Dividing Multi-Digit Numbers

Grade 4 Number and Operations in Base Ten: 4.NBT.B.6

In this lesson:

• Divide numbers up to four digits by a one-digit number
• Show your work and know what a remainder means

Learning Objectives for This Lesson

You will learn to:

1. Divide up to four digits by one digit
2. Use partial quotients to divide
3. Use multiplication to find or check a quotient
4. Model division with a rectangle
5. Explain what a remainder means

4,872 ÷ 8 — Which Is It?

• Two students divided 4,872 by 8. One got 6. One got 609.
• Both can't be right.

Hold onto this — you'll be able to settle it by the end of class.

Naming the Parts: Dividend, Divisor, Quotient

• In 846 ÷ 2: 846 is the dividend, 2 is the divisor.
• The answer is the quotient.

We'll actually solve this one in a few slides.

Sharing 69 Stickers: Splitting by Place Value

• 69 stickers shared among 3 friends.
• Split: 60 ÷ 3 = 20, and 9 ÷ 3 = 3.
• Combine: 20 + 3 = 23 each.

Solving 846 ÷ 2 by Place Value

846 ÷ 2 = (800 + 40 + 6) ÷ 2 = 400 + 20 + 3 = 423

Check: 2 × 423 = 846.

Predict First: Does This Split Work?

For 372 ÷ 3, does splitting into 300 + 70 + 2 let you divide each part evenly? Predict before advancing.

Choosing Your Decomposition More Flexibly

• 70 ÷ 3 and 2 ÷ 3 are not whole numbers — that split fails.
• Instead: 372 = 300 + 72. Both parts divide evenly by 3.
• 300 ÷ 3 = 100, 72 ÷ 3 = 24. Total: 124.

Four Digits, Same Idea: 4,236 ÷ 4

4,236 ÷ 4 = (4,000 + 200 + 36) ÷ 4 = 1,000 + 50 + 9 = 1,059

Check: 4 × 1,059 = 4,236.

Practice: Decompose, Divide, and Recombine

1. 484 ÷ 4 (straightforward place-value split)
2. 936 ÷ 3 (straightforward place-value split)
3. 636 ÷ 4 (needs regrouping across places)
4. 6,318 ÷ 3 (four digits, needs regrouping)

Write your decomposition and equation for every item, not just the quotient.

What If the Dividend Doesn't Split Nicely?

Place-value splitting works great when every part divides evenly.

What happens when it doesn't split into nice pieces at all?

Partial Quotients: Any Convenient Multiple

• Subtract any multiple of the divisor you're confident about.
• Record how many groups you removed.
• Repeat until nothing is left — then add up everything you recorded.

852 ÷ 6, Taking Big Bites

• 6 × 100 = 600. 852 − 600 = 252. (Recorded: 100)
• 6 × 40 = 240. 252 − 240 = 12. (Recorded: 40)
• 6 × 2 = 12. 12 − 12 = 0. (Recorded: 2)
• Total: 100 + 40 + 2 = 142.

852 ÷ 6 Again, Taking Small Bites

• 6 × 50 = 300. 852 − 300 = 552. (Recorded: 50)
• 6 × 50 = 300. 552 − 300 = 252. (Recorded: 50)
• 6 × 42 = 252. 252 − 252 = 0. (Recorded: 42)
• Total: 50 + 50 + 42 = 142.

Framing Every Step as "Divisor Times What?"

At each step, ask: "6 times what is close to the number I have left?"

Check the final quotient: 6 × 142 = 852.

Four Digits: 3,456 ÷ 8

• 8 × 400 = 3,200. 3,456 − 3,200 = 256. (Recorded: 400)
• 8 × 30 = 240. 256 − 240 = 16. (Recorded: 30)
• 8 × 2 = 16. 16 − 16 = 0. (Recorded: 2)
• Total: 400 + 30 + 2 = 432.

When Something Is Left Over: 5,000 ÷ 7

• 7 × 700 = 4,900. 5,000 − 4,900 = 100. (Recorded: 700)
• 7 × 14 = 98. 100 − 98 = 2. (Recorded: 14)
• 2 is less than 7 — we stop. That leftover is the remainder.
• Quotient: 714 R 2.

