How to Explain Multiplication to Fourth Graders

"Why do we put a zero there?"

If your fourth grader can't answer this question about the standard multiplication algorithm, they're following steps without understanding. And when steps are followed without understanding, errors multiply.

Fourth grade is when multiplication gets serious—multi-digit numbers demand strategies beyond memorized facts. Let's explore how to build real understanding.

Why Multi-Digit Multiplication Matters

In fourth grade, students tackle problems like 34 × 26 or 2,456 × 7. These aren't just bigger versions of basic facts—they require:

• Deep understanding of place value
• Knowledge of the distributive property
• Ability to organize multi-step calculations
• Estimation skills to check reasonableness

Students who master multi-digit multiplication with understanding will find division, fractions, and algebra much more accessible.

The Foundation: What Multiplication Really Means

Before tackling algorithms, ensure students understand multiplication as:

Repeated addition: 4 × 3 = 3 + 3 + 3 + 3 = 12

Groups of: 4 groups of 3 = 12

Array/area: A rectangle with 4 rows and 3 columns has 12 squares

This last interpretation—area—becomes crucial for multi-digit multiplication.

Method 1: The Area Model

The area model makes place value visible. It shows that multiplying 34 × 26 is really four separate multiplications combined.

How It Works

To multiply 34 × 26:

Break each number into place values:

• 34 = 30 + 4
• 26 = 20 + 6

Draw a rectangle divided into four parts:

           30         4
       ┌─────────┬────────┐
   20  │  30×20  │  4×20  │
       │  =600   │  =80   │
       ├─────────┼────────┤
    6  │  30×6   │  4×6   │
       │  =180   │  =24   │
       └─────────┴────────┘

Add all four products: 600 + 80 + 180 + 24 = 884

Why It Works

The area model shows that 34 × 26 equals:

• (30 × 20) + (4 × 20) + (30 × 6) + (4 × 6)
• This is the distributive property in action!

When to Use It

The area model is perfect for:

• Building initial understanding
• Visual learners
• Making place value explicit
• Connecting multiplication to geometry

Method 2: Partial Products

Partial products organize the same calculations in a list format.

How It Works

34 × 26:

Write out each partial product:

    34
  × 26
  ────
    24  ← 4 × 6 (ones × ones)
   180  ← 30 × 6 (tens × ones)
    80  ← 4 × 20 (ones × tens)
   600  ← 30 × 20 (tens × tens)
  ────
   884

The Connection to Area Model

Each line in partial products corresponds to one rectangle in the area model:

• 24 = small rectangle (4 × 6)
• 180 = bottom-left rectangle (30 × 6)
• 80 = top-right rectangle (4 × 20)
• 600 = large rectangle (30 × 20)

Advantages of Partial Products

• Shows all the "hidden" multiplications
• Reduces regrouping errors
• Easy to check each step
• Builds understanding before learning the shortcut

Method 3: The Standard Algorithm

The standard algorithm is a shortcut that compresses partial products.

How It Works

34 × 26:

    34
  × 26
  ────
   204  ← 34 × 6 (multiply by ones)
   680  ← 34 × 20 (multiply by tens)
  ────
   884

The Critical "Why"

Why do we add a zero (or shift left) when multiplying by the tens digit?

Because we're not multiplying by 2—we're multiplying by 20!

34 × 2 = 68, but we need 34 × 20 = 680

The zero (or leftward shift) accounts for the place value of the tens digit.

Teaching the Algorithm with Understanding

Don't just teach steps. Connect each step to meaning:

Step 1: 34 × 6 = 204

• This is the ones portion (34 groups of 6)

Step 2: 34 × 20 = 680

• This is the tens portion (34 groups of 20)
• The 0 or shift shows we're working with tens

Step 3: Add the partial products

• 204 + 680 = 884

Multiplying by a 1-Digit Number

Before 2-digit × 2-digit, master multiplying larger numbers by a single digit.

Example: 2,456 × 7

Using partial products:

  2,456
  ×   7
  ─────
     42  ← 6 × 7
    350  ← 50 × 7
  2,800  ← 400 × 7
 14,000  ← 2,000 × 7
  ─────
 17,192

Using the standard algorithm:

    3 4
  2,456
  ×   7
  ─────
 17,192

(Regroup the 4 tens from 6×7=42, then 3+35=38 from 5×7+4, etc.)

Common Errors

• Forgetting to add regrouped numbers
• Regrouping to the wrong place
• Writing regrouped digits in the wrong position

Fix: Have students write regrouped digits small and above, then cross them off as they add them.

