[0:03]So here's my attempt to explain the progression of multiplication. Well, most of us might think it starts in third grade, but in actuality, it starts in second grade, where students are expected to take a square or a rectangle and partition that square into no more than five rows or five columns.
[0:27]Here the understanding of repeated addition is established. 5 + 5 + 5 + 5 + 5 is 25. Then, in third grade, students take that understanding they've begin to develop in second grade and begin to construct arrays using square colored tiles or any manipulative for that matter. The important piece is once they've actually constructed that model, they draw a pictorial representation which matches their model. Here, we have four rows of six, or it could also be seen as six rows of four, depending on how you turn the model. And the big piece here is that we don't push an understanding on students, we let students explain what their model represents. The big piece is that the context will explain the equation, not the teacher. Then, we take that understanding and we come back to the biggest piece probably in third grade, which is that there's a relationship happening here. And that relationship is taking place between multiplication and division. Take this, for example, 12 divided by 4 is a number, 4 times what number is 12? Well, if we take 12 tiles and we put them in four rows until we run out, we actually see that we have three in each row. So there's a connection between multiplication and division, and arrays can be used for both. The context. Yep, let's go back to it. Here's a simple one: there were three packs of pencils, each pack had 40 pencils. Well, at the beginning part of multiplication, students might do something like this where they draw out three circles, and in each circle, they put in four ten rods, so 40, or they could actually go ahead and draw 40 little pencils. That's great, but the standards of mathematical practice and in particular number seven want us to be efficient, and one way to do that is we can begin to make use of structure, and when we do, we make sense of it. Here, we begin to create an area model for multiplication, very much like an array, just on a bigger scale with larger numbers. So, 3 x 40, we take it, build it out of 10 rods. We have 10 tens, which is a 100, plus 20 is 120, or it could also be seen as 12 tens. Really big understanding for students. Once students have established this, they should be diving into drawing pictorial models for what their concrete. Again, we don't want them to be sitting there with manipulatives all the time. So, we move them away, and they begin to draw a pictorial representation for what their concrete model looks like. Then, from there, we'll have them use numbers to represent their picture. All of the time connections are being made. Then, eventually, after multiplying all of these ones and tens, students begin to collect and group numbers together. Now, we're beginning to become efficient thinkers. The connections are being made all around because students are exploring. In fourth grade, students are expected to multiply two-digit by two-digit numbers. So what does this look like at with a concrete model? Well, it kind of looks the same, really. So, we have 23 across the top and 22 down the side. 10 times 10 is 100, and that's where those 100 plates come into play.
[3:57]Then, once students have gone ahead and actually built a model, they see that they have 400s, 10 tens, six ones, and they add that all together, 506. Then just like we did with smaller numbers, we expect that students should draw a pictorial representation, which matches their concrete model. You can see the 100 plates, the ten rods, and the six ones. There they are. Altogether, 506.
[4:34]And just like before, we'll ask that students go ahead and start using tens and numbers to actually represent their model. So what you're probably thinking right now is this is not efficient. There's too many tens going on here, too many ones, and students are going to get confused. And you know what? You're absolutely right, and it's at this point where kids say, this sucks. We need to find a shortcut. So what do they do? They begin to group and collect their numbers just like they did before. So here we have the box method, but in reality, it's partial products. So let's not call it the box method anymore and start calling it by its real name. Then, we take that partial product understanding, and we begin to formalize it into some kind of standard algorithm. Yep, this is a standard algorithm. It's a series of repeated steps. And you can see where each of the numbers come from. Then, we have the algorithm that we're all pretty much used to, but you see that zero? Now the connection is made, and students know why we magically put that zero there. But this, all this understanding happens at the end of fifth grade. Let's not rush them. There's so much happening here and so much to explore. Let's slow down a little bit.
