Adding with the Abax (1)

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Notes to teachers and parents

Adding with the Abax (Lesson 1)

Now that you understand counting with the abax, let's get reckoning. This lesson really teaches us about regrouping and carrying in the base- ten system (also known as the decimal system). If you want to know more about it, or need to explain it to a student, check out the Regrouping and Carrying lesson in the FAQs section of this site.

A great hint I can give you is to add from the highest column to the lowest. I know, I know... in school they always teach you to start with the ones column first. But let me ask you a question - if you buy something for 64 dollars, and something else for 31 dollars, and you want to know how much money you need altogether, do you start with the "chump change?" Do you care more about the 6 tens and the 3 tens which are $90, or do you care more about the 4 singles and the 1 dollar bill which are worth $5 first?

Don't worry, it is as easy to add from left to right as it is from right to left. It is just a little harder to teach that way. But I've got you covered on that one! Just stick with this till you get it, and you will be adding left to right pretty soon in "real life", too.

The figures in cells in the bottom row, (the ones with a border around them) are the only slightly difficult thing in learning the abax. You don't even need to do it this way, but if you learn it, you will have learned a powerful lesson about carrying, regrouping and tens-complements. All of these things are used over and over in MathMojo, and are some of the best lessons you will ever learn about math. They are part of the basis of separating those who really "get" arithmetic from those who can merely do it "okay." Please spend some time with them until you get them. If you have problems with them, go to the "contact" link and send me an e-mail and I will help you.

abax addition fig. 1
Adding 23 +11

This is the number 23. We want to add 11 to it.
That is a cinch. Place one more pebble in the tens column, and one more pebble in the ones column,and you are done. Now count the total, and you get...

abax addition fig. 2
Adding 23 +11 (continued)

  ...34.
  That was simple because we never had to fill any columns and do any trading one full column for a pebble in the next column.
  (If you don't know what I am talking about, go back to the lessons on counting on the abax).

abax addition fig. 3
Adding 37 + 46

  This one is going to be a little harder. Here we have 37. We are going to add 46 to this.
  Remember, we are going to add from the highest column first. Since we are adding 4 tens, we put 4 more pebbles in the tens row. That makes 7 in that row so far. But wait - when we look over to the ones, we see that when we go to add the 6 to that column, it will overflow. So we add one more to the tens column now (making the amount we add to the tens column 5 instead of 4), instead of carrying later.
  This leaves us the situation in the next figure.

abax addition fig. 4
Adding 37 + 46 (continued)

  We added the 4 pebbles in the tens column of the 37 and an extra one for the carry we knew we'd have to do in this step, and we got 87.
  But we are not done, yet, because we still have the six ones to add. Or do we? We already added one extra ten in the tens column. We really only needed to add six ones, but they wouldn't have fit in the ones column.
  Since we added ten, but really only needed to add 6, we have really added 4 too many. What do we do?
  We take those 4 away from the ones column! Then we will be done. See the situation in the next figure.

abax addition fig. 5
Adding 37 + 46 (continued)

Here we have taken away the 4 extra pebbles from the ones column, and have our final answer to 37+46, which is 83.

Basically, you can learn from this that whenever you see that you are going to have a total of more than 9 in any one column, you just add an extra pebble to the higher column, and subtract the extra from the column which would have overflowed.

Here is a hint - that "extra" amount which you will subtract will always be the tens complement of the higher digit in the column that you are working on. Subtract that ten-compliment from the lower digit in that column. In the case of the ones-column in this example. the 3 was the tens complement of the 7. (you use the tens-complement of the 7, because 7 is higher than 6).
So we subtract that 3 from the 6, and got 3 That 3 is the final answer in the ones column.

To go on to the next lesson, click here

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