Understand Fraction Division

Fifth Grade Mathematics

📚 Understanding Division and Fractions

Division and fractions are closely related! In fact, every fraction represents a division problem. Understanding this relationship helps us solve complex problems involving fractions.

Key Concept:

Fractions ARE Division!

🔄 Relate Division and Fractions

Core Formula:

a ÷ b = a/b

The dividend becomes the numerator
The divisor becomes the denominator

Examples:

Example 1: 3 ÷ 4 = 3/4

3 cookies divided among 4 people = 3/4 of a cookie per person

Example 2: 5 ÷ 8 = 5/8

5 pizzas divided among 8 people = 5/8 of a pizza per person

Example 3: 1 ÷ 2 = 1/2

1 cake divided among 2 people = 1/2 of a cake per person

📝 Fractions as Division: Word Problems

Problem-Solving Strategy:

Step 1: Identify what is being divided

This becomes the numerator (dividend)

Step 2: Identify how many groups/people

This becomes the denominator (divisor)

Step 3: Write as a fraction

Use the formula: dividend/divisor

Sample Word Problems:

Problem: 4 brownies are shared equally among 5 friends. How much does each friend get?

Solution: 4 ÷ 5 = 4/5
Answer: Each friend gets 4/5 of a brownie

Problem: 7 sandwiches are divided equally among 10 students. What is each student's share?

Solution: 7 ÷ 10 = 7/10
Answer: Each student gets 7/10 of a sandwich

➗ Divide Unit Fractions by Whole Numbers

Formula:

(1/a) ÷ b = 1/(a × b)

Multiply the denominator by the whole number

Using Models - Visual Understanding:

Example: 1/2 ÷ 3

Step 1: Start with 1/2 (one shaded half)

Step 2: Divide that half into 3 equal parts

Step 3: Each part is 1 out of (2 × 3) = 6 total parts

Answer: 1/2 ÷ 3 = 1/6

Example: 1/3 ÷ 4

Solution: 1/(3 × 4) = 1/12

Dividing 1/3 into 4 equal parts gives 1/12

Example: 1/4 ÷ 5

Solution: 1/(4 × 5) = 1/20

🔲 Divide Unit Fractions Using Area Models

Area Model Strategy:

Step 1: Draw a rectangle divided into parts (denominator)

Shade 1 part to show the unit fraction

Step 2: Divide the shaded part horizontally

Split into the number of groups (whole number divisor)

Step 3: Count total small rectangles

One small rectangle over total = answer

Example: 1/2 ÷ 4 Using Area Model

• Draw: Rectangle with 2 columns (for 1/2)
• Shade: 1 column
• Divide: Shaded part into 4 rows
• Result: Creates 2 × 4 = 8 total small rectangles
• Answer: 1 shaded rectangle out of 8 = 1/8

🔢 Divide Whole Numbers by Unit Fractions

Formula:

a ÷ (1/b) = a × b

Multiply the whole number by the denominator

Using Models - Visual Understanding:

Example: 3 ÷ 1/2

Question: How many halves are in 3 wholes?

Step 1: Each whole has 2 halves

Step 2: 3 wholes × 2 halves = 6 halves

Answer: 3 ÷ 1/2 = 6

Example: 4 ÷ 1/3

Solution: 4 × 3 = 12

There are 12 thirds in 4 wholes

Example: 2 ÷ 1/4

Solution: 2 × 4 = 8

There are 8 fourths in 2 wholes

📐 Whole Numbers ÷ Unit Fractions: Area Models

Area Model Strategy:

Step 1: Draw rectangles for the whole number

One rectangle for each whole

Step 2: Divide each rectangle

Into parts based on the unit fraction's denominator

Step 3: Count all the parts

Total parts = answer

Example: 2 ÷ 1/3 Using Area Model

• Draw: 2 rectangles (for 2 wholes)
• Divide: Each rectangle into 3 equal parts
• Count: 2 rectangles × 3 parts = 6 thirds
• Answer: 2 ÷ 1/3 = 6

📏 Divide Using Number Lines

Number Line Strategy:

For Unit Fraction ÷ Whole Number:

• Mark the unit fraction on a number line
• Divide that section into equal parts
• Count the size of one part

For Whole Number ÷ Unit Fraction:

• Draw a number line from 0 to the whole number
• Mark off jumps of the unit fraction size
• Count total jumps

Examples:

Example 1: 1/4 ÷ 2

Mark 0 to 1/4, divide into 2 parts = 1/8 each

Example 2: 3 ÷ 1/2

Count 1/2 jumps from 0 to 3 = 6 jumps

📊 Key Formulas Summary

Division Type	Formula	Example
Whole ÷ Whole = Fraction	a ÷ b = a/b	3 ÷ 4 = 3/4
Unit Fraction ÷ Whole	(1/a) ÷ b = 1/(a × b)	1/2 ÷ 3 = 1/6
Whole ÷ Unit Fraction	a ÷ (1/b) = a × b	4 ÷ 1/2 = 8
Reciprocal Rule	÷ (a/b) = × (b/a)	÷ 1/3 = × 3