 Alright, so let's take a look at some more algebraic problems, and again one of the most important goals in early mathematics education is the preparation for and the learning of algebraic thinking. Again, we don't learn basic arithmetic because basic arithmetic is something we have to do in everyday life. We learn basic arithmetic because it paves the way for doing algebra. And the other important thing to keep in mind is what this means is that it is an understanding of the concepts that is important. Digit pushing, not so important. If you understand why you're pushing the digits around, that's what is going to be useful. And another thing to keep in mind here is remember that we are going to be using a tape diagram. The idea here is we're not actually going to be solving our algebra problems in the future using tape diagrams, but we're using this as scaffolding to get to the generalized notion of algebra. And so, again, just as I can represent a concrete amount using a picture that looks something like that, or I can represent it using an abstract symbol like this, I can represent an algebraic expression concretely using a tape diagram, get a little bit later on, or represented abstractly as a set of symbols. But it is the understanding of what you're doing when you manipulate the tape diagrams that informs your understanding of how you do algebra. So let's take a look at that. So here's a situation. So Jeff, I start with 25 marbles, and I lose some of my marbles, and I have 18 left. And so I want to do is represent the situation using a tape diagram and solve the problem. So in a tape diagram, remember that the length of a tape corresponds to the magnitude of a quantity. And the new thing that we're going to introduce here is that an equation corresponds to having two tapes of equal length. So here it seems I have two quantities, the initial amount of marbles, 25 marbles that I started with, there's my original amount. And in the description of the problem, some of these are lost and there are 18 left. So what I might take into consideration here is I've lost some amount. So those are gone, but there's now 18 left. And what I now have is I have two tapes of equal length. So I have the original amount, 25, and then I have the original amount, except this portion is no longer there. I have a tape of length 25, I have a tape of length 18, and whatever was lost. And I went to find the amount that's lost, and I can do this using several different methods. A very primitive method is I might just count up from 18 to 25. And so that might look something like this, 18 plus 2 to 20 plus 5 to 25. So the number of marbles in this lost area must be 2 plus 5, must be 7. And again, the thing that's worth noting here is that I have solved an algebraic problem. And the only preliminaries I really need to be able to do this is I have to know how to count. And what that means is I can introduce this type of algebraic thinking very, very, very early in the curriculum. Again, research shows that, properly introduced, you can start to introduce algebra problems in the first or second grade. Well, let's take a look at some more complicated problems. So Anna and Bernard have some coins. If Anna has five, more than twice the number of coins Bernard has. And Anna and Bernard have 26 coins altogether. How many coins does each one have? Really standard algebra problems that some of us were probably introduced to in middle school, seventh, eighth grade, something like that. Again, third grade, second grade possibly with the correct scaffolding. So let's consider this as a tape diagram. So each of these tapes is going to represent a quantity. And it seems that we have two quantities. Well, we're looking for the number of coins each one of these people have. So it seems that our two quantities, the number of coins Anna has, the number of coins Bernard has. So I also know a relationship between the two quantities. Anna has five, more than twice the number of coins Bernard has. And that's going to be critical for forming our equation ultimately. But let's go ahead and draw this. So I have the first tape that represents how many coins Anna has, the second tape that represents how many coins Bernard has. The problem is that that relationship that I've described, Anna has five more than twice the number of coins Bernard has, doesn't seem to be reflected in this. As it's worded, it seems that Anna should have more coins than Bernard. And as I've drawn it, Bernard has more. Now it's not strictly speaking absolutely necessary to get those magnitudes correct in our drawing. This drawing here really is a way of keeping track of what we're doing. But it's nice to at least have some relationship of our drawing to what we're trying to describe. And so maybe I want to switch that around so that Anna does seem to have more coins than Bernard does. Actually, let's go ahead and take into account this particular description. Anna has five more than twice the number of coins has. So I should redraw Anna's tape to reflect this. So here's the amount of coins that Bernard has. So Anna has twice as many and five more. So maybe I'll draw Anna's tape looking something like this. So here Anna has twice what Bernard has and then five more. And the important thing here is this is not an equation. I do not have two tapes of equal length. And so I don't have any sort of equation yet. What I have is a representation of what Anna has and what Bernard has. Now what I do know is that Anna and Bernard have 26 coins all together. And what that suggests is if I put the two tapes together, so here's Anna's tape, here's Bernard's tape, if I put the two coins together then what I end up with is a tape that should have a length that is equal to a tape representing 26. So here is where my equation is. Now just in the interest of pointing something out, I don't actually need these two drawings initially here. I use them as part of the scaffolding for this problem, but really this is the thing that I need to draw what I have here, Anna's amount, Bernard's amount. And this is just footnotes that tell me how to get to here. But here's the actual equation. So let's try and figure this out. So what I would like to know is if I could figure out how big these blue pieces are, I can figure out how much each of the two have. So let's think about that. So I have two tapes of equal length here, this and this. And so what I might do is I might remove that section five from both of them and that leaves me, again, this was a tape of length 26. This must be 21. And while 21 is one, two, three of these blue pieces, so that says each of these pieces must be seven apiece. And I'll fill those in and that will give me the amounts that each of them have and I can just read it off my diagram at this point. And here's why having that little footnote at the beginning is actually helpful because now once I've filled in these pieces, then I know that Anna here has 7, 14, 19 and Bernard has seven. And I can just read that right off my tape diagram.