Beginning Multi-Digit Multiplication
After half a year of multiplication facts, addition, subtraction, and geometry, we’ve finally started our arithmetic unit on multi-digit multiplication. We are starting off with 2 digit by 1 digit numbers, with a few 2 digit by 2 digit thrown in. Instead of drilling the standard algorithm (that is, the method for multiplying multi-digit numbers that most adults learned in school), we have been exploring multiple models for understanding what is going on when we multiply larger numbers.We started at the beginning of the week with modeling multiplication problems, much like we did earlier in the year. Yesterday we looked at factoring to turn the problems into known elements. For example, 20 x 6 can be thought of as (2 x 10) x (6 x 1). Then we use the commutative and associative properties of multiplication to regroup as (2 x 6) x (10 x 1). These simplify as (12) x (10) which is easily solved as equaling 120. Here’s another example:
30 x 9 =
(3 x 10) x (9 x 1) =
(3 x 9) x (10 x 1) =
(27) x (10) =
Of course, those two examples both work with multiples of ten. We can carry the concept further by representing any whole number as the product of two factors. For example,
12 x 25 =
(3 x 4) x (5 x 5) =
(3 x 5) x (4 x 5) =
(15) x (20) =
(15 x 1) x (2 x 10) =
(15 x 2) x (1 x 10) =
(30) x (10) =
Now, yes, that is definitely a bulky way to solve a multiplication problem, and the standard algorithm is definitely faster, but factoring works and it allows the mathematician to break the problem down into easier parts that take advantage of the math facts we’ve been practicing all year long.
Today I introduced the idea of using area models to represent problems in a very similar way. Imagine you have a rectangle with a length of 28 inches and a width of 4 inches. You can “chunk” that rectangle into three parts: 10 x 4, 10 x 4, and 8 x 4. These are easy math facts that can be quickly determined (40, 40, and 28) and then summed (108). We did several examples like this today, drawing quick sketches of rectangles, labeling them, dividing them into manageable pieces, and then solving.
The best part of introducing these various models and strategies for solving multiplication problems is when students start creating their own problems that are far more complex than I have given them. I had some students who were generating 4 digit by 4 digit multiplication problems, then drawing rectangles, dividing them into manageable chunks, and solving! I love that my students are applying these concepts and challenging themselves! We will be exploring more models over the next few days before tackling the standard algorithm. Some of my students already know this method, but I am glad that they are willing to explore different models that challenge their thinking and get them focusing on what is actually going on when they multiply numbers, rather than just quickly doing computations without any understanding. This approach takes more time, but I feel it is worth it to establish more in-depth comprehension!