Monday, August 20, 2012

Two-Digit Subtraction without Regrouping

Math is something that Will and I tend to move a little slowly through, because it's really important to me that she have the concepts not just memorized, but ingrained. To that end, we've also done a lot of sensorial work in math, building numbers with Base 10 blocks or Cuisenaire rods, for instance, instead of necessarily speeding through computation concepts.

Having a good knowledge of those Base 10 blocks and Cuisenaire rods, though, makes learning the computation so much easier, I think, especially because, even when handling big numbers, I want the girls to understand not just what they're doing on paper, but what they're actually doing with actual numbers.

For instance, I'm not sure if Willow would have really understood what was actually happening with two-digit subtraction without our Base 10 blocks and, most importantly, the Montessori-style Base 10 number cards that we use with them. 

Case in point for how slowly we move through math: I explained how to build big numbers using Base 10 blocks and Montessori number cards back in January, as preparation for Willow learning how to add with carrying, and she STILL doesn't add with carrying! But she has her addition math facts memorized up to 10+10, and she can add multi-digit numbers in her head by rounding. And now we're going to do subtraction for a while, and we'll just come on back to carrying later.

To subtract two-digit numbers without regrouping, at the same time modeling each problem to demonstrate how it physically works, drag out all the Base 10 blocks and Montessori number cards. I wrote a set of problems in Willow's notebook for her, so for each problem, she first built the subtrahend (in blocks and with the cards), then built the minuend the same way, then lined them up vertically:

We often start math by building a couple of numbers like this, simply because it reinforces the concept that 75 is made up of 7 tens and 5 one units, however you play with them and rearrange them and fiddle them about.

For the first couple of problems, Will physically subtracted the Base 10 blocks--first the units, then the tens. Then, when she counted the units back up to find her answer, I noted that you could also find the answer by subtracting the units, then the tens, just using the Montessori number cards: 

This is a good path towards the simple pencil-and-paper computation, since 1) we've already spent so many hours keying these Montessori number cards to the Base 10 blocks that the representation is well ingrained, and 2) the cards are stacked in such a way that you can unstack them and easily see the units and tens that they're made of; physically stacking them to build the number reinforces this, and helps you remember the concept.

When it looks like Will understands the procedure and, more importantly, the concept underlying the procedure, I let her finish up her set using just her pencil and paper:

We'll likely end up doing this exact same lesson again in the near future; as we were working, I discovered that Willow doesn't have all of her subtraction math facts up to 10-10 memorized, nor does she have the complete vocabulary of subtraction memorized, so we'll be doing flash cards and simple subtraction drills for a bit of review this week. We may then have to repeat this particular subtraction lesson afterwards, but Willow will be better equipped to master it and move forward with that basic grammar of math memorized.

Will's Singapore 2B workbook wants her to be able to both mental addition and subtraction of multi-digit numbers before she moves on to the next unit; I don't know if we'll choose their roundabout rounding method for subtraction, as we did with addition, or if Will will actually choose borrowing as her method, but she'll be able to choose for herself after she's learned both methods.

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