Division | Fifth Grade
Complete Notes & Formulas
1. Division Patterns Over Increasing Place Values
Definition: When dividing by 10, 100, or 1,000, we can use patterns and place value understanding to solve problems quickly without doing full calculations.
📐 Key Patterns:
Dividing by 10 → Move decimal point 1 place LEFT
Dividing by 100 → Move decimal point 2 places LEFT
Dividing by 1,000 → Move decimal point 3 places LEFT
✏️ Examples:
5 ÷ 1 = 5
50 ÷ 10 = 5
500 ÷ 100 = 5
5,000 ÷ 1,000 = 5
Pattern: The answer is always 5!
2. Divide Numbers Ending in Zeros
Definition: When both the dividend and divisor end in zeros, we can cancel out equal zeros to simplify the problem.
🔑 Strategy - Cancel Equal Zeros:
Cancel the same number of zeros from dividend and divisor
✏️ Examples:
Example 1: 2,400 ÷ 30
Cancel one zero from each: 240 ÷ 3
240 ÷ 3 = 80
Answer: 80
Example 2: 5,000 ÷ 50
Cancel one zero: 500 ÷ 5 = 100
Answer: 100
3. Divide Numbers Ending in Zeros: Word Problems
Definition: Apply the zero-canceling strategy to solve real-world problems involving division.
✏️ Example Problem:
A school has 1,000 brochures to distribute equally among 20 classrooms. How many per classroom?
Solution:
1,000 ÷ 20
Cancel one zero: 100 ÷ 2 = 50
Answer: 50 brochures per classroom
4-5. Estimate Quotients (2-Digit Divisors)
Definition: Estimating quotients involves rounding numbers to make division easier and get an approximate answer.
📝 Steps to Estimate:
- Step 1: Round the divisor to the nearest ten
- Step 2: Round the dividend to a compatible number (multiple of rounded divisor)
- Step 3: Divide the rounded numbers
✏️ Example:
Estimate: 525 ÷ 46
Round 46 → 50
Round 525 → 500 (compatible with 50)
500 ÷ 50 = 10
Estimated Answer: About 10
6. Divide Multi-Digit Numbers by 1-Digit Numbers
Definition: Long division is the standard algorithm for dividing larger numbers by smaller numbers.
📝 Long Division Steps (Divide, Multiply, Subtract, Bring Down):
- Divide: How many times does divisor go into first digit(s)?
- Multiply: Multiply quotient digit by divisor
- Subtract: Subtract the product from the dividend
- Bring Down: Bring down the next digit
- Repeat until all digits are used
✏️ Example: 456 ÷ 3
4 ÷ 3 = 1 R1
Bring down 5 → 15 ÷ 3 = 5
Bring down 6 → 6 ÷ 3 = 2
Answer: 152
7. Divide by 1-Digit Numbers: Interpret Remainders
Definition: A remainder is what's left over after division. How you interpret the remainder depends on the context of the problem.
🔑 Three Ways to Interpret Remainders:
1. Round Up (Need More):
Use when you need enough for everyone
Example: 17 students, 4 per car → Need 5 cars (not 4 R1)
2. Drop the Remainder (Ignore Extra):
Use when you can't use partial amounts
Example: 23 cookies, 4 per bag → Can fill 5 bags (3 left over)
3. Use as a Fraction/Decimal:
Use when you can split things evenly
Example: $25 split among 4 people → $6.25 each (or 6 ¼)
8. Divide Multi-Digit Numbers by 1-Digit: Word Problems
Definition: Apply long division to solve real-world problems involving sharing, grouping, and partitioning.
✏️ Example:
A bakery made 384 cookies. They pack 6 cookies in each box. How many boxes?
384 ÷ 6 = 64
Answer: 64 boxes
9-11. Divide by 2-Digit Numbers (Methods)
Definition: There are multiple strategies for dividing by 2-digit numbers: estimation, models, and partial quotients.
