## Saturday, October 17, 2009

### M4N3 - Developing multiplication algorithms (7)

M4N3. Students will solve problems involving multiplication of 2-3 digit numbers by 1 or 2 digit numbers.

So far, we have discussed the following:
(1) extending the multiplication table to 10x10
(2) multiplying multiples of 10 and 100 by 1-digit numbers
(3) multiplying 2- and 3-digit number by 1-digit numbers
(4) multiplying by multiples of 10.

Now, we are ready to tackle multiplication of 2- and 3-digit numbers by 2-digit numbers. Before we get started, I wanted to say that, to me, teaching of an algorithm means helping students make their own strategies into written procedures instead of imposing a specific algorithm upon students. Of course, that doesn't mean "anything goes." Rather, teachers must think carefully about how to influence students' thinking naturally. Moreover, it may be possible for teachers to sequence students' experiences in such a way that the algorithm students develop "naturally" is something very similar to, or exactly the same as, the conventional algorithm. For that purpose, the area model of multiplication can play a very important role. Therefore, the use of the model along with base-10 blocks before reaching this point is an integral part of the process. So, how do we help students expand their written methods into multiplication of 2- and 3-digit numbers by 2-digit numbers?

Let's think about 12x23 first. How can students use what they have learned so far to think about ways to calculate this problem? There are at least three possible ways. At the most abstract level, students might be able to think of 12x23 as 12x20+12x3 - i.e., 23 groups of 12 can be split into 20 groups of 12 and 3 groups of 12. Then, each of 12x20 and 12x3 are already discussed. If students can think about this way, they can record the process using the vertical notation,
or .

The notation on the right is basically the standard algorithm for multiplication.

Another possibility is for students to go back to the area representation of multiplication. 12x23 means that we are making a rectangle with the dimension of 12 units by 23 units. The product is represented by the area of this rectangle. So, if you construct this rectangle using base-10 blocks, and using the fewest number of blocks (i.e., use large blocks whenever possible), you can make a rectangle like this one:

By examining the arrangement, we see that there are 1 by 2 rectangles made of flats (200), 2 by 2 rectangles of longs (40), 1 by 3 rectangles made of longs (30) and 2 by 3 rectangles of units (6). So, the product is 200+40+30+6=276. After students have become comfortable with the area model representation with base-10 blocks, you may want to encourage students to move toward drawing instead of using actual base-10 blocks. Sometimes you can make this transition simply by giving students multiplication problems with larger factors. Students will realize that actually making rectangles using base-10 blocks is too tedious.

When students become comfortable with drawing rectangles, they might realize that it is still rather tedious. This is when you may be able to suggest if they could use an adaptation of a notation that we used when we were multiplying 2- or 3-digit number by 1-digit number. Some students may be able to start at this point, without going all the way back to using base-10 blocks. That judgment must be made by teachers, using their knowledge of students. Anyway, the notation might look like this for 12x23:

Again, after students have become fluent with this notation, you might want to bring their attention to the four products (in the example here, 200, 40, 30, and 6). Noticing that these are the products of the two tens digits, the tens digit and the ones digit (in both direction) and the two ones digits. So, you can introduce a new notation that records the same information as this diagram does:

You can then negotiate with your students a consistent order in which you calculate these four products (typically called "partial products) so that we can make sure that we have accounted for all of them. If you really want students to understand the conventional multiplication algorithm, you will start with the ones digit of the multiplier (the bottom number) and multiply the ones and then the tens digits of the multiplicand (the top number). You will then multiply the tens digit of the multiplier with the ones and then the tens digits of the multiplicand. So, this problem would look like this:

If you combine the first two partial products and the last two partial products, you will have:

Note that the example we used, 12x23, did not involve any re-grouping. In a way, this is the most "basic" situation. As students move from one notation to another, you may want to consider moving back to a basic situation. Once students become comfortable with the notation (area model, symbolic notation, or whatever), then you want to look at other situations such as those involving re-grouping and a 0 in the factor/product.

When extending the multiplicand to 3-digit numbers, for example, 587x34, you may want to go back to the diagram notation - it will be rather difficult to actually model these multiplication with base-10 blocks. From the diagram, you can move to the notation that will explicitly record all partial products, then eventually to the conventional algorithm.

As usual, you do want to pay close attention to the numbers (factors) you use. Some students have difficulty with 0's - either in the factors or in the product/partial products, so you want to pay particular attention to those situations.