Adding with the Abax (Lesson 3)In Lessons 1 and 2 you learned how to add from left-to-right with the abax. You probably suspected that you could do the rows in any order. If you did, you would have been right.

Never forget about adding using the tens-complements.Just for fun, on this page we are going to add the columns out of order. There is a great lesson in doing this. If you want to know a little about the mathematical reasoning behind why this works, click here to learn about The Commutative Law of Addition.

	abax addition fig. 12 This abax represents the number 429. We are going to add 394 to it. Let's see what it would look like if we started by adding the tens columns first.	+ 394
abax addition fig. 13 Adding the tens column first If you add 9 tens (of the 394) to the 2 tens (of the 429) (remembering to do it by subtracting the tens-complement of the 9 from the 2), you end up with one in the tens column, and adding one to the hundreds, which gives you 519.	abax addition fig.14 Adding the ones column second Now add 4 ones (of the 394) to the 9ones (of the 519) using the tens-complements. So you will end up with 3 in the ones column, and one more in the tens. That means a total of 523, so far.	abax addition fig. 15 Adding the hundreds column last Finally, add the 3 hundreds (of the 394) to the 5 hundreds (of the 523) and get 823.
or...
abax addition fig. 16 Adding the hundreds column first  That's easy enough. Just add 3 to the 4 in the hundreds, and get 729. (You didn't need to add with tens-complements, because the digits didn't add to higher than 9.)	abax addition fig. 17 Adding the ones column second Now add 4 ones (of the 394) to the 9 ones (of the 519) using the tens-complements. So you will end up with 3 in the ones column, and one more in the tens. That means a total of 733, so far.	abax addition fig. 18 Adding the tens column last Finally, add the 9 tens (of the 394) to the 3tens (of the 733) (remembering to do it by subtract the tens-complement of the 9 from the 2), you end up with 2 in the tens column, and adding 1 to the hundreds, which gives you 823. This is the same answer as above.
or...
abax addition fig. 16 Adding the ones column first Add 4 ones (of the 394) to the 9 ones (of the 429) using the tens-complements. You will end up with 3 in the ones column, and one more in the tens. That gives you 433, so far.	abax addition fig. 17 Adding the hundreds column second Add the 3 hundreds (of the 394) to the 4 (of the 433)hundreds and get 733.	abax addition fig. 1 Addingthetens column last Finally, add the 9 tens (of the 394) to the 3tens (of the 733) (remembering to do it by subtract the tens-complement of the 9 from the 2), you end up with 2 in the tens column, and adding 1 to the hundreds, which gives you 823. This is the same as both of the above answers.

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These are just three of the combinations or orders you could add 2 three-digit numbers in. You could also have done one of the other orders listed below.

Of course, if you had to add 429 + 394, you could have started by putting the 394 in the abax, and added the 429 to it. You could have done that addition with the columns in any order you wanted, too. So you see, there are many ways to do addition, not just the way they want you to do it in school.

Do you realize how many ways you have learned in just the last few lessons? Just for out of curiousity, lets see how many ways we already know how to add 429 + 394.

You could have done 429 + 394 like this:

• ones first, tens second, hundreds third
• ones first, hundreds second, tens third
• tens first, ones second, hundreds third
• tens first, hundreds second, ones third
• hundreds first, ones second, tens third
• hundreds first, tens second, ones third

That makes 6 different ways right there. But, don't forget that you could have done those with tens-complements, or the old way. That makes 12 ways. (Actually, you could have done some columns with tens-complements and some without, which would give you dozens more combinations, but let's keep it fairly simple for now.)

Naturally, you could have used those same 12 combinations if you added them in reverse, doing 394 + 429, instead of the other way around. That gives you a minimum of 24 ways to do this problem! Now that's what I call freedom! You get to chose which way makes most sense for you. Imagine that!

How many ways were you taught in school? 429 + 394 or 394 + 429. Just two ways. And maybe you were told that you could do them either way because of the Commutative Law of Addition. They probably stopped there, and didn't explain the Commutative Law any further. Learn to look deeper at everything and you will learn math so much better than ever before. You will also get more fun out of it. Isn't that true about life in general?

I feel it is most effective to start with the highest columns and work you way down, because the larger numbers are usually the most important (think in terms of dollars and cents). It is also great to start with the higher columns because you will get an immediate estimate of the entire answer just by doing the highest column first.

Using the tens-complements seems to make the most sense to me, because with a little practice, you can do it quickly, with almost no conscious thought, freeing your mind up for other things. The only problem with it for some people is that we have learned the old way first, and we are used to it. Sometimes you have to make an effort to learn something better, though.

If you are a parent or teacher helping young students, PLEASE teach them about tens-complements and have them use them as their main system. They may seem like a "curiousity" to you, but that is because you were never adequately taught them when you were young. Do not perpetuate that inadequacy in your own teaching.

Remember, though, it is up to you how you want to do it. Practice lots of different ways before you make your decision.

Math Mojo is part of Magic and Learning - a company that uses methods of magicians to teach thinking skills.