Wednesday, August 22, 2012

Middle Zeros

I had an opportunity to spend some time with a very enthusiastic group of third grade students at a local school. They were subtracting three digit numbers by regrouping and were doing quite a spectacular job. That is, until we reached the dreaded Middle Zero.

The problem was 601 - 347. The following is an actual transcript of the dialogue that ensued.

Me: Does anyone know how to do this problem?
Student 1: You can't take away 7 from 1 so you have to make the zero a 9.
Me: You mean I can just turn a 0 into a 9? How can I do that?
Student1: Well, you make the 6 into a 5 and then the zero becomes a 9.
Me: I still don't understand how the zero becomes a 9. And why does the 6 become a 5?
Student2: There aren't any tens so you have to borrow a ten from the hundreds column.
Me: Ok, that makes sense. How many tens are in the hundreds column?
Student2: 6

and so on...

It was fairly obvious that they had been introduced to the mechanics of regrouping but they had not achieved a true understanding of the process. The beauty of mathematics and the amazing "aha!" moments are revealed only when we investigate the math beneath the surface.

Third grade students are generally very competent when it comes to regrouping tens to ones in the case of 85 - 29 or 371 - 158. Problems arise when there are no tens in the tens column. This requires a visit to the hundreds column whereupon one of the hundreds is regrouped as 10 tens.

For example, in the problem:

402 - 138

the number 402 would have 3 hundreds, 10 tens and 2 ones. Now it's possible to regroup one of the tens into 10 ones and add those to the ones column. The final version of the number 402 is 3 hundreds, 9 tens, and 12 ones.

While this is no doubt trivial and rather obvious to you as a teacher or parent, third grade students find this incredibly confusing. In fact, I'm convinced that even with the benefit of base ten blocks and dozens of practice problems, students are still not really sure what's going on here.

What if we approach problems with middle zeros a little differently. First, help your students see numbers in terms of tens. If you use the number 80, ask how many tens that is. If the number is 527, help them see not only that there are 5 hundreds, 2 tens, and 7 ones but that there are 52 tens. Once that is understood, you can explain regrouping numbers with middle zeros more easily.

Let's revisit the problem, 402 - 138. A ten is needed to make 12 ones. Turn this into a single step process by having the students think "40 tens". Now they can use one of those tens to change the number of ones from 2 to 12 and they'll have 39 tens left. It's much simpler, there's less crossing out, and students really get it.

This method works with numbers containing more than three digits as well.

How do you teach regrouping with middle zeros?