 A tape diagram is a really convenient way of organizing information about numerical and later on algebraic quantities. So what this corresponds to is our concrete representation of a number. So let's suppose we're working in base 10 for right now, and if I want to determine the number of objects in a collection working base 10, well, I have a bunch of things and what I'll do is I'll look for sets of 10, there's one, and I'll bundle them. And I might as well bundle the other one just for convenience. And I'll trade, again I don't really need to do the physical trade, but the important thing is this should not belong in the 1's place, it should actually be over in the 10's place. And I have this as my bundling trade. Now if I drop the place value chart and join the objects together, I still have a representation of the same numerical quantity. But now it looks a little bit more organized and this corresponds to what a tape diagram is. And so for example I might draw a tape diagram representing the number 37. And I could draw a set of 37 boxes and maybe it'll look something like this, but maybe I want it to be a little bit more organized. And I certainly don't want to have to draw 37 boxes, so maybe I'll just outline the units and the sets of units and I'll label these amounts. So I have 37, that's 3 10's, or 30. And then that last set of boxes, there's 7 1's there, that's going to be 7 boxes. And there's a tape diagram for 37. Now there's many other ways we could have drawn this if we were a little bit more sophisticated. We would draw maybe 30 and 7 and we could draw a tape diagram that way. One of the nice things is that these tape diagrams are very useful to giving us a visual representation of a lot of different arithmetic and algebraic operations. So for example I'll draw a tape diagram representing the addition of 34 plus 23, and then I'll use it to find the sum. So again it's helpful to remember the set definition of addition, which defines the sum as the cardinality of the union of two sets. And we're going to put two sets together, one of them with 34 elements. So again I can draw that as a set of 10, a 10, a 10, and a 4. So there's my 34 elements there, and another set with 23 elements. So here's my set with 23, that's 10, 10, and 3. And so I'm going to join those two sets together, I'll glue them together like that. And it's worth noting that I can rearrange the pieces. So maybe I'll rearrange them like that. And then at this point I can just figure out how much I have. I have 10, 20, 30, 40, 50, and 4, 3. I have 57 pieces altogether, and so my sum is going to be 57. Now the thing to realize is what we did there is counting really more than anything else. We didn't actually do any addition. Once we get comfortable with the actual process of addition, we can move to larger pieces for our tapes. So I might draw my tape representing 34 as a 30 tape and a 4 tape. And likewise that 23 tape is a 20 tape and a 3 tape. And again if I have some facility with addition, I really don't need to redraw the tape diagram. I just need to add the pieces that I see. And so here's a 30 and 20 piece, and here's a 4 and a 3 piece. So I can add those pieces together, the 30 and 20, that's 50, the 4 and 3, that's 7. And so all together I have 57 as my sum.