The Japanese Multiplication Trick as it is called has been circulating social media sites again and with it have been a lot of questions about why it works and if it is worth teaching to students. If you take the time to play around with multiplication problems this way it’s easy to see that it is really another way to transition from the the concrete to the abstract. It’s a way to draw or represent the partial products with a slight rotation of the complete rectangle. The separation of dots or kinds or place values is not unlike the split and shift we do with the blocks in the Mortensen Math program in order to facilitate counting like kinds. If taught as more than a trick it can have its place as a useful tool. It would be a shame to teach it as being a trick with no understanding of what all its parts truly represent.
In the examples below the first factor represents the across and the second factor represents the up of the rectangle. Red dots, numbers, and blocks represent hundreds; blue dots, numbers, and blocks represent tens; and green dots, numbers, and blocks represent ones or units. In the first example the drawing and the rectangle have been tilted to the right in order to see how it would generally be presented in the Japanese model. In the second example the drawing and rectangle are upright which is generally how they would be presented in a Mortensen Math model. If one is going to draw a picture using the Japanese model it would be better to tilt the rectangle as it makes it easier to see how you get the partial products and from where – this is equivalent to the split and shift used in the Mortensen Math models. It is easy to see that both the Japanese model and the Mortensen model are synonymous.