Explaining Finger Multiplication

Date: 02/13/2002 at 09:36:51
From: erika
Subject: Finger multiplication

The following technique was widely used in medieval Europe.  

Knowing how to multiply two numbers less than 6, you can multiply two 
numbers between 5 and 10 as follows. Open both palms toward you. To 
calculate 7 x 9, put 7 - 5 = 2 fingers down on the left hand and put 
9 - 5 = 4 fingers down on the right hand. Count the number of down 
fingers (4 + 2 = 6) and multiply together the number of up fingers 
(3 x 1 = 3) and put the two answers together (63). Does this work, and 
why?

I have started by looking at all the different ways to multiply the 
numbers 5-10. I found that some of them you can simply combine and 
others you have to add, but you have to place a 0 at the end of the 
number you get on your left hand and then add. Please help me figure 
out a way to explain to my students how this works and why. I think 
this is a wonderful problem and I would love to use it in my 
classroom.

Thank you
Erika

Date: 02/13/2002 at 13:15:10
From: Doctor Peterson
Subject: Re: Finger multiplication

Hi, Erika.

You've made a good start in actually trying to use the method and 
finding that it is not quite stated correctly; I might have failed to 
do that. "Putting the two answers together" may involve a carry, as 
you discovered.

Here's how I can prove the method algebraically. Suppose the given 
numbers, both greater than 5, are 5+a and 5+b, so that you have a 
fingers down (and 5-a up) on one hand, and b down (and 5-b up) on the 
other. Then their product is

    (5+a)(5+b) = 25 + 5a + 5b + ab

The sum of the "down" fingers is a+b; the product of the "up" fingers 
is (5-a)(5-b). Putting these together, as you found, really means 
using the sum as the tens and the product as the ones of a number, 
with carries as needed. We can express this by multiplying the sum by 
10, then adding:

    10(a+b) + (5-a)(5-b)

This can be simplified to

    10a + 10b + 25 - 5a - 5b + ab = 25 + 5a + 5b + ab

which is just what we found the product we are looking for to be. So 
the method works.

Now, you haven't said what grade you have, but I suspect they don't 
know algebra, so this won't work as an explanation for them. If you 
don't want to just present it as a piece of math magic, how can you 
make it clear that there is a reason for it? I can imagine a pictorial 
way to show the algebra. Let's try it. I'll make a 5+a by 5+b 
rectangle:

    +-----------+---+
    |           |   |
    |           |   |
    |    5*5    |5*b|5
    |           |   |
    |           |   |
    +-----------+---+
    |           |   |
    |    5*a    |a*b|a
    |           |   |
    +-----------+---+
          5       b

I've marked the size of each portion (the number of objects in them, 
if you think of the rectangle as made up of 5+a rows of 5+b objects). 
The product we are looking for is the area (number of objects) in the 
whole rectangle.

Now how do we get the number of fingers UP into the picture? That's 
how many it takes to get from each number up to 10:

    +-----------+---+-------+
    |           |   |       |
    |           |   |       |
    |    5*5    |5*b|  5*d  |5
    |           |   |       |
    |           |   |       |
    +-----------+---+-------+
    |           |   |       |
    |    5*a    |a*b|  a*d  |a (left fingers down)
    |           |   |       |
    +-----------+---+-------+
    |    5*c    |b*c|  c*d  |c (left fingers up)
    +-----------+---+-------+
          5       b     d
                right  right
                down    up

Now, the whole square is 10 by 10; ten times the sum of the "down 
fingers" is a rectangle 10 by a plus another 10 by b. I'll mark those 
rectangles using "/" and "\" shading respectively, because we'll have 
overlapping pieces at first:

    +-----------+---+-------+
    |           |\\\|       |
    |           |\\\|       |
    |           |\\\|       |5
    |           |\\\|       |
    |           |\\\|       |
    +-----------+---+-------+
    |///////////|XXX|///////|
    |///////////|XXX|///////|a
    |///////////|XXX|///////|
    +-----------+---+-------+
    |           |\\\|=======|c
    +-----------+---+-------+
          5       b     d

The middle piece is doubly shaded. I also marked with "=" the product 
of the "up fingers," which is the lower right piece, c*d.

Now we're claiming that the sum of all the marked parts, including the 
middle part twice, is the product we are looking for. Well, take the 
lower right four parts (including just one of the two copies of the 
middle rectangle) and slide them up into the 5*5 at the left, since 
they are also a 5-by-5 square:

    +-----------+---+-------+
    |///////////|\\\|       |
    |///////////|\\\|       |
    |///////////|\\\|       |5
    |///////////|\\\|       |
    |\\\\=======|\\\|       |
    +-----------+---+-------+
    |///////////|\\\|       |
    |///////////|\\\|       |a
    |///////////|\\\|       |
    +-----------+---+-------+
    |           |   |       |c
    +-----------+---+-------+
          5       b     d

The shaded part is now exactly the 5+a by 5+b rectangle we started 
with.

I don't know whether this visual explanation helps, but it was fun to 
figure out!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/