Divide Whole Numbers - Sixth Grade

Complete Notes & Formulas

What is Division?

Division is splitting a number into equal groups or finding how many times one number is contained in another.

Dividend ÷ Divisor = Quotient (Remainder)

Example: 20 ÷ 4 = 5

Key Vocabulary

Dividend: The number being divided (e.g., in 20 ÷ 4, 20 is the dividend)

Divisor: The number dividing (e.g., in 20 ÷ 4, 4 is the divisor)

Quotient: The result of division (e.g., in 20 ÷ 4 = 5, 5 is the quotient)

Remainder: The amount left over when division doesn't result in a whole number

Division-Multiplication Relationship

Division is the Opposite of Multiplication

If a ÷ b = c, then b × c = a

Example: If 20 ÷ 4 = 5, then 4 × 5 = 20

1. Divisibility Rules

Divisibility rules help you quickly determine if one number can be divided evenly by another without actually dividing.

Divisible by	Rule	Example
2	Last digit is 0, 2, 4, 6, or 8 (even)	124, 56, 1,000
3	Sum of all digits is divisible by 3	123 (1+2+3=6)
4	Last two digits form a number divisible by 4	316 (16÷4=4)
5	Last digit is 0 or 5	45, 120, 595
6	Divisible by both 2 AND 3	42, 126, 3,000
8	Last three digits divisible by 8	2,016 (016÷8=2)
9	Sum of all digits is divisible by 9	729 (7+2+9=18)
10	Last digit is 0	50, 100, 1,230

Example: Check Divisibility

Problem: Is 234 divisible by 3?

Rule for 3: Sum of digits must be divisible by 3

Sum = 2 + 3 + 4 = 9

9 ÷ 3 = 3 ✓

Answer: Yes, 234 is divisible by 3

2. Division Patterns with Zeros

Shortcut for Dividing Numbers Ending in Zeros

Steps to Divide with Zeros:

1. Cancel equal zeros from dividend and divisor

2. Divide the remaining numbers

Example 1: Both Have Zeros

Problem: 2,400 ÷ 60

Method 1 (Cancel zeros):

Both have 1 zero minimum

2,400 ÷ 60 = 240 ÷ 6

240 ÷ 6 = 40

Method 2 (Powers of 10):

2,400 = 24 × 100

60 = 6 × 10

(24 × 100) ÷ (6 × 10) = (24 ÷ 6) × (100 ÷ 10) = 4 × 10 = 40

Answer: 40

Example 2: Patterns

80 ÷ 10 = 8

800 ÷ 10 = 80

8,000 ÷ 10 = 800

Pattern: Dividing by 10 removes one zero

Quick Tip: Dividing by 10, 100, or 1000 removes that many zeros!

3. Estimate Quotients

Why Estimate?

Estimating helps you check if your answer is reasonable and allows for quick mental calculations.

Steps to Estimate Quotients

Step 1: Round the dividend to a compatible number

Step 2: Round the divisor to make division easy

Step 3: Divide the rounded numbers

Example 1: Estimate Quotient

Problem: Estimate 478 ÷ 22

Step 1: Round dividend to nearest hundred: 478 → 500

Step 2: Round divisor to nearest ten: 22 → 20

Step 3: Divide: 500 ÷ 20 = 25

Estimated Quotient: About 25

(Actual: 478 ÷ 22 = 21.7...)

Example 2: Compatible Numbers

Problem: Estimate 364 ÷ 57

Use compatible numbers (numbers that divide easily):

364 → 360 (close and divisible by 60)

57 → 60 (compatible with 360)

360 ÷ 60 = 6

Estimated Quotient: About 6

(Actual: 364 ÷ 57 = 6.4)

4-5. Divide Whole Numbers (2-Digit & 3-Digit Divisors)

Steps for Long Division

Step 1: DIVIDE - How many times does the divisor fit into the first digits?

