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Relate multiplication with area models to the standard algorithm

CCSS Math: 5.NBT.B.5

Video transcript

- [Instructor] 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 two or we could think of it as two 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 80, so plus 80, and then we have one one, so plus one. And you might have be familiar with multiplying like this in the past, setting up this grid. And it's essentially we're applying the distributive property. We're gonna take the two and multiply it times 400 plus 80 plus one. So we're gonna multiply two times each of those numbers, and actually I'm gonna just draw some quick lines here. So we have that, and then we'll have let me 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. 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 two times 400 plus 80 plus one. So two times 400 is 800. Two times, let me do it in that same blue color, so this is 800. Two times 80 is 160, and then two times one is two. And then we can multiply 50 times these. So what's 50 times 400? Well five times four is 20, and then we have another one, two, three zeros. So one, two, three, so that's 20,000. 50 times 80, five times eight is 40, and then we have two zeros, just like that. And then we have 50 times one which is of course going to be equal to 50. And then we go to the 300, which we will distribute and multiply times each of these, each of these numbers. 300 times 400, three times four is 12, and then we have four zeros. One, two, three, four. We get 120,000. 300 times 80, three times eight is 24, and then we have one, two, three zeros. One, two, three, so we get 24,000. And then 300 times one is of course equal to 300. 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. 800 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 100 and, 144,300? 144,300. 120,000 plus 24,000 is 144,000 plus 300, there you have it. And then you would add up these numbers just like that to get your final answer. Now I'm gonna 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've said, 481, and this is sometimes called the standard algorithm, 481 times 300 and 50, let me do the same colors, 352. And in the standard algorithm, the way that we do it is we start with this two in the ones place and then we multiply it times 481. So two times one is two, two times eight is 16. So we put the six here and then we sometimes we will say we carry the one, but really regrouping that as hundreds. That's 10 tens which is hundreds. And then two times four is eight, which is really 800, plus one, so that's nine or really 900. 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 one, times the 80, times the 400. We saw that over here, and then we just added 'em 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 saying all right what's five times one? It's five. What's five times eight? It is 40. We regroup the four. Let's delete this from before. What's five times four? Well that's 20 plus four 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 gonna happen when we take this three 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. And when I say algorithm it just means a method of doing something. And so we'll say three times one is three. Three times eight is 24. And then three times four is 12 plus two is 14. And so notice, I have 144,300. And the standard way of doing it at this point we just add 'em all up. So whether we're doing it here or here, we just add everything up. So two plus zero plus zero is two. Six plus five is 11. Regroup that one. So one plus nine plus three is 13. And then one plus four plus four is nine. Two plus four is six. Then we have a one right over there, so we get 169,312. And so when you just learn this method, the standard algorithm we some people might call it, it might seem like hey this just seems 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 one to get this. Then you're multiplying 50 times 400 plus 80 plus one, and then 300 times 400 plus 80 plus one, exactly what we did right over here.