Students typically learn to add and subtract fractions and mixed numbers in upper elementary school. The denominators must be the same when adding or subtracting fractions, so students must first learn to find the “least common denominator.” The least common denominator (LCD) is the smallest number that both denominators will divide into evenly. Although denominators in math books are kept fairly simple at first, the numbers students are expected to work with begin to increase in sixth and seventh grades, making it more challenging to determine the LCD. As a middle-school math teacher, I know it is important to show students methods that work for both large and small denominators. If a student doesn’t know how to find a common denominator, he is headed for confusion in future math classes.
Even in middle school, some students still confuse numerators and denominators. Give them a memory trick: “down” and “denominator” both begin with the letter “d.” The denominator is “down” on the bottom, under the fraction bar.
You can always simply multiply the denominators together to find a common denominator. The trouble with this method is that the result is often not the least common denominator, so the denominator may be too large to work with comfortably.
If the fractions you’re adding are 5/6 and 1/3, you can multiply the denominators, 6 and 3, and find a common denominator of 18. That number can be used; however, it will require reducing the final answer. It’s best to find the least common denominator – in this case, 6.
Match Least Common Multiples
This is the predominant method used – especially in elementary school – to find the LCD.
To find a number’s multiples, multiply the number by 1, 2, 3, and so on.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28 …
To find the least common denominator (LCD) for two or more fractions, list the denominators’ non-zero multiples until you find a match. That match is the least common denominator.
Find the LCD for 1/6 and 1/8:
Multiples of 6: 6, 12, 18, 24…
Multiples of 8: 8, 16, 24…
The lowest number that’s a multiple of both 6 and 8 is 24; therefore, the LCD for 1/6 and 1/8 is 24.
Practice finding least common multiples/denominators with the entire class by putting IXL’s least common denominator program on an overhead device. Create a customized practice sheet for finding the LCD/LCM at Math Worksheet Site.com.
Use the Greatest Common Factor
Multiply the denominators together and then divide by their greatest common factor (the greatest common factor or GCF is the largest number that will divide into the denominators evenly).
Find the least common denominator for 1/12 and 2/9.
Multiply the denominators: 12 x 9 = 108.
Divide 108 by 3, the GCF of the two denominators: 108/3 = 36.
The least common denominator for 1/12 and 2/9 is 36.
Another form of this method is to divide one of the denominators by its GCF and then multiply the other denominator by that quotient.
12 divided by the GCF, 3, is 4.
9 times 4 is 36.
Choose the denominators you wish to use and create a customized worksheet at Math Teacher’s Assistant.
Large Numbers: Use Prime Factorization
Students need to understand how to find the prime factorization of a number in order to use this method for finding the least common denominator. This method is definitely the way to go when working with larger denominators. Click here for tips on how to teach prime factorization.
- Factor each denominator into primes.
- Count the number of times each prime number appears in each factorization.
- Take the largest of these counts for each number and write it down.
- The least common denominator is the product of those prime numbers you have written.
Find the least common denominator for 1/5, 1/9 and 1/12.
9: 3, 3
12: 2, 2, 3
The most 2 appears on any factor list is two times (for the denominator 12); the most 3 appears is two times (for the denominator 9); the most 5 appears is one time (for the denominator 5).
Multiply: 2 x 2 x 3 x 3 x 5 = 180
The least common denominator for 1/5, 1/9 and 1/12 is 180.
Create your own practice sheet at WorksheetWorks.com to practice using prime factorization to find least common denominators. Download this free worksheet and answer key from MathX.net for practice working with large denominators.
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