Visualise Maths Problems With the Japanese Multiplication Method

Visualise Maths Problems With the Japanese Multiplication Method
Graphic: Evannovostro, Shutterstock

In the age of smartphone calculators and voice assistants, multiplying large numbers by hand may seem like a quaint and entirely unnecessary skill. However, you never know when you’ll need to do quick maths, and the Japanese multiplication method (also called multiply-by-lines) can help you figure out the answer simply by counting. All you’ll need is a piece of paper and a pen. (It can help to have three different colours of ink, but that’s not essential.)

Here’s how the trick works.

First, pick a multiplication problem. YouTuber MindYourDecisions has an excellent explanatory video, so to follow along with their example we’ll go with 12 x 13.

Next, draw out your lines. You’ll need one line to represent each “tens” place and a parallel set of individual lines to represent the numbers in the “ones” place. This will give you a box shape formed by one line + one line + two lines + three lines. The “tens” are always on the left, and the box is rotated 45 degrees.

Screenshot: Emily Long Screenshot: Emily Long

(If, for easier understanding, you want to colour-code your lines, this is where your multi-coloured pens come in handy. Again, though, this isn’t necessary to use this method.)

Once you have your lines laid out, all that’s left to do is draw dots anywhere lines intersect and then count up all of your dots.

Screenshot: Emily Long Screenshot: Emily Long

In the right corner of your box for 12 x 13, you have six dots (the intersections of the two lines representing the black 2 and the three lines representing the blue 3). This gives you 6 for your “ones” place in your final number.

In the bottom corner, you have two dots representing the intersection of the 10 in 13 and the black 2. In the top corner, you have three dots representing the intersection of the 10 in 12 and the blue 3. Add these together, and the resulting 5 (representing 50 actual intersections) gives you your “tens” place in your final number.

Lastly, the left corner is a single dot (1), but it represents 10 times 10, or 100. This 1 gives you your “hundreds” place in your final number.

Screenshot: Emily Long Screenshot: Emily Long

It can be helpful to draw curved vertical lines to separate out each digit place, but from here it’s as easy as dropping each digit into its correct spot: 12 x 13 = 156.

Of course, this trick also works for larger numbers with more digits. Here’s a handy sketch for how to solve 213 x 13.

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