Finding Greatest Common Factors (GCF) Using T-Charts

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SWBAT determine common factors and the GCF of two numbers less than or equal to 100 by listing all the factors of both numbers

Big Idea

There is always a greatest number that divides evenly into each of two whole numbers. Sometimes that number is 1.

Think About It

7 minutes

Students work in pairs on the Think About It  problem.  After 3 minutes of work time, we debrief the problem as a class.  I ask for students to share how the decided to attempt this problem.  Some students may have used a guess and check strategy, while others might have used an organized list.  If any student used a t-chart to organize the factor pairs, I highlight that strategy and show it on the doc cam in my classroom.

I frame the lesson by telling students that in this lesson we are going to use what we know about factors and divisibility to determine the greatest common factor of two numbers.  We are going to t-charts to help us show our factors and common factors.  

Intro to New Material

10 minutes

This lesson builds off of the Performance Task lesson taught the day before.  In that lesson, students explored ways to create the largest number of equal groups of items.  In this lesson, students strengthen their ability to use organized lists of factor pairs to identify the GCF.  Students will be asked to determine the common factors and/or GCF of two numbers less than or equal to 100. 

To start the Intro to New Material section, I have students fill in the definitions at the top of the page:

Common factors – factors that given numbers have in common.

Greatest common factor – the largest number that is a factor of given numbers.  

I have a student read the first example aloud.  Once we've annotated the problem, I guide students through the steps below:

Steps for Finding Factors of  a Number

1)      Create a small T-Chart and write the number on top.

2)      On the left side of the chart start with the first factor, 1. On the right side of the chart across from the 1, next to the 1 write what number you multiply 1 by to get the original number, which will be that original number.

3)      Continuing in numerical order, try each number mentally (or by dividing off to the side), and write each factor pair.

4)      Stop when a factor pair repeats itself.


Steps for Finding the GCF

1)      Find the factors of all numbers involved in separate t-charts using the steps for finding factors

2)      Starting with 1 in both columns, go through each number and see if it appears in both columns.  If it does, it is a common factor and circle it.

3)      Once you are done, identify the largest number you have circled.


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 pair.  I am looking for:

  • Are students explaining their thinking to their partner?
  • Are students annotating each problem?
  • Are students writing their work in the work space?
  • Are students working systematically to find all of the factors?
  • Are students circling the common factors and clearly identifying the GCF?
  • Are students answering the question in a full sentence?


I am asking:

  • Explain you determined the factors. 
  • How did you know these were the common factors?
  • How did you know this was the GCF?
  • What does this mean, given the context of the problem?
  • What numbers is ___ divisible by?  How do you know?  Is it divisible by ______?  How do you know it isn’t?


Because this is the second lesson I teach in the school year, I am also very publicly praising groups during this time for quiet voices (we call them 'restaurant style' voices at our school) and on-task conversation.



Independent Practice

15 minutes

Students work on the Independent Practice problem set.

In this lesson, students often miss factor pairs.  As I circulate, I make note of the students who are struggling with divisibility rules or who are not fluent with their multiplication facts.  I'll use this information to plan cumulative review and interventions for the kids who need it the most.

I'll also model the thinking I want kids to go through as they work.  For example, if I notice a student does not write 2 and 18 in a t-chart for 36, I will say to that student (loud enough for kids around to also hear): "I notice 36 is even.  I know that every even number can be evenly divided by 2.  I like to think of money.  If I had $36, and you and I were going to split it, how much would we each get?"  If a student is not sure if a number is a factor as they work systematically, I prompt them to list the multiples of the number they want to test.  I'll 'roll' the multiples with them, if they need some scaffolding:  "Is 3 a factor of 36?  Let's roll our 3s to find out.  3, 6, 9, 12....36.  Oh, we said 36.  3 is a factor of 36!  How many multiples did we list? 12.  3 x 12 is 36"

I want students to build the strategies they can access when they need to think about divisibility rules.  I do provide more scaffolding in this lesson than I would later in the year so that students start to hear multiple strategies the can use if ever they get stuck.  

Closing and Exit Ticket

10 minutes

After independent work time, I bring the class back together for a conversation about Problem 4.  I ask the class to turn and talk with their partners about this question: What did we have to know about the context, in order to make sense of the problem?  I want students to talk about squares, and what that means for the dimensions.  I highlight drawing a picture as a strategy that would help them to make sense of this problem.

Students then work on the Exit Ticket independently to close the lesson.  My key illustrates the work that I expect to see from students on their exit tickets.