Division | Fourth Grade

Complete Notes & Formulas

1. Division Facts to 10

Definition: Basic division facts from 0 ÷ 1 to 100 ÷ 10 (division tables).

📊 Division Terminology:

Dividend ÷ Divisor = Quotient

• Dividend: The number being divided

• Divisor: The number dividing

• Quotient: The answer/result

• Remainder: Amount left over (if any)

✏️ Key Division Facts:

0 ÷ any number = 0 (Example: 0 ÷ 5 = 0)
Any number ÷ 1 = same number (Example: 8 ÷ 1 = 8)
Any number ÷ itself = 1 (Example: 7 ÷ 7 = 1)
Division is inverse of multiplication (Example: 20 ÷ 4 = 5 because 4 × 5 = 20)

2. Division Facts: Word Problems

Definition: Real-world problems using basic division facts.

📋 Types of Division Problems:

1. Equal Groups (Sharing):

"Share 24 cookies equally among 6 children"

2. Measurement (Grouping):

"How many groups of 4 can you make from 28?"

💡 Keywords:

3. Properties of Division

Definition: Rules that govern how division behaves with different numbers.

📐 Four Key Properties:

1. Division by 1

Any number ÷ 1 = the number itself

a ÷ 1 = a

Example: 57 ÷ 1 = 57

2. Division by Itself

Any number ÷ itself = 1 (except 0)

a ÷ a = 1

Example: 99 ÷ 99 = 1

3. Zero Divided by Any Number

0 ÷ any number = 0

0 ÷ a = 0

Example: 0 ÷ 25 = 0

4. Division by Zero

Any number ÷ 0 = UNDEFINED (NOT POSSIBLE)

a ÷ 0 = Undefined

You cannot divide by zero!

4-5. Divide Larger Numbers (Long Division)

Definition: Dividing 2-digit, 3-digit, or 4-digit numbers by 1-digit or 2-digit divisors using long division.

📝 Long Division Steps (DMSB Method):

D - Divide: How many times does divisor fit into dividend?
M - Multiply: Multiply quotient by divisor
S - Subtract: Subtract result from dividend
B - Bring Down: Bring down next digit and repeat

✏️ Example: 156 ÷ 12

       13
    ________
12 | 156
     12↓    (12 × 1)
     ___
      36
      36    (12 × 3)
     ___
       0

Answer: 156 ÷ 12 = 13

🔑 Formula:

Dividend = (Divisor × Quotient) + Remainder

6. Complete the Division Table

Definition: Filling in missing values in division tables using division facts and patterns.

📊 Sample Division Table:

÷	2	3	4	5
20	10	?	5	4
30	15	10	?	6

Use: Dividend ÷ Divisor = Quotient

7. Interpret Remainders

Definition: Understanding what to do with the remainder based on the context of the problem.

📝 Four Ways to Interpret Remainders:

1. Use Only the Quotient (Ignore Remainder)

When whole items are needed

Example: 23 students, 4 per car → 23 ÷ 4 = 5 R3 → Answer: 5 cars needed

2. Round Up (Quotient + 1)

When all items must be included

Example: 23 students need cars → 23 ÷ 4 = 5 R3 → Answer: 6 cars needed

3. Use Only the Remainder

When the question asks "how many left over"

Example: 23 candies, 4 per bag → 23 ÷ 4 = 5 R3 → Answer: 3 candies left

4. Write as a Fraction or Decimal

When items can be divided into parts

Example: 23 pizzas for 4 people → 23 ÷ 4 = 5 R3 → Answer: 5¾ pizzas each

8. Choose Numbers with Particular Quotient

Definition: Selecting numbers from a set that divide to give a specific quotient.

🎯 Strategy:

Identify target quotient
Use: Dividend ÷ Divisor = Quotient
Or: Dividend = Quotient × Divisor
Test different combinations

✏️ Example:

Given: 48, 72, 6, 8, 12

Target Quotient: 6

Try: 48 ÷ 8 = 6 ✓

Or: 72 ÷ 12 = 6 ✓

Answers: (48,8) or (72,12)

9. Divide Numbers Ending in Zeros

Definition: Quick method for dividing numbers with zeros (like 600 ÷ 30).

📝 Steps:

Cancel equal zeros from dividend and divisor
Divide the remaining numbers
If zeros remain in dividend, keep them in quotient

✏️ Examples:

Example 1: 600 ÷ 30

Cancel one zero from each: 60 ÷ 3

Divide: 60 ÷ 3 = 20

Answer: 20

Example 2: 8,000 ÷ 40

Cancel one zero from each: 800 ÷ 4

Divide: 800 ÷ 4 = 200

Answer: 200

10. Estimate Quotients: Word Problems

Definition: Finding approximate quotients by rounding before dividing.

📝 Steps to Estimate:

Round dividend to nearest 10, 100, or 1000
Round divisor to nearest 10 or 100
Divide rounded numbers (use compatible numbers)
Write answer with ≈ symbol

✏️ Example:

Problem: 387 students, 19 buses. About how many per bus?

Round: 387 ≈ 400, 19 ≈ 20

Divide: 400 ÷ 20 = 20

Estimated Answer: ≈ 20 students per bus

💡 Compatible Numbers:

Numbers that divide easily: 200÷10, 300÷50, 600÷30, etc.

11-12. Divisibility Rules

Definition: Quick tests to check if a number is divisible by another without dividing.

📐 Divisibility Rules Chart:

Divisible By	Rule	Example
2	Last digit is 0, 2, 4, 6, or 8	124 ✓
3	Sum of digits divisible by 3	123 (1+2+3=6) ✓
4	Last 2 digits divisible by 4	316 (16÷4) ✓
5	Last digit is 0 or 5	125 ✓
6	Divisible by both 2 AND 3	126 ✓
9	Sum of digits divisible by 9	729 (7+2+9=18) ✓
10	Last digit is 0	340 ✓

13. Division Patterns Over Place Values

Definition: Patterns when dividing by 10, 100, 1000, etc.

🔢 Pattern Examples:

Pattern: 560 ÷ 7 = 80

560 ÷ 7 = 80
5,600 ÷ 7 = 800
56,000 ÷ 7 = 8,000

📐 Division by Powers of 10:

Dividing by 10, 100, 1000:

450 ÷ 10 = 45 (remove 1 zero)
4,500 ÷ 100 = 45 (remove 2 zeros)
45,000 ÷ 1,000 = 45 (remove 3 zeros)

🔑 Rule:

When dividing by 10, move decimal point 1 place left
When dividing by 100, move decimal point 2 places left
When dividing by 1000, move decimal point 3 places left

Division Quick Reference Chart

Concept	Key Formula/Rule
Basic Division	Dividend ÷ Divisor = Quotient
Check Division	(Divisor × Quotient) + Remainder = Dividend
Division by 1	a ÷ 1 = a
Division by Itself	a ÷ a = 1
Zero Divided	0 ÷ a = 0
Division by Zero	a ÷ 0 = Undefined
Divisibility by 2	Last digit is 0, 2, 4, 6, or 8
Divisibility by 3	Sum of digits divisible by 3
Divisibility by 5	Last digit is 0 or 5
Divisibility by 10	Last digit is 0
Division Patterns	÷10 remove 1 zero, ÷100 remove 2 zeros
Long Division	DMSB: Divide, Multiply, Subtract, Bring down