How cool would that be?īase Ten Block Arrays & Japanese Multiplication IS the Standard Algorithm ![]() 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. So while many might consider this to be a pretty cool “trick”, it is much more powerful if students can articulate where procedures like these come from and why they work.īetter yet, after students have a thorough understanding of arrays with base ten blocks, I’d much rather challenge them to see if they could come up with an easier way to visually represent their two-digit multiplication on paper without having to draw a bunch of rectangles and squares. You’ll notice that the 10 one’s must be swapped out for a ten rod. Why do I always see the lines in the Japanese multiplication method on a diagonal?Īs you can see above, an opportunity to circle back to place value and the importance of understanding that in base ten, we cannot have any number greater than 9 in any place value column. Japanese Multiplication: Why Diagonal Lines? In particular, students should have the opportunity to spend a significant amount of time working with concrete materials like square tiles and base ten blocks to build arrays in order to build strong multiplication fluency prior to pushing students to an iconic or visual representation like drawing the base ten blocks or using a more abstract representation like drawing intersecting lines. In the case of Japanese multiplication, I would argue that it is only a multiplication trick if you are teaching this method without students having had the opportunity to work with the conceptual underpinnings that make it work flawlessly. I’d like to think that if students have built a deep conceptual understanding prior to moving towards procedures and algorithms, it is likely that they will better understand how to use the procedure efficiently and will also be able to get themselves out of a jam if troubles ever arise. Understanding “how many times bigger” one “piece” of a fraction is than another is very important prior to simply giving students a tool like cross multiplication to simply “get to an answer” as fast as possible. In the case of solving proportions, students should be able to solve a proportion using opposite operations and their understanding that the equivalent relational quantities are multiples of each other. It is my belief that there is no such thing as a “trick” in math class when a deep conceptual understanding is constructed prior to introducing procedural fluency. I have to be careful here because I’m not suggesting that cross multiplication or sum and product are bad methods to use in math it is more about when and how they come about in math class.ĭeveloping a Deep Conceptual Understanding Will Lead to Procedural Fluency While I’ve taught many math tricks such as cross multiplying for solving proportions and sum and product for factoring in the past, these past few years I have completely abandoned this approach from my teaching. This statement is only true if you never seek out to understand why it works. I’ve seen a bunch of posts floating around social media suggesting that Japanese multiplication is a multiplication trick or some sort of “ magic” or “ voodo trick“. Japanese Multiplication Is Only a Trick If You Don’t Know Why It Works! Now that you’ve had a chance to experience using base ten blocks through this post or more in-depth here, you can probably visualize the base ten blocks sitting between the lines that are used in the Japanese multiplication method. Looking at both the array with base ten blocks or Japanese multiplication, both methods are automatically chunking our factors of 12 and 15 to make use of the distributive property 12 = 10 + 2 and 15 = 10 + 5. Finally, in the bottom right corner, we have 5 units multiplied by 2 units to give 10 units or 10 intersection points.In the bottom left corner, we have a ten-rod multiplied by 2 units to give 2 ten-rods or 2 intersection points to represent 20. ![]()
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