ILLINOIS TEACH & TALK

Compute Fluently and Find Common Factors and Multiples

This is the first grade level in which students will be formally taught about LCM and GCF.  For the most part, strategies, will remain the same but using and applying the distributive property is a major focus as well.  

 

Students will find the greatest common factor of two whole numbers less than or equal to 100. For example, the greatest common factor of 40 and 16 can be found by

1)   listing the factors of 40 (1, 2, 4, 5, 8, 10, 20, 40) and 16 (1, 2, 4, 8, 16), then taking the greatest common factor      (8). Eight (8) is also the largest number such that the other factors are relatively prime (two numbers with      no common factors other than one). For example, 8 would be multiplied by 5 to get 40; 8 would be    

     multiplied by 2 to get 16. Since the 5 and 2 are relatively prime, then 8 is the greatest common factor. If  

     students think 4 is the greatest, then show that 4 would be multiplied by 10 to get 40, while 16 would be 4    

     times 4. Since the 10 and 4 are not relatively prime (have 2 in common), the 4 cannot be the greatest

     common factor.

2)  listing the prime factors of 40 (2 • 2 • 2 • 5) and 16 (2 • 2 • 2 • 2) and then multiplying the common factors

     (2 • 2 • 2 = 8).

 

Students also understand that the greatest common factor of two prime numbers will be 1.

Students find the least common multiple of two whole numbers less than or equal to twelve. For example, the least common multiple of 6 and 8 can be found by

1)  listing the multiplies of 6 (6, 12, 18, 24, 30, …) and 8 (8, 26, 24, 32, 40…), then taking the least in common  

     from the list (24); or

2)  using the prime factorization.

 

Step 1: find the prime factors of 6 and 8.

6 = 2 • 3

8 = 2 • 2 • 2

 

Step 2: Find the common factors between 6 and 8. In this example, the common factor is 2

 

Step 3: Multiply the common factors and any extra factors: 2 • 2 • 2 • 3 or 24 (one of the twos is in common; the other twos and the three are the extra factors.

 

Examples:

* What is the greatest common factor (GCF) of 24 and 36? How can you use factor lists or the prime factorizations to

   find the GCF?  

Solution: 22•3 = 12. Students should be able to explain that both 24 and 36 have 2 factors of 2 and one factor of 3, thus 2 x 2 x 3 is the greatest common factor.)

 

* What is the least common multiple (LCM) of 12 and 8? How can you use multiple lists or the prime factorizations

   to find the LCM? 

Solution: 23• 3 = 24. Students should be able to explain that the least common multiple is the smallest number that is a multiple of 12 and a multiple of 8. To be a multiple of 12, a number must have 2 factors of 2 and one factor of 3 (2 x 2 x 3).  To be a multiple of 8, a number must have 3 factors of 2 (2 x 2 x 2). Thus the least common multiple of 12 and 8 must have 3 factors of 2 and one factor of 3 ( 2 x 2 x 2 x 3).

 

* Rewrite 84 + 28 by using the distributive property. Have you divided by the largest common factor? How do you

   know?

 

* Given various pairs of addends using whole numbers from 1-100, students should be able to identify if the two

  numbers have a common factor. If they do, they identify the common factor and use the distributive property to       rewrite the expression. They prove that they are correct by simplifying both expressions.

27 + 36 = 9 (3 + 4)

63 = 9 x 7

63 = 63

 

31 + 80

 

There are no common factors. I know that because 31 is a prime number, it only has 2 factors, 1 and 31. I know that 31 is not a factor of 80 because 2 x 31 is 62 and 3 x 31 is 93.