Lesson 6

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Make sure you know you know your basic multiplications of one-digit number (like 8 x 7) with “Numbers Jugging - Times without the Tables.”

Introducing the concept of “safe” digits. Multiplying numbers made up of safe digits (1, 2, 4, 5, 7, 8 and 9) by 3, from left to right, mentally.

Multiplication by 3 of numbers made up of digits from 0 to 3 are easy, because there is no carrying involved.

Once you get to the digits that are 4 or higher, that is another story. When you multiply 3 times 4, you will have to carry. Normally, this would be handled just like the method you learned for multiplication by 2. But there is another factor. We’ll get to it in this lesson.

Today we’re going to learn how to multiply long numbers that are made up of “safe” digits by 3, from left to right, mentally.

I will explain what “safe” digits are in a bit, but first we’re going to explore the Idea of carrying more than a “1″.

You remember that when we multiplied any digit by 2, you’d never have to carry more than 1, because no digit times two gives you a product with more than 1 in the tens-column. That’s not the case with multiplication by the other digits.

That doesn’t have to be a problem though, if we take it slow and learn each lesson rock solid. Of course this means you have to know your single-digit multiplications rock solid, too. If you don’t, you will be a bit frustrated. Once you have them down, this lesson will lock them in for you, tight as bear-trap.

When you consider the multiplication of all the digits by 3, you’ll notice that zero through six multiplied by three will still only have you carry a maximum of 1. It’s the digits seven through nine that will make you carry a 2. No big deal. But it does introduce the concept of “safe” digits.

What are “safe” digits”? They are digits that, when they are multiplied, will never be forced to change the tens-digit of their products.

Let me give examples to make it clear:

When you you are multiplying, say, 3 x 1, you will (naturally) get 3 as the product. If that digit 1 is part of another number, let’s say 47,154, it will have another digit behind it (in this case that digit is a 5). When it comes time to multiply that digit, it may have a carry, which in this case will change the 3 (the three that was the product of 3 x 1) into a 4. That is simple, it’s just like when you were multiplying by 2.

Here’s the thing, though - when we were multiplying by 2, no numbers could ever cause you to have to change the carry of the number you were working on. That’s not the case with multiplication by 3 or higher. When you look at the multiplications by 3 you will see:

0*3 = 0
1*3 = 3
2*3 = 6
3*3 = 9
4*3 = 12
5*3 = 15
6*3 = 18
7*3 = 21
8*3 = 24
9*3 = 27

Notice that if you have to carry the 1 from 12, 15, or 18 to the 9 (in 3*3) you will have to changed the 3 into a 10. You are going to end up having to carry the 1 from that 10 “further up the chain.”

Something similar happens if you have to carry a 2 (from 21, 24, or 27 ) to that 9. You’ll be changing it to an 11.

And if you have to carry 21, 24, or 27 to the product of 6*3 (which would be 18) you would have to turn the 18 into a 20, thus changing the carry from 6*3 into a 2, instead of the normal 1. This gets a little confusing.

Good news, though - we won’t be dealing with that until tomorrow!

For now, we’ll only multiply 3 by the “safe” numbers - those are 1, 2, 4, 5, 7, 8, and 9. Each of those numbers, when multiplied by 3, will give you a product who’s ones-column consists of a digit which is two low to be affected by any carries of any other digits. So we proceed exactly like multiplication by 2, except, of course, that we multiply by 3!

So get our your deck of cards and get to work multiplying 3 by 1, 2, 4, 5, 7, 8, and 9.

Or, if you would like a worksheet of large “safe” numbers to print out and practice multiplying by three with, click here. The answers are included on the sheet, so fold it over in half (on the dotted line) to hide the answers as you do the problems.

For a complete transcript of this podcast, click here.
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