## Learning Math: Number and Operations

# Fractions and Decimals Part C: Ordering Fractions (25 minutes)

There are several ways to compare fractions, many of which use benchmarks or intuitive methods and do not require computation of common denominators or converting to decimal form. See Note 3 below.

When ordering fractions, use 0, 1/2, and 1 as benchmarks for comparison. That is, first determine whether the fraction is more or less than 1. If it is less than 1, check to see if it is more or less than 1/2. Then further refine the comparisons to see if the fraction is closer to 0, 1/2, or 1.

**Problem C1
**

**a.**What quick method can you use to determine if a fraction is greater than 1?

**b.**What quick method can you use to determine if a fraction is greater or less than 1/2?

**Problem C2
**Organize the following fractions according to these benchmarks: 0 to 1/2, 1/2 to 1, greater than 1:

After you organize fractions by benchmarks, you can use these intuitive methods:

**• **Same denominators: If the denominators of two fractions are the same, just compare the numerators. The fractions will be in the same order as the numerators. For example, 5/7 is less than 6/7.

**• **Same numerators: If the numerators of two fractions are the same, just compare the denominators. The fractions should be in the reverse order of the denominators. For example, 3/4 is larger than 3/5, because fourths are larger than fifths.

**• **Compare numerators and denominators: You can easily compare fractions whose numerators are both one less than their denominators. The fractions will be in the same order as the denominators. (Think of each as being a pie with one piece missing: The greater the denominator, the smaller the missing piece, thus, the greater the amount remaining.) For example, 6/7 is less than 10/11, because both are missing one piece, and 1/11 is a smaller missing piece than 1/7.

**• **Further compare numerators and denominators: You can compare fractions whose numerators are both the same amount less than their denominators. The fractions will again be in the same order as the denominators. (Think of each as being a pie with x pieces missing: The greater the denominator, the smaller the missing piece; thus, the greater the amount remaining.) For example, 3/7 is less than 7/11, because both are missing four pieces, and the 11ths are smaller than the sevenths.

**• **Equivalent fractions: Find an equivalent fraction that lets you compare numerators or denominators, and then use one of the above rules.

**Problem C3
**Arrange these fractions in ascending order:

**a. **

**b. **

**c. **

**d. **

**e. **

**Problem C4
**Use benchmarks and intuitive methods to arrange the fractions below in ascending order. Explain how you decided. (The point of this exercise is to think more and compute less!):

### Notes

**Note 3
**Computing and then comparing common denominators can be extremely tedious, as is changing everything to decimals and then comparing the decimals. Try these common-sense methods — you’ll like them!

**Note 4
**It is important to be able to picture fractions in your mind (that’s one reason using manipulatives is important) — at least the ones with single-digit denominators. For example, if you use a pie to visualize, let’s say, 2/3 and 4/5, it’s pretty easy to quickly tell which one is larger.

### Solutions

**Problem C1
a. **If the numerator is larger than the denominator, the fraction is greater than 1.

**b.**If twice the numerator is larger than the denominator, the fraction is greater than 1/2. If twice the numerator is smaller than the denominator, the fraction is less than 1/2.

**Problem C2
Here are the fractions within each range:
**0 to 1/2: 17/35, 2/9

**•**

**1/2 to 1: 4/7, 14/15**

**•****Greater than 1: 25/23**

**•****Problem C3
**

**a.**These fractions have the same denominator. The order is 4/17, 7/17, 12/17.

**b.**These fractions have the same numerator. The order is 3/8, 3/7, 3/4.

**c.**These fractions all have numerators that are one less than their denominators. The order is 3/4, 5/6, 7/8.

**d.**These fractions all have numerators that are five less than their denominators. The order is 1/6, 8/13, 12/17.

**e.**These fractions all have numerators that are one less than their denominators. The order is 2/3, 5/6, 10/11.

**Problem C4
**First divide the list into fractions larger and smaller than 1/2:

• |
Smaller than 1/2: 2/5, 1/3, 1/4 |

• |
Larger than 1/2: 5/8, 3/4, 2/3, 4/7 |

For those smaller than 1/2, 2/5 is larger than 1/4 (numerators three less than denominators), 1/3 is larger than 1/4 (same numerator), and 2/5 is larger than 1/3 (compare using equivalent fractions 6/15 and 5/15). The order is 1/4, 1/3, 2/5.

For those larger than 1/2, convert 3/4 and 2/3 to fractions where the numerator is three less than the denominator. (Or you may be able to visualize that 3/4 is greater than 2/3.) The list becomes 5/8, 9/12, 6/9, 4/7. This makes it easy to put the list in order: 4/7, 5/8, 6/9, 9/12.

The complete list is 1/4, 1/3, 2/5, 4/7, 5/8, 2/3, 3/4.