 So welcome, you guys, to your grades three to five, continuing the conversation with Kathy here. It's a continuation from your face-to-face session, and we hope that you are able to interact and enjoy the webinar atmosphere if you're new to this. I think I know Barhead, you were here before, so welcome back again. And then I'm going to turn this over to Harry who's going to say hello, introduce Kathy, and away we go. So over to you, Harry. Thanks very much, Juan. I'd like to welcome everyone to the webinar this afternoon, and to thank Kathy for offering the webinar and that opportunity to talk with her. We are a relatively small group, and so I'm going to encourage people to use Kathy and use her expertise because this is a wonderful opportunity to ask questions and to probe a little more deeply into some of the things that we've been working on in mathematics, so please feel free to do that. And Kelly, I know that you mentioned that you have to leave at five, so we'll just go along as long as it makes sense, and as long as people still have questions of Kathy. And so welcome to all of you here. I want to just end this introduction with a little quote that I have sitting up on my wall written by Kathy, and I use it to inspire the work that we're doing at the RLC, and it says, our work is driven by the desire to transform classrooms into communities of mathematicians, places where children, example, are interesting problems, and like mathematicians, engage in crafting solutions, justifications, and proofs of their own making. So that's something that we use at the RLC to inspire our work and our planning, and I know that today is going to be a great session and a great opportunity to talk with Kathy, because that's the sentiment that she wrote. So, Kathy, over to you. Oh, hey, everyone. This is going to be a lot of fun. We've got such a nice small group here. Very cozy. So hopefully we can get into this nice and deeply, and we have a good time together tonight. I did one yesterday. I don't know if you heard. I did a webinar yesterday with the folks that were in the K2 session, and I started that one with this slide, too. So, Keith, I know you and the bar head group saw it yesterday, but Kelly and Leslie haven't seen it, and so let's just take a minute. If Kelly and Leslie, if you want to look at it, and, yeah, tell me what you're seeing. Can you start us off, turn your mic on, and tell us what you're seeing in this slide? Is your mic working? I just had a great answer and reply to that, and the mic wasn't on, so I guess now we need to remember it all over again. We were just saying we actually downloaded the PowerPoint from the link in the e-mail, and we were quite struck by this picture, and I said that I want to get a copy of that. Where did you find it? We love it. Do you see the Sudoku in it? Do you know Sudoku puzzles? Yeah. Yeah. Yeah. Yeah. Okay, so it's actually a painting of mine. I think when I was there doing the face to face with you all, I mentioned that my side hobby is painting, I do oral painting, and so this is a Sudoku that I was sort of playing around with the idea of, of course, in a Sudoku, the numbers need to be one through nine, needs to be a cross, right? Every digit goes across one through nine, and every digit also goes one through nine down, and then in the square, every square that's a three by three, one through nine also has to appear. So I had this idea that where the number needed to be mathematically is also where it should be aesthetically, and that if I painted a really finished piece of mathematics, the finished Sudoku puzzle, that it should work aesthetically, because every three, you know, it can appear once and only once in certain places in the painting, and if it is all the same color every time it appears, it should be balanced, so that's what you're looking at, and, you know, maybe a way to sort of see patterns all around us, even in art, but anyway, we'll go to the next slide, because, of course, tonight we're not going to focus on Sudoku's, we're going to focus on mini lessons for multiplication and division, and I know that, yes, yes. I just posted a little question there on the side. I didn't see it. I wonder if there's any truth here. Oh, I see it now, keep you. Thanks. I was so busy looking at my slides that I missed the chat room there. I did not do the pictures in the context for learning math investigation books and posters. I will tell you, however, that when I first started writing, I thought I might at least do the big read-alouds, and I did two illustrations for measuring for the art show, took me longer than it took me to write the unit. I thought, I'll never finish this curriculum if I try to do all the illustrations, too, so no, I didn't. We are sending Heinemann and everyone were gracious enough to get good illustrators, so I didn't. Okay, Susan, I see that you've joined us, and can you hear okay? Yeah. Yeah, looks like you just sent Wanda, no, and you can hear okay. Great. Okay. So, I'll go on to the next slide. This is a warm-up mental math string that I wrote for you. What I'd like you to do to start us off is take a look at the numbers, expressions that I've put in this string. Of course, if we were actually doing this string in a classroom together, I would be putting up one problem at a time, but I've given you my whole string to analyze. What I'd like you to think about is what strategy does this string support, and what big ideas underlie the strategy. So, take a minute and look at it and be thinking about those two questions, and give me a green check when you feel like you're ready to discuss it. Hi, Pat. I see that you've joined us. Great. What we're doing, Pat, if you can see the slide that's up there, is we're discussing the two questions we're thinking about. We haven't started the discussion yet, but we're thinking about the two questions on the right, and I've asked people to give me a green check when they're ready to discuss it. Good. I see that you've jumped right in, Pat. Great. So, Kelly, how are you doing, Kelly, and