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### 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
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(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/
```
Associated Topics:
Elementary Multiplication

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