Multi-Digit Multiplication Using Area Models

Lest I accidentally promote the false conclusion that I have an eager, happy math learner here, I should tell you right now that Will does not currently enjoy math. So many things come so easily to my bright kid that she is very much disliking the many-stepped process of learning multi-digit multiplication. As she can do with so much else, she wants simply to look at the problem and immediately know its answer. She does not want to compute with pencil and paper, and she certainly does not want to compute for a problem that has more than one digit! She does not want to go through the bother of writing down numbers that carry, and she does not want to also go through the additional bother of crossing out those numbers once she's used them. This, of course, leads to many clerical errors, and she very, very, very much does not want to have to erase and re-compute when she's worked a problem incorrectly because of a clerical error.

Our school days can be greatly extended these days, as what could have been thirty minutes of math is drawn out with many philosophical discussions that consist mainly of "But WHY do I have to do [insert current step here]?!?" And then when I bring out the Base 10 blocks to once again demonstrate why a certain step works the way that it does (it occurs to me that I should have showed you how to use Base 10 blocks to do multi-digit multiplication prior to this. If the kid ever lets me get them out with her again, I will photograph the process and show you), I hear, "NOOO!!! NOT the Base 10 blocks!" because, of course, the Base 10 blocks make working the problem take even longer--not as long as a problem takes when you spend all your time whinging about it instead of working it, mind you, but unlike the whinging, you get to blame the blocks on ME. And anyway, the question isn't directed at confusion by how the calculations work, but at despair that this multi-stepped method is the only one that whatever idiot who must be in charge of Math could come up with.

Nevertheless, I do like to spend some time with Will each week exploring different concrete manifestations of how multi-digit multiplication works, knowing that although some of her fuss about the computations is just laziness, some of it must also be a lack of confidence, and a fear that she's not "getting" it, quick as she usually is.

Area models are a great concrete manifestation of how the math works, and an understandable extension for the kid, since by now she should have played plenty both with arrays and with area models for single-digit multiplication, back when she was learning her facts.

To do this, you need the same manipulatives that I am forever going on about--a huge set of Base 10 blocks, and plenty of centimeter-sized graph paper. If you're printing your graph paper, as I do, make sure that the "fit to page" box is unchecked; just the other day, I ruined a giant, 1,000-centimeter ruler that I was making Will to use as a multi-digit multiplication and division manipulative by not doing this, and then only discovering after I was done that the damned Cuisenaire rods didn't line up correctly with it--GRRR!!!

The only other tricks to this are 1) make sure that your multiplication problem doesn't exceed the size of your graph paper, and 2) if your graph paper isn't square, make sure that your kid knows to draw the problem going the correct way (our graph paper is something like 20x30, so if I wrote a problem that was 17x27, for instance, I had to make sure that Will drew it just like that. It'll make more sense in a minute).

1. Write your problem for the kid. I wrote mine at the top of each piece of graph paper (you'd save paper if you had a dry erase board set up with a centimeter graph--this is on my "to buy" list!), and gave Will five total problems to solve for this lesson.

2. Have the kid draw a box that represents the problem. She should be able to figure out that 17x27, for instance, means 17 rows with 27 units in each row:

3. Here's where the Base 10 blocks come in, and it's really cool. First, have the kid fill up the box as much as she can with the hundred flats:

4. Then, have her fill up as much as she can with the ten bars:

5. Finally, have her fill in all the rest of the space with units.

6. All she has to do is count the Base 10 blocks to find her answer! This is a great time to encourage her to skip count by hundreds, then continue to skip count by tens, then add on by units. Stretch that smart little brain!

The next time we play around with this concept, in the next week or so, I plan to focus more on the area aspect of it, by drawing an irregular polygon directly on the graph paper, having Will use the Base 10 blocks to find its area, then having her invent a way to compute that area with only pencil and paper (by writing multi-digit multiplication problems that require the correct order of operations to solve, thus making her complicit in the usage of both the mathematical procedures that currently offend her, mwa-ha-ha!).