Subtraction with Borrowing across Zeroes using Base Ten Blocks

With an exciting title like that, how could this activity NOT be fun?!?

Last week Syd faced a review worksheet of subtraction with borrowing across zeroes with such hysteria that it was as if she'd never seen problems like that before. She has, of course, completely learned how to subtract with borrowing across zeroes before, but never mind--one of the many good things about homeschooling is that there's plenty of room to pause one's math journey to relearn subtraction across zeroes.

Here's how to do that:

If you don't own Base Ten blocks, get them. I don't know how I'd manage without them. Get extra thousand cubes, so that you've got at least nine of them, and preferably more than ten. Same with the hundred flats--you need at least ten. Get as many of the ten bars and unit cubes as you can stand, but remember that those will also work interchangeably with the Cuisenaire rods, which you should also own.

Anyway, get out your Base Ten blocks, and make up a sheet of subtraction problems, all the way up to the thousands, if you want. Make them look really hard, because then your kid will see how easy this method is.

Set all your supplies up in a large space--the blocks, a big piece of butcher paper, the subtraction problems, and a pencil. I'm probably alone in this, but every time I ask the children to get a pencil, I have to say, "Go get a sharp pencil with a good eraser." If I don't say these exact words, I swear to you that the child will every single time come back with a pencil with a broken lead and a non-existent eraser. Every. Single. Time!

Now, your kid should already be quite comfortable using Base Ten blocks, including the key feature, the fact that units are units, tens come in ten bars, hundreds come in hundred flats, and thousands come in thousand cubes; if they want to make 20, for instance, they should automatically go for two ten bars, and not for 20 unit cubes. If this doesn't come automatically, no problem--set them up building big numbers with Base Ten blocks for a couple of lessons, and then carry on.

Have the kid read the first subtraction problem, and then build the minuend out of Base Ten blocks, labeling each place value, as well. In this problem, for instance, Syd is writing a 1 above the thousand cube, an 8 above the hundred flats, and so on:

The subtrahend, of course, is what is to be subtracted from the blocks that make up the minuend. If your kid is really a beginner at this, let her do it however she likes, but encourage her to keep the place values in their separate spots so that she can more easily see what she's doing, and also encourage her to continually notate her process, also so that she can more easily see what she's doing.

If you let her do that over the course of several lessons, then she'll either come up with the traditional method on her own (or a better one!) or, after a while, you can say something like, "Hey, interested in seeing a kind of shortcut?" and introduce her in that way to the following method:

Moving from units on up, physically take the subtrahend away from the minuend. If there isn't enough of one place value--unit cubes, say--to physically take away the number of units called for in the subtrahend, then take a piece from a higher place value, break it up, and move it down. Notate what you've done. For instance, in the following example, Syd was clearly required to subtract something from the ones place, but had zero ones. She took a ten bar from the tens place, broke it into units, put it into the ones place, and recorded what she had done:

Now she has some units to subtract!

She can do the same, of course, if it's tens or hundreds that she's needing, each time physically removing a block from the higher place value, physically breaking it down, physically putting it into the lower place value, and recording her work:

If there's a zero in the next higher place value then you have to keep going, of course. You've got to break a hundred into tens, for instance, so that you have a ten to break into units: 

Notice how the notations that the kid makes are exactly the notations that she'll make when she's doing this problem only with pencil and paper.

My younger kid REALLY likes manipulatives, and really likes repeating something once she feels that she's good at it, so I like to let her keep doing her subtraction this way for as long as she likes, but this method also takes a REALLY long time, and after the first couple of lessons, I don't reduce the number of problems that she has to complete. Once she's faced with 30 problems, it doesn't take her long to take me up on my offer to teach her a "shortcut" that doesn't require the blocks. That shortcut, of course, is the subtraction with borrowing algorithm, and as I show her, I point out how it looks exactly the same as the notations that she made while physically subtracting using the blocks. That's because it IS the same!

Now since Syd is doing this same lesson for the second time in twelve months, it clearly didn't stick with her the first time. However, every time you relearn something, the process gets quicker and easier. I barely had to show Syd how to subtract using the blocks this time before she was off and doing it, and I know that last time, she spent several lessons getting comfortable with the process.

The other trick with Syd is that she LOVES repeating things when they're easy for her, so I have to let her stick with a concept until she's reluctant to move on. When I suggest starting fractions, and she protests, then I know that she's totally mastered subtraction with borrowing... again.