Comparing and Ordering Fractions

Key Concepts

When comparing and ordering fractions, we focus on understanding the magnitude of fractions and arranging them in a specific sequence. The key concepts include:

• Finding a Common Denominator
• Comparing Numerators
• Ordering Fractions

Finding a Common Denominator

To compare fractions, it is often necessary to find a common denominator. This means converting the fractions so they all have the same denominator. The common denominator is usually the least common multiple (LCM) of the denominators.

For example, to compare 1/3 and 2/5, find the LCM of 3 and 5, which is 15. Convert each fraction:

1/3 = 5/15 (multiply numerator and denominator by 5)

2/5 = 6/15 (multiply numerator and denominator by 3)

Comparing Numerators

Once the fractions have a common denominator, compare their numerators. The fraction with the larger numerator is the larger fraction.

For example, comparing 5/15 and 6/15:

Since 6 > 5, 6/15 > 5/15, so 2/5 > 1/3.

Ordering Fractions

Ordering fractions involves arranging them in ascending or descending order. Ascending order means from smallest to largest, and descending order means from largest to smallest.

For example, to order the fractions 1/4, 3/8, and 5/16:

Find the LCM of 4, 8, and 16, which is 16. Convert each fraction:

1/4 = 4/16 (multiply numerator and denominator by 4)

3/8 = 6/16 (multiply numerator and denominator by 2)

5/16 remains 5/16

Now compare the numerators: 4, 6, and 5.

Ascending order: 4/16, 5/16, 6/16, which corresponds to 1/4, 5/16, 3/8.

Descending order: 6/16, 5/16, 4/16, which corresponds to 3/8, 5/16, 1/4.

Examples and Analogies

Imagine you have three pies of different sizes. You cut the first pie into 4 equal pieces, the second into 8 equal pieces, and the third into 16 equal pieces. To compare the sizes of the pieces, you need to make sure they are all the same size by finding a common denominator.

Another analogy is comparing the lengths of ribbons. If you have ribbons cut into different lengths, you can compare them by converting them to the same unit of measurement, just like finding a common denominator for fractions.

Practice Exercise

Try ordering the following fractions in ascending and descending order:

Fractions: 2/3, 3/4, 5/6

Find the LCM of 3, 4, and 6, which is 12. Convert each fraction:

2/3 = 8/12 (multiply numerator and denominator by 4)

3/4 = 9/12 (multiply numerator and denominator by 3)

5/6 = 10/12 (multiply numerator and denominator by 2)

Ascending order: 8/12, 9/12, 10/12, which corresponds to 2/3, 3/4, 5/6.

Descending order: 10/12, 9/12, 8/12, which corresponds to 5/6, 3/4, 2/3.