Why does japanese multiplication work

Draw sets of parallels, perpendicular to the first sets of parallels, corresponding to each digit of the second number the multiplier. Put dots where each line crosses another line. On the left corner, put a curved line through the wide spot with no points. Do the same with the right. Count the points in the right corner. Count the points in the middle. Some might use sticks for base ten blocks and maybe, just maybe, someone in your class might come up with something similar to this stick method.

How cool would that be? Oh, and before you go, you should know that using base ten blocks or the Japanese multiplication method is a great way to explain why partial products and the standard algorithm for multiplication works. If we have a look at the array and the standard algorithm, side by side we can clearly see each step of the algorithm.

Check it out:. Concreteness fading is a theory suggesting that mathematical concepts are best learned in three stages; the enactive stage, where students use concrete manipulatives that represent the mathematical concept they are working on. Over time, after students have had enough experience physically working with the concrete manipulatives, they move to the iconic stage, where they begin to often naturally draw a visual representation of the concrete manipulative instead of having to physically hold and manipulate the object in their hands.

As students become increasingly comfortable with the iconic or visual representations, does it make sense for them to begin using symbols that represent the meaning behind the previous visual and concrete representations.

This stage is thought to be the most abstract of the three stages because now numbers and symbols are used as a more efficient way to represent the work and experiences that have been developed in the previous stages.

So what does the multiplication we just explored today look like relative to the three stages of concreteness fading? When it comes to single digit by single digit multiplication using individual unit tiles as we did at the beginning of this post, the stages might look like this:. As we move to two digit by one digit or two digit by two digit multiplication, the stages of concreteness fading might look like this:.

Finally, another possibility might be:. I hope the time and effort spent at least has you thinking about how we might work to deepen our student understanding of multiplication in conjunction with Concreteness Fading. I strongly believe that as we are exposed to more ways to represent concepts in mathematics, our understanding of those concepts will continue to deepen and produce more and more connections over time.

I am living proof that this is true, because I am shocked routinely at the new connections that seem to present themselves to me with less and less effort with each passing day.

There are lots of ways to multiply numbers. At first it seems like something out of a magic show. But math should never feel mystical to the point of confusion.

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? 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:. 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. If they could break problems into actual physical pieces they could arrange and count, that was easier, faster, and required less brute force memorization even in the short term. Contrast this opacity with the chisanbop method from the s or the Russian way to multiply , both well documented for decades. Vedic math technically means the kinds of math used by people in ancient India during a time known as the Vedic period, but the term was coopted by a s book that claimed the secrets of all mathematics were unlocked from some previously misunderstood ancient texts.

Perhaps someone made this method up more recently as an interesting and cool visualization of how long multiplication works. Type keyword s to search. Today's Top Stories. A viral TikTok video shows an old, unique way to multiply using sticks.