My Math Tools: Lattice Multiplication

Multiplication is a brilliant shortcut for addition. Long multiplication, on the other hand, can seem like tough math love because the most popular method, “block form” or the standard algorithm, is tedious and prone to arithmetic mistakes. That is where lattice multiplication comes in to save the day.
lattice multiplication 3 digit example

The exact origins of lattice multiplication are unknown, though scholars argue that it was developed in the Middle East. No matter who or where its creator was, the concept seems to predate the widely used standard algorithm. Their astounding similarities even suggest that lattice multiplication may be its unwitting parent. I first learned about this highly manual, ingenious strategy from a middle school student who absolutely refused to use the standard algorithm – he had never learned it! Lattice multiplication was the only way he performed long multiplication. Then a college student, taking a math course designed for aspiring elementary school math teachers, showed me how her professor wanted her to practice this method. After some experimentation and soul searching, I was converted.


First, construct the boxes.

You need as many boxes as you have digits. For example, if you are multiplying two 2-digit numbers, you need 4 boxes. If you are multiplying a 2-digit number by a 3-digit number, you need 6 boxes, and if you multiply two 3-digit numbers together, you need 9 boxes.


first step for lattice multiplication


You can place either number on the top or side. Remember that only 1 number goes to each column or row, and you must only number the right side, not the left side.
Then draw a diagonal in each box. Each diagonal must go from the bottom left corner to the top right corner. This is done to prepare for the addition we do near the end of the process, adding from right to left.


second step for lattice multiplication


Next, start filling in your boxes.

Each row or column acts like a times table. For example, the 5 column multiplies each box in the intersecting rows by 5, while the 8 row multiples each box in the intersecting columns by 8. You can only have 1-digit or 2-digit numbers in the boxes. If the product has 2 digits, both fill the box. If there’s only 1 digit, you place a “0” in the left-hand triangle of the diagonal and the actual number in the right-hand triangle.

third step for lattice multiplcation


The hardest step is adding the diagonals.

The bottom right corner of the grid always drops down, and if you don’t carry over to the top left corner, that number also drops down unless it is “0.” If a sum of a diagonal is more than 9, you do have to “carry over” the extra digit to the next diagonal – just like in the standard algorithm, where you carry over to the next column during long addition.

final answer lattice multiplication


The beauty of lattice multiplication is that it makes so much sense! Notice that in the standard algorithm, the very first row matches the very last row of your grid, and that is an unbreakable rule for lattice form. Each row above the last row in the lattice grid matches the next row underneath the very first row in standard algorithm multiplication. In other words, lattice multiplication is the key to standard algorithm multiplication and a bona fide shortcut!


block form versus lattice multiplication
compare block form lattice multiplication


Another function is to multiply decimals easily and quickly. The steps are no different than for whole numbers. Keep in mind how many decimal places you need to apply at the end, after you have reached your final product, and move the invisible but ever present decimal point after the last digit from right to left until all the decimal places are accounted for.


decimals lattice multiplication
lattice multiplication decimal answer


Not only a quick and great mnemonic, lattice multiplication is a lifesaver on tests where calculators are not allowed. I’ve taught this method to children taking entrance exams and adults taking the ASVAB and CBEST. Doesn’t drawing a grid and filling in boxes take more time than the habitual standard algorithm? Actually, it takes less time! The genius of lattice multiplication is how the more you practice using it, the faster it goes. I can beat students who use the standard algorithm and be finished long before them. Try lattice multiplication today, for school, test prep or everyday life, and leave the hassle of long multiplication behind!

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Published by Natalie Gorna

Bookworm, artist, musician. Dreamer and cynic. Writing is my everything.

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