 All right, let's summarize all of our ways, or at least some of our ways, of doing division, and let's take a single division problem and see how many ways we can perform the problem. So here's 50 ways of dividing. So let's take the problem 495 divided by 15, and this time we'll show five different methods. So we know many different ways of solving this problem, so let's be a little bit creative here. And one thing we might start off with 15 is a number that we can factor. We can decompose it into two different numbers. So the divisor 15 is 5 times 3. So that means if I want to do the division 495 divided by 15, I could divide by 5 and then divide by 3. So 495 divided by 5, that's 99 divided by 3 is 33. I could also decompose the dividend. So this is 495, and I'm going to break this apart into a bunch of different pieces. And keeping this in mind, because I'm dividing it by 15, what I want to do is I want to break it into pieces that can be divided by 15. So, well, you could think about clock time. What are the 15 minute increments on the clock? 30, 45, 60, which is one hour. So one thing I might notice here is 495. Well, there's a 150 I can break off really easily. That's obviously divisible by 15 plus leftovers. I could break off another 150 and have leftovers again, and one more 150 and then I have 45 leftover. So now when I want to divide by 15, I'll divide each of these pieces by 15, 150 by 15, three times, and then my last one, and that's 10, 10, 10, and 3. And so my quotient is going to be 33. How about repeated subtraction? Remember, that's all the division really is. And so I'm going to take 495 and I'm going to subtract 15 as many times as I can manage to do so. And I might take a block of 10, 15s, 495 minus 150, and I can take another block. This time maybe I'll go for 20, 15s. I have 45 left. And now I can subtract out 3, 15s, and now I have zero. And altogether I subtracted 10, 20, 30, and 3, 33, 15s all together. How about an area model? Again, remember that any method of multiplication can double as a method of division. So let's go ahead and represent this division using an area model. And so that means I'm going to take an area. It's going to be 495. I'm going to make one side equal to 15 and then figure out what the other side has to be. So let's take a chunk. How about a chunk of size 10 that has area 150? Not quite enough. I'll take out another chunk of size 10, also area 150. So far what do we have? 150, 300. Still got area to account for. So I'll take out another chunk of size 10. That's 450. That leaves me 45 left over. And if this area is 45, this side is 15. How do you figure out what this side is? Well, you can do the division 45 by 15 if you remember your timetable. Or, again, we can view 15 and decompose the divisor into 5 and 3. So 45 divided by 5 is 9. 9 divided by 3 is 3. And there's our other length. And so my length of this side, 10, 20, 30, and 3 as my quotient. Again, still an area model. Well, depending on how much we know and how much we want to work with, we can make this a little bit more efficient. So let's take a look at a different area model. Same approach, just using a different version of the same model. So again, I have an area of 495. One side's going to be 15. And maybe I'll start out by taking a big chunk, 30 by 15. That has area 450. And then I have some amount left over, 45 again. And so, again, the question is, if I have an area 45 left over, one side is 15. What's the other side going to be? And I can figure that out. It's got to be 3 once again. And my quotient is the length of the other side, 30 and 3. Okay, so let's take a look at Long Division. Now, it's possible you may have gotten the impression from earlier videos in this sequence that I might not like the Long Division algorithm. Well, I have to say that if you believe that you're actually correct, the standard Long Division algorithm is actually a really badly conceived algorithm that exists for one purpose, which is to save paper. And if paper and ink were so expensive that saving paper was a worthy thing to do, then maybe the standard Long Division algorithm is worth remembering. But paper is cheap, ink is cheap. And so the rationale for the standard Long Division algorithm doesn't really exist. So let's go ahead and see how that might work in a more effective manner. So we'll go ahead and set up our quotient 495 divided by 15. And again, the standard Long Division algorithm assumes that I can make perfect guesses without hesitation every time. So I'll make a perfect guess. 15 into 495 goes around 30 times. And now 30 times 15 gets me 450. I'll subtract that off. And again, I can make a perfect guess without hesitation every time 15 into 45 goes 3 times. I'll write that up there. 3 times 15, 45 subtract and remainder is 0. And my quotient 30 and 3. And again, the standard Long Division algorithm that saves that really expensive paper and horrifically costly ink shifts the parcel quotients over so that they're lined up with the dividend. Again, no rational reason why this has to be. And, well, writing those zeros is so expensive and time-consuming that it's worth dropping them out. And then we collapse the partial quotient into a single line to save a whole one line of space. And there's your standard Long Division algorithm versus an alternative form, which is relying on partial quotients.