Multi-Digit Multiplication
Multiply 2-digit by 2-digit numbers with the standard algorithm, understand why each step works through partial products, and multiply by powers of 10 in your head.
1 What multi-digit multiplication really is
Multiplication is repeated addition of equal groups. 23 × 14 means 23 groups of 14 — or, by place value, (20 + 3) groups of (10 + 4).
Partial products: break it apart
Split each number by place value, multiply every part by every part, then add:
Those four pieces — 200, 80, 30, 12 — are called partial products. The standard algorithm you write on paper is just a shortcut for adding them up.
The standard algorithm, line by line
For 23 × 14: multiply 23 by the ones digit (4), then multiply 23 by the tens digit (1 ten = 10). Write a 0 placeholder first on the second row, because that row is really multiplying by 10. Then add the rows.
23 × 10 = 230
92 + 230 = 322
2 See it: the area model
Draw 23 × 14 as a rectangle 23 wide and 14 tall. Splitting the sides by place value splits the area into the four partial products.
3 Worked examples
Same pattern every time: ones row, tens row (with the 0), add.
- Thinking: ones row first: 23 × 4 = 92.
- Tens row: 23 × 1 ten = 230 — write the 0 placeholder first, then 23.
- Add the rows: 92 + 230 = 322.
- Check: 23 × 14 is about 20 × 15 = 300. 322 is close, so it is reasonable.
- Ones row: 46 × 8 = 368 (8 × 6 = 48, write 8 carry 4; 8 × 4 = 32, plus 4 = 36).
- Tens row: 46 × 20 = 920 — 0 placeholder first.
- Add: 368 + 920 = 1288.
- Check: about 50 × 30 = 1500. 1288 is in range.
- Thinking: multiplying by 100 shifts every digit two places left — just append two zeros.
- 35 × 100 = 3500.
- Check: 3500 ÷ 100 = 35. Correct.
- Ones row: 67 × 3 = 201.
- Tens row: 67 × 40 = 2680 — 0 placeholder first.
- Add: 201 + 2680 = 2881.
- Check: about 70 × 40 = 2800. 2881 is close.
4 Common mistakes
Two traps catch almost everyone. Spot them here and they will never cost you points.
23 × 4 = 92
23 × 1 = 23
92 + 23 = 115. Way too small!
23 × 4 = 92
23 × 10 = 230
92 + 230 = 322.
5 Key vocabulary
- Factor
- A number being multiplied. In 23 × 14, both 23 and 14 are factors.
- Product
- The answer to a multiplication.
- Partial product
- One piece of the total, found by multiplying parts of the factors (e.g. 20 × 10 = 200).
- Placeholder zero
- The 0 written at the start of the tens row because that row multiplies by tens.
- Regrouping (carrying)
- Trading 10 ones for 1 ten (and so on) inside a row.
- Estimate
- A quick nearby value — round first, then multiply — used to check reasonableness.
6 Quick check
Try these, then reveal the answers.
Key points to remember
- Break numbers by place value: 23 × 14 = (20 + 3) × (10 + 4).
- Partial products add up to the final product: 200 + 80 + 30 + 12 = 322.
- In the standard algorithm, write the 0 placeholder first on every new row.
- Multiplying by 10, 100, 1000 just appends zeros.
- Estimate before you compute: 48 × 21 ≈ 50 × 20 = 1000.