 Alright, so let's take a look at some methods of subtraction. Now it's important to keep in mind as we look at these methods that the goal is not to learn how to subtract, rather the goal is to be able to subtract regardless of what level of knowledge we currently have. And so in this case, the simplest level of knowledge we might have is we might, at this point, only know how to count. Does this mean we will be unable to do subtraction? Well, let's take a look at it. And suppose we only know how to count at this point. Well, then one thing we can do, we can still do subtraction based on an approach called counting up, and this is based on the relationship between addition and subtraction. We have to understand what subtraction means, but if we understand what it means, then we can use what knowledge we have about counting in general to allow us to subtract. So, again, here's a problem that we might consider, 31 minus 25. Now from our definition of subtraction, remember that 31 minus 25 is whatever it's equal to, if and only if 25 plus that number is equal to 31. And what this means is I can solve my subtraction problem by asking, what do I have to add to 25 to get to 31? Well, if I can count, I can figure that out. So I might just count by ones. So I'm at 25, and I want to count to 31. So that's, let's see, 25, 26, 27, 28, 29, 30, 31, and there I am. And so what can I conclude from this? Well, how far did I have to count? I had to count one, two, three, four, five, six numbers. 31 is the sixth number after 25, which tells me that 25 plus 6 is 31, and 31 minus 25 is going to be 6. Well, a little knowledge goes a long way. If I know a little bit more than how to count by ones, I can actually make this algorithm much more efficient. So, for example, I might use the adds to 10 approach. And remember that if I want to go from a 5 to the nearest 10, I need 5 more. In other words, the first step can be to add 5 to get to the next 10. I'm starting at 25. I want to get to 31. So my first step, I might say, well, if I count up by 5, that takes me to 30. And I need to get to 31, so I have to take one more step to get to 31. And altogether, I've gone 5 plus 1 steps. I've gone 6 numbers. And so once again, I get 31 minus 25 is equal to 6. Well, all we need to do is know how to count. And the nice thing about this is if you only know how to count, you can still do subtractions like 112 minus 78. And we'll do this using the arrow notation to record our work. And we'll actually do this three different ways to reflect different levels of knowledge of how much we might know of basic arithmetic. So let's use counting up. And so again, this question is how much do we have to go up from 78 to get to 112? And a lot of basic arithmetic is bookkeeping. And so what we're going to do is we're going to keep track of our work using our arrow notation. So how can we start this? Well, one possibility is we might start at 78 and just start counting. So 78, 79, 80, and well, here's something useful. I'm now at a benchmark number. I'm at 80. So I could continue to count up by once, but let's go ahead and count up by larger amounts. Since 80 is a 10, I can count up by 10. So 80 to 90 to 100 to 110. I want to get to 112, so I have to slow down a little bit here. So I'll slow down and take the last couple of doubts very easily, 111, 112. And altogether, I've gone 10, 20, 30, 31, 32, 33, 34. I've gone 34 numbers up from 78 to get to 112. So that tells me 112 minus 78 is equal to 34. Well, that's one method. And again, you don't really need to know very much at this point to apply this method. You need to know how to find the next number, the next number, and then how to add 10 and slow down a little bit towards the end. So here's something that you can do with a very limited level of knowledge of basic arithmetic facts. On the other hand, if you know a little bit more, there's several things that we can do. So for example, we might consider that it's always easy to add 10 to a number. So instead, I might just count up by 10s from 78. So I might start at 78 and 10 gets me to 88 and 10 gets me to 98 and 10 and 10 and 10. Oh, wait, that's too far. Wait, I only want to get to 112. So I went a little bit too far. So let's get rid of this last step. At this point, I do want to get to 112. I've got to slow down and count up by ones. So here's one more and again and again and again. And altogether, I've gone up by 10, 20, 30, 31, 32, 33, 34. And so once again, 112 minus 78 is equal to 34. And again, this is something we can do if we know very little beyond how to find the next number. Well, if we actually know some addition facts, we can much improve our counting up algorithm. So for example, I'm at 78, I want to get to 112. So the thing I might do is I might count up to the nearest 10 by adding 2. So 78 plus 2 takes me to 80. Now I want to get up to 112. So again, if I know my addition facts, I know that 80 plus 20 is 100. And that's useful because that's a benchmark number. And I want to get up to 112. So I can go 12 more to take me to 112. And altogether, I've gone 20 plus 2 plus 12, I've gone 34. And so there again is the third method I have of doing the subtraction, 112 minus 78. And again, at this point, all I am relying on is the fact that I know what the next number is and maybe some additional addition facts.