Lattice multiplication puts the problem in a grid. Each square holds the product of one digit from the top and one digit from the side, split by a diagonal into a tens half and a ones half. Adding along the diagonals produces the answer. The student never has to decide where a digit belongs, because the grid decides.
The method is centuries old and travelled through Arabic and Italian arithmetic texts long before it reached modern classrooms. It is included in elementary curricula today because it separates two skills that usually get tangled: knowing the multiplication facts, and keeping track of place value.
How the method works
- Draw a grid with one column per digit of the first factor and one row per digit of the second.
- Write the first factor across the top and the second down the right side.
- Draw a diagonal through every square, from top right to bottom left.
- Fill each square with the product of its column digit and its row digit. Tens go above the diagonal, ones below.
- Add along each diagonal, starting at the bottom right. If a diagonal totals ten or more, carry into the next diagonal.
- Read the answer down the left side and along the bottom.
A worked example
Take 27 x 43. The grid is two columns by two rows.
Filling the squares
2 x 4 = 08 7 x 4 = 28 2 x 3 = 06 7 x 3 = 21
Adding the diagonals, bottom right first
diagonal 1: 1 -> 1 diagonal 2: 2 + 6 + 8 = 16 -> 6, carry 1 diagonal 3: 0 + 2 + 8 = 10, +1 -> 1, carry 1 diagonal 4: 0 + 1 -> 1
Each diagonal collects halves from different squares. The third one, for instance, holds the tens half of the lower left cell, the tens half of the upper right cell, and the ones half of the upper left cell. Its total of ten, plus the one carried in from the diagonal below it, makes eleven: write 1 and carry 1 again.
reading the edges: 1 1 6 1 -> 1161
A second grid
58 times 36
cells: 5 x 3 = 15 8 x 3 = 24 5 x 6 = 30 8 x 6 = 48 diagonals from the bottom right: 8 4 + 0 + 4 = 8 3 + 2 + 5 = 10 -> 0, carry 1 1 + 1 = 2 answer: 2088
Why it works
Every square in the grid is one partial product, the same partial products used in partial-products multiplication. The diagonals are what handle place value. Squares that sit on the same diagonal always represent the same power of ten, so adding along a diagonal is adding like with like. Carrying from one diagonal to the next is the ordinary carrying of column addition, rotated forty five degrees.
Common mistakes
Writing a one-digit product as one digit
A product like 2 x 4 must be written as 08, with a zero above the diagonal. Writing a bare 8 in the wrong half moves it onto the wrong diagonal and changes the answer.
Forgetting to carry between diagonals
A diagonal that totals ten or more behaves exactly like a column that totals ten or more. The extra ten moves left.
Drawing the diagonals the wrong way
Diagonals run from top right down to bottom left in every square, without exception. One square drawn the other way silently scrambles the place values.
Check yourself
34 x 26
cells: 3x2 = 06 4x2 = 08 3x6 = 18 4x6 = 24 diagonals: 4 2 + 8 + 8 = 18 -> 8, carry 1 1 + 0 + 6 = 7, +1 = 8 0 answer: 884
63 x 47
cells: 6x4 = 24 3x4 = 12 6x7 = 42 3x7 = 21 diagonals: 1 2 + 2 + 2 = 6 4 + 1 + 4 = 9 2 answer: 2961
Why do the diagonals, and not the columns, carry the place value?
Because of where each square sits. Moving one square to the left multiplies a partial product by ten, and moving one square up does the same. A diagonal is exactly the set of squares that those two moves cancel out on, so every square along one diagonal stands for the same power of ten.
See also
Sources
- Lattice multiplication, also called the gelosia method, reached European arithmetic through Arabic sources and appears in Renaissance texts. It is taught today in Everyday Mathematics among other programmes. Grids and examples on this page are our own.