The area model finds every partial product. The written method below does the same work in a shorter way: one column at a time, carrying as you go.
Key idea
Multiply the ones, then the tens, then the hundreds. When a product is 10 or more, write its ones digit and carry the rest to the next place. In the next place, multiply first, then add the carried amount.
Why it works
In $346 \times 7$, the ones give $6 \times 7 = 42$ ones: 4 tens and 2 ones. The 4 tens join the tens column: $4 \times 7 = 28$ tens, plus the 4 tens makes 32 tens. These are the same partial products as the area model, $2{,}100 + 280 + 42$, added as you go.
Worked example: Carrying twice
Find $346 \times 7$.
- Ones: $6 \times 7 = 42$. Write 2, carry 4.
- Tens: $4 \times 7 = 28$, plus 4 carried is 32. Write 2, carry 3.
- Hundreds: $3 \times 7 = 21$, plus 3 carried is 24. Write 24.
- Check with the area model: $2{,}100 + 280 + 42 = 2{,}422$.
Answer: $2{,}422$
Worked example: Four digits
Find $1{,}508 \times 6$.
- Ones: $8 \times 6 = 48$. Write 8, carry 4.
- Tens: $0 \times 6 = 0$, plus 4 carried is 4. Write 4.
- Hundreds: $5 \times 6 = 30$. Write 0, carry 3.
- Thousands: $1 \times 6 = 6$, plus 3 carried is 9. Write 9.
Answer: $9{,}048$
Common mistake
Forgetting to add the carried amount, so $346 \times 7$ comes out as $2{,}182$. Another slip is adding the carry before multiplying ($4 + 4 = 8$, then $8 \times 7$). Multiply first, then add the carry. Estimate to check: $350 \times 7 = 2{,}450$, close to $2{,}422$.