K-12 Teaching and Learning From the UNC School of Education

About this article

This article was adapted from a presentation at the 2005 conference of the North Carolina Council of Teachers of Mathematics by Vicki Thomas, a fifth-grade math and social studies teacher at River Dell Elementary in Johnston County, North Carolina.

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Lattice multiplication is a method of multiplying large numbers using a grid. It is algorithmically equivalent to regular long multiplication, but the lattice method breaks the multiplication process into smaller steps, which some students find easier. Digits to be carried are written within the grid, making them harder to miss.

Multiplying on a lattice

After the lattice is constructed, a round of multiplication is followed by a round of addition, much as in traditional multiplication of large numbers.

Setting up the lattice

First, draw a grid that has as many rows and columns as the multiplicand and the multiplier. The grid shown here is for multiplying a 4-digit number by a 3-digit number.

blank lattice multiplication diagram

Next, draw a diagonal through each box from upper right corner to lower left corner. Continue the line a short way past the grid. (Ms. Thomas describes the process to her students as “corner to corner and out the door.”)

lattice multiplication diagram

Write one factor across the top and the other down the right side, lining up the digits with the boxes.

lattice multiplication diagram

Multiplication

The multiplication is performed by multiplying the digits at the head of each row and column. Fill in each square of the grid with the product of the digits above and to its right, recording the products so that the tens are in the upper (diagonal) half of the square and the ones are in the lower half. If the product does not have a tens digit, record a zero in that triangle.

In the example shown here, the highlighted row and column give us 1×5=5, so we write 0 in the upper half of the square and 5 in the lower half.

lattice multiplication diagram

Addition

Now add the numbers in the grid along the diagonals, starting from the lower right corner. (Ms. Thomas describes this step to her students as “ride the slide.”) Carry any tens into the top of the next diagonal.

In this example, the highlighted diagonal gives us 4+2+5+1=12, so we write 2 at the bottom of the diagonal and carry the 1 to the top of the next diagonal to the left.

lattice multiplication diagram

To find the answer, read the digits starting down the left of the grid and continuing across the bottom. Here, the answer to 2314×157 is 363,298.

lattice multiplication diagram

Lattice multiplication of decimals

Lattice multiplication can easily be extended to multiply decimal fractions. Suppose, instead of the above problem, we wanted to multiply 2.314 by 1.57. We would proceed as before, but draw lines from the decimal points down and to the left until they meet, then follow the diagonal to the left or bottom of the grid. The point where this diagonal emerges from the grid is the position of the decimal point in the answer.

lattice multiplication diagram with decimals

Here, the result of the multiplication is 2.314 × 1.57 = 3.63298.

Creating a lattice in Microsoft Word

Although most students can draw lattices accurately by hand, some may have difficulty. Printed lattices may also be helpful when students are first learning the process.

Drawing the lattice

  1. Use a large font, such as 24-point.
  2. On the top toolbar, click on Table and then select Draw Table. This will open the Tables and Borders toolbar.
  3. Click on Insert Table. Select the desired number of columns and rows (one more each way than the number of digits you plan to multiply). Set table height to 0.5”. Select Finish.
  4. Click on Draw Table on the Tables and Borders toolbar. Use the pencil cursor and left mouse button to draw diagonal lines.
  5. Highlight the outside left row. Use the Border button to remove the outside border and horizontal lines. Repeat with the bottom row to remove the outside border and vertical lines.

Manipulating the lattice

Copying. To copy a table, select the entire table, right click, and select Copy. Move the cursor to another part of the page, right click, and select Paste.

Moving. To place tables side by side, rest the cursor on the upper-left corner of the table until the table move handle appears. Right click and drag the table using this handle.

Enlarging. Rows and columns can be inserted to increase lattice size. Click on Table, then Insert, and choose either rows or columns.