Multiples, LCM, and GCF

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Objective

SWBAT: • Define multiple, LCM, and GCF • List the first 10 multiples of a given number • Find the LCM of two numbers less than 12 • Find the GCF of two numbers less than 50

Big Idea

What are multiples? How do they compare with factors? Students work to answer these questions and develop an understanding of greatest common factor and least common multiple.

Do Now

10 minutes

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Students love doing anything with their birthday J.  Here, students must decide where their day fits in the Venn diagram.

To share about the do now, I have students think-pair-share.  See my Think Pair Share video in my Strategy Folder for more details.  I ask for volunteers to share where the given numbers belong in the diagram.  I ask students, “What other numbers, besides 2, are even and prime?”  None, of course :)

Greatest Common Factor

10 minutes

I read the objectives for the day and mention that greatest common factor is a skill that will help us when we work with fractions.  Students take notes and look over my example.  I have students work independently on the examples on the next page.

If students struggle with multiplication facts, I give them a multiplication chart or the factors reference sheet.  This way these students are still able to access the task. 

I ask volunteers to show and explain their work for number 1 and 2 on the document camera.

Multiples

10 minutes

Students take notes on multiples.  Many students struggle to differentiate between factors and multiples.  I try to use the phrase “factor pair” often, which emphasizes the fact that factors are two numbers that result in a product when you multiply them.  I talk about skip counting in connection with multiples.  If something is a multiple of 10, you can skip count by tens and eventually say the number.

I have students work on the practice page in pairs.  If students successfully complete the page, I have them work on create dot diagrams (see my lesson Brownies & Factors) that are higher than 49. 

Once most students have completed the practice page, we come together.  I ask students what they notice about multiples of 9.  I have students think-pair-share about this question. I am looking for students to notice that if you add the digits of any multiple of 9 you get a number that is a multiple of 9.  For example, 63 is a multiple of 9 and 6+3 = 9.  Or 279 is a multiple of 9 and 2+7+9 = 18, which is a multiple of 9.  Students should then be able to decide that 113 is not a multiple of 9 because 1+1+3= 5, which is not a multiple of 9.  Students can prove this by dividing 113 by 9.  Students should notice a similar rule with multiples of 3.  These questions have students engaging in MP 7: Look for and make use of structure and MP8: Look for and express regularity in repeated reasoning.   

Least Common Multiple & Practice

15 minutes

Students take notes on least common multiple and we work through problems 1 and 2 together. 

For number 3 I have students participate in a Think Write Pair Share. A common misconception is that you can always find the least common multiple by multiplying the two numbers together.  I am looking for students to use #1 as a counterexample.  6x9 = 54, which is a common multiple of 6 and 9, but it is not the least common multiple.  The least common multiple is 18.  I ask students how we could write a new statement that is correct.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

Students work independently on the practice page. I have these answers posted around the room, so students can check their work.  See my Posting A Key video in my Strategy Folder for more details.  I am walking around to monitor student progress.  I am looking to see student responses for #3, 11, and 12.  If students are confusing LCM and GCF I have them go back to their notes and review.

If students successfully complete the practice, they can work on the challenge question about ferris wheels.

Closure and Ticket to Go

15 minutes

 See my Closure video in my Strategy Folder for more details.  I have students participate in a think-pair-share about #3 on the practice page.  I want students to prove their answer.  For example, 7 is a factor of 7 because 7x1=7.  Or, 7 is a multiple of 7 because when you skip count by 7, the first number to say is 7.  I want students to recognize that both answers are correct.

I ask students how they keep GCF and LCM separate in their heads.  I call on students to share out strategies.

I pass out the Ticket to Go for students to complete independently.  

If you have extra time, go back to the dot diagram.  Have students look at the last column of numbers.  What do they notice?  All of these numbers have a common factor of 7.  See if students can identify numbers that have common factors, just by looking at the dots.  Students are engaging in MP 7: Look for and make use of structure and MP8: Look for and express regularity in repeated reasoning.   

Reflection + Student Work

I collected the tickets to go to see what students understood about factors and multiples and what gaps in understanding they had.  I corrected the tickets to go and grouped them in the following way:

Novice:  These students struggled to distinguish between factors and multiples.  This particular student was able to list the factors of 10 and the first five multiples of 10, but was unable to use that knowledge to answer problems 3 and 4 correctly.  Furthermore, this student uses multiples to find the greatest common factor and least common multiple of 8 and 6.  This student does not understand the difference between factors and multiples and how they can be used. Unit 1.7 Student Work N.jpg

Approaching Mastery:  These students were able to distinguish between factors and multiples, the problem came when they needed to find the greatest common factor and least common multiple of 8 and 6.  This particular student was able to correctly find the greatest common factor of 6 and 8 by listing out the factors.  For problem 6, this student found the least common factor (which will always be 1) instead of finding the least common multiple.  A number of other students made a similar mistake, so I will address this misconception briefly during the next block. Unit 1.7 Student Work AM.jpeg

Proficient:  These students were able to distinguish between factors and multiples.  They were also able to apply these skills in order to correctly find the greatest common factor and least common multiple.  They showed their work. Unit 1.7 Student Work P.jpeg

For this situation, I did not include an “advanced” category because the content did not require students to explain or analyze their work.   Most students were able to distinguish between multiples and factors, but made a careless mistake while finding the greatest common factor or least common multiple.  For the few students who were novices, I will be pulling them to work with me on the review lesson before the quiz on factors, multiples, GCF and LCM.