Leslie? Still thinking? We're just discussing the links between each, so, for example, the first three are obviously linked, and you figure out the first two and add them together. Then, with the 99, you can think of 100, and then just take away 13 and get it easily. Great. So, you've started the conversation, and that's exactly where I want to go with this. So, thanks. So, you've noticed that the first three are linked. Did everybody notice that? Can you just give me a, you know, a yellow face if you did notice that the first three were linked? Great. Good. Okay. So, the first one is the helper, and I'm just going to notate that so that we're noticing that as we go through these strings. The first one, I'm assuming everybody knows that it's easy, and I'm assuming actually that the second one is also a helper. So, those two I expected, without any problem, people to be able to do very quickly. The third one is the challenge, but if people notice that the first two can be used to solve the third one, then actually that third one becomes a rather easy problem, too. And so, the string is designed to support the development of the noticing of this relationship, and then the next one is also a challenge. But the first problem right here, let me grab my, oops, I lost my little hand there. There we go. Okay, let me just grab that for a minute. The first problem right here can be very helpful for this problem. So, the helper once again can come back in, and then when we go to this problem, we could really use this problem to help with this problem. But now we are faced with yet another challenge, and the challenge is do I add 99, do I add 14, do I add 13, do I take away a group, what do I do here? So, I've flipped it on purpose commutative, I used the commutative property to change the 13 times 99, I could have written that, to 14 times 99, and I moved the 99 to the second place, and moved the 13, which became a 14 to the first place. And that's what's causing people to have to think about what am I going to add here. And then when we go to this problem. Now, this one is a real challenge, because this one, I need to look, so I'm going to move this over here, and go back to this. This one right here actually is a real challenge where there's no helper anywhere in the string that can help with that problem. And so, I end the string and the writing of a string with a problem that now you've got to make your own helper. So, I've put these two questions up, what strategy does the string support? So, as you think, landscape, can anyone name that strategy? And is there a big idea that underlies it, the strategy that's being supported? If you want to stick a green check up when you're ready to respond, if you can name the strategy, and if you know the big idea that underlies it. All right, Keith, great. I see that you're still alone in that room. You want to turn your mic on and tell us what you're thinking, and let us know whether there's anyone else in the room with you there, too. I'm all alone. Just the distributor property over addition. Okay. Distribute, thanks, Keith, thanks for turning my mic off. We don't get that backlash, great. The distributor property for multiplication over addition, yes. And we see that as the big idea in part of the string. Anyone see another form of the distributor property, too? Susan, how about you? You want to respond? Can you turn your mic on, Susan, and tell us what you're thinking? Your mic's still not on, Susan. You want to click in the bottom left on that square. It says control plus F2, with the icon of the mic is. Oh, okay. So, Susan, you have no mic. Okay. So, let's see, Kelly or Pat, how about either of you, or Susan, if you want to tell in a chat room, do you see another form of the distributor property anywhere? Keith said the distributor property for multiplication over addition. And if you look in the first, second, and third problems right here, first, second, and third, the first two can be added together to do the third. So, the multiplication is being distributed over addition. Well, what's happening here? Is that over addition? Or here? I guess is everybody probably would have done 200 times 34. Wouldn't you take a group away? And wouldn't you take a group away here? Right. Distributive for multiplication over subtraction. Yeah. So, the string is designed with two big ideas in mind. Distributive property for multiplication over addition and over subtraction. And if we want to name the strategy, we could say one uses the distributor property. But we could also say partial products. So, I'm just going to jot that down for us right here. Partial products, because that's really the strategy that we're using. And then the big idea of the distributor property. Okay. So, let's go on and let's go down to third grade, because I know I got an email from someone, I don't remember who sent it to me. But someone sent me an email through, I think it came to Harry and Harry forwarded it on to me that you wanted to spend some time in third grade. So, I'm going to do this whole first portion of the webinar on some third grade mini-lessons. And I'd like you to take a look at two pictures. We might use one one day the fruit, and we might use another one another day the patios. And what I'd like you to do is analyze each of these as a separate mini-lesson. And they're considered strings too, because we might do just the apples first, and then the lemons. So, there's a, we can think about is there going to be a relationship. And with the patios, we might do this first, let me just, this first one right here. And as the first mini-lesson, and then a second, mini-less, sorry, a second problem. And then a third problem, and then a fourth problem. So, we have four problems that actually would be given on the same day with the same picture. So, it still is a string of related problems. So, what I'd like you to do is think about now what strategies would you expect children to use. And let's save the second question for a minute, the representation question. Let's first just talk