Lattice Method — Definition, Formula & Examples
The lattice method is a way to multiply multi-digit numbers using a grid of boxes split by diagonal lines, where you fill in single-digit products and then add along the diagonals to get the final answer.
The lattice method (also called lattice multiplication or the Italian method) is a multiplication algorithm that arranges partial products within a rectangular grid. Each cell is divided by a diagonal, separating the tens digit from the ones digit of each single-digit product. The final result is obtained by summing digits along diagonal columns from right to left, carrying as needed.
How It Works
Start by drawing a grid with as many columns as digits in one factor and as many rows as digits in the other. Draw a diagonal line through each cell from the top-right corner to the bottom-left corner. Write one factor's digits across the top and the other factor's digits down the right side. Multiply each column digit by each row digit, placing the tens digit above the diagonal and the ones digit below it inside the corresponding cell. Finally, add the numbers along each diagonal strip running from upper-right to lower-left, carrying any tens into the next diagonal. Read the digits around the outside of the grid to get your product.
Worked Example
Problem: Use the lattice method to multiply 36 × 14.
Step 1: Draw a 2×2 grid (2 digits × 2 digits). Write 3 and 6 across the top. Write 1 and 4 down the right side. Draw a diagonal in each cell from top-right to bottom-left.
Step 2: Fill in each cell by multiplying the column digit by the row digit. Top-left cell: 3 × 1 = 03 (0 above diagonal, 3 below). Top-right cell: 6 × 1 = 06 (0 above, 6 below). Bottom-left cell: 3 × 4 = 12 (1 above, 2 below). Bottom-right cell: 6 × 4 = 24 (2 above, 4 below).
Step 3: Add along each diagonal, starting from the bottom-right corner. Diagonal 1 (far right): 4. Diagonal 2: 6 + 2 + 2 = 10, write 0 and carry 1. Diagonal 3: 0 + 1 + 0 + 1 (carry) = 2. Diagonal 4 (far left): 0.
Step 4: Read the digits from top-left down and across the bottom: 0, 5, 0, 4. The product is 504.
Answer: 36 × 14 = 504
Another Example
Problem: Use the lattice method to multiply 23 × 7.
Step 1: Draw a 2×1 grid (2 digits across for 23, 1 row for 7). Write 2 and 3 across the top and 7 on the right side. Draw a diagonal in each cell.
Step 2: Left cell: 2 × 7 = 14 (1 above diagonal, 4 below). Right cell: 3 × 7 = 21 (2 above diagonal, 1 below).
Step 3: Add along diagonals from right to left. Diagonal 1: 1. Diagonal 2: 4 + 2 = 6. Diagonal 3: 1.
Step 4: Read the result: 1, 6, 1.
Answer: 23 × 7 = 161
Why It Matters
The lattice method is commonly taught in elementary math programs like Everyday Mathematics because it reduces multi-digit multiplication to a series of single-digit problems, making it easier for students who struggle with traditional long multiplication. It also reinforces place value understanding, since each diagonal represents ones, tens, hundreds, and so on. Many students find it more visual and less error-prone when working with larger numbers.
Common Mistakes
Mistake: Placing the tens and ones digits on the wrong side of the diagonal.
Correction: Always write the tens digit above (or to the left of) the diagonal and the ones digit below (or to the right of) it. For a product like 7 × 8 = 56, the 5 goes above and the 6 goes below.
Mistake: Forgetting to carry when a diagonal sum is 10 or more.
Correction: When a diagonal adds up to 10 or more, write only the ones digit for that diagonal and carry the tens digit to the next diagonal to the left — just like carrying in regular addition.