This is the fourth in a series of posts in which I am discussing the development of multiplication algorithms. Up to this point, students were calculating mentally. The focus has been more on consolidating students' understanding of our number system and the meaning of multiplication by using those understanding to figure out multiplication beyond the basic facts. Today's standard is the first step toward developing paper-and-pencil algorithms. As I begin my post, let me emphasize that teaching of an algorithm for any operation should focus on helping students develop the algorithm on their own. In other words, we need to move away from the show-and-tell approach where teachers show students how to multiply using the multiplication algorithm and then have them practice over and over. Practice is important, but students should first develop the algorithm themselves. Of course, that does NOT mean that we just leave students on their own. Rather, teachers must plan carefully to guide students' thinking.
One useful idea in developing a multiplication algorithm is the area model of multiplication. In Grade 3, students learn about area of rectangles and squares. When students cover a rectangle with unit squares, they notice that they are arranged rows and columns of equal sizes. Because all rows (or columns) are equal, we can use multiplication to efficiently determine the area. This idea can be used to model multiplication where the two factors are represented by the two dimensions of a rectangle and the product is represented by the area. So, for example, 4x6=24 can be modeled as shown below.
Notice that since you can turn the rectangles around without changing the area, this is also a useful model to show why the commutative property of multiplication is true. It is also useful to model the distributive property of multiplication.
When you model multiplication problems like 14x7 using base-10 blocks, you can certainly try to make 7 groups of 14 (1 long and 4 units). However, we want to encourage students to organize the model more systematically using the area model. The area model representation will make it much easier to determine the product by observation if the same type of blocks are grouped together.
Eventually, we want to help students move beyond modeling with actual base-10 blocks. One useful approach to do so is to have students draw what they would have done with base-10 blocks. Thus, drawing the picture like the one above. Grid papers can be very helpful in that process. However, as they become comfortable with drawing pictures, they realize that drawing can be rather tedious. Given our goal is to determine the product, what we want to know is how many longs and how many units we have. Thus, we can model the multiplication explicitly showing only the information we need. Here is an example for 14x7.
Once students become comfortable with modeling multiplying 2-digit number by 1-digit number this way, we can ask if they can think of a way to represent this model using a vertical notation like we did with addition and subtraction. Here are two possibilities:
Students can see that 70+28 and 28+70 are the same. Thus, we can write it either way. At this point, it is ok to suggest that we agree to write the product of the ones digit first. Again, after students practice this notation, they might notice that the ones digit for the partial product of the tens digit on the multiplicand and the multiplier is always 0. Therefore, the ones' digit of the product is always the ones digit of the partial product of the ones digit of the multiplicand and the multiplier, 8 in the example above. Then, we have to add the tens digits of the partial products to find the product. This process can be combined if you use a notation like this:
This may be a slightly different notation than some of us are used to, where the tens digit of the partial product above the tens digit of the multiplicand. That notation sometimes causes students to add the re-grouped digit and the tens digit of the multiplicand before multiplying by the multiplier - that is students end up doing (2+1)x7 instead of 1x7+2. Writing the re-grouped digit below the horizontal bar (the equal sign) might minimize that error.
In the next post, I will discuss how this procedure may be extended to multiplying 3-digit numbers