about what strategies would you expect to see. And we'll talk about the fruit first. So, if you would just give a green check when you feel like you have some ideas of what children in third grade class might say. Kelly and Leslie are still thinking? Let me just put a red check up if you're puzzled or still thinking. See, people are writing up on the PowerPoint already. Terrific. So, kids might say, let me go back so I see your notes there. You're noting that some kids would say double the apples. So, in one tray there's six, and so one might look at this as two sixes and double, yes, six plus six. So, they might skip count, six, 12, they might do repeated addition, six plus six. Anything else they might say? They could skip count. Was that you, Kelly? It wrote skip count. You want to turn your mic on, Kelly and Leslie? No, we were thinking about the doubling, but I'll gladly take credit for the skip counting. Okay. Okay. Take credit for it and tell us how they might skip count, because there's a variety of ways here. So, tell us how kids might skip count, Kelly and Leslie. Once you put your mic back on and tell us how they might skip count. Sorry, I keep forgetting about the mic. Well, with the apples, it's set up very nicely to skip count by twos, whereas the lemons are set up nicely for your skip counting by threes. Okay. But there might be some other ways to skip count, we're at the apples, too. One could skip count, for example. Oh, by threes, three, that's the other way, that's right. Exactly, yes. So, you could skip count by twos, you could skip count by threes, you could skip count by sixes, right? I mean, there's only two trays of sixes. One might see the, kids might see the sixes in columns, three and three, or they might see the six rows of two. So, let's just stick with these lemons, I'm sorry, with the apples now for a minute, and imagine that kids might count, all right, by ones. They might skip count in all the ways we just talked about. They might do grouping and then do repeated addition. So, given they can skip count by twos, they might group by twos, and do two plus two plus two plus two plus two plus two, or they might do three plus three plus three plus three, or they might do six plus six. But now what they might do, if they're going to group, imagine that kids group by say threes, all right, three plus three plus three plus three. There's an opportunity here for a teacher to say, oh, you see four threes. But another, Wanda, would you mind, could you type for me when I just write this, when I say this? It will save me having to type at the same time as describing off what I want to talk about. So, because if one looks at this right here and sees six and then uses it for this as well, then we have an opportunity to talk about that as two times six, right? Could you write that for me, Wanda, two times six? Thanks. But if one sees it this way, what if another kid says the way I solved it is I see it as three plus three, plus three, plus three, and so now we have an opportunity to talk about this as four threes. So, we could write it with an equal sign right here next to the two times six. We could write equals four times three. Let's put an equal sign in between the two expressions that we have thus far. Yup. And we can also put it over here. This also equals two times six again, as well as it equaling four times three. Because we really have grouped, we really have grouped the two threes together to make a six. So, there's not only grouping and repeated addition and skip counting that can occur, but there's regrouping that we now have an opportunity to bring up to discuss. And this, of course, then introduces yet another strategy. This strategy of two times six equals four times three. Can anyone name that? It is regrouping, yes. The big idea is that one big idea is that groups can be regrouped. But can anyone name the regrouping that occurred here? Yeah, double in having. Look, the two became a four and the six became a three. And so, even in third grade, just with pictures, some of the strategies that are going to get used as you'll see tonight when we go up into fourth and fifth grade working with much larger numbers for multiplication can even be introduced in a very informal way with nice discussion, just simply honoring what children see and what they notice in terms of how they decide to organize in these pictures. There's also the potential here for a big idea to come up. Let's go back again to the fruit. When kids say, I see four groups of three, the four is being used to name the number of groups, the three is being used to name the objects that are in the group. Anyone remember off the landscape what that big idea is? Can anyone name it, type in the big idea, unitizing sound familiar? Yeah, the number of groups, number of objects, the four is being used to name the number of groups, to count the number of groups, but the same numerical terms. In other words, numbers are being used to count objects in the group. So the objects in the group are now becoming a group of three that's being named and being counted as a one. So the group itself is becoming one. And I think when I was with you doing the face to face, we named that as, oops, sorry, unitizing sound familiar? Making the group a unit that can be counted, a very big idea on the landscape for, you know, kids in second and third grade, that idea that numbers can be used in two ways, not just counting objects, but to count groups. All right. So the question, how would you represent them, could be done just as I did here too with circles, you can circle whatever kids say, which then gives you an opportunity with the children you're working with to start discussing the relationships that are down here that we've written about on the bottom down here, all of these, the doubling and having and all of these expressions can be written by the teacher and named. But it's really coming out of the discussion up here that we've been talking about in terms of children thinking. All right. Let's go to the next slide. So that's