 Well, let's consider another way of adding, and this relies on something that's called adds to 10, and this comes as follows. Basic arithmetic facts can be derived, can be classified into two sets. We have the things that we have to memorize, and we have the facts that we derive. Derived facts are usually retained better and used much more easily than memorized facts. One of the nice things about mathematics is you can derive almost everything. There's very little you have to memorize, and arguably there's very little you should memorize. The focus of mathematics should not be on memorized facts, it should always be on derived facts. How do I get there from here? So if I take a look at addition facts, I can classify them into three groups. They're what are called the under ten facts. These are numbers where I add and I get a number less than ten. The ten facts where I add two numbers and I get ten, and the over ten facts where I add two numbers and I get a number larger than ten. And for decades, math education in the United States has been focused on memorizing all of these addition facts. So you drill A plus B, 5 plus 7, 4 plus 2, 5 plus 9, and so on and so forth, and you memorize all of these sums. Well if we look at how other countries do it, notably those countries whose children routinely outperform US children on international mathematics tests, the most common way of approaching addition is to use these first two strategies to derive this third strategy. We do not, in other countries, see memorization of 9 plus 8 equals 17. What we actually see is an approach to addition that uses the under ten facts and the ten facts to figure out what 9 plus 8 is. And this is a strategy that's known as adds to 10. So let's take a look at that. Find 9 plus 7 using adds to 10. Well why not just remember 9 plus 10 equals 16? Well first of all you have to memorize it, but secondly the memorization doesn't introduce a key concept which is understanding the connection between what you're doing and how you're getting there. So here we're looking for add ends that add to 10. And so what we might do here is we might say, well I have 9, I'm adding 7. And the question that I want to think about is 9 plus what gives you 10? I find that 9 plus 1 gives you 10. And so what I'm going to do is I'm going to split my 7 into 1 and whatever's left over. It turns out to be 6. So 9 plus 7, well 7 is a 1 and a 6. And the thing that I can rely on is I know 9 plus 1 is 10 and 6 and 10 plus 6, well that's going to be 16. And so there's my classic adds to 10 approach. Well again you might ask the question, well why don't I just memorize 9 plus 7 equals 16? Well the reason is that the adds to 10 and the under 10 facts, we can do multi term additions through what's called composition and decomposition. And essentially what we're going to do is we're going to break numbers apart and put them together in ways that they will either add to 10 or I'll have some left over numbers that I can then add pretty easily. For example let's consider a sum like this 8 plus 3 plus 6 plus 9 plus 5 and try to add this in your head mentally. And if you're like most people you're going to add 8 plus 3 9 and that's 11 plus 6 plus 9 and you're going to add it left to right and you're going to figure out what the value is. But here's a slightly different way of approaching it. The first add end is 8 so I might ask well what's going to add to 10 with 8? And well 8 plus 2 is 10. So where can I get a 2 from? Well I can get a 2 from this 3. I can split this 3 into a 2 and a 1. So there's my 10 and the other thing I might notice here is this 1 combines with this 9 to make another 10 so I've taken care of the 1 and then I have these last two terms. The next add end is 6 which I need a 4 to make 10. I'm already using this 9 here so I'm going to split the 5 into a 4 and a 1. And so what do I have? I can make 10s with what I have. 8 and 2, there's a 10, 9 and 1, there's another 10, 6 and 4, there's another 10 and then there's this 1 left over which gives me my total 31. Now the natural question that often emerges is don't have to write all of that and there's a quick answer. Well yes if you're asked to show all of your work but in practice when you're actually adding the process goes something like this and we might do this. If I were to add this I might do something like this. 8 plus 3, I know that's 11 plus 9 is 20, 6 and 5 is 11 so I have 20 plus 11 is going to be 31 and if you take apart what I just described what you see is I went through this process 8 and 3, that's 10 and 11, 20, 6 and 5, 10 and 1 and that gives me my answer 31. This decomposition adding to 10 and composition of numbers really is a valuable way of doing complex mental arithmetic and to see an example of why that works well let's consider a more complicated problem and the real value shows up when we look at multi-digit sums. So again let's consider the sum 38 plus 73 plus 46 and again try to do this in your head using whatever method that you want and you'll probably find it's kind of difficult to do that. But if I break these numbers apart into what they are, well 38 is 30 and 8, 73 is 70 and 3, 46 is 40 and 6 and what I can do is I can add the 10s fairly easily. That's 140 and then I can add the other things together that's 10, 11, 17 and then I can add the numbers together 140 plus 17, 157 and again mentally what this goes how this works is something like this 38, 73, 46, 30 plus 70 plus 40 is 140, 8 plus 3 plus 6 is 17 that gets me 140 plus 17, 157 and with very little effort it becomes easy to add multi-digit sums here mentally without ever having to write anything down.