Step 2: MULTIPLY - Multiply the quotient digit by the divisor

Step 3: SUBTRACT - Subtract the product from the dividend

Step 4: BRING DOWN - Bring down the next digit

Step 5: REPEAT - Repeat until no more digits remain

Remember: "Dad, Mom, Sister, Brother" or "Divide, Multiply, Subtract, Bring down"

Example 1: Two-Digit Divisor

Problem: 756 ÷ 36

Step-by-step:

1. 36 goes into 75 two times (2 × 36 = 72)

2. 75 − 72 = 3, bring down 6 → 36

3. 36 goes into 36 one time (1 × 36 = 36)

4. 36 − 36 = 0

Answer: 21

Example 2: Two-Digit with Remainder

Problem: 487 ÷ 23

Answer: 21 R4 (or 21 remainder 4)

Example 3: Three-Digit Divisor

Problem: 3,456 ÷ 128

       27
128 | 3,456
     256↓ (128×2)
     -----
      896
      896 (128×7)
      -----
        0

Answer: 27

6. Division Word Problems

Key Division Keywords

• Split

• Share

• Divide

• Per (each)

• Average

• Quotient

• How many groups

• How many in each group

Example 1: Equal Groups

Problem: A bakery has 432 cookies to pack equally into 24 boxes. How many cookies will be in each box?

Given:

Total cookies = 432

Number of boxes = 24

Operation: Division (keyword: "equally")

Solution:

432 ÷ 24 = 18

Answer: 18 cookies in each box

Example 2: Division with Zeros

Problem: A warehouse has 3,600 books. They need to pack them in boxes with 60 books each. How many boxes are needed?

Solution using zero shortcut:

3,600 ÷ 60

Cancel one zero from each: 360 ÷ 6 = 60

Answer: 60 boxes

Example 3: Multi-Step Problem

Problem: A school bought 896 pencils for 32 classrooms. Each classroom gets the same number. How many pencils does each classroom receive?

896 ÷ 32

Use long division: 896 ÷ 32 = 28

Answer: 28 pencils per classroom

Example 4: Remainder Interpretation

Problem: 125 students are going on a field trip. Each bus holds 30 students. How many buses are needed?

125 ÷ 30 = 4 R5

This means 4 buses are full, but 5 students remain

We need another bus for the remaining 5 students

Answer: 5 buses needed (must round up!)

Quick Reference: Division Facts

Concept	Formula/Rule
Division	Dividend ÷ Divisor = Quotient
Check Division	Quotient × Divisor + Remainder = Dividend
Divide by 10	Remove one zero
Zero Rule	0 ÷ any number = 0

💡 Important Tips to Remember

✓ Divisibility rules help you check division quickly

✓ When dividing by 10, 100, 1000: remove zeros

✓ Estimate first to check if your answer makes sense

✓ Long division: Divide, Multiply, Subtract, Bring down

✓ Check your work: Quotient × Divisor + Remainder = Dividend

✓ Use compatible numbers for easier estimation

✓ 0 ÷ any number = 0

✓ Cannot divide by 0 (undefined)

✓ In word problems, remainders need interpretation

✓ Division is the opposite of multiplication

🧠 Memory Tricks & Strategies

Long Division Steps:

"Dad, Mom, Sister, Brother"

Divide, Multiply, Subtract, Bring down

Divisibility by 2:

"Even numbers end in 0, 2, 4, 6, or 8!"

Divisibility by 3:

"Add up the digits, if divisible by 3, the whole number is too!"

Divisibility by 5:

"End in 0 or 5, and you're alive!"

Divisibility by 9:

"Add digits, divisible by 9 = whole number is fine!"

Division with Zeros:

"Cancel equal zeros, divide what's left, simple as that!"

Check Your Work:

"Multiply back, add remainder, should equal where you started!"

Master Division! ÷ 📊 🔢

Practice makes perfect - divide every day!