basically what we've just been saying. But now let's take a look at the lemons. Kids might see two lines there. But I've written next to it on the PowerPoint right here. Another way that kids might think about it, because they might see it as a three by three, and say I have two three by three, or they might say I have two nines, or some kids might say I have three, six, nine, 12, 15, 18, so skip counting comes back in. But if you look across the rows, you also have six rows of three. So if kids have skip counted by threes, you have the opportunity now to really talk about how this is all related. Two nines becomes three groups of three twice, and we also can see six rows of three. Now I'm not saying that you bring this up for discussion. You want it to come from the children. But the underlying big idea then is the associative property. Because take a look at how really what's going on up here is that one can look at this as either two nines, and the nine came from the parentheses around the three by three, if kids see it that way. But one could also put the parentheses around the two by three. And then, of course, depending on how you're seeing it, you might see that you've got six threes. So again, this becomes a big underlying idea that potentially can come up for children just depending on how they start organizing and what they see. So are there any questions on that, thoughts you're having, comments you'd like to make? If you want to click on the blue hand in the square box, then I know that you'd like to comment. All right, then we'll go on. I don't see any clicks there, but feel free along the way as I wanted to say it. You can click on that blue hand with the green arrow, and then I know that you want to comment. Okay? So let's go to the next slide. All right, so here's the patio one. And it's designed a bit differently. I told you in the slide right here that I've built in a constraint on purpose. So this is the tool the teacher has really too. You can build constraints into your pictures, and by doing so, it may support the development of another strategy. So I told you it's the distributive property, but I'd like to hear some responses from you. What constraints do you see that I've built in, and how are these four patios in any way going to support the distributive property? If you could think about that question and then let me know with the blue hand when you'd like to respond. Well, we think that for lots of kids, they would go with the first chunks that they can see. So for example, in the bottom left-hand corner, they would figure out the four groups of five first, and then move on from there. All right, let me ask you a question, Kelly and Leslie. Of course, kids could do that. You're saying that they might see something like this, because the beach umbrella is covering the rest. If one were to do this picture right here, first, without the top left, the top right, you're probably right. But notice that the top left is the first thing. Yeah, that's much better. Wanda, thanks. Can you just, if you'll move that for me. That one is the first problem that we give in the mini-lessons. So that's the helper. The second problem with the little pool in the flippers, that's the second problem. Note the one above that one, yeah, that one with the pool in the flippers. There, that's the second problem. The one with the beach umbrella is the third problem given, and then the one with the chaise lounge is the fourth problem given. And given that we show one at a time, we get discussion, and then we show the second. I mean, the image is shown all at once. But we're only, we might cover up the other three and only show the upper left, the helper, and then we might cover the bottom two and only show the second problem. So given that, what relationships might children see? We can go back to you Kelly and Leslie if you want to respond, or Susan is giving us a little note. She doesn't have a mic. She's saying they would just remove a row from the helper to the second one. Yes, exactly. So this now is four fives where the first problem was five fives. I'm calling it a helper because some kids may just know that five fives is 25. Other kids, again, this is third grade, may have no other strategy here except to count by ones. So some kids may be at that level on the landscape. Other kids may notice this five in a row and they skip count by fives. But no matter how they do it, they get 25 for this. Maybe you could write for me, Blanda, up on the top, five times five equals 25. So the helper, 25 has come up. So when they get to the second problem now and they notice it's not five fives, it's only four fives, can that be used to help, can the helper problem be used to help with the second problem? And I asked if you noticed a constraint. Any thoughts on why I might have drawn the second patio with a pool on it in slippers? Yeah, bar head, tell us why you think the pool in the slippers might be there. So that they cannot do the ones one counting? Exactly, exactly. In other words, we're pulling the rug right from, out from underneath the counters. So the early third graders that solved the helper problem by counting by ones can't count by ones with the second problem. Well, of course, kids are pretty cagey, aren't they? And they do find ways to do that. But we've tried to make it a little bit harder where counting by ones now isn't as easy to do as it was in the helper. And so they may use the helper problem and take away a group of five. And so by providing this constraint, covering some of the tiles with the pictures, we're stopping kids from counting and skip counting and encouraging them to use the relationships between the problems and the relationships of the number of rows and columns. So what do you think they might do now? I'm going to go back to Leslie and Kelly on this one because I've notated with the highlighter on the third problem what you initially said. So now that you see that the beach umbrella is a constraint, it's stopping kids from counting. And now that you understand that the