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Sample lesson - Least Common Multiple and Denominator

SAMPLE LESSON: Basic Math: Least Common Multiple and Denominator

Students find the least common multiple (LCM) and least common denominator (LCD) of two and three numbers.

Definition of Term LEAST COMMON MULTIPLE: The least common multiple of two or more numbers is the smallest number (excluding zero) that is a multiple of all of them.

Example: Find the LCM of 6 and 8
The multiples of 6 are 6, 12, 18, 24, 30, etc.
The multiples of 8 are 8, 16, 24, 32, 40, etc.
The smallest multiple that appears in both sets is 24
24 is the least common multiple (LCM) of 6 and 8.

NOTE: An interesting characteristic of the least common multiple (LCM) of two or more numbers is that it is the smallest whole number that can be evenly divided by each of the numbers.

The LCM of 2 and 3 is 6.
The LCM of 2 and 6 is 6.
The LCM of 2, 3, and 5 is 30.
The LCM of 3, 4, 6, and 12 is 12.

*If the largest number is a multiple of the other numbers, it is the least common multiple.

TIP: To find the LCM, list the first few multiples of each number and select the smallest multiple that appears in (or is COMMON to) both lists.

FIND the LCM of 2 and 3:
Multiples of 3 are 3, 6, 9. . .
Multiples of 2 are 2, 4, 6. . .
The smallest multiple that appears in both lists is 6.
Thus, 6 is the LCM of 2 and 3.

Using PRIME FACTORIZATION to find the LCM: Find the LCM of 12 and 15

1. Factor each of the numbers into its prime factors
12 = 2 x 2 x 3
15 = 3 x 5

2. For each prime, find the MOST number of times it is used in any one factorization.
12 = 2 x 2 x 3
15 = 3 x 5
2 appears twice, 3 appears once, and 5 once.

3. Multiply these prime factors together to get the LCM.
LCM = 2 x 2 x3 x 5 = 60.

NOTE: The Least Common Multiple (LCM) is the same as the least or Lowest Common Denominator (LCD).

The Lowest Common Denominator (LCD) is used to rename or change unlike fractions to like fractions in the addition and subtraction of fractions and mixed numbers.

RULE: To find the Least Common Denominator (LCD) of two fractions, find the Least Common Multiple of their denominators.

Example: Find the Least Common Denominator for 2/3 and 3/4.
Multiples of 3 are 3, 6, 9, 12. . .
Multiples of 4 are 4, 8, 12, 16. . .
The LCD is 12 because 12 is the Least Common Multiple of the denominators 3 and 4.

Comments

Luis P.'s explanation of Least Common Multiples as well as Denominators is a great example of logical thinking. I really enjoyed revisiting the land of LCMs and LCDs. His sample lesson reminded me to pay attention to prime factors of each number as well as denominators in fractions and mixed numbers.
Mary M, Thank you for your encouraging note. The real trick is making one's own enjoyment and appreciation of the beauty of math to children who sometime have little interest in the subject. That's when you have to draw on all your 'soft' or human skills as well as your inner "Mr. Spock"!