Number and Operations Session 7, Part C: Ordering Fractions

Session 7, 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. Note 3

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. Note 4

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!):

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