腦洞數學 第四週 Fun Math Week 4 日本人發明的魔幻乘法

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

https://www.youtube.com/watch?v=85Vd0NpL32k

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 color). So for the number 12 we get:

日语乘法方法如何工作？

在日语乘法中，我们只需画几条线并计算相交点就可以完成乘法问题。 听起来好得令人难以置信，对吧？

让我们以12×32为例。 请记住，数字是用位值表示的：12表示一十和二，32表示三十和二。

然后，我们绘制对应于十的对角线，并在留出间隙后，平行绘制更多的线来表示该对角线（这有助于使用不同的颜色）。 因此，对于数字12，我们得到：

                                   10 + 2

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:

我们要做的就是将熟悉的数字的位置值表示形式并使之可视化。 现在让我们做32号，除了这次我们要往相反的方向走。 您应该保持粗糙的菱形形状，并且在拐角处相交：

要计算乘积，我们只需要计算所有线相交的次数并将每个数字写在菱形下即可。

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

                                        12x32=384

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:

首先将交点垂直分组。 也就是说，在最接近左侧（蓝色和橙色线相交的位置）的一组相交处绘制一个循环。 然后开始向右移动。 在中心相交处画一个圈（红色和蓝色，橙色和绿色）。 最后，在最靠近右侧（绿线和红线相交的地方）的相交处画一个圈。 您实际上所做的是计算产品中的百，十和一的数量：

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:

为什么日语乘法方法有效？

考虑如何使用长乘法的标准方法计算12×32。 在计算过程中，會產生四個乘積：

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 hundred).

日文乘法方法实际上只是表示这四个步骤的视觉方式。 每个交集对应于将两个数字相乘的四个较小乘积之一（例如，左边的3×1乘积为300或300）。

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!

日语乘法方法有用吗？

非常！ 在表示形式之间切换是您的孩子测试他们对特定方法的理解的一种好方法。 知道如何执行程序（如长乘法）是一回事，但这仅在您的孩子知道该方法为何有效时才有用。 一旦他们在符号方法和视觉方法之间建立了联系，他们就可以在不同情况下应用其完整的程序工具包。

您的孩子将学习评估哪种方法最适合给定的问题。 例如，日语乘法方法在处理小数时非常有效–只需尝试9×8，突然您发现自己在计数72个不同的交点。 效率不及其他乘法方法！

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

位置值的可视化还使我们能够探索一些重要的属性。 例如，我们可以从字面上看到一个列中的数字如何组合在一起形成下一个。 这是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 maths as a bunch of mysterious rules. Instead, they will appreciate that maths is full of interesting patterns that connect to one another in logical ways.

我们可以算出右边的十个相交点，对应于十个相交点，下一相交处又是十个。 我们将额外的十个加到已经存在的七个中，总共八个。

还有许多其他可用的方法-将每种方法都视为孩子武器库中的另一种工具。 一旦他们掌握了这些“技巧”背后的理由（原因以及方式），他们就不必将数学视为一堆神秘的规则。 相反，他们将意识到数学充满了有趣的模式，这些模式以逻辑方式相互连接。

（以上材料來源：https://www.whizz.com/blog/how-the-japanese-multiplication-method-works/）

Fun Math Homework 腦洞數學作業：

Please use the Japanese magical multiplication method you learned today to calculate the following calculation problems. Please attach your Japanese calculation sheet(picture)too. thank you all!

請用今天學習的日本魔幻乘法計算出以下計算題，請必須附上你的日本計算圖。謝謝大家！

Grade 2-4年級

1. 12 x 32

2. 22 x 14

3. 34 x 31

4. 11 x 21

5. 42 x 23

6. 24 x 24

Grade 5-8年級

1. 123 x 321

2. 222 x 214

3. 134 x 31

4. 256 x 118

5. 17 x 19

6. 413 x 161