helper and the second problem would have been given first, any thoughts on what kids might do with the third problem, Kelly and Leslie? Well, obviously the kids are going to go into the idea of adding another group of five on. Exactly, yeah, exactly. So the distributive property comes up. And so they might do five times five again, but the helper's up there. They don't have to count because the answer is left up there. As a teacher, once you've discussed it, no matter how kids got it, you leave the 25 up there and you've written five times five, even if they only counted by ones or skip counted. But when you get to the second and the third problem, now hopefully some kids begin to make use of that and you've got the distributive property coming up. Same with the whole thing, yeah. Just wondering, you never mentioned the distributive property in problem number two, but you would address that with kids too if they came up with or recognizing that there's five lefts and where did that five come from? It happens to be one row or five. That's actually. I don't remember who it was that said it now, but whoever offered us a solution about kids or strategy of what they thought kids might say. Was it you, Pat? I forget. You did say that it was one group of five left and so they are using the distributive property for multiplication over subtraction. If your question, Keith, is would I label it the distributive property at this point? Probably not. This is beginning third grade. I'm just trying to get the grouping. I'm trying to move them think landscape. Early on some kids are counting by ones. They're skip counting, but I'm trying to get them trying to put a constraint in to urge them to not count by ones and to start making groups. Yes, it's partial products, but I'm not going to label it as such here in a formal way. I just want kids. I'd make a big thing out of, oh, so if it's, it's got four rows of five and you solved that with the helper problem, you took one group away. So you used five groups of five and you did four groups of five by taking five away and I'd not take that. I might even write four times five equals five times five minus five. So I would not take whatever it is that the children say. Does that answer your question, Keith? Is that what you were really asking? Would I introduce the, yeah, okay, good. All right, so on this, on this last one, Kelly and Leslie, what do you think they would do on the fourth problem? We're going to keep adding groups of five. Well, they could. That might be a kind of low level strategy. Can you find them? I'm going to notate it. Let's see if we can find the helper problem in here. All right, so there's five by five, okay. So right up and across is the helper problem. So if kids can, after the discussion on the second problem and the third problem, kids might begin now to start trying to find groups within the group and use a group they know. And they know five by five is 25 and it's written up there. What's left? Five by four, right? Any, anything that we've discussed so far in this mini lesson that kids might use? Great, Pat, tell us what you're thinking. Can you turn your mic on, Pat, and tell us what you're thinking? She's written in the, okay, Pat's typing instead. She says they also know four times five from the second problem. Absolutely, so the commutative property could even come up here. It's four fives, now we've got five fours, maybe the array can actually be turned around and one can begin to see how four times five can become five times four. And now the two partial products we already know can be used to solve this last problem. And so partial products is again up. This is not something you tell children. You try to have them notice it and you support them to start. You can even say, you know, fun things like, wonder if there's any patio up there we've discussed so far that could be helpful with this, because, boy, that shape lounges in the way. It's really hard to count, isn't it? Anybody have an idea of something we've discussed so far that we might be able to use? So you're thinking like that, you're putting ideas forth like that for the children to consider, but we want it to be their ideas. All right, let's go to the next slide. Anyone have a comment or a question they want to bring up about the patios? All right then. So here's the landscape, and I've clicked on the portion of it. I've clicked the bottom part of it because for entering third graders, when you first begin to introduce multiplication, you may not even be using the X yet. You may just be showing the pictures and asking kids what they see and asking them how many lemons there are, how many apples there are. And you may, you're just getting them to talk about their strategies. So here's the bottom of the landscape. If you could use your highlighter and pick any color you want and highlight some of the things that are on the landscape that are strategies or big ideas that might have come up in these mini lessons. Great, repeated additions can be regrouped, unitizing, yeah. Also look at the very bottom. Think about where some of your entering third graders might be. Again, kids are going to be all over this landscape. Yup, you might see some kids counting by ones. Absolutely. How about using partial products? Do you see if they're above doubling and halving? Dealing out, whoever's using the green, you've highlighted dealing out or counting all. Can you just put a check next to your name so I know who's using green? Okay, Susan. The picture's given, so there's no dealing out that has to be given, that has to be done. So the dealing out refers more to division rather than multiplication. For example, if you gave a third grade or a problem, I've got 12 cookies and three kids. How many cookies is each kid going to get? Children will often take unifix cubes or multilinks and they will literally count out 12 cubes and then they will deal them out. One for each kid and then they will keep dealing until they've dealt all 12 out and then they've made their groups and then they will count