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

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