﻿ Fractions and Decimals: Lesson 1

 Reduction of fractions and quotients as fractions. Lesson 2-3 Least common denominator. Lesson 4-5 Least common denominator of three fractions. Lesson 6 Comparing the sizes of fractions. Lesson 7 Relationship between a fraction and a decimal. Lesson 8 Comparing the sizes of a fraction and a decimal. Lesson 9 Practicing the operations and relationships between fractions and decimals. Lesson 10

# EQUIVALENT FRACTIONS

## A. Objectives

Students will be able to:

1. Understand equivalent fractions.
2. Understand that the value of a fraction does not change if the denominator and the numerator are divided by a common divisor.
3. Find equivalent fractions.

## B. Instructional Content and Activity

### Equivalent fractions

To better understand equivalent fractions, students will compare the sizes of actual objects and express the relationships as fractions.

Equally divide two rectangular pieces of cardboard or colored paper; shade some of the parts as shown below. Tell students that the rectangles represent two same-sized garden patches, with equally divided areas set aside for particular crops.

Get students to observe that the cabbage section and the radish section are the same size. Help them realize that therefore the fractions representing each section are also the same size.

`4/10 = 2/5`

Next, have students compare the sizes of the two fractions using number lines.

Explain again that 4/10 = 2/5.

### Dividing numerator and denominator by the same non-zero number

Here, students understand that fractions with equal value but different names (equivalent fractions) can be created by dividing the numerator and the denominator by the same non-zero number.

Help students work through the following problems:

 6/15 = 6 ÷ 3 = 2 8/20 = 8 ÷ 4 = 2 10/25 = 10 ÷ 5 = 2 12/30 = 12 ÷ 6 = 2 15 ÷ 3 5 20 ÷ 4 5 25 ÷ 5 5 30 ÷ 6 5

Be sure they note that the answers are all the same. Each starting fraction was reduced to 2/5 by dividing its numerator and denominator by the same non-zero number. Since all the starting fractions are equivalent to 2/5, they are also equivalent to each other. That is,

`2/5 = 4/10 = 6/15 = 8/20 = 10/25 = 12/30 = ...`

Students should observe that all these fractions have the same value but different names.

71

### Creating fractions equivalent to 16/24

Deepen students' understanding of equivalent fractions by having them find different fractions equal to 16/24.

First, have students divide both the numerator 16 and the denominator 24 by 2. This results in the fraction 8/12. Help them see that 8/12 is equal to 16/24.

Next have them divide the numerator and denominator of 16/24 by 4. The answer is 4/6, which is also equal to 16/24.

Repeat these steps, dividing the numerator and denominator of 16/24 by 8. The result is 2/3, another fraction equal to 16/24.

 16/24 = 16 ÷ 2 = 8/12 = 2/3 24 ÷ 2 16/24 = 16 ÷ 4 = 4/6 = 2/3 24 ÷ 4 16/24 = 16 ÷ 8 = 2/3 24 ÷ 8

Help students to realize that, because 8/12, 4/6, and 2/3 are all equal to 16/24, they are also equal to each other.

Remind students that dividing both the numerator and the denominator of any fraction by the same non-zero number (2, 4, or 8 in the exercises above) produces another fraction equal in value to the first. These are called equivalent fractions.

### Creating fractions equivalent to 36/60

As with the previous example, and starting with the fraction 36/60, have students divide both the numerator and the denominator by various numbers (2, 6, and 12). Be sure they understand that the resulting fractions are all equal to 36/60.

 36/60 = 36 ÷ 2 = 18/30 = 3/5 60 ÷ 2 36/60 = 36 ÷ 6 = 6/10 = 3/5 60 ÷ 6 36/60 = 36 ÷ 12 = 3/5 60 ÷ 12

When the numerator and the denominator of the fraction 36/60 are divided by 2, 6, and 12, the results are fractions equivalent to 36/60. Furthermore, dividing the numerator and the denominator by any non-zero number results in fractions that are the same size as (equivalent to) the original fraction.

Get students to express the general principle: If the numerator and the denominator of a fraction have a common divisor, they can be divided by that number to produce another fraction equivalent to the original fraction.

### Finding fractions equivalent to 3/5

Have students examine the presented fractions (5/3, 6/10, 6/5, 9/15 , and 15/25). They should notice that in the starting fraction (3/5), the denominator 5 is larger than the numerator 3. Help them realize that therefore, any fraction whose denominator is smaller than its numerator cannot be equivalent to 3/5. They can therefore rule out 5/3 and 6/5 in their search for fractions equivalent to 3/5.

For each of the remaining fractions, students should examine the numerator and denominator and identify a common divisor. They should then divide by that number to get an equivalent fraction.

 6 ÷ 2 = 3 9 ÷ 3 = 3 15 ÷ 5 = 3 10 ÷ 2 5 15 ÷ 3 5 25 ÷ 5 5

As shown, the fractions equivalent to 3/5 are 6/10, 9/15, and 15/25.

## C. Additional Points to Consider in Teaching

• Equivalent fractions are fractions that are equal in value but expressed with different denominators.
• Teaching equivalent fractions aims at helping students have deeper understanding of fractions of the same value.
• Therefore, it is very important that students clearly understand the meaning of equivalent fractions.