Lessons 4-5
LEAST COMMON DENOMINATOR
A. Objectives
Students will be able to:
- Change fractions with different denominators to fractions with the same denominator.
- Know the meaning of common denominator and least common denominator.
- Find the least common denominator of two fractions, using least common multiples.
B. Instructional Content and Activity
Finding the common denominator
In the previous lesson, students studied changing fractions to equivalent fractions and reducing fractions to simplest form. Now they will discover how to change two fractions with different denominators to fractions with the same denominator.
First, have students examine the two lists of fractions and notice that they are made up of equivalent fractions.
- 2/3 = 4/6 = 6/9 = 8/12 = 10/15 = 12/18 = ...
- 1/4 = 2/8 = 3/12 = 4/16 = 5/20 = ...
Next, have students point out the fractions with the same denominator (8/12 and 3/12). Get them to remember how to create the equivalent fractions 8/12 and 3/12 from the starting fractions 2/3 and 1/4 (that is, by multiplying both numerator and denominator by the same number):
| 2/3 = | 2 × 4 | = 8/12 | 1/4 = | 1 × 3 | = 3/12 | |
| 3 × 4 | 4 × 3 |
Therefore, 2/3 = 8/12 and 1/4 = 3/12.
(2/3, 1/4) => (8/12, 3/12)
Let students know that the denominator that is the same in both fractions is called the common denominator. In the example above, the common denominator is 12.
In changing fractions with different denominators to fractions with the same denominator, students are finding the common denominator(s). It is very important for students to understand why this process is necessary, as they will be more effective in finding the common denominator when they know the reason for doing so. Let them know that finding common denominators is a necessary process for adding, subtracting, and comparing the sizes of fractions.
Understanding the least common denominator
Students have learned the meaning, purpose, and method for finding common denominators. In this section, they discover that a pair of fractions can have many common denominators, and learn the meaning and importance of the least common denominator.
Start by listing equivalent fractions for 3/4 and 1/6:
3/4 = 6/8 = 9/12 = 12/16 = 15/20 = 18/24 = 21/28 = 24/32 = 27/36 = 30/40 = 33/44 = ...
1/6 = 2/12 = 3/18 = 4/24 = 5/30 = 6/36 = 7/42 = 8/48 = ...
Get students to identify the common denominators (12, 24, and 36).
That is, the fractions with the same denominator are as follows:
| (3/4, 1/6) | => | (9/12, 2/12) |
| (18/24, 4/24) | ||
| (27/36, 6/36) | ||
In this way, students come to realize that there are many sets of fractions with the same denominator, as the numerator and denominator can be multiplied indefinitely.
Explain that, for calculations that require changing to a common denominator, it is easiest to use the smallest number out of all the given fractions' common denominators. This is known as the least common denominator. The least common denominator for the fractions 3/4 and 1/6 is 12.
| (3/4, 1/6) | => | (9/12, 2/12) |
Finding the least common denominator using least common multiples
In this section, students learn a method for finding the least common denominator of two fractions by determining the least common multiple of the denominators.
Using the starting fractions 5/8 and 7/12, show students how to break the denominators down in order to determine the least common multiple. (In this and all similar problems throughout the unit, students should first be sure the starting fractions are in simplest form, to avoid a false result.)
| 2 ) | 8 | 12 |
| 2 ) | 4 | 6 |
| 2 ) | 2 | 3 |
| 1 | 3 | |
The least common multiple is
2 × 2 × 2 × 1 × 3 = 24
Thus, the least common denominator for the fractions of 5/8 and 7/12 is the least common multiple of their denominators, or 24.
Next, have students convert 5/8 and 7/12 to equivalent fractions with the denominator 24. They must figure out what number 8 is multiplied by to get 24 and then multiply the numerator by the same number (3). The same process is followed for 7/12 (multiply by 2).
| 5/8 = | 5 × 3 | = 15/24 |
| 8 × 3 | ||
| 7/12 = | 7 × 2 | = 14/24 |
| 12 × 2 |
Thus, using the least common denominator, (5/8,7/12) is changed to (15/24, 14/24).
To increase their understanding, have students use the method of least common multiple to find the least common denominator for 5/6 and 9/10.
| 2 ) | 6 | 10 |
| 3 ) | 3 | 5 |
| 1 | 5 | |
The least common multiple is
2 × 3 × 1 × 5 = 30
Get students to realize that the least common denominator for 5/6 and 9/10 is the least common multiple of their denominators, or 30. Then have them convert the starting fractions to equivalent fractions with the least common denominator.
| 5/6 = | 5 × 5 | = 25/30 |
| 6 × 5 | ||
| 9/10 = | 9 × 3 | = 27/30 |
| 10 × 3 |
(5/6, 9/10) => (25/30, 27/30)
Reinforce the idea that finding the least common multiple of the denominators, when the starting fractions have different denominators, is the fastest way to find their least common denominator.
C. Additional Points to Consider in Teaching and Evaluation
Teaching considerations
- Particular emphasis should be given to using the least common multiple for finding the least common denominator. The method of simply multiplying the two denominators should be avoided in most cases.
- However, when the two denominators are both prime numbers, multiplication of denominators is appropriate. For example, in the case of starting fractions 1/2 and 1/3, the denominators 2 and 3 are prime numbers, so the two are multiplied to get the least common denominator. The fractions are then converted using the common denominator 6.
| 1/2 = | 1 × 3 | = 3/6 | 1/3 = | 1 × 2 | = 2/6 | |
| 2 × 3 | 3 × 2 |
(1/2, 1/3) => (3/6, 2/6)
Evaluation considerations
-
If you find that a student is simply multiplying the denominators and getting results like the following
- (5/8, 7/12) => (60/96, 56/96)
- (5/6, 9/10) => (50/60, 54/60)
discovered from the evaluation process the student should be given remedial instruction.