the number of groups left. I'm sorry, they'll count the number in the group to know how many, they'll count the groups and they'll count the objects. So that's really what's meant by this one right over here, that's more for division. I think what you're referring to is that some kids may actually count by ones in, for example, with the lemons or with the fruit and with that first patio, but we're building constraints and to support kids to move away from counting by ones. And then they may skip count. We see a pathway here, it's a developmental pathway. The counting by ones often comes first, skip counting later. Repeated addition and then they begin to notice that repeated additions can be regrouped, we talked about that, we talked about doubling and doubling and halving, so this is a developmental pathway. Partial products might happen with the patio or also with the lemons. We saw two groups of nines. We're just talking about this sheet and I love this landscape sheet, and when I'm working with kids, I've had a system where we take anecdotal notes up until now, but we love that this gives us a structure for saying where the kids are on the landscape and looking at creating groups to work together. But a quick question that I have for you, with the proportional reasoning, Susan had mentioned earlier with the patio pictures of using two, two, five groups of five and then minus of five. Would that be considered proportional reasoning? Because she's working with the two groups of five, the two, two groups of twenty-five? I know what you mean. No, it's not. Proportional reasoning we're actually going to see later in a clip, but I can remind you also, I don't know if you remember, but in the face-to-face, we looked at that clip. We know all the work that we did with the turkey problems in the face-to-face institute. And there were two little girls that we looked at who were figuring out the cost of the turkey. So, for example, if a pound of turkey is $1.25, and if a kid puts and is trying to figure out what ten pounds would be, if they do, well, $1.25, $1.25, I know that two is going to be 250, then if two pounds is 250, then four pounds would be $5. I'm scaling up. And that's proportional reasoning. So it can start with repeated addition. And you even may see it in your third graders. And it even, you even may see it, I don't think we've seen it in those pictures right there. I think what you're referring to is more the use of partial products that you talked about this one right here. You talked about figuring out four fives by using five fives and taking away a five. And the big idea related to that is the distributive property for multiplication. So, again, it's the distributive property, but it was over subtraction. The distributive property for multiplication, the multiplication is being distributed over subtraction. Do you want to respond more on that, more questions about that? Did that make any change? No, that clarifies that for me. Thank you. I was thinking of it in terms of, she had two, Susan had written two, five groups of five, so it was two, 25. Uh-huh. And, but I see more clearly now. Thank you for clarifying that. I think it's primarily because it's two, kids probably solve it with just the repeated addition, 25 plus 25. Keith, do you want to join in? Well, I just, I made a comment there about the poster with Antonio, where that seems to lead to the discussion about times as many. Absolutely. Absolutely. Yep. Yep. That's a nice noticing, Keith. The unit grocery stamps and measuring strips is designed. It's a, it's an early third grade unit to introduce multiplication. And it's designed to really early on, there are a lot of pictures like the ones we've been looking at in the beginning. But then it moves to the stamps, which promotes unitizing. Because kids begin to talk about, I have three fives, and what they really mean is I have three five-cent stamps. But they start calling it three fives, and there's the unitizing. And then in that unit, you next come to the investigation with Antonio. And the tree might be the question is, how tall is the tree? We know that Antonio is four feet tall. And so if the tree is three Antonio's, then we're scaling up the four three times. So if we think about proportional reasoning, kind of a scaling, which really gets to the heart of multiplication. But it's now multiplicative, it's not additive, it's no longer repeated addition. It's really a scaling up, I need three of them, it's three times in it. That's proportional reasoning. And so the pathway begins with coning and skip coning and repeated addition and moves to doubling, and moves to groups can be regrouped, and then doubling and having and the use of partial products, and then up into proportional reasoning. You mentioned, Kelly and Leslie, that you like using the landscape as an assessment tool. That's a great use of it. You know, it's actually a very sophisticated use of it. So the fact that you're even trying to do that, I applaud you. Really fabulous that you're trying to do that. Most of our teachers don't start off like that. They, in New York, they start off just even themselves trying to understand it and notice the ideas in children's work, and themselves trying to get an idea of what these are, what these labels are. Of course, the graphic is very complex, but it really represents, I think, a much truer picture of the complexity of the network of relations children are developing. I would love to put things on this nice line and just say one skill comes first and then the next and then the next. And that's the way that we were taught to think learning occurred, but we now know in research it's far more complex than that. And it depends on the problems that children encounter and the conversations that happen in classrooms, and it's really a whole network of relationships that are occurring as sort of neural pathways of synapses. I like thinking about it. So I don't know any other way to graphically represent it except with the complexity of these interwoven pathways, which to