Decomposition in Math Subtraction by Gail Englert

A response to the question:

Could you give ideas for teaching the decomposition method for subtraction?

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I am trying to teach my daughter the decomposition method for subtraction-where the bottom number is higher than the top number. Using base 10 materials is working. I am surprisingly having a lot of success with her verbalizing the process. We are only working on 2 digits at the moment.

My problem is that when I started to use this method in 5 and 6 digit numbers with lots of zeros, I couldn't do it with success very often. I especially came into trouble with lots of zeros in a row.

You help is greatly appreciated.

Margaret

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Dear Margaret,

I believe that "decomposition algorithm" refers to the standard algorithm used in many American schools for subtraction. Decomposing means breaking down, and that is what is done when you subtract, in a way.

Here is a step leading up to that algorithm that might help you understand what is happening:

When you want to subtract 987 and 123, what you are really doing is subtracting each place value separately. So you could look at the problem

```      this way:               or this way:

987            9 hundreds   8 tens   7 ones
-123           -1 hundred    2 tens   3 ones
____           ______________________________
```

that is pretty easy to solve.

But what about this one: 907 - 123. Now there is a problem you can't subtract easily in the tens so you have to "regroup" what you have:

```       907     9 hundreds   0 tens   7 ones     8 hundreds  10 tens   7 ones
-123    -1 hundred    2 tens   3 ones    -1 hundred    2 tens   3 ones
____    _____________________________   ______________________________
```

That works because 8 hundreds 10 tens and 7 ones is the same thing as 9 hundreds 0 tens and 7 ones.

Here is another: 6007 - 451

```6 thousands 0 hundreds 0 tens 7 ones  5 thousands 10 hundreds 0 tens 7 ones
-           4 hundred  5 tens 1 ones    -          4 hundred  5 tens 1 ones
_____________________________________   ____________________________________
```

But that still doesn't fix all the places for us, so lets "regroup" again:

```5 thousands 10 hundreds 0 tens 7 ones   5 thousands 9 hundreds 10 tens 7 ones
-            4 hundred  5 tens 1 ones    -          4 hundred   5 tens 1 ones
_____________________________________   ____________________________________
```

That works because 5 thousands 9 hundreds 10 tens 7 ones is the same thing as 6 thousands 0 hundreds 0 tens 7 ones.

If you take it one step at a time, it should be easier for your daughter to figure out. You might want to try writing the numbers out by place values for a while, and then going back to the "shorthand" method we learned in school.

If you think she is still having trouble with this "regrouping", there are some chip trading games that can help her feel more comfortable with that skill. Just write back and let us know. Good luck!

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Dear Gail,

When you refer to the shorthand method we used at school, do you mean the borrowing method? Or do you just mean writing the decomposition method with just numbers?

I'm interested to know why we would go between two different methods for children - wouldn't that confuse them?

Thanks again,

Margaret

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Dear Margaret,

I am glad the explanation helped. What I meant by shorthand was that we don't usually write out the place values for numbers. Instead, we let the place they are in tell that.

So, where we would ordinarily write 359 I wrote out 3 hundreds 5 tens 9 ones.

We definitely wouldn't want to continue to use this method of writing out the place values in words, because it takes a long time, and isn't really as easy to work with. Would we want to switch back and forth between the two methods? Well, if a child is having trouble with the way we are trying to solve a problem, sometimes it is beneficial to back up and show some of the underlying meanings (which is what we are doing if we write the number words out), and then try to move back into something more abstract. If it is still confusing, we may have to back up again.

As for confusing children, I think they are much more adaptable than we give them credit for. They really need to recognize that there are multiple ways to solve problems, and that there are many ways to write amounts... for example, the number we were using before, 359, can be considered as 359 ones, or 35 tens and 9 ones, or 3 hundreds and 59 ones, or 3 hundreds 5 tens and 9 ones. It is all dependent upon which of those ways of looking at that amount is most helpful to us at the moment. Helping children see multiple ways is a way of empowering them to be problem solvers. (Now I shall step back down off my soapbox, before it tips me onto my ear!)

-Gail, for the T2T service

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