatest common factor and least common multiple.

Big Idea

Clearly understanding the difference between factors and multiples builds a foundation for applications and problem solving using GCF and LCM.

Curriculum Reinforcer

5 minutes

In this lesson's Curriculum Reinforcer, I am checking my students' ability to retain information from previously taught standards (NS.3). The tasks is also an opportunity for them to practice using arithmetic to solve real world problems.

David is building a doghouse and needs bolts and washers from the hardware store. How much will David pay for 16 bolts that are 4 inches long and a washer for each?

3-inch bolts

$0.87 each

4-inch bolts

$0.92 each


$0.12 each


My expectation is that my students will have little to no difficulty when solving this problem. I am interested in seeing how they interpret the given information and write a number sentence to model the problem (MP4). 

Instruction and Direct Modeling

5 minutes

I find that my students often confuse Factors and Multiples. In order to help overcome this issue, I take a relatively direct approach. I show this Factors or Multiples Power Point. It begins with definitions, then moves on to comparative identification. It is straight-forward, direct teaching, but it helps my students to be more aware of the difference between these two types of number sets.

My GCF & LCM video presents some of the teaching strategies that I use to explain these concepts to students if they have questions. 

Problem Solving

10 minutes

Once we have clarified the difference between the factors of a number and the multiples of a number, we are ready to move on to some problem. I have two tasks planned for this lesson. 

The first task is called The Promoter Task. For this task one of my students will volunteer to be the promoter and all other students will line up as if they are in line to receive Promotional Tickets for a concert. I provide the Promoter with three types of Tickets: Backstage Passes, Floor Seats, and Free Album Download. The Promoter will then pass out tickets based upon the mathematical criteria described in the problem.

At the end of the exercise one (and only one) lucky student in line will have all 3 types of Promotional Ticket. That student represents the LCM. I will use strategic questioning to lead students to discover that the student with all 3 tickets represent the LCM. Then all students will sit down. Then, I will write the scenario out on the board mathematically so that student can easily see the correlation between the task and the mathematics involved.

Next, I will have my students pretend to be florists as they work on the The Florist Task. For this task they will have three different types of flowers. Working together, students will need to figure out how to arrange the given number of flowers in vases without having any flowers left over. The criteria that they must follow are included in the problem statement. As the students work, I will use strategic questioning to help students discover that the number of vases represents the GCF.

After the students solve the Florist Task, I will write the scenario out on the board mathematically so that the students can easily see the correlation between the task and the use of factors to identify a solution. Students then experience the power that the concept and its application give them as problem solvers. My students typically have a lot of questions during this demonstration. I encourage them to take notes so that they can repeat the process themselves later. 




Guided Practice

10 minutes

Now that we have defined factors and multiples and had some fun solving problems using them, I will give my students a Guided Practice worksheet with four problems that target the application of GCF and LCM. I will work with my students to complete the first two problems on the worksheet. While I am completing these problems, I will ask my students to follow along with me, taking careful notes on their paper.

After we have gone over the first two problems carefully, I will ask my students to complete problems 3 and 4. I will give them one minute and thirty seconds to complete each problem, hoping that this will motivate them to work quickly. As they are completing the problems, I will be traveling the room checking for accuracy as well as any common misconceptions that I may need to address before I can release my students to work independently.

Once the three minutes have elapsed, I will go over the two problems to pointing out some of the things that I observed students doing well or performing incorrectly. 

Group Exploration

20 minutes

To explore today's concepts in a little more depth,  I will organize my students into homogeneous groups of four for this section of the lesson. I want my higher achieving students working together so that I can provide them with some enrichment activities if they finish the task quickly. I also want to avoid the situation where they take over and complete the work for their team members in a heterogeneous group. The concepts of factors and multiples are important throughout secondary math, so I try to provide all students with the opportunity to work at their own pace and process the information that we have covered today.

The tasks that the students will be completing are called:

  1. The Finding Common Factors Task
  2. The School Supplies Task

Each group will complete one of the two tasks and make a presentation about their work. I have chart paper and markers available for students to make a poster for their presentation. Students will check their work with me before they are allowed to go get the supplies necessary for their presentation. 


Closing Summary

20 minutes

To close out today's lesson, students will present their work from the Group Exploration portion of this lesson. The manner in which this will happen is as follows:

  1. The students will be called up according to the task that they completed. 
  2. They will display their chart paper. 
  3. Each group that completed the task being will talk about the process that they went through to solve their scenario. 
  4. This sequence will be repeated for the other task. 

While one group of students is making presentations the other half of the class should be taking notes, asking questions, making comments, and critiquing the work of their peers.

The Homework was included on the bottom of the Guided Practice worksheet.