me kind of represents the brain. We like that it's kind of messy, just like learning. Yeah, yeah, yeah, great. Glad, I'm glad you like that because sometimes people are so afraid of it and say, oh my gosh, it's so complex that it doesn't help me. But of course, learning is complex, great. Okay, let's go into a third grade classroom. And this is, I think I may have shown this clip. Can you run it for us, Wanda? We'll take a look at a teacher using one of these pictures. It's a short clip. I think I may have shown some of it in the face-to-face screenshot. I can't remember, but it's a short clip anyway. It's only a couple of minutes. So we'll take a look at it together. I think now that we've been looking at these pictures, it's kind of nice to revisit this again. Absolutely, and I'll just jump in for a quick second. What you guys will see is a minute. You'll see a little link up here in the chat area. And if you will, please click on that link. It'll take you right to the video. You will have to click play in the middle of the video. And then if you don't mind, if you guys would, when you're finished watching the video, you could close that down. Don't close on your illuminate site, but close on the video. And click on the green check mark when you come back, because everyone will watch at a slightly different rate, and then we'll know when you'll finish the video. And if you have any problems at all, please click the red check mark. So we only have to click, Wanda, on the text. We don't, on the link. We don't have to copy, paste it, and put it in Google or anything. Exactly. That link should take you directly there. Then just click play. Great. Great. That works too, Wanda. Thank you so much. That was easy viewing. All right. So, oops. Sorry. So, what are your thoughts on it? What strategies did you see, any big ideas? We'll go back to that landscape. All right. I'm going to go back one. There's the landscape again. In that mini lesson, what's going on? What strategies did you hear? Any underlying big ideas? Yeah. Patch has community property. The kids talking about five times four equals four times five. They're congruent. Isn't that lovely? The kind of language children are using. It's come up earlier in geometry in that class. And they're really seeing the array can be turned. I'm unitizing. Yeah. Keith, I'm turning the mic on. Tell us where you saw the unitizing. Well, just when they were saying we have four groups of five. There you go. Yep. Great. All right. Did you see skip counting? Anyone see skip counting? Early on, Kea. Early on, kids were skip counting, right? So, you see that the full landscape here, it's not just like one idea, it's really the landscape that we see in front of us. But the conversation is really facilitated. And the picture itself is designed to support movement along the pathway. Do anyone see the distributive property? Do you see any partial products in the distributive property? Yeah. Kelly and Leslie, go ahead. A little bit at the end when they were going with the two groups of 20. Exactly. Great. And that's also why the picture is designed the way it is. The picture is broken up with the five fours and another five fours. So, when the question is asked, what about when it's all together, we now have 10 fours, then it can be solved with five fours plus five fours. So, again, this is third grade work all done with pictures. And I'm wondering if you would just put a green check up. If any of you did try many lessons like these with, yeah, Keith is saying all with doubling, right? That's doubling to Keith, correct? Did anyone try many lessons like these with pictures? Oh, terrific. Good. A lot of you. Did anybody go to magazines and try to find any pictures in magazines beyond the ones provided in my materials? Why don't you put your hand up, applauding if you did, if you find anything in magazines. Here's one I found. It's a lovely place to go, you know? You don't have to be dependent just on my materials. You know, I started this with the Sudoku because you start noticing patterns all around when you really start mathematicizing your reality. And this is just a potato advertisement that I found in a magazine. But I immediately saw the basket and just noticed the way that the colors are laid out. And, you know, I started thinking to myself, gosh, one might see this as a group of six. There's a group of six, right? And if you look at sixes going down, of course, this row right here, the third row, has one extra potato. But if one sees it, then the whole picture can be solved as four sixes plus one extra potato. Or one could look at it this way. We've got four threes and another four threes and one potato. So lots of ways to group. So I encourage you after this session tonight, start looking in magazines. A friend of mine uses a camera and takes tons of pictures in the environment all around her. And encourages kids even to take pictures and bring their pictures in and talk about how they're mathematicizing the things in the groups. Think of window panes on windows. And you can walk around the environment and take, you know, snapshots of big buildings. And kids can figure out window panes. And it's all about introducing multiplication. All right. So let's go on now and go up into grades four and five. So here's a string for you. Analyze it. It's a grade four string. Think about the focus of the string. And I'm going to ask you to vote A, B, C, or D. What do you think this string is about? It was crafted to support the development of A, B, C, or D. And I guess we could use E if you want none of the above. Or if you check E, I'll know you're thinking none of the above or you might use E for all of the above or whatever. So feel free to use an E, too, if you want to. You can tell us why you used it. Susan, you're still thinking? You want to put a, yeah, okay. Good. Last one made you think. Good. Still thinking, Keith? You know, you stepped away for a minute, but it looks like you're back. Oh, very nice, Susan. Interesting. Tell us why you did E. What are you thinking? I