 Alright, well let's take a look at all of our methods of subtraction so far. And again, it's important to keep in mind that what we want here is to build adaptive expertise to be able to use the most efficient algorithm for any particular task. And that's going to change. You do not butter a slice of bread with a chainsaw, and you don't cut down a tree with a butter knife. So we want to be able to switch between the tools that we have available. And we can look at a single problem and try to see if we can find the solution in any number of ways. So let's take the subtraction 317 minus 85, and let's try to be, yeah, we could do this 50 different ways, but let's make it a little bit easier. Let's look at it in five different ways. So let's think about that. Counting up, that's our basic method of subtraction. It's predicated on nothing more than the ability to count. You don't have to know any real arithmetic facts, you just know how to count. So I'm going to try to count up from 85 to 317. And so I'm going to count up 5 to 90, 10 to 100, 200 to 300, and 17 up to 317. And so if I only know how to count and how to add, well I can figure out what the subtraction is. I had to go up by, let's see, 200, 210, 227, 232. So the difference there is going to be 232. Well let's also, we can also do counting down. So this is going to count down from 317 down to 85. So now I start at 317 and I go backwards a bit. I want to stop at 85 so maybe I'll start by going back 200, that takes me down to 117. Maybe I'll go back 17 to 100. I'll go back 10 to 95, back to 85. And again, how far have I gone back? That's 200, 217, 227, 232. And so my difference is going to be again 232. Now we don't have to count down this way. We can actually count down by a different method. Note that this is still the method of counting down, but maybe I go back 17 to 300. That's easy. 15, back to 285, and then down 200 to 85. So here I am counting down still, but I'm going through a different sequence of counts. I still get a total backwards distance 200, 15, 232. I'm counting past. So again, the idea here is we might go too far and then work our way backwards. And so what I might notice here is that I'm subtracting 85. So, well, 100 is a convenient benchmark number that's a little bit more than 85. So I will subtract too much and then I'll return what I shouldn't have subtracted. So I'm going to start at 317. I'll take away 100. That's too much. I've got to return 15. That takes us back to 32. How about equal add-ins? So equal add-ins and counting past are very similar methods based on the same sort of idea that 85 is close to a nice benchmark number that's easy to work with. So if I add 15, I get 100. I have to add to both, get 332. And if I do the subtraction, my difference is going to be 232. All right, well, how about decomposition? 317 minus 85, that's 317 minus 80, then minus 5. So I'll decompose the subterhend, I'll subtract 80, then I'll subtract 5, and there's again my remainder. And then we have the standard algorithm. 317 minus 85 is the only tool you have as a hammer that the only thing you can do to a problem is to bash it to pieces. So we'll apply the standard algorithm. Let's see, that's 317 minus 85, and let's see, I can subtract 5 from 7, that's not a problem. I can't subtract 8 from 1, so I have to trade one of these for 10 more in the next place so that 3 is going to become a 2 and an 11, 8 from 11 is 3, and then 2 minus, well, there's nothing there, that's going to be a 2 there. One quick observation to keep in mind, again, this is all about adaptive expertise and switching back and forth between different problems. The standard algorithm requires that we rewrite the problem. We're given the problem 317 minus 85. The thing we have to do with the standard algorithm is we have to rewrite it vertically before we can even perform that particular operation. And one of the things that this means is that we have to spend a little bit of time writing the problem down and making sure we actually put it down correctly, and then, after we've done that, then we can actually perform the computation. And again, adaptive expertise may be given that the problem is expressed this way, there may be an easier thing to do than to transcribe the problem so we can apply the standard algorithm and then perform the standard algorithm on it. My inclination for a problem like this, if I were confronted with a problem like this, I would probably either do some variation of decomposition 317 minus 80 minus 5, or possibly I might do something like counting past 317 minus 100 is 217 plus 15 is 232. So depending on how I happen to feel I might do this in a couple of different ways, I would probably not use the standard algorithm on this unless I had to because it involves taking some extra steps.