Partial Quotients, With and Without Remainders

1. 936 ÷ 6
2. 4,185 ÷ 9
3. 6,341 ÷ 5 (this one has a remainder)

For one problem, find two different sets of partial quotients that both work.

Division as an Unknown Side Length

When you multiplied, you built a rectangle — one side times the other gives the area.

In division, you know the area and one side — you need the other.

The Area Model for 648 ÷ 3

Total width: 200 + 10 + 6 = 216. So 648 ÷ 3 = 216.

Four Digits in the Area Model

• Section 1: 6 × 400 = 2,400 (width 400).
• Section 2: 6 × 10 = 60 (width 10).
• Section 3: 6 × 6 = 36 (width 6).
• Total width: 400 + 10 + 6 = 416.

Reading the Model: Sections and Strips

• Each section's width is a partial quotient.
• The total width is the full quotient.
• A leftover strip too narrow for another full column is the remainder.

A Remainder Inside the Area Model

Width so far: 236. Leftover area: 3 — too narrow for another column. Quotient: 236 R 3.

Partition the Rectangle, Read the Quotient

1. 536 ÷ 4 (draw and partition the rectangle)
2. 3,915 ÷ 5
3. 2,690 ÷ 8 (this one has a remainder — draw the leftover strip)

Write the matching equation beside every model.

Three Strategies, One Compressed Notation

You now have three ways to think about division.

Long division compresses all three into one compact notation.

Answering the Opening Question: 6 or 609?

• Estimate first: 8 × 600 = 4,800 — close to 4,872.
• The answer is in the hundreds, so it has three digits.
• The student who wrote "6" skipped the hundreds place entirely. Correct: 609.

Long Division Is Partial Quotients, Compressed

Same subtractions, same answer — just written in a more compact form.

852 ÷ 6 in Long Division Notation

• 8 hundreds ÷ 6 = 1 hundred (6×1=6). Subtract: 8−6=2 hundreds left.
• Bring down 5 tens: 25 tens ÷ 6 = 4 tens (6×4=24). Subtract: 25−24=1 ten left.
• Bring down 2 ones: 12 ones ÷ 6 = 2 (6×2=12). Subtract: 12−12=0.

Four Digits, With a Remainder: 5,738 ÷ 7

• 57 hundreds ÷ 7 = 8 hundreds (7×8=56). Left: 1 hundred.
• Bring down 3 tens: 13 tens ÷ 7 = 1 ten (7×1=7). Left: 6 tens.
• Bring down 8 ones: 68 ÷ 7 = 9 (7×9=63). Left: 5.

The Check Equation, Every Time

Verify every completed division with this equation before moving on.

Predict First: Is Remainder 11 Ever Valid?

Dividing by 6, is a remainder of 11 ever a correct final answer? Predict before advancing.

One of These Two Is Actually Wrong

• 754 R 5: check gives 6×754+5 = 4,529. Remainder 5 < 6 — valid.
• 753 R 11: check gives 6×753+11 = 4,529. Remainder 11 > 6 — invalid.
• Both pass the check equation. Only one is actually correct.

Find the Error in This Quotient

A student divides 4,218 by 7 and gets 62 R 4. What place value got skipped?

Four-Digit Division, Check Equation Required

1. 5,436 ÷ 4
2. 7,208 ÷ 9 (remainder)
3. 6,015 ÷ 8 (remainder)
4. 3,467 ÷ 6 (remainder)

Write the check equation for every answer, and confirm each remainder is smaller than its divisor.

Four Common Mistakes to Double-Check

Every quotient place needs a digit, even a zero.

Check each subtraction with addition as you go.

The remainder must be smaller than the divisor.

Estimate the quotient's size first to know where to start.

Key Takeaways From This Lesson

✓ Splitting a dividend by place value works when the parts divide evenly.
✓ Partial quotients let you subtract any convenient multiple, piece by piece.
✓ The area model builds the quotient as an unknown side length.
✓ Long division compresses all three into one compact notation.

Coming Up Next: Division in Word Problems

Real problems don't just ask for a quotient and remainder — they ask what the remainder actually means.