 One way we can use this idea of division being performed by repeated subtraction is to do what's called division by partial quotients. And again, it's worth keeping in mind what we are presumed to understand at this point. The assumption is we know how to multiply. We know how to subtract. We don't necessarily know any division facts. We don't necessarily know the standard algorithm, but we do know that division is a matter of finding how many times a divisor will add to the dividend, and so we can answer it by finding how many times we can subtract the divisor. So one thing we can do to improve our process is to subtract large multiples. But unfortunately, identifying these multiples can be difficult. But one of the things we can do is we don't have to find the right multiple. We don't have to find the perfect multiple, but we can use what's called chunking to avoid a multiple that works. And this is the basis of our division by partial quotients. So for example, let's take the problem 3010 divided by 86. And again, part of the thing to keep in mind here is we're introducing this problem long before we introduce anything called the standard algorithm for division. And so how do you do this? Well, what is this? Well, I want to subtract 86. I want to figure out how many times I can subtract 86 from 3010. And so what I might not want to do is subtract 86 over and over and over again. But here's some easy multiples I can find. 86 times 10 is 860, and 86 times 100 is 1,600. Well, I obviously can't subtract this much. This is way too big. But I can certainly subtract 860. And so what I'm going to do is I'm going to keep track of how many 86s I've subtracted. So I have my initial dividend, and I'm going to subtract 860. That's 1086s, and I have this left. And well, how about that? I can subtract another 860, and that's another 10. Well, third time's a charm. I can subtract 861 more time, and I am left with 430. Now, at this point, I can't subtract 860, but that's OK. I can start by subtracting 86. Now, here's where it's useful to do things like remember how we can find subtractions. 86, well, I might say this is 100, subtract 100, and then return 14. Because from 86 to 100 is 14, so I'll subtract 100, add back 14. My next subtraction, I can still subtract 86, I'll subtract 100, and add back 14. And I can do those subtractions relatively quickly. And my quotient is just going to be how many 86s I've subtracted altogether. And that's going to be recorded in this column. So what is that? 10, 20, 30, 31, 32, 33, 34, 35, 86 is all together. And there's my quotient. Well, let's do a three-digit number. Again, we don't know yet at this point how to apply the standard algorithm. This is a four-digit dividend divided by a three-digit divisor. I don't know how to apply the standard algorithm. I haven't yet learned that at this point, but that's OK because I know that division is a repeated subtraction. So I'm just going to subtract 173, or better yet, multiples of 173. Now one thing I might start with here, I could make my process a little bit more efficient. I could construct a multiplication table. So I know 10 times 173. Well, that's easy. And maybe I know how to multiply. Well, if I double to get 20 173s, well, that's this many, well, I can definitely subtract that. Well, what if I double it again? Let's see. So again, doubling again, that's 40 of these. That's 69 20. Well, I could definitely subtract this many. And if I go up to 80, I get this, and that's too much. I can't subtract this many. So I can either ignore that particular entry, and the only values I need are going to be these entries here. So let's see. Well, I can definitely subtract 6920. So let's do that. Again, that's 40 173s. And now I have 865 leftover. Now something that's useful, once we've actually formed this multiplication table, let's go ahead and take advantage of it. These were 10 times, 20 times, and 40 times. Well, if I drop the zeros, I don't have the zeros there. That's one times, two times, and four times. And so I have all those same values. So I can either have the zeros and have the numbers there. Or if I don't have the zeros, I have the numbers there. More pragmatically, I might actually have started with this table and noted that if this is one times, then 10 times is 1730, 100 times 70,000, and so on. But let's see. Well, I definitely have enough to subtract 692, so let's do that. Oh, and good. I have enough to subtract 173 one more time. And altogether, I have subtracted 40, 44, 45, 173s. And so there's my quotient.