Multiplying Three-Digit Numbers

Lesson overview

What you will learn

Three-digit multiplication extends the distributive decomposition: split one factor into hundreds, tens, and ones, multiply the other factor by each chunk, sum the partial products. Three hundred twelve times twenty-four: 312 × 20 = 6,240; 312 × 4 = 1,248; total 7,488.

Choosing the factor to decompose matters more at this scale. Decompose the factor with more zeros or simpler digits — 205 × 132 is easier as 205 × (100 + 30 + 2) because 205 × 100 = 20,500 is immediate.

Partial products can be checked independently against estimates. Three hundred twelve times twenty should be near 300 × 20 = 6,000; getting 6,240 confirms the first chunk. Catching a error in one chunk prevents propagating it.

This stage integrates skills from two-digit multiplication, powers of ten, and addition of large numbers. Weakness in any prerequisite shows up as slowdowns or errors in specific chunks.

Why it matters

Build the idea before the speed

Chunking keeps large multiplication transparent and allows partial results to be checked independently.

In adult life

Where this skill shows up

Construction and manufacturing quotes often multiply unit costs by three-digit quantities: 312 units at $24 each is a real bid calculation where mental verification catches spreadsheet formula errors.

Distance and time problems scale similarly: traveling 312 miles per day for 24 days is 7,488 miles — a figure worth estimating before trusting a GPS log summary.

The method

Step by step

1. 1Choose the factor with the easiest decomposition.
2. 2Multiply the other factor by each place-value chunk.
3. 3Add partial products from largest to smallest.

Worked examples

See the structure

Avoid these traps

Common mistakes

• Forgetting one of the three partial products when decomposing into hundreds, tens, and ones.
• Misplacing place value in partial products, especially for the ×20 or ×200 chunk.
• Decomposing both factors into four or six partial products unnecessarily, overwhelming working memory.
• Skipping independent estimates on each chunk and losing the ability to localize errors.

Practice tips

• Decompose the factor that makes the largest partial product easiest — usually the one with round hundreds.
• Estimate the full product first: 312 × 24 ≈ 300 × 25 = 7,500.
• Compute and register each partial product before moving to the next; do not chain silently.
• Verify the final sum by comparing to your opening estimate.

Check your understanding

1. Why is 205 × 132 computed as three partial products rather than two?
2. What independent estimate confirms that 312 × 20 = 6,240 is reasonable?
3. How would doubling-and-halving apply — or not apply — to 487 × 103?

Your turn

Interactive practice

Answer one problem at a time. Each problem has a fresh 60-second timer, immediate correction, and a final accuracy score. Aim for understanding first; speed comes after accuracy.