It’s not magic, it’s math – how the Japanese multiplication method works

There are lots of ways to multiply numbers. One approach that has caught people’s eye of late is the Japanese multiplication method. At first it seems like something out of a magic show. But math should never feel mystical to the point of confusion. And while magicians may never reveal their tricks, we think it’s essential to lift the lid on just why these strange methods work. It’s the only way to fully appreciate them!

 

How does the Japanese multiplication method work?

In the Japanese multiplication method, we are able to complete a multiplication problem by merely drawing a few lines and counting the points of intersections. Sounds too good to be true, right?

Let’s take 12×32 as our example. Remember that numbers are represented using place value: 12 means one ten and two ones, 32 means three tens and two ones.

We then draw diagonal lines corresponding to the tens and, after leaving a gap, we draw more lines in parallel to represent the ones (it helps to use a different colour). So for the number 12 we get:

All we’re doing is taking the familiar place value representation of numbers and making it visual. Now let’s do the number 32, except this time we’ll go in the opposite direction. You should be left with a rough diamond shape, with the lines crossing at the corners:

To calculate the product, we just need to count how many times all of the lines intersect and write each number under the diamond.

Begin by grouping the intersections vertically. That is, draw a loop around the group of intersections that is closest to the left side (where the blue and orange lines intersect). Then begin moving right. Draw a loop around the center intersections (the red and blue, and the orange and green). Finally, draw a loop around the intersections that are closest to the right side (where the green and red lines intersect). What you’ve actually done is calculated the number of Hundreds, Tens and Ones in the product:

So the 12×32 is 3 hundreds, 8 tens and 4 ones – in other words (or symbols, rather!) it is 384.

 

Why does the Japanese multiplication method work?

Think about how you would calculate 12×32 using the standard method for long multiplication. There are four smaller products you calculate along the way:

The Japanese multiplication method is really just a visual way of representing those four steps. Each cluster of intersections corresponds to one of the four smaller products that go into multiplying two numbers (for instance, the left cluster, 3×1, is what gets you the 300 – or 3 hundreds).

 

Is the Japanese multiplication method helpful?

Very! Switching between representations is a great way for your child to test their understanding of a particular method. It’s one thing to know how to carry out a procedure (like long multiplication), but this is only useful when your child knows why that method works. Once they make these connections between symbolic and visual methods, they’ll be able to apply their full toolkit of procedures in different situations.

Your child will learn to evaluate which method is most appropriate for a given problem. For example, the Japanese multiplication method becomes very efficient when dealing with small numbers – just try 9×8 and suddenly you find yourself counting 72 different intersections. Not nearly as efficient as other multiplication methods!

The visualisation of place value also lets us explore some important number properties. For example, we can literally see how numbers in one column group together into the next. Here is 12×15:

We can count the ten intersections on the right, corresponding to ten ones, which goes into the next column as one more ten. We add this extra ten to the 7 tens already there to make 8 tens in total.

There are so many other methods available – think of each one as another tool in your child’s arsenal. Once they master the reasoning behind these ‘tricks’ (the why as well as the how), they won’t have to see math as a bunch of mysterious rules. Instead, they will appreciate that math is full of interesting patterns that connect to one another in logical ways.

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