Lesson 7

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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.”

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

Make sure you have your playing cards, or pencil and paper out in order to learn this. You won’t need them once you get the hang of it, but you will during the learning process.

First, let me remind you that when you multiply by 2, you never get a product that ends in a digit higher than 8 (because 2 x 9 = 18). And you never had to carry more than 1 (for the same reason). Therefore, no product could ever end in a digit higher than 9.

The new factor involved when you multiply by 3 is that there is a situation in which you will have to carry in more than one column at a time. Let me give and example:

234 x 3. When you start with the 2, you would normally look over to the right and see that when you multiply the 3 in the tens column by 3, you will get 9, which wouldn’t cause you to carry. But wait! When you eventually get to the ones column, you will multiply 3 times 4 and get 12. You will have to carry the 1 to the tens column, which would then cause the 9 to become a 10, which then WILL create a carry over to the hundreds column.

So when you see the digit 3 in the multiplicand it signals you that its product, which normally wouldn’t cause you to have to carry anything, might cause you to after all, if the digit to the right of it would cause its product (normally a 9) to become a 10 or 11. So when you see a three to the right of the digit you are working on, you can’t automatically assume that you won’t be carrying anything to the product of the digit you are working on. You have to keep looking to the right of the 3, to see if the digit after that will cause the product of the three to have a carry.

I call numbers like the 3 “ambiguous” digits. They may cause a carry, they may not. When you multiply by three, there are only two ambiguous digits; they are 3 and 6.

Definition: (Remember, this is only my definition. This is not an “official” math word that you will see in textbooks. I made up this method, and I made up this word for this part of the method. Although I seriously suspect that plenty of other people and cultures have used this, without this specific word. And this word is only used in this sense when multiplying from left to right.):

An ambiguous digit is a digit that may or may not cause you to have to increase the amount that you will have to carry to the digit that is to the left of it, depending on whether or not the product of its tens-column will be increased when the carry from the digit to the right of it is added to it.

When multiplying by 3, there is a simple rule for each ambiguous digit.

This is the rule for the ambiguous digit 3:
When you see a three, keep looking to the right of it.

• If the digit to the right is higher than 3, you will have to carry a 1 higher up the chain.
• If the digit to the right is lower than three, there will be no carry.
• If the digit to the right is 3, you must keep looking right to see if the next digit is higher or lower than 3,
• and if that digit is a 3, you must keep looking, and so on, until you find a digit that is higher or lower than three.

There is normally no carry with 3*3, but if it’s followed by a number higher than three, the carry will be 1.

Let’s take another example: 9,335 * 3

We look at the nine, and see it is followed by a three. Normally, 3*9 is 27, which would normally mean 2 in the ten-thousands column of the product, and 7 in the thousands column.

But we always have to look over to the right to see if we will have to carry anything into the thousands column. When we look to the right, we see a 3. Normally 3*3=9 wouldn’t cause us to carry anything, but that 9 may turn into a 10 or 11 when we multiply the number to the right of it.

So we keep looking to the right. There is another 3. That doesn’t help us yet, because we don’t known if the product of that 3*3 will be a 9, or eventually turn into a 10 or 11.

When we look further to the right we see a 5. We know that 3*5=15, which means we will have to carry the 1 up to the tens-column, turning it’s product from a 9 into a 10, which in turn will turn the hundreds-column’s product from a 9 to a ten, which in turn will turn the thousands-column (the one we are working on) from a 7 (because 9*3=27) into an 8.

Whew! So we can to ahead and write the 2 in the ten-thousands column, and the 8 in the thousands.

It seemed like a lot of work, but as we did it, we also determined that there will be zeros in the hundreds-column and the tens-columns (because their products were each bumped up from 9s to 10s, and we have already carried the 1s to their next highest columns.

All we need to do now is write 5 as the product of the ones-column, and we are done.

Yes, I know, this all seems extreme, just to multiply by three. But you are developing a skill that most people don’t have. Not only will you be able to amaze anyone you show it to, but it is useful in everyday life.

Imagine you want to buy 3 cds that come to $17.52 each including tax. With just a little bit of practice you be able to freak cashiers out by telling them, “That’s fifty-two, fifty-six” before they can ring it up.

Teachers will love (or hate, depending on their own self-esteem) you.

Now get out your deck of cards and get to work multiplying 3 by 1, 2, 4, 5, 7, 8, 9 and 3. Use one of each of the cards, except for the 3s; use all four of them.

Or, if you would like a worksheet of large numbers (with lots of threes and no sixes) 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.

The podcast for this lesson is not up yet. If you click on the podcast icon, you will get an error message. The podcast will be up by the end of the first week of March.



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