Dividing Multi-Digit Numbers: Standard Algorithm

Grade 6 Mathematics — 6.NS.B.2

In this lesson:

• Use the four-step algorithm for any division problem
• Extend to 2-digit divisors using estimation
• Interpret remainders in context

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. Explain the four-step algorithm in terms of place value
2. Fluently divide multi-digit numbers by a 1-digit divisor
3. Fluently divide multi-digit numbers by a 2-digit divisor using estimation
4. Interpret the remainder as a leftover, round-up signal, or fraction

From Small Division to Large Numbers

You know: 42 ÷ 6 = 7 and 56 ÷ 8 = 7

What about these?

• 1,428 ÷ 6 = ?
• 1,596 ÷ 38 = ?

Same idea — but we need a structured method

The Four-Step Cycle: How the Algorithm Works

D → M → S → B, then repeat for each quotient digit

Worked Example: 756 ÷ 3

Hundreds: 7 ÷ 3 → 2. 2×3=6. 7−6=1. Bring down → 15

Tens: 15 ÷ 3 → 5. 5×3=15. 15−15=0. Bring down → 6

Ones: 6 ÷ 3 → 2. 2×3=6. 6−6=0

756 ÷ 3 = 252 ✓ Check: 252 × 3 = 756

Worked Example: 1,428 ÷ 6

Step 1: 14 ÷ 6 → 2. 2×6=12. 14−12=2. Bring down → 22

Step 2: 22 ÷ 6 → 3. 3×6=18. 22−18=4. Bring down → 48

Step 3: 48 ÷ 6 → 8. 8×6=48. 48−48=0

1,428 ÷ 6 = 238 ✓ Check: 238 × 6 = 1,428

First Digit Smaller Than the Divisor

216 ÷ 4 — 4 doesn't fit in 2 → combine: 21

• 21 ÷ 4 → 5 (tens). 5×4=20. 21−20=1. Bring down → 16
• 16 ÷ 4 → 4 (ones). 4×4=16. 16−16=0

216 ÷ 4 = 54 — 2-digit quotient

Estimate: 200 ÷ 4 = 50 ✓

Division with a Remainder: 431 ÷ 5

Step 1: 43 ÷ 5 → 8. 8×5=40. 43−40=3. Bring down 1 → 31

Step 2: 31 ÷ 5 → 6. 6×5=30. 31−30=1

Remainder = 1 (less than divisor 5 ✓)

Check: (86 × 5) + 1 = 431 ✓

Three Interpretations of the Remainder

25 ÷ 4 = 6 R1 — same math, three answers:

Ask first: what does the remainder mean here?

Check-In: Try 372 ÷ 4

Label all four steps for each position.

1. Divide: how many 4s in ___?
2. Multiply: ___ × 4 = ___
3. Subtract: ___ − ___ = ___
4. Bring down: new partial dividend = ___

Check your answer by multiplying back.

Try before the next slide.

Answer: 372 ÷ 4 = 93

Step 1: 37 ÷ 4 → 9. 9×4=36. 37−36=1. Bring down 2 → 12

Step 2: 12 ÷ 4 → 3. 3×4=12. 12−12=0

Result: 93 — Check: 93 × 4 = 372 ✓

Missed it? Check: Did you bring down the 2? Is remainder < 4?

2-Digit Divisors: Same Steps, New Challenge

Already know: Divide → Multiply → Subtract → Bring Down

The difference: No table for "how many 38s in 159"

The solution: Estimation

• Round divisor to nearest ten
• Estimate the trial digit
• Try → check → adjust if needed

Estimation Strategy for 2-Digit Divisors

• Round: 38 → 40
• Estimate: 159 ÷ 40 ≈ 4
• Try: 4 × 38 = 152 → remainder 7 → 7 < 38 ✓

Worked Example: 1,596 ÷ 38

Partial dividend 159: 38→40. Try 4: 4×38=152. 159−152=7. 7<38 ✓ → 4. Bring down → 76

Partial dividend 76: 38→40. Try 2: 2×38=76. 76−76=0. 0<38 ✓ → 2

Trial Too Large: 261 ÷ 43

43 → 40. Try 7: 7×43 = 301 > 261 ✗

Try 6: 6×43 = 258. 261−258 = 3. 3 < 43 ✓

Rule: product > partial dividend → decrease trial by 1

Trial Too Small: 186 ÷ 23

23 → 20. Try 7: 7×23 = 161. 186−161 = 25. 25 ≥ 23 ✗

Try 8: 8×23 = 184. 186−184 = 2. 2 < 23 ✓

Rule: remainder ≥ divisor → increase trial by 1

Guided Practice: Solve 952 ÷ 34

Step 1 — Divide 95 by 34:
Round 34 → 30. Estimate . Try : ×34=
Remainder: 95−=. Is ___<34? Bring down 2 → ___

Step 2 — Divide the new partial dividend by 34:
Round, estimate, try, check.

Estimate: 950 ÷ 30 ≈ 31 — expect a 2-digit answer

Answer: 952 ÷ 34 = 28

Step 1: Try 3: 102>95 ✗. Try 2: 2×34=68. 95−68=27<34 ✓. Bring down → 272

Step 2: Try 8: 8×34=272. 272−272=0<34 ✓. Digit: 8

Step 1 adjusted: trial 3 too large → reduced to 2

Key Takeaways: Long Division Algorithm

✓ Repeat for every digit: Divide → Multiply → Subtract → Bring Down

✓ Each quotient digit placed above its place value position

✓ 2-digit divisors: round → estimate → try → check → adjust

✓ Remainder always less than the divisor — or trial is too small

✓ Word problems: decide before computing what the remainder means

Skip Bring Down → wrong digit count Remainder ≥ divisor → increase trial

What's Next: Decimal Division (6.NS.B.3)

The same algorithm extends to decimals:

• (integers — done today)
• — same steps, track the decimal point

The four-step cycle stays the same — only the decimal point moves.

Next lesson: dividing with decimal numbers