How to Explain Multi-Digit Multiplication to Fifth Graders

"Why do I have to put a zero there?"

If your fifth grader has asked this about the standard multiplication algorithm, they're ready to understand what they're really doing—not just following steps.

Why Multi-Digit Multiplication Matters

In fifth grade, multiplication expands dramatically. Students move from multiplying by single digits to handling problems like 345 × 67. This skill is essential for:

• Real-world calculations (costs, measurements, quantities)
• Understanding area and volume problems
• Preparing for division with larger numbers
• Building toward algebra and proportional reasoning

The Standard Algorithm: What It Really Means

Let's decode what's actually happening when we multiply 345 × 67.

Step-by-Step Breakdown

      345
    ×  67
    -----

Step 1: Multiply 345 by 7 (the ones digit)

      345
    ×  67
    -----
     2415   (345 × 7)

Step 2: Multiply 345 by 60 (the tens digit)

      345
    ×  67
    -----
     2415
   20700    (345 × 60)

Step 3: Add the partial products

      345
    ×  67
    -----
     2415
   20700
   -----
   23115

Why the "Zero" Appears

That zero (or shift) in 20700 exists because we're multiplying by 60, not 6!

• 345 × 6 = 2,070
• But we need 345 × 60 = 20,700

The zero is a placeholder that shows we're in the tens place.

Two Methods That Build Understanding

Method 1: Partial Products (Expanded Form)

Break each number into place values:

345 × 67 = 345 × (60 + 7)

345 × 7  =    2,415
345 × 60 =   20,700
            -------
             23,115

Or expand even further:

345 = 300 + 40 + 5
67 = 60 + 7

300 × 60 = 18,000
300 × 7  =  2,100
40 × 60  =  2,400
40 × 7   =    280
5 × 60   =    300
5 × 7    =     35
         -------
          23,115

This shows every multiplication that happens!

Method 2: Area Model (Box Method)

Visualize multiplication as finding area:

              300        40         5
         +--------+--------+--------+
    60   | 18,000 |  2,400 |   300  |
         +--------+--------+--------+
     7   |  2,100 |    280 |    35  |
         +--------+--------+--------+

Add all boxes: 18,000 + 2,400 + 300 + 2,100 + 280 + 35 = 23,115

This visual model shows exactly what partial products represent.

Building from Simpler to Complex

Level 1: Two-Digit × One-Digit (Review)

    47
  ×  6
  ----
   282

(6 × 7 = 42, write 2, carry 4)
(6 × 4 = 24, plus 4 = 28)

Level 2: Two-Digit × Two-Digit

    47
  × 23
  ----
   141   (47 × 3)
   940   (47 × 20)
  ----
  1081

Level 3: Three-Digit × Two-Digit

    234
  ×  56
  -----
   1404   (234 × 6)
  11700   (234 × 50)
  -----
  13104

Level 4: Larger Numbers

    1,234
  ×    45
  -------
    6,170   (1,234 × 5)
   49,360   (1,234 × 40)
  -------
   55,530

The Estimation Checkpoint

Always estimate first!

For 345 × 67:

• 345 ≈ 350
• 67 ≈ 70
• 350 × 70 = 24,500

Our answer (23,115) is close to 24,500. ✓

If a student gets 2,311 or 231,150, the estimate catches the error immediately.

Quick Estimation Strategies

Round to easy numbers:

• 489 × 52 ≈ 500 × 50 = 25,000

Use compatible numbers:

• 312 × 48 ≈ 300 × 50 = 15,000

Hands-On Activities

Graph Paper Multiplication

Use graph paper to literally show area:

• Draw a rectangle 23 squares by 14 squares
• Count total squares = 23 × 14 = 322
• Or split into sections and add

Building with Base-10 Blocks

For smaller problems like 24 × 13:

• Make 13 groups of 24 (or 24 groups of 13)
• Trade and count

The Lattice Method Alternative

Some students love the lattice method:

For 34 × 52:

    3    4
  +----+----+
  |1 /|2 /| 5
  | / | / |
  |/ 5|/ 0|
  +----+----+
  |0 /|0 /| 2
  | / | / |
  |/ 6|/ 8|
  +----+----+

Add diagonals: 8, 0+6+0=6, 2+5+0=7, 1=1
Answer: 1,768

Multiplication War

Use playing cards (face cards = 10, Ace = 1):

• Each player draws 3 cards for a 3-digit number
• Then draws 2 cards for a 2-digit number
• Multiply. Largest product wins the round!

