ecy's multiplication galaxy: a simpler, faster, and scalable method of "how asians...
DESCRIPTION
Of late, there have been viral videos - of all things - on Mathematics. In particular, these viral videos deal with a visual method of multiplying numbers. Some people refer to this visual multiplication approach as "How Chinese Multiply." Other people claim that the visual multiplication approach, which involves drawing lines to represent numbers, originated from India. These latter people say the visual multiplication method is a tool of the controversial "Vedic Math." In order not to be caught in the middle, I'm referring to the visual multiplication approach that uses lines as "How Asians Multiply (HAM)." Yesterday, I came across the HAM approach on Facebook, where the presenter used lines to multiply the following numbers: 123 x 321. Although the method looks novel and intriguing, the HAM approach has several weaknesses. First, it is TIME-CONSUMING to draw lines when you are dealing with big numbers that have 7, 8, or 9; for instance, 78x56. Second, the HAM Method is inherently complex. Although one can easily observe the rules when the HAM Method is being demonstrated, the method is quite complex to describe and generalize. As the numbers get bigger, the rules of the HAM Method get more confusing. Due to this complexity, the HAM Method would mainly be used for novelty or entertainment purposes. For long multiplication problems in real life, other approaches are more efficient. Finally, due to increasing complexity as the HAM Method scales, users would be prone to making errors especially when counting intersections or nodes. Such errors limit the application of the HAM Method in real life. And one more thing ... the HAM Method is not spatially efficient; it occupies a lot of space especially when multiplying large and/or long numbers. For the reasons given above, I decided to briefly explore the HAM Method with a view to making it simpler, faster, and more scalable. "Ecy's Multiplication Galaxy (EMG)" is the result. For those who don't know, Ecy is our lovely daughter and it is to her that I dedicate my discovery. Ecy is an aspiring "Mathlete" and she already finds interesting the Multiplication Galaxy. And why is the new visual multiplication approach called the "Multiplication Galaxy"? Well, Ecy gave me the name. In our family, a "Galaxy" refers to a 3x3 fractal user interface that I invented about seven years ago. Such a Galaxy can be used to visually organize and present information. I guess Ecy is right in applying the name of Galaxy to the approach. After all, what I've done is develop a visual language for better organizing and presenting information when multiplying two numbers. Thanks, Ecy. And may you find the Multiplication Galaxy fun as well as useful. Hey and don't forget that you can create Multiplication Galaxy Games for individuals as well as teams. Have some Galactic fun ... Warmly, Rod. http://goo.gl/kcPH8aTRANSCRIPT
32 x 21 =
x 2 1
2 4 2
3 6 3
7
2
6
7 2 6 210 x 231 =
x 2 3 1
0 0 0 0
1 2 3 1
2 4 6 2 1
0
5
8 5 4
8 4
0 1
2x2 Mul,plica,on 3x3 Mul,plica,on
A Simpler, Faster, and Scalable Method of “How Asians Mul;ply (HAM)”
5 Steps for Using Ecy’s Mul,plica,on Galaxy 1. Draw a square grid: 3x3, 4x4, 5x5, … 2. Write the digits to be mul,plied in the
direc,on indicated by the x and y axes 3. Fill in cells by mul,plying digits 4. Add result of cell along each diagonal arrow
to obtain a total for each circle 5. Product is number defined by a sequence of
digits going in an an,-‐clockwise direc,on
……. x ….… =
x
…..… x ……. =
x
2x2 Mul,plica,on 3x3 Mul,plica,on
5 Steps for Using Ecy’s Mul,plica,on Galaxy 1. Draw a square grid: 3x3, 4x4, 5x5, … 2. Write the digits to be mul,plied in the
direc,on indicated by the x and y axes 3. Fill in cells by mul,plying digits 4. Add result of cell along each diagonal arrow
to obtain a total for each circle 5. Product is number defined by a sequence of
digits going in an an,-‐clockwise direc,on
A Simpler, Faster, and Scalable Method of “How Asians Mul;ply (HAM)”
1263 x 5974 = 7,545,162
x 5 9 7 4
3 15 27 21 12
6 30 54 42 24
2 10 18 14 8
1 5 9 7 4
6
2
1
25,781 x 12,457 = 321,153,917
x 1 2 4 5 7
1 1 2 4 5 7
8 8 16 32 40 56
7 7 14 28 35 49
5 5 10 20 25 35
2 2 4 8 10 14
1
7
9
5 4 5 7
4x4 Digit Mul,plica,on 5x5 Mul,plica,on
2 3 5 5 Steps for Using Ecy’s Mul,plica,on Galaxy 1. Draw a square grid: 3x3, 4x4, 5x5, … 2. Write the digits to be mul,plied in the
direc,on indicated by the x and y axes 3. Fill in cells by mul,plying digits 4. Add result of cell along each diagonal
arrow to obtain a total for each circle 5. Product is number defined by sequence
of digits going in an,-‐clockwise direc,on
3
1 1
A Simpler, Faster, and Scalable Method of “How Asians Mul;ply (HAM)”
………… x ……….. = ………………
x
………….. x ………….. = …………..…………..
x
4x4 Digit Mul,plica,on 5x5 Mul,plica,on
A Simpler, Faster, and Scalable Method of “How Asians Mul;ply (HAM)”
5 Steps for Using Ecy’s Mul,plica,on Galaxy 1. Draw a square grid: 3x3, 4x4, 5x5, … 2. Write the digits to be mul,plied in the
direc,on indicated by the x and y axes 3. Fill in cells by mul,plying digits 4. Add result of cell along each diagonal
arrow to obtain a total for each circle 5. Product is number defined by sequence
of digits going in an,-‐clockwise direc,on
Ecy’s Mul8plica8on Galaxy Is a Discovery and Presenta,on by
Dr. Rod King ([email protected])
This Discovery is Dedicated to Our Daughter, Ecy King
x 6
30
2 4
x 2 1
9 6
2
3
2x2 Mul,plica,on 3x3 Mul,plica,on
Challenge: Complete, without using a calculator, the Mul;plica;on Galaxy to obtain the product of the original two numbers.
25 x 6 = 2 x =
5 Steps for Using Ecy’s Mul,plica,on Galaxy 1. Draw a square grid: 3x3, 4x4, 5x5, … 2. Write the digits to be mul,plied in the
direc,on indicated by the x and y axes 3. Fill in cells by mul,plying digits 4. Add result of cell along each diagonal arrow
to obtain a total for each circle 5. Product is number defined by a sequence of
digits going in an an,-‐clockwise direc,on
………. x ………. = …………….
x
6 3 3
12 8 4
3 1 1
4 2 2
………… x …………. = ………………..
x
2 1 4 3
8 0 4
0 2 8
6 12 9
1 4
4x4 Digit Mul,plica,on 5x5 Mul,plica,on
5 Steps for Using Ecy’s Mul,plica,on Galaxy 1. Draw a square grid: 3x3, 4x4, 5x5, … 2. Write the digits to be mul,plied in the
direc,on indicated by the x and y axes 3. Fill in cells by mul,plying digits 4. Add result of cell along each diagonal
arrow to obtain a total for each circle 5. Product is number defined by sequence
of digits going in an,-‐clockwise direc,on
Challenge: Find, without using a calculator, the original two numbers that are mul;plied as well as the product for the Mul;plica;on Galaxies below.