➗ Understand Division - Grade 3

What is Division?

Division is splitting a number into equal parts or groups! It's the opposite of multiplication.

\(\text{Dividend} \div \text{Divisor} = \text{Quotient}\)

🔵 Dividend - The number being divided (total)
🔵 Divisor - The number of groups OR number in each group
🔵 Quotient - The answer (result)

👥 Divide by Counting Equal Groups

Two Types of Division Problems:

Type 1: Equal Sharing

You know: Total items and number of groups
Find: How many in each group?

Example: Share 12 cookies among 3 friends
\(12 \div 3 = 4\)
Each friend gets 4 cookies!

Type 2: Equal Grouping

You know: Total items and number in each group
Find: How many groups can you make?

Example: 12 cookies, put 3 in each bag
\(12 \div 3 = 4\)
You can make 4 bags!

Steps to Divide by Counting Equal Groups:

Start with the total number (dividend)
Make groups (or distribute into groups)
Share items equally one by one into each group
Count how many are in each group OR how many groups you made
That's your answer! (quotient)

Visual Example: \(15 \div 3\)

Problem: Divide 15 stars into 3 equal groups

Step 1: We have 15 stars total ⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐⭐

Step 2: Make 3 groups
Group 1: ⭐⭐⭐⭐⭐
Group 2: ⭐⭐⭐⭐⭐
Group 3: ⭐⭐⭐⭐⭐

Step 3: Count how many in each group: 5

Answer: \(15 \div 3 = 5\) ✓

✍️ Write Division Sentences for Groups

What is a Division Sentence?

A division sentence is a complete math statement showing division!

Parts of a Division Sentence:

\(\underbrace{20}_{\text{Dividend}} \div \underbrace{4}_{\text{Divisor}} = \underbrace{5}_{\text{Quotient}}\)

Read as: "Twenty divided by four equals five"

How to Write Division Sentences:

Identify the total (dividend) - write this first
Write the division symbol (\(\div\))
Write the number of groups OR number in each group (divisor)
Write the equal sign (=)
Write the answer (quotient)

Examples:

Example 1: Equal Sharing

Picture: 18 apples shared among 6 children

Total: 18 apples
Groups: 6 children
Each child gets: 3 apples

Division sentence: \(18 \div 6 = 3\)

Example 2: Equal Grouping

Picture: 24 crayons, put 4 in each box

Total: 24 crayons
In each box: 4 crayons
Number of boxes: 6

Division sentence: \(24 \div 4 = 6\)

🔗 Relate Multiplication and Division for Groups

The Big Connection!

Multiplication and division are opposite operations! They undo each other!

If \(a \times b = c\), then \(c \div b = a\) and \(c \div a = b\)

Fact Families

A fact family is a group of related multiplication and division facts using the same three numbers!

Example: Fact Family for 3, 4, and 12

Multiplication Facts:
\(3 \times 4 = 12\)
\(4 \times 3 = 12\)

Division Facts:
\(12 \div 3 = 4\)
\(12 \div 4 = 3\)

All four facts use the numbers 3, 4, and 12! ✓

Visual Example with Groups:

Picture: 5 groups with 3 flowers each
🌸🌸🌸 🌸🌸🌸 🌸🌸🌸 🌸🌸🌸 🌸🌸🌸

Multiplication:
\(5 \times 3 = 15\) (5 groups × 3 in each = 15 total)
\(3 \times 5 = 15\) (3 in each × 5 groups = 15 total)

Division:
\(15 \div 5 = 3\) (15 total ÷ 5 groups = 3 in each)
\(15 \div 3 = 5\) (15 total ÷ 3 in each = 5 groups)

Same picture, different ways to think about it!

How to Use Multiplication to Check Division:

Division Problem: \(28 \div 4 = 7\)

Check with Multiplication:
\(7 \times 4 = 28\) ✓
OR
\(4 \times 7 = 28\) ✓

If multiplication gives you the original dividend, your division is correct!

📐 Write Division Sentences for Arrays

What is an Array?

An array is an organized arrangement of objects in equal rows and columns!

🔵 Rows - Go across (horizontal) →
🔵 Columns - Go up and down (vertical) ↓
🔵 Arrays help us see both multiplication AND division!

Division with Arrays - Two Ways!

\(\text{Total} \div \text{Number of Rows} = \text{Items in Each Row}\)

OR

\(\text{Total} \div \text{Items per Row} = \text{Number of Rows}\)

Example: Array with 12 Objects

Array:
⭐ ⭐ ⭐ ⭐
⭐ ⭐ ⭐ ⭐
⭐ ⭐ ⭐ ⭐

What we see:
• Total objects: 12
• Number of rows: 3
• Objects in each row: 4

Division Sentence 1:
\(12 \div 3 = 4\)
(12 total ÷ 3 rows = 4 in each row)

Division Sentence 2:
\(12 \div 4 = 3\)
(12 total ÷ 4 in each row = 3 rows)

Same array, two different division sentences! ✓

Steps to Write Division Sentences from Arrays:

Count the total objects - this is the dividend
Count the rows
Count the columns (or items in each row)
Write two division sentences:
- Total ÷ Number of rows = Items in each row
- Total ÷ Items in each row = Number of rows

🔗📐 Relate Multiplication and Division for Arrays

Arrays Show BOTH Operations!

The same array can be used to show multiplication AND division! They're connected!

Multiplication: Rows × Columns = Total
Division: Total ÷ Rows = Columns
OR
Total ÷ Columns = Rows

Complete Example: \(4 \times 6 = 24\)

Array: 4 rows × 6 columns
🔵 🔵 🔵 🔵 🔵 🔵
🔵 🔵 🔵 🔵 🔵 🔵
🔵 🔵 🔵 🔵 🔵 🔵
🔵 🔵 🔵 🔵 🔵 🔵

Multiplication Facts:
\(4 \times 6 = 24\) (4 rows × 6 columns = 24 total)
\(6 \times 4 = 24\) (6 columns × 4 rows = 24 total)

Division Facts:
\(24 \div 4 = 6\) (24 total ÷ 4 rows = 6 in each row)
\(24 \div 6 = 4\) (24 total ÷ 6 in each row = 4 rows)

All four facts come from ONE array! ✓

Another Example: \(3 \times 5 = 15\)

Array: 3 rows × 5 columns
🟢 🟢 🟢 🟢 🟢
🟢 🟢 🟢 🟢 🟢
🟢 🟢 🟢 🟢 🟢

Fact Family for 3, 5, and 15:

Multiplication:
\(3 \times 5 = 15\)
\(5 \times 3 = 15\)

Division:
\(15 \div 3 = 5\)
\(15 \div 5 = 3\)

Use the array to remember all four facts! ✓

Why This Matters:

✓ If you know multiplication facts, you know division facts!
✓ Arrays help you visualize the relationship
✓ You can check division answers with multiplication
✓ Understanding one operation helps you understand the other!

📝 Important Formulas Summary

Basic Division:

\(\text{Dividend} \div \text{Divisor} = \text{Quotient}\)

Multiplication and Division Relationship:

If \(a \times b = c\), then:
\(c \div a = b\) and \(c \div b = a\)

Division with Equal Groups:

\(\text{Total} \div \text{Number of Groups} = \text{Items in Each Group}\)

OR

\(\text{Total} \div \text{Items per Group} = \text{Number of Groups}\)