Multiplying using area models and the standard algorithm

What we're going to do in this video is multiply the numbers 352 and 481, and we're gonna do it in two different ways. But realize that the underlying ideas are the same.

So first, let's just appreciate that 352 can be rewritten as 300 plus 50 plus 2, or we could think of it as 2 plus 50 plus 300 plus 300. You add these three numbers together, you're going to have 352.

And same idea, 481. That's four hundreds, four hundreds plus eight tens, which is eighty, so plus eighty, and then we have one, one, so plus one. You might have been familiar with multiplying like this in the past, setting up this grid. It's essentially we're applying the distributive property. We're going to take the 2 and multiply it times 400 plus 80 plus 1.

So we're going to multiply 2 times each of those numbers. And actually, let me just draw some quick lines here so we have that. And then we'll have, we do it like this, then we have this. And then, let me set up my grid. I'm having trouble drawing straight lines. Okay, there we go, and then one more in this direction. There you go. And then in this direction, let me draw some horizontal lines to have a neat clean grid here. There we go.

Now first, we'll multiply 2 times 400 plus eighty plus one. So, two times four hundred is eight hundred. Two times, I'm doing that same blue color, so this is eight hundred. Two times eighty is a hundred and sixty, and then two times 1 is 2.

And then we can multiply 50 times these. So what's 50 times 400? Well, 5 times 4 is 20, and then we have another 1, 2, 3 zeros. So, 1, 2, 3. So that's twenty thousand. Fifty times eighty, 5 times 8 is forty, and then we have two zeros, just like that. And then we have fifty times one, which is, of course, going to be equal to fifty.

And then we go to the 300, which we will distribute and multiply times each of these numbers. 300 times 400, 3 times 4 is 12, and then we have four zeros, 1, 2, 3, 4. We get a hundred twenty thousand.

Three hundred times eighty, 3 times 8 is 24, and then we have 1, 2, 3 zeros, 1, 2, 3, so we get twenty-four thousand. And then three hundred times one is, of course, equal to three hundred.

And then what we wanna do is add up all of these numbers. So let's actually add up the rows first. So if we add up the rows, let me draw another line going straight down like that. And so if we sum this up, this is going to be 962. Eight hundred plus 160 is 960 plus two, so this is 962.

This right over here is 24 050, 24 050. And then this right over here is what? A hundred and a hundred and forty-four thousand, three hundred. One hundred forty-four thousand, and three hundred. Hundred twenty thousand plus twenty-four thousand is hundred forty-four thousand plus three hundred, there you have it.

And then you would add up these numbers, just like that, to get your final answer. And I'm going to hold off doing that for a second, as we see the other way of multiplying these numbers.

So the other way of doing it, we could have said 400. And this is sometimes called the standard algorithm, 481 times 350. We do the same colors, 352.

And in the standard algorithm, the way that we do it is we start with this 2 in the ones place, and then we multiply it times 481. So 2 times 1 is 2. 2 times 8 is 16. So we put the 6 here, and then we sometimes we'll say we'll carry the 1. We're really regrouping that as the hundreds. That's 10 tens, which is the hundreds.

And then 2 times four is eight, which is really eight hundred plus one, so that's nine, or really nine hundred. Do you see a pattern here? This 962 is the exact same thing as that 962 right over there. Why? Well, because we multiplied the 200 times the 1 times the 80 times the 400. We saw that over here, and then we just added them all together to be 962. That's all the standard algorithm did just now.

And then we move over to the five, but this is really five tens, or 50, and that's why in the standard algorithm, we put a zero here before that, before saying all right, what's 5 times 1? It's 5. What's 5 times 8? It is 40. We regroup the four.

Let's delete this from before. What's 5 times 4? Well, that's 20 plus 4 is 24. Notice 24 050, that's exactly what we had over here. And it makes sense because we're taking a 50 and we're multiplying it times 481, which is exactly what we did right over there.

And so you might guess what's going to happen when we take this 3 and we multiply it times 481. That's really a 300 times 481. Let me delete that so I don't get confused.

So because it's a 300 in the standard algorithm, we put two zeros here first. When I say algorithm, it just means a method of doing something. And so we'll say 3 times 1 is 3, 3 times 8 is 24, and then 3 times 4 is 12 plus 2 is 14.

And so notice I have forty-four thousand three hundred. And the standard way of doing it, at this point, we just add them all up. So whether we're doing it here or here, we just add everything up. So 2 plus 0 plus 0 is 2, 6 plus 5 is 11, regroup that one.

So one plus nine plus three is thirteen, and then one plus four plus four is nine. Two plus four is six, and we have a one right over there. So we get a hundred sixty-nine thousand three hundred and twelve.

And so when you just learn this method, the standard algorithm we, some people might call it, it might seem like, hey, this just seemed like a little bit of magic. But all you're doing is you're going to each of these places, and you're distributing it. You're multiplying it times 400 plus 80 plus 1 to get this, then you're multiplying 50 times 400 plus 80 plus 1, and then 300 times 400 plus 80 plus 1, exactly what we did right over here.