🔑 Method 1: Estimate and Adjust
- Estimate the quotient by rounding
- Try the estimate
- Adjust if too high or too low
🔑 Method 2: Partial Quotients
Subtract multiples of the divisor until you reach 0 or a remainder less than the divisor.
Example: 285 ÷ 15
285 - 150 (10 × 15) = 135 → Partial quotient: 10
135 - 150 won't work, try 75 (5 × 15) = 60 → Partial quotient: 5
60 - 60 (4 × 15) = 0 → Partial quotient: 4
Add partial quotients: 10 + 5 + 4 = 19
Answer: 19
12-13. Divide 2-Digit & 3-Digit Numbers by 2-Digit Numbers
Definition: Use long division or partial quotients to divide smaller numbers by 2-digit divisors.
✏️ Examples:
96 ÷ 12
Think: 12 × ? = 96
12 × 8 = 96
Answer: 8
456 ÷ 24
Estimate: 24 ≈ 25, 456 ≈ 450
450 ÷ 25 = 18 (approx)
Check: 24 × 19 = 456 ✓
Answer: 19
14-15. Divide 4-Digit Numbers by 2-Digit Numbers
Definition: Apply long division algorithm to divide larger 4-digit numbers by 2-digit divisors.
📝 Steps for Long Division:
- Look at first 2 digits of dividend
- Estimate how many times divisor goes in
- Multiply, subtract, bring down
- Repeat for each digit
✏️ Example: 1,824 ÷ 32
18 ÷ 32 doesn't work, use 182
32 × 5 = 160, 182 - 160 = 22
Bring down 4 → 224
32 × 7 = 224, 224 - 224 = 0
Answer: 57
16. Adjust Quotients
Definition: Sometimes your first estimate is too high or too low. You need to adjust and try again.
🔑 When to Adjust:
- Too High: Product is greater than dividend → Try smaller quotient
- Too Low: Remainder is larger than divisor → Try larger quotient
✏️ Example:
147 ÷ 21
Try 8: 21 × 8 = 168 (too high!)
Try 7: 21 × 7 = 147 ✓
Answer: 7
17. Relate Multiplication and Division
Definition: Multiplication and division are inverse operations. You can use one to check the other.
🔑 Key Relationship:
If a ÷ b = c, then b × c = a
If a × b = c, then c ÷ b = a
✏️ Examples:
72 ÷ 8 = 9
Check: 8 × 9 = 72 ✓
15 × 6 = 90
Related: 90 ÷ 6 = 15 or 90 ÷ 15 = 6
18. Complete the Division Sentence: 2-Digit Divisors
Definition: Fill in missing numbers in division equations using your understanding of division relationships.
✏️ Examples:
___ ÷ 12 = 8
Think: 12 × 8 = 96
Answer: 96
144 ÷ ___ = 12
Think: 144 ÷ 12 = 12 or 12 × 12 = 144
Answer: 12
19. Choose Numbers with a Particular Quotient
Definition: Select numbers that when divided give a specific quotient. This requires understanding of multiples and division facts.
📝 Strategy:
If you want quotient = Q and divisor = D, then dividend = Q × D
✏️ Example:
Find a number that divided by 15 gives quotient 8
Dividend = 15 × 8 = 120
Check: 120 ÷ 15 = 8 ✓
Answer: 120
Division Quick Reference Chart
| Strategy | When to Use |
|---|---|
| Cancel Zeros | Both numbers end in zeros |
| Estimate | Quick approximate answer needed |
| Long Division | Standard method for any division |
| Partial Quotients | Breaking down into easier steps |
| Multiplication Check | Verify your division answer |
💡 Division Memory Aids:
Long Division Steps
Divide, Multiply, Subtract, Bring Down
Check Division
Quotient × Divisor = Dividend
🔑 Key Formulas:
Dividend ÷ Divisor = Quotient (+ Remainder)
To Check: Quotient × Divisor + Remainder = Dividend
📚 Fifth Grade Division - Complete Study Guide
Master these division concepts for math excellence! ✨