Multiplying multi-digit numbers comes down to breaking the problem into smaller, single-digit multiplications and then combining the results. Whether you use the traditional method taught in most classrooms, a visual grid, or a lattice, the underlying idea is the same: multiply piece by piece, keep track of place value, and add everything together at the end.
The Standard Algorithm
This is the classic method you’ll see in most math classes. Write the larger number on top and the smaller number directly below it, lining up the digits by place value (ones under ones, tens under tens, and so on). Draw a line underneath and you’re ready to start.
Begin with the ones digit of the bottom number. Multiply it by each digit of the top number, working right to left. When a single multiplication gives you a two-digit result, write the ones digit in the answer row and “carry” the tens digit above the next column to the left. For example, if you’re multiplying 92 × 7 and you start with 2 × 7 = 14, write the 4 below the line and carry the 1 above the tens column. Then multiply 9 × 7 = 63, add the carried 1 to get 64, and write 64 to the left of the 4. Your answer is 644.
When the bottom number has more than one digit, you repeat this process for each digit, but you shift your answer one place to the left each time. That shift accounts for place value. If you’re multiplying by the tens digit, your partial product starts in the tens column. Some people place a zero as a placeholder to keep things lined up. For the hundreds digit, you’d place two zeros, and so on. Once you’ve written out all the partial products, add them together to get the final answer.
Worked Example: 253 × 46
Write 253 on top and 46 below it. Start with the 6 in the ones place:
- 6 × 3 = 18. Write 8, carry 1.
- 6 × 5 = 30, plus the carried 1 = 31. Write 1, carry 3.
- 6 × 2 = 12, plus the carried 3 = 15. Write 15.
Your first partial product is 1,518.
Now move to the 4 in the tens place. Place a 0 in the ones column as a placeholder, then multiply:
- 4 × 3 = 12. Write 2, carry 1.
- 4 × 5 = 20, plus the carried 1 = 21. Write 1, carry 2.
- 4 × 2 = 8, plus the carried 2 = 10. Write 10.
Your second partial product is 10,120. Add the two partial products: 1,518 + 10,120 = 11,638.
The Area Model (Box Method)
The area model makes place value visible by splitting each number into its parts and organizing the multiplication inside a grid. It’s especially helpful if carrying digits feels confusing, because every piece of the problem is laid out in front of you.
To multiply 6 × 5,432, break 5,432 into 5,000 + 400 + 30 + 2. Draw a rectangle divided into four sections, one for each part. Label the top of each section with one component (5,000, 400, 30, 2) and write 6 along the side. Now multiply to fill each box:
- 6 × 5,000 = 30,000
- 6 × 400 = 2,400
- 6 × 30 = 180
- 6 × 2 = 12
Add the four results: 30,000 + 2,400 + 180 + 12 = 32,592.
For two multi-digit numbers, you break both of them apart. To multiply 34 × 52, split 34 into 30 + 4 and 52 into 50 + 2. Draw a 2 × 2 grid. Label the columns 30 and 4 across the top, and the rows 50 and 2 down the side. Fill each of the four boxes: 50 × 30 = 1,500, 50 × 4 = 200, 2 × 30 = 60, 2 × 4 = 8. Add them: 1,500 + 200 + 60 + 8 = 1,768.
Lattice Multiplication
Lattice multiplication uses a grid with diagonal lines to organize carrying automatically. It looks unusual at first, but once you’ve tried it a few times, many people find it faster and less error-prone than the standard algorithm for larger numbers.
Here’s how to multiply 48 × 36:
Draw a 2 × 2 grid (two digits by two digits). Write 4 and 8 across the top of the two columns. Write 3 and 6 down the right side of the two rows. Then draw a diagonal line from the top-right corner to the bottom-left corner of each box.
Multiply each column digit by each row digit and write the result in the corresponding box. The tens digit goes above the diagonal, and the ones digit goes below it. For instance, 4 × 3 = 12, so write 1 above the diagonal and 2 below it. Do this for all four boxes: 8 × 3 = 24, 4 × 6 = 24, 8 × 6 = 48.
Now read the answer by adding along the diagonals, starting from the bottom-right corner. The first diagonal (ones place) contains just the 8 from 48. The next diagonal (tens) contains 4 + 4 + 2 = 10; write 0 and carry 1 to the next diagonal. The hundreds diagonal has 2 + 1 + 2 = 5, plus the carried 1 = 6. The final diagonal (thousands) is just 1. Reading from top-left to bottom-right gives you 1,728.
Checking Your Answer with Estimation
A quick estimate before or after you multiply can catch big mistakes. The idea is to round each number to something simple, multiply those rounded numbers in your head, and see if your detailed answer is in the right neighborhood.
Round each factor to its leading digit. For 253 × 46, round 253 to 250 (or even 300) and 46 to 50. Then 250 × 50 = 12,500. Your exact answer of 11,638 is reasonably close, so it checks out. If you’d accidentally gotten 116,380, the estimate would immediately flag the error.
A useful trick for mental math: break one of the rounded numbers into smaller pieces. To compute 250 × 50 in your head, think of it as 250 × 5 × 10. That’s 1,250 × 10 = 12,500. This “break it up” approach works well whenever one factor ends in zeros.
Choosing the Right Method
All three methods give the same answer. The standard algorithm is the most compact and the fastest once you’re comfortable with carrying. The area model is the clearest for understanding why multiplication works, because you can see each partial product tied to a specific place value. Lattice multiplication handles the carrying for you inside the grid, which makes it a strong choice for three-digit-by-three-digit problems or larger, where keeping track of carried digits in the standard algorithm gets tricky.
If you’re learning for the first time, try the area model to build intuition, then practice the standard algorithm for speed. If you keep making carrying errors, give the lattice method a shot. The best method is whichever one you can execute accurately and confidently.