Estimation: The Reasonableness Check

Before and after calculating, estimate to check reasonableness.

Front-End Estimation

Round both numbers to their leading digit:

• 34 × 26 ≈ 30 × 30 = 900

Our answer of 884 is reasonable!

Is 34 × 26 = 8,840 Reasonable?

No! 30 × 30 = 900, so the answer should be near 900, not near 9,000.

Teaching estimation catches place value errors before they become habits.

Hands-On Activities

Base-10 Block Multiplication

Use base-10 blocks to physically build area models:

• Flats (100) represent the largest rectangle
• Rods (10) represent the medium rectangles
• Units (1) represent the smallest rectangle

Build 23 × 14 with blocks, then count to find the product.

Grid Paper Area Models

Draw area models on grid paper where students can count squares:

• For 23 × 14, draw a 23 × 14 rectangle
• Divide into parts by place value
• Count or calculate each section

Multiplication War

Play with a deck of cards (face cards = 10, ace = 1):

• Each player draws 4 cards and arranges as a 2-digit × 2-digit problem
• Players calculate their products
• Larger product wins all cards

Real-World Problems

Connect multiplication to life:

• "A pack of 24 markers costs $6. How much for 12 packs?"
• "There are 365 days in a year. How many days in 4 years?"
• "The auditorium has 28 rows with 32 seats each. How many seats total?"

Connecting Methods

Help students see that all three methods give the same answer because they're doing the same math—just organized differently.

Show the connections:

Area Model:           Partial Products:    Standard Algorithm:
  30    4               34                     34
┌────┬────┐           × 26                   × 26
│600 │ 80 │20          24                    204
├────┼────┤           180                    680
│180 │ 24 │6           80                    884
└────┴────┘           600
600+80+180+24         884
    = 884

All three organize the same four multiplications: 30×20, 30×6, 4×20, and 4×6.

Common Mistakes and How to Fix Them

Mistake 1: Not Shifting on the Second Line

Wrong:

  34
× 26
────
 204
  68  ← Should be 680!
────
 272  ← Wrong!

Fix: Ask "What are you multiplying 34 by?" If they say "2," point out that the 2 is in the tens place, so it represents 20. Use partial products to show why the shift matters.

Mistake 2: Regrouping Errors

Wrong: 34 × 6 = 184 (forgot to add the regrouped 2)

Fix: Write regrouped digits clearly above, use a consistent system, and check each step.

Mistake 3: Basic Fact Errors

Multi-digit multiplication falls apart if basic facts aren't automatic.

Fix: If basic facts are shaky, practice them while also using calculators for multi-digit work. Don't let weak facts prevent conceptual growth, but do address them separately.

Mistake 4: Place Value Misalignment

Wrong: Writing 600 as 60 because they forgot what place they're in.

Fix: Use graph paper to keep digits aligned. Have students write place value labels above each column.

Building Mental Math Skills

Strong mental math supports written algorithms.

Breaking Apart Strategy

For 25 × 12:

• 25 × 12 = 25 × 10 + 25 × 2
• = 250 + 50
• = 300

Using Friendly Numbers

For 19 × 6:

• 19 × 6 = 20 × 6 - 1 × 6
• = 120 - 6
• = 114

Doubling and Halving

For 25 × 16:

• Double 25 → 50, halve 16 → 8
• 50 × 8 = 400

Connecting to Future Concepts

Multi-digit multiplication prepares students for:

Long Division

Division is the inverse of multiplication. Understanding place value in multiplication makes division algorithms sensible.

Algebra

The distributive property (30 + 4)(20 + 6) = FOIL in algebra: (a + b)(c + d) = ac + ad + bc + bd

Fraction Multiplication

Multiplying fractions uses the same principles—multiply numerators, multiply denominators.

Practice Ideas for Home

Daily Estimation

See a product in the store? "About how much would 24 of those cost?" Practice quick estimation.

Recipe Multiplication

"This recipe serves 4, but we have 12 people. How much of each ingredient do we need?"

Distance and Time

"We drive 55 miles per hour for 3 hours. How far do we go?"

Money Problems

"If you save $15 per week, how much will you have after 52 weeks?"

The Bottom Line

Multi-digit multiplication is more than an algorithm to memorize. It's an opportunity to deeply understand place value, the distributive property, and how our number system works.

Start with models that show the "why"—area models and partial products. Build toward the standard algorithm as a shortcut that makes sense because students understand what it's shortcutting.

When your fourth grader can explain why we "add a zero" when multiplying by the tens digit, they don't just know how to multiply—they understand mathematics.