Long division: A systematic method for dividing large numbers by breaking the process into smaller steps: Divide, Multiply, Subtract, Bring Down, Repeat.
- Divide: How many times does the divisor go into the first digit(s)?
- Multiply: Multiply the divisor by the quotient digit
- Subtract: Subtract the product from the current number
- Bring Down: Bring down the next digit
- Repeat: Continue until all digits are processed
Write 4 ) 84 with the divisor (4) outside and dividend (84) inside
How many times does 4 go into 8? Answer: 2 times
Write 2 above the 8
2 × 4 = 8
Write 8 below the 8 in the dividend
8 - 8 = 0
Write 0 below
Bring down the 4 to make 04
How many times does 4 go into 4? Answer: 1 time
Write 1 above the 4
1 × 4 = 4
4 - 4 = 0
No remainder
Check: 21 × 4 = 84 ✓
84 ÷ 4 = 21
• DMBS Rule: Divide, Multiply, Subtract, Bring Down
• Place value: Each digit's position affects its value in calculations
• Verification: Always check by multiplying quotient by divisor
Long division with remainder: When the final subtraction doesn't result in 0, the remaining number is the remainder.
Write 6 ) 97
How many times does 6 go into 9? Answer: 1 time
Write 1 above the 9
1 × 6 = 6
Write 6 below the 9
9 - 6 = 3
Write 3 below
Bring down the 7 to make 37
How many times does 6 go into 37? Answer: 6 times (6 × 6 = 36)
Write 6 above the 7
6 × 6 = 36
37 - 36 = 1
Since 1 is less than 6, this is the remainder
Check: (6 × 16) + 1 = 96 + 1 = 97 ✓
97 ÷ 6 = 16 R1
• Remainder rule: Remainder must be less than the divisor
• DMBS sequence: Follow the order of operations carefully
• Verification: Use Dividend = (Divisor × Quotient) + Remainder
Place value: The value of a digit based on its position in the number. In 156, the 1 represents 100, the 5 represents 50, and the 6 represents 6.
In 156, we have 1 hundred, 5 tens, and 6 ones. Starting with the highest place value ensures we find the largest possible groups first.
How many times does 3 go into 1? Answer: 0 times
So we consider 15 (tens place)
How many times does 3 go into 15? Answer: 5 times
Write 5 in the tens place of the quotient
3 × 5 = 15
15 - 15 = 0
Bring down 6
How many times does 3 go into 6? Answer: 2 times
Write 2 in the ones place
156 ÷ 3 = 52
Check: 52 × 3 = 156 ✓
We start with the highest place value because it ensures we find the largest possible groups first, which gives us the correct quotient. If we started with the ones place, we would get an incorrect answer. The process builds the quotient digit by digit from left to right.
• Place value rule: Process digits from left to right (highest to lowest value)
• Systematic approach: Ensures accuracy and prevents missing larger groups
• Digit-by-digit processing: Builds the quotient correctly
Dividend: The number being divided (the whole).
Divisor: The number by which the dividend is divided (the size of each group).
Quotient: The result of division (the number of equal groups).
Remainder: What's left over after division that cannot form another complete group.
Place value: The value of a digit based on its position in the number.
- Standard algorithm: Divide, Multiply, Subtract, Bring Down, Repeat
- Partial quotients: Break division into smaller, easier divisions
- Chunking method: Repeated subtraction in chunks
- Area model: Visual representation of division
• Process digits from left to right (highest to lowest place value)
• Remainder is always less than the divisor
• Use the DMBS sequence: Divide, Multiply, Subtract, Bring Down
• Always verify your answer using multiplication
• Align numbers properly in columns to avoid computational errors
Analysis: The chart shows how long division breaks down a complex problem into simpler steps.
- Each step reduces the problem size
- Place value determines the quotient digit position
- The process continues until all digits are processed
Word problems with long division: Real-world situations that require division to solve. Pay attention to whether you need to round up for practical purposes.
We need to divide 248 students by 8 students per bus: 248 ÷ 8
8 ) 248
How many times does 8 go into 2? Answer: 0 times
Consider 24 (first two digits)
How many times does 8 go into 24? Answer: 3 times
Write 3 above the 4
3 × 8 = 24
24 - 24 = 0
Bring down 8 to make 08
How many times does 8 go into 8? Answer: 1 time
Write 1 above the 8
248 ÷ 8 = 31
Since there's no remainder, exactly 31 buses are needed
Check: 31 × 8 = 248 ✓
31 buses are needed for the trip.
• Problem-solving strategy: Identify total amount, group size, and what to find
• Real-world interpretation: In this case, no rounding up needed since there's no remainder
• Verification: Always check your calculation with multiplication
Zeros in quotient: When the divisor doesn't go into the current number being considered, we write 0 in the quotient and continue with the next digit.
Write 4 ) 408
How many times does 4 go into 4? Answer: 1 time
Write 1 above the 4
1 × 4 = 4
4 - 4 = 0
Bring down 0 to make 00
How many times does 4 go into 0? Answer: 0 times
Write 0 in the tens place of the quotient
Bring down 8 to make 08
How many times does 4 go into 8? Answer: 2 times
Write 2 in the ones place of the quotient
408 ÷ 4 = 102
Check: 102 × 4 = 408 ✓
When the divisor doesn't go into a part of the dividend, we write 0 in that position of the quotient. In this problem, 4 doesn't go into 0 (the tens place), so we write 0 in the tens place of the quotient, giving us 102.
• Zero in quotient: Write 0 when divisor doesn't go into current number
• Place value maintenance: Keep track of positions even when writing zeros
• Continue process: Bring down next digit and continue division
Long division: A systematic method for dividing large numbers by breaking the process into smaller steps: Divide, Multiply, Subtract, Bring Down, Repeat (DMBS).
Dividend: The number being divided (the whole).
Divisor: The number by which the dividend is divided (the size of each group).
Quotient: The result of division (the number of equal groups).
Remainder: The amount left over after division that cannot form another complete group.
Place value: The value of a digit based on its position in the number.
- Set up the problem: Write divisor outside, dividend inside the bracket.
- Divide: Determine how many times divisor goes into the first digit(s).
- Multiply: Multiply divisor by the quotient digit.
- Subtract: Subtract the product from the current number.
- Bring Down: Bring down the next digit of the dividend.
- Repeat: Continue until all digits are processed.
- Check: Verify using multiplication.
• Process digits from left to right (highest to lowest place value)
• Remainder is always less than the divisor
• Use the DMBS sequence: Divide, Multiply, Subtract, Bring Down
• When divisor doesn't go into current number, write 0 in quotient
• Always verify your answer using multiplication
f₁(x) = Steps needed for x ÷ 4 (for x = 10-99)
f₂(x) = Steps needed for x ÷ 4 (for x = 100-999)
f₃(x) = Steps needed for x ÷ 4 (for x = 1000-9999)
Analysis: The chart shows how the number of steps in long division increases with the number of digits.
- More digits require more steps in the division process
- Each digit typically adds one more iteration of DMBS
- Complexity grows linearly with the number of digits