 Well, let's see if we can develop an approach to finding the products that doesn't rely on having to do repeated addition. And so again, we'll start by using a concrete representation of our process. So let's consider the problem find 4 of 235 base 6. So again, here we're working in base 6, and what that means is that 6 of something will be the next larger unit. And so I can read this problem as saying I'm going to take 4 235 base 6's and I'm going to add them all together. Well, let's go ahead and set down our place value chart. Again, I don't actually need to draw out what each of the units looks like, but it's just a convenience. And I have my number 235 base 6, that's 2 of these, 3 of these, 5 of these. So let's go ahead and make a concrete representation of our number. So there's 2 of these, 3 of these, 5 of these, and there's our number 235 base 6. Now I want 4 of those. I'm taking 4 of these, so I'm just going to set down 3 more. So there we go. Here's our 1, 2, 3, 4 conveniently color coded, although it doesn't really make a difference. All right, so I'm going to run them all together. Here's my sum, addition is a compilation of objects. And again, I don't need those colors anymore, they were just there to keep track of things. And because I'm working in base 6, I need to bundle and trade, and I'm going to look for groups of 6. So here they are, group of 6, group of 6, group of 6, a couple of things left over. So I'll bundle and trade them. And trading, these are not where they should be. I'm going to move them over to here, and that puts them there. And there's my first bundle and trade. And I'm going to keep doing this. So again, bundle again, so here's a group of 6, here's a group of 6. I'm going to package them up for shipment, and I'm going to ship them out. And again, bundle, and package them up for shipment, and ship them out. And I can't do anything else at this point, so I can record the product by using our abstract symbols. And so what do I have here? Well, I have one of these, four of these, three of these, and two of these. So my product, 1, 4, 3, 2, base 6. There's my product in base 6. Now that's a lot of drawing out, so let's see if we can move on to a more abstract representation of the problem. So here's a different one, find 3 times 2, 5, 4, this time working base 8. So again, this is just a repeated addition. 3 of 2, 5, 4, base 8 is 2, 5, 4, base 8, plus 2, 5, 4, base 8, plus 2, 5, 4, base 8. And I can just do this as a repeated addition, and I'll set up my place value table. So I'm adding these, these, and these. Now again, arithmetic is mostly bookkeeping. And so what we're going to do is we're just going to keep track of how many of these we have, how many of these we have, how many of these we have, and we'll bundle and trade as necessary. And the important thing is that we want to stay within each of our places, because as soon as we move to another place, we can only move to another place through some amount of bundling and trading. We can stay within the columns, not a problem. Two of these, and two of these, and two of these. Well that's six of whatever these things are. Five of these, five of these, five of these. I'm going to abuse notation slightly and write down what we would record this as. 5, 5, and 5 is 15. Even though we're working base 8, this doesn't mean anything. This number 1, 5 doesn't mean anything in base 8. It's something that we understand what it means, but we'll have to do something with it at the end of the problem. And then 4, 4, and 4, that's going to be 12. And again, 12 doesn't mean anything in base 8. So we'll have to do something with it a little bit later on. And well, now is a good time since we've done all of our additions. So remember the last thing we have to do is we have to bundle, trade, and combine as necessary. And so since we're working base 8, then I want to look for sets of 8. So here, I have 12 things all together. And so I know I can find a bundle of 8, and then I have a 4 left over. So that 12 is now gone. I have a bundle of 8 and 4 left over. And I can trade this bundle of 8 for one thing in the next column over. And I'll combine those two. And now I have 16. Well, that's actually two bundles of 8. So I'll bundle and trade. So again, each of these becomes one thing in the next place over. And now I have a bundle of 8. And actually, I have this bundle of 8 very nicely right here. And so that's going to become one thing in the next place over. And my last step is I'm going to record the fact that I don't have anything of these, any of these, but I do have four of these. And one of whatever this largest unit is going to be. And so my final product is going to be 1004 base 8. Now there's a slight shortcut we can make that'll become an important step in developing our standard algorithm for multiplication. And no surprise, it's based on our definition of multiplication. So again, we did this as a repeated addition. And let's look at our repeated addition. What we had to do is we had to add 4, and 4, and 4, and get 12, 5, and 5, and 5, and got 15, 2, and 2, and 2, and got 6. Well, wait a minute. That's just a repeated addition. And what's our notation for repeated addition? Well repeated addition is a multiplication. So I added 3, 4 is together. That's 3 times 4. I added 3, 5 is together. And I added 3, 6 is together. And so what this suggests is that if I want to do this product, 3 times 2, 5, 4 base 8, how I can start is I can take each digit of my number and multiply it by 3. So here I get 3 times 4. I get 3 times 5. I get 3 times 2. And those are going to be the values, not necessarily the digits, because we have to do the bundling and trading, but those will be the values in each of our places. Hey, let's try that out. 7 times 2, 5, 7 base 8. So again, as a repeated addition, what I'd do is I'd add together 7, 2, 5, 7 base 8. And so in this first column, I'd get 7, 7s. Well, if I know something about multiplication, I can figure out what that is. In the next column, I get 7, 5s. And again, I can at least write that down, even if I don't know what that is. And then finally, I have 7, 2s in that third column. And again, if I know something about multiplication, I can figure out what those are, 7 times 7, and so on. And again, these numbers don't really mean anything because I'm working base 8. I do know I have 49 of whatever this is, 35 of whatever this is, 14 of whatever this is, but as a base 8 number, I'm not really done. However, it's worth noting what I have at this point is what you would call correct but not complete. It tells me how much I have. I have 49 ones, 35, 8s, and 1464s. So this is a actual value base 8. It's just not written in the correct manner. It is correct, but it is not complete. You have a few more things to do, and we can do that by bundling and trading. So this 49 has a whole bunch of 8s in it. In fact, it's got, I can take 48. That's a group of, that's a group of a bunch of 8s, and a 1 left over. And I can bundle and trade for 6 more in the next place. I'll combine those. This 41 has a bunch of 8s in it. There's my bundle, which I trade and combine. This 19 has a couple of 8s in it. There's my bundle and trade, and my number 2, 3, 1, 1, base 8, and that's my final answer.