I have come across some fun math problems (yes, math CAN be fun!) through the internet and friends. Here is a video of Japanese Multiplication. First watch the video, and then see if you can figure out how they are multiplying before you read my explanation below!
Well, did you figure it out? Let’s take a closer look at the first example:
- The first set of lines, the green ones in my image above, represent 2 sets of 10 , or 20
- The second region with one line, which is orange in my image, represents 1 set of 1
- Together these sets of lines, read top to bottom, represent 2×10 + 1×1 which is expanded notation for 21
Let’s look at the next set of lines that were drawn perpendicular to these lines:
- The first region that contains the 1 blue line represents 1 set of 10 or 1×10
- The second set of lines that were drawn, the three red lines, represent 3 sets of 1
- Together these 4 lines, read left to right, represent 1×10+3×1 = 13 in expanded notation.
Now for the tricky part!
Here is a quick Algebra example to remind you
First = x * 2x = 2x^2
Outer= x*5 = 5x
Inner = 3*2x = 6x
Last = 3*5 = 15
Then, 2x^2 + 5x +6x +15 = 2x^2 + 11x +15 (the Outer and Inner were “like” terms, so could be added together)
Now back to the arithmetic. If you look at the product 21×13 by separating out each factor by its place values, you have:
(20 + 1)(10 + 3) and now you can FOIL out the values, just like in the Algebra problem!
First = 20×10 = 200
Outer = 20×3 = 60
Inner = 1×10 = 10
Last = 1×3 = 3
The 200 is represented by the 2 sets of crossing lines circled in yellow on the image above- that location on the paper represents the hundreds place value, so having a 2 in the hundreds location represents 2×100 = 200. In the video a 2 is placed as the first digit of the product, which will be the hundreds place.
The 60 is represented by the 6 sets of crossing lines in green on the top right
The 10 is represented by the 1 set of crossing lines in green on the bottom left
Together the 60+10 gives 70. In the video, the areas circled in green on the image above both represent the tens place value, so they are adding up the 6 crossed marks and the 1 crossed mark to get 7 sets in the tens place, or 7×10=70. They then place a 7 to the right of the 2 in the product (placing it in the tens place)
The 3 is represented by the 3 crossed marks in the lower right (circled in red on the image above). This area of the paper represents the ones place, so we have 3×1 = 3. They then place a 3 to the right of the 7 in the product, placing the 3 in the ones place.
This gives the final product of 200+60+10+3 or 200+70+3 = 273
~Now look at the second product in the video and see if you can figure out how it works!