Adding and Subtracting Multi-Digit Numbers

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

In this lesson:

• Add and subtract large numbers with the standard algorithm
• Check your answers so mistakes never slip by

Learning Objectives for This Lesson

You will learn to:

1. Add multi-digit numbers with the standard algorithm
2. Subtract multi-digit numbers with the standard algorithm
3. Explain why carrying and borrowing work
4. Subtract across zeros using chain borrowing
5. Check answers with estimation and inverse operations

52,347 − 18,693 = 43,654 — Right?

• A student computed this subtraction and got 43,654.
• It looks reasonable at a glance. Is it actually correct?

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

Two Quick Recalls Before We Start

• 10 ones make how many tens? 10 tens make how many hundreds?
• Round 47,694 to the nearest thousand.

Review: 456 + 378 — Naming Every Trade

6 + 8 ones = 14 ones = 1 ten and 4 ones. Write the carry clearly above the tens column.

Before We Add: Estimating 24,358 + 47,694

• Round each number to the nearest thousand: 24,000 and 48,000.
• Estimate: 24,000 + 48,000 = 72,000.

Now let's find the exact answer — and check it against this.

24,358 + 47,694, Column by Column

What If You Forget the Carry?

In the hundreds column, 3 + 6 + 1 = 10 hundreds. What answer would you get if you wrote the 0 but forgot to carry the 1 to the thousands column?

Six Digits, Same Procedure: 385,476 + 247,859

Add: Four Digits, Then Five, Then Six

1. 3,847 + 2,965
2. 36,284 + 47,916
3. 528,374 + 296,847

Estimate each first. One of these has a column that sums to exactly 10.

Addition Carries Up, Subtraction Borrows Down

Addition combines two numbers and carries extra amounts up to bigger places.

Subtraction is the mirror image — it borrows amounts down from bigger places.

Review: 834 − 378 — Naming Every Trade

4 ones isn't enough to subtract 8. Borrow 1 ten, making 14 ones: 14 − 8 = 6.

Student A: Just Subtract the Smaller?

For 7,324 − 2,856, Student A subtracts the smaller digit from the larger in every column: 6−4, 5−2, 8−3, 7−2. Is that a valid method? Predict before advancing.

Why That Doesn't Work — Real Answer: 4,468

Student B: Forgot to Reduce?

For the same 7,324 − 2,856, Student B adds 10 to borrow in each column but never reduces the column that lent it: 14−6, 12−5, 13−8, 7−2. Is that answer, 5,578, correct? Predict before advancing.

Cross Out, Reduce, Then Subtract

• Student B's error: never subtracted 1 from the lending column.
• Rule: cross out the lending digit and write the reduced digit immediately.
• Check: 5,578 + 2,856 = 8,434, not 7,324 — the check catches it.

63,421 − 27,856: Four Borrows in a Row

Subtract: Regrouping, None, and Mixed

1. 8,652 − 3,412 (no regrouping needed)
2. 91,634 − 47,758 (regroups in every column)
3. 5,742 − 2,381 (regroups in only one column)

Check each answer with the inverse operation.

What If There's Nothing to Borrow From?

Every borrow so far has come from a column with something to lend.

What happens when that neighboring column is also zero?

5,000 − 1,237: Where Do We Borrow From?

• Ones column: 0 − 7. We need to borrow — but the tens digit is 0.
• The hundreds digit is also 0. Where do we actually go?

Think it through before advancing.

The Chain of Trades, in Expanded Form

The Pattern: Every Zero Becomes a 9

• Every zero the borrow passes through becomes a 9; the final column gets 10.
• Write the regrouped digits separately — the last one is 10, not one digit.

80,000 − 34,567: The Pattern Again

• Top row regrouped: 7, 9, 9, 9, 10.
• 10 − 7 = 3, 9 − 6 = 3, 9 − 5 = 4, 9 − 4 = 5, 7 − 3 = 4.
• Answer: 45,433. Estimate: 80,000 − 35,000 = 45,000 — reasonable.

7,403 − 2,658: A Zero in the Middle

• Ones needs to borrow, but tens is 0 — reach past it to hundreds.
• 4 hundreds → 3 hundreds; 0 tens → 10 tens; 1 ten moves down to make 13 ones.
• Continuing the chain gives 4,745.

More Practice Subtracting Across Zeros

1. 6,000 − 2,483
2. 40,000 − 17,529
3. 9,006 − 3,478 (zero in the middle)
4. 300,000 − 145,672

Estimate first. Write out the regrouped digits before subtracting.

Back to That Subtraction From Class

Remember 52,347 − 18,693 = 43,654 from the start of class?

You now have everything you need to check it for real.

Diagnosing 52,347 − 18,693 = 43,654

• Estimate: 52,000 − 19,000 = 33,000. The answer is about 10,000 too high.
• Inverse check: 43,654 + 18,693 = 62,347, not 52,347. It fails.
• The error: the ten-thousands digit wasn't reduced from 5 to 4. Correct answer: 33,654.

The Full Fluency Routine, Step by Step

1. Estimate before computing.
2. Compute using the standard algorithm.
3. Check using estimation or the inverse operation.
4. Compare the check to your original answer.

Modeling the Full Cycle: 34,567 + 28,945

• Estimate: 35,000 + 29,000 = 64,000.
• Compute: 34,567 + 28,945 = 63,512.
• Check: 63,512 − 28,945 = 34,567. Matches the original — confirmed.

Now Subtract 28,945 Back Out

You just found 34,567 + 28,945 = 63,512.

Subtract 28,945 from that result. What should you get?

Your Turn: A Mixed Set, Fully Independent

1. 372,846 + 158,937
2. 604,213 − 287,459
3. 45,608 + 96,375
4. 500,231 − 263,847

Estimate, compute, and check every single one — on your own.

Two More Zero Crossings to Practice

1. 50,000 − 23,456
2. 8,004 − 3,675

Both need an estimate and a check — no exceptions, even under time pressure.

Four Common Mistakes to Double-Check

A dropped carry cascades into every later column.

Cross out and reduce the lending column the moment you borrow.

Don't ignore which number a digit belongs to — regroup instead.

Zeros aren't a wall — keep going left, and each becomes a 9.

Key Takeaways From This Lesson

✓ Carrying and borrowing are the same trade, forward and backward.
✓ A zero column has nothing to lend — keep going left.
✓ Estimate before you compute, and check after.
✓ Fluency means speed and accuracy together, not speed alone.

Coming Up Next: Multi-Digit Multiplication

Multiplying large numbers breaks into smaller partial products.

Adding those partial products together uses the exact addition skill you just built.