Division
Master long division and understand remainders
What You'll Learn
Understanding Division
Division is the inverse operation of multiplication. It involves splitting a quantity into equal groups or determining how many times one number fits into another.
Key Division Terminology
Dividend
Number being divided
12 ÷ 3 → 12
Divisor
Number dividing by
12 ÷ 3 → 3
Quotient
Result of division
12 ÷ 3 = 4
Remainder
What's left over
13 ÷ 4 = 3 R1
Important Properties
Inverse of Multiplication
Division undoes multiplication
If 4 × 3 = 12, then 12 ÷ 3 = 4 and 12 ÷ 4 = 3
Non-Commutative
Order DOES matter in division
12 ÷ 3 = 4 but 3 ÷ 12 = 0.25
Identity Property
Any number divided by 1 equals itself
7 ÷ 1 = 7
Long Division Algorithm
Example: 784 ÷ 4
196 ------ 4 | 784 4↓ (4 × 1 = 4) --- 38 (bring down 8) 36 (4 × 9 = 36) --- 24 (bring down 4) 24 (4 × 6 = 24) --- 0
Answer: 196
Division with Remainders: 157 ÷ 6
26 R1 ------- 6 | 157 12↓ (6 × 2 = 12) --- 37 (bring down 7) 36 (6 × 6 = 36) --- 1 (remainder)
Answer: 26 remainder 1 (or 26 R1)
Checking Division with Multiplication
Always verify your division by multiplying the quotient by the divisor (and adding any remainder):
(Quotient × Divisor) + Remainder = Dividend
Example 1: No Remainder
156 ÷ 12 = 13
Check: 13 × 12 = 156 ✓
Example 2: With Remainder
89 ÷ 7 = 12 R5
Check: (12 × 7) + 5 = 84 + 5 = 89 ✓
Division by Zero
Important Rule
Division by zero is undefined. You cannot divide any number by zero.
Undefined ✗
- 8 ÷ 0 = undefined
- 0 ÷ 0 = undefined
- 1,000,000 ÷ 0 = undefined
Valid ✓
- 0 ÷ 5 = 0
- 0 ÷ 100 = 0
- Zero divided by any non-zero = 0
Mental Division Strategies
Using Multiplication Facts
Think: "What times the divisor equals the dividend?"
63 ÷ 9 → "What × 9 = 63?" → 7
Breaking Apart (Chunking)
Divide parts of the number separately
84 ÷ 4 = (80 ÷ 4) + (4 ÷ 4) = 20 + 1 = 21
Halving for Division by 2
To divide by 2, find half of the number
86 ÷ 2 = half of 86 = 43
Using Patterns
Recognize patterns in division
Any number ÷ 10 → move decimal left one place
240 ÷ 10 = 24
Interpreting Remainders in Word Problems
In word problems, you must interpret what the remainder means in context:
Round Up
When you need whole units (buses, boxes, etc.)
25 students ÷ 4 per car = 6 R1
Answer: Need 7 cars (can't leave 1 student behind!)
Round Down
When you can only use complete groups
$25 ÷ $4 per item = 6 R1
Answer: Can buy 6 items (not enough for 7th)
Use the Remainder
When the remainder itself is the answer
17 cookies ÷ 5 people = 3 R2
Answer: 2 cookies left over
Practice Problems
Test your understanding
Basic Division
1. 48 ÷ 6 = ? ▼
8
2. 81 ÷ 9 = ? ▼
9
3. 144 ÷ 12 = ? ▼
12
With Remainders
1. 47 ÷ 5 = ? ▼
9 R2
2. 83 ÷ 7 = ? ▼
11 R6
3. 125 ÷ 8 = ? ▼
15 R5
Word Problems
Problem 1: Distributing Supplies
A teacher has 156 crayons to distribute equally among 12 students. How many crayons will each student receive?
156 ÷ 12 = 13 crayons per student
Problem 2: Field Trip Groups
There are 94 students going on a field trip. If each chaperone can supervise 8 students, how many chaperones are needed?
94 ÷ 8 = 11 R6
12 chaperones needed (round up!)
Problem 3: Grading Papers
A teacher needs to grade 168 papers. If she can grade 14 papers per hour, how many hours will it take?
168 ÷ 14 = 12 hours
Common Mistakes to Avoid
- • Remainder Errors: Remainder must be less than divisor
- • Place Value Mistakes: Keep digits aligned in long division
- • Forgetting to Bring Down: Remember to bring down the next digit
- • Division by Zero: Remember this is undefined
- • Not Checking: Always verify by multiplying
Key Takeaways
- ✓ Division is the inverse of multiplication
- ✓ Check: (Quotient × Divisor) + Remainder = Dividend
- ✓ Division by zero is undefined
- ✓ Interpret remainders based on context
- ✓ Use multiplication facts to divide mentally
Related Lessons
Congratulations!
You've completed basic operations! Ready to explore decimals?
