Multiplication: place value and area models
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Multiplying with distributive property
In this video, I'm going to multiply 87 times 63. But I'm not going to do it just by using some process, just showing you some steps. Instead, we're just going to use the distributive property to actually try to calculate this thing. So first, what I'm going to do-- let me rewrite 87. So this is the same thing as 87. But instead of writing 63 like that, I'm going to write 63 as 60 plus 3. Now, what is this going to be equal to? Well, 87 times 60 plus 3, that's going to be the same thing as-- and let me actually copy and paste this. So this is going to be the same thing as 87 times 60 plus 87 times 3. You could say that we've just distributed the 87. We're multiplying 87 times 60 plus 3. That's 87 times 60 plus 87 times 3. I could put parentheses here to make it a little bit clearer. Well, fair enough. But then how do you calculate what this is? Well, now we can rewrite 87 as 80 plus 7. So let's rewrite that. So this is the same thing. Actually, let me write it this way. I can swap them around. So this is the same thing as 60 times 87. But I'll write that as 60 times 80 plus 7. We could do it like this. 80 plus 7 plus 3 times 80 plus 7, or 3 times 87. Let me just copy and paste that, so I don't have to keep switching colors. Plus 3 times 80 plus 7. So copy and then let me paste it. And then you have it just like that. So all I did, just to be clear-- all of what you see right over here, 87 times 60, well, that's the same thing as 60 times 87, which is the same thing as 60 times 80 plus 7. All that you see here, 87 times 3, that's the same thing as 3 times 87, which is the same thing as 3 times 80 plus 7. That's just that over here. But look, we can distribute again. We can distribute the 60 times 80 plus 7. So this is going to be 60-- I'm going to do that same color. Color changing is hard. This is 60 times 80 plus 60 times 7 plus 3 times 80 plus 3 times 7 right over here. So notice what we really did is we thought about what each of these digits represent. 8 represents 80. 7 represents 7. 6 represents 60, because it's in the tens place. The 8 was in the tens place, as well. This 3 is in the ones place, so it's just 3. And we just multiplied them all together. We multiplied the 80 times the 60. We multiplied the 80 times the 3. We multiplied the 7 times the 60 right over here. We multiplied the 7 times the 3. And then we add them all up together. And this will actually give us our product. So for example, this right over here, 6 times 8 is 48. But this isn't six 8's. This is 60 80's. So this is going to be 4,800. We've got two 0's right over here, so 48 followed by the two 0's. This right over here, 60 times 7, is 420. 6 times 7 is 42. But it's going to be 10 times as much, because this is a 60. And then 3 times 80-- well, same logic. 3 times 8 is 24. So this is going to be 240. And then, finally, 3 times 7 is 21. And then to get the product, we can add these two together. And you might be saying, hey, Sal, I know faster ways of doing this. But the whole reason I'm doing this is to show you that that fast way you knew how to do it, it's not some magical formula or some magical process you're doing. It just comes out of really the distributive property and, hopefully, a little bit of common sense. So what is this going to be equal to? Well, we could add them all up. 4,800 plus 420 plus 240 plus 21. Well, you're going to get a 1 here. Let's see. 20 plus 40 plus 20 is 80. Let's see. 800 plus 400 is 1,200 plus 200 more is 1,400. And so you get 5,481. It's equal to 5,481. And you might say, gee, this was a bit of a pain to have to do the distributive property over and over again. Is there a simpler way to maybe visualize this? And there is. You could actually write this as a grid. So we could say we're multiplying 87 times 63. We could write it like this. We could say it's 80 plus 7 times 60 plus 3. And then you can set up a grid like this. So let me set up a little box here. It's 2 digits by 2 digits. So it's going to be a 2 by 2 grid, 2 rows and 2 columns. And then you just have to calculate. Well, what's 60 times 80? Well, we already calculated that. That's 4,800. What is 60 times 7? Well, that's going to be 420. What is 3 times 80? We already calculated that. That is 240. And I want to do that same color-- 240. And finally, what is 3 times 7? 21. You add them all together. You get 5,481. And I encourage you to now just do this same multiplication problem, the same 87 times 63, the way that you might have traditionally learned it. And look at the different steps and why they are making sense and why, at the end of the day, you really are doing the same thing that we just did in this video. You're just doing it in a different way. And the whole point of this whole exercise, this whole video, is so you're not just blindly doing some type of steps to find the product of two numbers. But you can actually understand why those steps work and how those numbers relate to each other.