Lattice multiplication — also called the grid or gelosia method — turns a long multiplication into a table of single-digit products. It was used by Arab, Indian and Italian mathematicians centuries ago and is still taught in many US elementary classrooms because it separates the multiplying from the carrying. This calculator draws the lattice, fills it in and adds the diagonals for you.
How to use the lattice multiplication calculator
- Enter the first number; its digits run across the top of the grid.
- Enter the second number; its digits run down the right side.
- The product appears on the tape, and the drawing shows the full lattice. Tens digits sit above each diagonal, ones digits below it. The blue digits around the left and bottom edges are the diagonal sums, and small “+1” marks show carries.
- The steps list every box and every diagonal addition. Whole numbers up to 10 digits each are supported; a negative sign is handled with the usual sign rule.
Steps of the lattice method
- Draw the grid with one column per digit of the first number and one row per digit of the second.
- Draw a diagonal in every box from the top-right corner to the bottom-left corner.
- Fill each box with its column digit times its row digit. Write the tens digit above the diagonal and the ones digit below; use 0 for the tens of a one-digit product (3 × 2 = 06).
- Add along the diagonals, starting at the bottom right. If a diagonal sums to 10 or more, write the ones digit and carry the rest into the next diagonal.
- Read the answer down the left side and then along the bottom.
Worked example: 345 × 27
The grid has 3 columns (3, 4, 5) and 2 rows (2, 7). The six boxes hold these products, written as tens digit / ones digit:
| 3 | 4 | 5 | |
|---|---|---|---|
| 2 | 0/6 | 0/8 | 1/0 |
| 7 | 2/1 | 2/8 | 3/5 |
Adding the diagonals from the bottom right:
- 5 → write 5
- 0 + 8 + 3 = 11 → write 1, carry 1
- 8 + 1 + 1 + 2 + 1 (carried) = 13 → write 3, carry 1
- 6 + 0 + 2 + 1 (carried) = 9 → write 9
- 0 → leading zero, dropped
Reading from the top left: 9,315. Check with long multiplication: 345 × 27 = 6,900 + 2,415 = 9,315 ✓
Why the diagonals work
Every box sits at a particular place value. The ones digit of the bottom-right box is worth 1; moving one box up or one box left multiplies the value by 10. A diagonal collects every digit that has the same place value, so adding along a diagonal is the same as adding a column in long multiplication — the lattice simply keeps the digits from drifting out of line.
Tips
- Always write two digits in each box, even if the tens digit is 0. Skipping it is the most common source of errors.
- Draw the diagonals long enough to extend past the grid so the sums are easy to place.
- Know your facts: every box is a fact from the multiplication table, so fluency with 0–9 is all the method requires.
Frequently asked questions
How does lattice multiplication work?
Write one number across the top of a grid and the other down the right side. Fill each box with the product of its column and row digits, tens above the diagonal and ones below. Then add the numbers along each diagonal from the bottom right, carrying into the next diagonal, and read the digits around the left and bottom edges.
Why is the lattice method easier for some students?
All the single-digit multiplications happen first and need no carrying. The only carrying happens once, when you add the diagonals, so there is less to keep track of at any one time.
Does lattice multiplication give the same answer as long multiplication?
Yes. Both methods split the problem into the same digit-by-digit products and add them by place value. Each diagonal of the lattice is one place value: ones, tens, hundreds and so on.
What if the answer starts with a zero?
The top-left diagonal can sum to 0, for example in 12 × 34. That leading zero is dropped, so the product is 408, not 0408. The calculator shows it in gray.