Compare Fractions | Fifth Grade

Complete Notes & Formulas

1. Graph and Compare Fractions on Number Lines

Definition: A number line helps visualize fractions and compare their values. Fractions further to the right are greater, and fractions further to the left are smaller.

📝 Steps to Graph Fractions on a Number Line:

Step 1: Draw a number line with two whole numbers (usually 0 and 1)
Step 2: Divide the space between 0 and 1 into equal parts based on the denominator
Step 3: Count from 0 according to the numerator and mark that point
Step 4: Label the point with the fraction

🔑 Comparing on Number Lines:

Fractions on the RIGHT are GREATER →

← Fractions on the LEFT are SMALLER

✏️ Example: Compare 2/5 and 3/5

On a number line divided into fifths:

0 ----[1/5]----[2/5]----[3/5]----[4/5]---- 1

• 2/5 is to the left of 3/5

• 3/5 is to the right of 2/5

Therefore: 2/5 < 3/5 or 3/5 > 2/5

2. Compare Fractions Using Benchmarks

Definition: Benchmark fractions are common fractions (like 0, 1/4, 1/2, 3/4, and 1) used as reference points to estimate and compare other fractions.

📐 Common Benchmark Fractions:

0 --- 1/4 --- 1/2 --- 3/4 --- 1

🔑 How to Use the 1/2 Benchmark:

If...	Then...	Example
Numerator < Half of Denominator	Fraction < 1/2	3/8 < 1/2 (3 < 4)
Numerator = Half of Denominator	Fraction = 1/2	4/8 = 1/2
Numerator > Half of Denominator	Fraction > 1/2	5/8 > 1/2 (5 > 4)

✏️ Example: Compare 3/10 and 7/8 using benchmarks

Step 1: Compare each to 1/2

• 3/10: Is 3 more or less than half of 10?

Half of 10 = 5, and 3 < 5

So 3/10 < 1/2

• 7/8: Is 7 more or less than half of 8?

Half of 8 = 4, and 7 > 4

So 7/8 > 1/2

Step 2: Compare: 3/10 < 1/2 < 7/8

Therefore: 3/10 < 7/8

💡 Quick Benchmark Checks:

Close to 0: Numerator is very small (Example: 1/10, 1/12)
Close to 1/4: Numerator is about 1/4 of denominator (Example: 2/9, 3/13)
Close to 1/2: Numerator is about half of denominator (Example: 5/9, 6/13)
Close to 3/4: Numerator is about 3/4 of denominator (Example: 6/8, 9/12)
Close to 1: Numerator almost equals denominator (Example: 9/10, 11/12)

3. Compare Fractions and Mixed Numbers

Definition: When comparing fractions and mixed numbers, there are several strategies depending on the types being compared.

📝 Strategies for Comparing:

Strategy 1: Compare Whole Numbers First

When comparing mixed numbers, look at whole numbers first.

Example: 3 1/2 > 2 5/6 (because 3 > 2)

Strategy 2: Same Whole Numbers? Compare Fractions

If whole numbers are equal, compare the fraction parts.

Example: 4 3/5 vs 4 2/3 → Compare 3/5 and 2/3

Strategy 3: Convert to Common Form

Use common denominators for fractions
Convert mixed numbers to improper fractions
Convert to decimals

✏️ Example 1: Compare 5 2/3 and 5 3/4

Step 1: Whole numbers are the same (both are 5)

Step 2: Compare fractions: 2/3 vs 3/4

Step 3: Find LCD: LCD of 3 and 4 = 12

• 2/3 = 8/12

• 3/4 = 9/12

Step 4: 8/12 < 9/12

Answer: 5 2/3 < 5 3/4

✏️ Example 2: Compare 7/8 and 1 1/4

Method: One is a fraction, one is a mixed number

• 7/8 < 1 (proper fraction is less than 1)

• 1 1/4 > 1 (mixed number is greater than 1)

Answer: 7/8 < 1 1/4

4. Put Fractions in Order

Definition: Ordering fractions means arranging them from least to greatest (ascending) or greatest to least (descending).

📐 Methods to Order Fractions:

Method 1: Common Denominator

Find LCD (Least Common Denominator)
Convert all fractions to equivalent fractions with LCD
Compare numerators (smaller numerator = smaller fraction)
Write original fractions in order

Method 2: Convert to Decimals

Divide numerator by denominator for each fraction
Compare decimal values
Write original fractions in order

Method 3: Benchmark Comparison

Compare each fraction to benchmarks (0, 1/2, 1)
Group fractions by benchmark ranges
Order within each group

✏️ Example 1: Order from least to greatest: 3/4, 1/2, 5/6

Using Common Denominator Method:

Step 1: Find LCD of 4, 2, and 6 → LCD = 12

Step 2: Convert to equivalent fractions:

• 3/4 = 9/12

• 1/2 = 6/12

• 5/6 = 10/12

Step 3: Compare numerators: 6 < 9 < 10

So: 6/12 < 9/12 < 10/12

Answer: 1/2 < 3/4 < 5/6

✏️ Example 2: Order from greatest to least: 2/5, 3/10, 1/2

Using Decimal Method:

Step 1: Convert to decimals:

• 2/5 = 0.4

• 3/10 = 0.3

• 1/2 = 0.5

Step 2: Order decimals: 0.5 > 0.4 > 0.3

Answer: 1/2 > 2/5 > 3/10

💡 Quick Tips for Ordering:

Same Denominator: Compare numerators (larger numerator = larger fraction)
Same Numerator: Compare denominators (smaller denominator = larger fraction)
Unit Fractions (numerator = 1): Smaller denominator = larger fraction
Near 1: If numerator and denominator are close, fraction is close to 1

Comparison Symbols Guide

Symbol	Meaning	Example
<	Less than	1/4 < 1/2
>	Greater than	3/4 > 1/2
=	Equal to	2/4 = 1/2

💡 Memory Trick:

The symbol "opens" toward the BIGGER number!
Think of it like a hungry alligator eating the larger number 🐊

Quick Reference Chart

Method	When to Use	Key Steps
Number Line	Visual comparison needed	Right = Greater, Left = Smaller
Benchmarks	Quick estimation	Compare to 0, 1/4, 1/2, 3/4, 1
Common Denominator	Precise comparison	Find LCD, convert, compare numerators
Decimals	Calculator available	Divide, then compare decimals