Learning to Multiply Three-Digit Numbers

Date: 07/03/98 at 10:51:18
From: Jonathan W
Subject: Multiplying big numbers

Dear Dr. Math,

I am going into second grade. My class is only doing multiplication of 
easy numbers like 2 x 4. I would like to learn how to do 
multiplication of bigger numbers like 120 x 743. Could you please 
help me?

Thank you,
Jonathan

Date: 07/06/98 at 12:43:05
From: Doctor Peterson
Subject: Re: Multiplying big numbers

Hi, Jonathan. I love to help people who want an extra challenge! I 
don't know how well I can teach you about this in a short message, but 
I think I can show you some ways to understand multiplication so that 
you can work out the rest on your own.

The first step is to learn how to multiply a two-digit number by one 
digit. Here's an example:

      24
    x  3
    ----
      72

Here's how it works: Think of 24 as 20 + 4, and multiply each part 
by 3:

      20      4
    x  3    x 3
    ----    ---
      60     12

If you can see why 20 x 3 is 60, you know enough to figure everything 
else out. 20 x 3 is just two tens times three, which is two-times-
three tens, or six tens, which is 60. Everything else I'll be telling 
you depends on that kind of thinking.

Now we just add the 60 and the 12 together to get 72, which is the 
answer. I could write it all together like this:

      24
    x  3
    ----
      12
    + 60
    ----
      72

I could also write it this way, just putting an empty space instead of 
the zero, so that the 6 goes in the same column as the 2 it comes 
from:

      24
    x  3
    ----
      12
    + 6 
    ----
      72

What we usually do is to write down the 2 as part of the answer, and 
the 1 as a carry, just as you do when you add big numbers:

      1
      24
    x  3
    ----
      72

Once you've learned to do that, the next step is to multiply two 
digits by two digits. It's really the same thing, but there's more to 
keep track of, so we usually write out a bit more:

      24
    x 13
    ----
      72
     24
    ----
     312

Here I've multiplied the 24 first by 3, as before, and then by 1. But 
since the 1 really means 10 (since 13 = 10 + 3), I move the 24 left 
one space as I did with the 6 in an earlier example. Here's a way to 
see what we are really doing:

          10    3
        +----+----+
    20  |200 | 60 |
        +----+----+
     4  | 40 | 12 |
        +----+----+

This means that 20 + 4 times 10 + 3 is the sum of:

    20 x 10 = 200
    20 x  3 =  60
     4 x 10 =  40
     4 x  3 =  12
              ---
              312

You may be interested in an old method of multiplication called the 
lattice method, which works just this way. I think that understanding 
how to do that can help a lot in seeing what you are really doing when 
you multiply. 

Here's a place to learn about that:

  http://mathforum.org/dr.math/problems/susan.8.340.96.html   

I hope this will help you discover how to multiply. If I've told you 
more than you can follow yet, just take it one step at a time and ask 
someone to help you with it. It's really a lot of fun once you get 
used to it.

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/