Common Mistakes and How to Fix Them

Mistake 1: Forgetting to Shift/Add Zero

Wrong:

    345
  ×  67
  -----
   2415
   2070   ← Missing the shift!
  -----
   4485   ← Way too small

Fix: Use place value language: "Now I'm multiplying by 60, which is in the tens place, so my answer goes in the tens place." Write the 0 first as a reminder.

Mistake 2: Carrying Errors

Wrong:

    345
  ×   7
  -----
   2115   ← Carried the 3 but forgot to add it

Fix: Write carries clearly above the problem. Check: 7 × 5 = 35 (write 5, carry 3). 7 × 4 = 28, + 3 = 31 (write 1, carry 3). 7 × 3 = 21, + 3 = 24.

Mistake 3: Multiplying Digits in Wrong Order

Wrong: Multiplying 345 × 67 by starting with the 6 instead of the 7.

Fix: Always start with the ones digit of the bottom number. Some students write "Start here →" by the ones digit.

Mistake 4: Losing Track in Long Problems

Fix: Use lined paper turned sideways to keep columns aligned. Or draw vertical lines between place values:

    |3|4|5|
  × | |6|7|
  ---------

Mistake 5: Not Checking Reasonableness

Wrong: Getting 231,115 for 345 × 67 and not noticing.

Fix: Make estimation mandatory. 350 × 70 = 24,500. Is 231,115 close to 24,500? Not at all!

Mental Math Strategies

Doubling and Halving

For 25 × 48:

• Half of 48 = 24
• Double 25 = 50
• 50 × 24 = 1,200

Breaking Apart

For 99 × 34:

• 99 = 100 - 1
• 100 × 34 = 3,400
• 1 × 34 = 34
• 3,400 - 34 = 3,366

Using Friendly Numbers

For 45 × 20:

• 45 × 2 = 90
• 90 × 10 = 900

Practice Ideas for Home

Real-World Problems

"If we drive 65 miles per hour for 4 hours, how far do we travel?"
"A box has 144 crayons. How many crayons are in 12 boxes?"
"Movie tickets cost $14 each. What's the cost for 23 people?"

Speed Drills with a Twist

Instead of timing basic facts, time multi-digit problems with estimation:

• Write estimate
• Calculate
• Check if answer is reasonable

Error Analysis

Give a worked problem with an error. Can they find it?

    267
  ×  45
  -----
   1035   ← Error somewhere!
  10680
  -----
  11715

(Error: 267 × 5 = 1,335, not 1,035)

Create Word Problems

Have your child write word problems that require multi-digit multiplication. This builds deeper understanding than just solving.

Connecting to Future Concepts

Decimal Multiplication

When students multiply 3.45 × 6.7, they'll use the same algorithm—then adjust for decimal places:

• 345 × 67 = 23,115
• Count decimal places: 2 + 1 = 3
• Answer: 23.115

Polynomial Multiplication (Algebra)

The area model for 34 × 25:

    30 + 4
    20 + 5

Becomes the FOIL method for (3x + 4)(2x + 5) in algebra!

Proportional Reasoning

Understanding that 20 × 345 = 10 × (2 × 345) = 10 × 690 builds proportional thinking.

The Bottom Line

Multi-digit multiplication isn't about memorizing a mysterious procedure. It's about understanding that when we multiply 345 × 67, we're really finding:

• 345 groups of 7, plus
• 345 groups of 60

The standard algorithm is simply an efficient way to organize that work.

When your fifth grader understands why they add that zero and why the partial products line up the way they do, they're not just following steps—they're doing real mathematics.

And that understanding will carry them through decimals, algebra, and beyond.