Multi-Digit Multiplication: The Standard Algorithm

Fluently Multiply Multi-Digit Whole Numbers

In this lesson:

• Connect the area model to the standard algorithm
• Execute carrying and placeholder zeros correctly
• Use estimation to verify every answer

What You Will Learn Today

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

1. Execute the standard algorithm accurately
2. Explain why each partial product represents a place value
3. Use estimation to verify reasonableness
4. Connect partial products to the area model
5. Identify and correct common errors

Review: Partial Products You Already Know

From Grade 4: You used the area model to multiply two-digit numbers.

• Break each factor into place-value parts
• Multiply every combination
• Add all the partial products

Today: Same math, faster format — the standard algorithm

Area Model: Visualizing 24 Times 13

Four partial products: 200 + 60 + 40 + 12 = 312

Algorithm Groups Four Products Into Two

The standard algorithm for 24 × 13 produces two partial products:

• First: 24 × 3 = 72 (combines 60 + 12)
• Second: 24 × 10 = 240 (combines 200 + 40)
• Sum: 72 + 240 = 312

Same four multiplications, grouped by the second factor's digit

Connecting Area Model to Algorithm

• The "3 column" sections combine into the first partial product
• The "10 row" sections combine into the second partial product

Quick Check: Area Model to Algorithm

For 36 × 27, the area model gives:

Which area sections combine into the first partial product (36 × 7)?

Think before you advance...

How Carrying Manages Place Value

When a column product exceeds 9, regroup:

• Write the ones digit in the current column
• Carry the tens digit to the next column left
• Multiply first, then add the carry

Example: 6 × 7 = 42 → write 2, carry 4 tens

Worked Example: 47 Times 36

Step 1: Estimate first

• 47 is close to 50
• 36 is close to 40
• 50 × 40 = 2,000

Expect an answer near 1,700 to 2,000

Estimation is the first step of every multiplication

First Partial Product: 47 Times 6

Multiply 47 by 6 (the ones digit of 36):

• 7 × 6 = 42 → write 2, carry 4
• 4 × 6 = 24, plus carried 4 = 28 → write 28

First partial product: 282

"Multiply, then add the carry" — say it each time

Second Partial Product: 47 Times 30

Clear carries. Start fresh. Write placeholder zero.

• Placeholder 0 in ones column (multiplying by tens)
• 7 × 3 = 21 → write 1, carry 2
• 4 × 3 = 12, + 2 = 14 → write 14

Second partial product: 1,410

Add and Verify: 47 Times 36

Step 3: Add partial products

Step 4: Check against estimate

• Estimate was ~2,000
• Answer is 1,692
• Close enough — reasonable!

Common Error: Carrying Across Partial Products

• Each partial product is a separate multiplication
• Clear all carried digits before starting the next line
• Use a different pencil color or cross out carries after each line

Your Turn: Compute 58 Times 43

Estimate: 60 × 40 = 2,400

Now compute:

• First partial product: 58 × 3 = ?
• Second partial product: 58 × 40 = ?
• Sum = ?

Pause and compute before advancing

Answer Revealed: 58 Times 43

• First partial product: 58 × 3 = 174
• Second partial product: 58 × 40 = 2,320
• Sum: 174 + 2,320 = 2,494
• Check: 2,494 vs. estimate 2,400 — reasonable!

From Computing to Checking Your Work

You can now execute the algorithm with carrying and placeholder zeros.

Next question: How do you know your answer is reasonable?

Answer: Estimate before you compute, then compare.

Three Strategies for Estimating a Product

Round both: 36 × 27 → 40 × 30 = 1,200

Round one, adjust: 36 × 27 → 36 × 25 = 900

Compatible numbers: 36 × 27 → 40 × 25 = 1,000

All give a ballpark near 1,000 (exact: 972)

Error Detective: Is This Answer Reasonable?

Problem: 63 × 47 = 29,610

Estimate: 60 × 50 = 3,000

The given answer is 29,610 — nearly ten times too large!

The real answer: 2,961 — the student added an extra digit

Which of These Answers Are Reasonable?

Use estimation to decide:

1. 52 × 38 = 1,976
2. 63 × 47 = 29,610
3. 85 × 29 = 465
4. 74 × 56 = 4,144

Estimate each product, then flag any unreasonable answers

Estimation Catches the Errors Instantly

1. 52 × 38 = 1,976 — Reasonable (~2,000)
2. 63 × 47 = 29,610 — Too large (actual: 2,961)
3. 85 × 29 = 465 — Too small (actual: 2,465)
4. 74 × 56 = 4,144 — Reasonable (~4,200)

Scaling Up: Larger Factors, Same Steps

You can multiply two-digit numbers confidently.

The standard algorithm scales to any size:

• 3-digit × 2-digit → two partial products
• 3-digit × 3-digit → three partial products
• The structure never changes — just more steps

Extending the Algorithm: 348 Times 56

Estimate: 350 × 60 = 21,000

348 × 6:

• 8 × 6 = 48 → write 8, carry 4
• 4 × 6 = 24, + 4 = 28 → write 8, carry 2
• 3 × 6 = 18, + 2 = 20

First partial product: 2,088

Completing the Product: 348 Times 56

348 × 50 (placeholder zero):

• 8 × 5 = 40 → write 0, carry 4
• 4 × 5 = 20 + 4 = 24 → write 4, carry 2
• 3 × 5 = 15 + 2 = 17

Estimate 21,000 — reasonable

Three Partial Products: 245 Times 163

• 245 × 3 = 735 (ones)
• 245 × 60 = 14,700 (tens — one placeholder zero)
• 245 × 100 = 24,500 (hundreds — two placeholder zeros)

Estimate: 250 × 160 = 40,000 — close!

Placeholder Zeros Follow a Place Value Pattern

Each line shifts one place left for the next power of 10.

Your Turn: Compute 276 Times 43

Estimate: 280 × 40 = 11,200

Compute:

• First partial product: 276 × 3 = ?
• Second partial product: 276 × 40 = ?
• Sum = ?

Estimate, compute, and verify before advancing

Answer Revealed: 276 Times 43

• First partial product: 276 × 3 = 828
• Second partial product: 276 × 40 = 11,040
• Sum: 828 + 11,040 = 11,868
• Estimate: 11,200
• Check: 11,868 is close to 11,200 — reasonable!

Apply the Algorithm to a Word Problem

A school orders 165 boxes at $47 each. Total cost?

• Estimate: 170 × 50 = $8,500

1. 165 × 7 = 1,155
2. 165 × 40 = 6,600
3. Sum: 1,155 + 6,600 = $7,755
4. Check: $7,755 vs. $8,500 — reasonable

Key Takeaways for Multi-Digit Multiplication

• Algorithm organizes the same partial products as the area model
• Carrying: multiply first, then add the carry
• Placeholder zeros reflect place value — not optional
• Clear carries before each new partial product
• Estimate, Compute, Check — every time

Fluent = Accurate + Efficient + Flexible

What Comes Next in Your Learning

You can now:

• Multiply multi-digit whole numbers using the standard algorithm
• Verify your answers with estimation

Coming up:

• Dividing multi-digit whole numbers (5.NBT.B.6)
• Multiplying and dividing decimals (5.NBT.B.7)