don't think it's A, so maybe E. Okay. A random guess, huh? Doubling and then having, all right, but not both, just anyone, but a lot of other people checked A. So let's get some discussion going here. Could you use the blue hand and let us know if you want to discuss? And then I'll just ask some people to do that or shall I just pick people? Pat, you were up first with the A, I think, really quickly. You want to tell us why you picked A? Pat says, does it have to be both? You want to turn your mic on, Pat, and tell us what you're thinking? Don't forget lower corner control F2 of the mic. You have to click on that. Oh, Pat's micless, that's right. Okay, how about Kelly and Leslie? You want to tell us what you're thinking? And Pat, you feel free to jump in and write if you want to. We see doubling and having in all of them, but we're stuck on the last one. So remember in that very first string, I'm going to go back to the beginning, here's the first string I started off with, and remember how we analyzed it. There we are, remember? So we analyzed it as helper-helper in the beginning, then there was a challenge, a challenge, when we got to the end of the string, there were no helpers, kids had to make their own helper, right? So remember that, now let's go to the one we're talking about now. So there is no helper for 18 times 50, but if we look at the relationships in the string, 6 times 8, 12 times 8 is just a double, no doubling and having yet. So 6 times 8 to helper, 12 times 8 only requires doubling. I'm assuming that if kids know 6-8, they can use that to double to make 12-8, because it's only going to require doubling. And now let's look at 12-4, third problem. That's another helper. It's only going to require having, right? If you look at the second problem, 12 times 8, 12 times 4 is half of it, but isn't that interesting? 12 times 4 is going to have the same answer as 6 times 8. So the doubling and having is going to come up. We don't expect children to use the strategy. If they already know the strategy, they don't need the string. So I've designed the string for kids who aren't using doubling and having, don't know it, and I'm trying to develop kids' ability to start thinking about it. I want to ensure that I get it up for conversation. So problem one and problem three, even if kids have only had to get problem three, or even if some kids are still skip counting by fours to try to figure it out, the answer they get is going to be the same as the answer to the first one. And that gives us an interesting place to have a conversation. Which then, when we get to 24 times 2, they might use it. So they might use the 24 double, the 12 double, the 4 halves, and that one is doubling and having. And now let's take a look at 3 times 16. Now this is kind of interesting. Which problem, you see the relationship of 3 times 16, and the other problem in the string that they might, yeah, exactly, 6 times 8 is doubled, the 8 has doubled, and the 6 has halved. And so they might use doubling and having again. What's kind of interesting here is now take a look, even though they might use doubling and having again, take a look at the relationship between the third problem, 12 times 4, and 3 times 16. All right, these two. So that's not doubling and having, but the answer is going to be the same, right? So actually the 12 has been quartered. And the 4 has been quadrupled. So one might think, oh, there might be something a little more interesting happening here besides just doubling and having. There may, in fact, be a suggestion that there may be the generalization here of it may move to the associative property. Just because now there's a lot of relationships that may, in fact, where the answer may be the same, take a look at 24 times 2 and 3 times 16. The 24th, actually, we divided by 8, and the 2 has been times 8. Not so much proportional reasoning, no, because it's not scaling up. It's really the factors being associated. We could write this as 3 times 8 times 2. And if you think about the association of what's being associated, if I use parentheses around the 3 times 8, I get 24 times 2. But if instead I put the parentheses around, oops, I didn't mean to erase my text box there. If I do the parentheses around 3 times 8 times 2. Can you type that for me? Wanda, I'm losing the text tool for some reason or other. There we go. Yeah, 3 times 8 times 2. Then I can instead put the parentheses this way, and the problem becomes 3 times 16. So doubling and having is the strategy. But there is the potential here for the associative property to come up for discussion. And really what's going on is that a factor is being associated differently. And now the last problem has no helper. Anybody have, you've got to make your own helper here. So anyone have a nice way to solve it? Given the conversation that might have come up on doubling and having and possibly the associative property. Does anyone have a nice strategy? Oh, beautiful. Look at that. Isn't that nice? Who's using green? Can you just put a check so I know who's using green? Yay, applause to you, Keith. Super, right? Isn't that a cool way to do it? Just double and have. 9 times 100, you got the answer in your head. Anyone have another way? I have one. Here's one. I like Keith's way better. But we could do 6 times 150. Right? I divided the 18 by 3, and I times the 50 by 3. Oh, nice, huh? Yeah, look at that. Times 10 divided by 10. Beautiful. So you see, this strategy can become really powerful. Really powerful. Okay. So now we move to this in the last part of the time we have together. We're going to shift to really looking at representation because representing on an array is tricky. So to think about how one, would you take just a minute, try to draw a picture? Let's suppose, for example, that a kid had 6 times 8, and when he did 12 times 4, he doubled in half. Take a sheet of paper. See if you can. I'm not going to have you do it on the whiteboard because I'm going to show you a video clip of a teacher doing it. But if you could just do 6 times 8 and then see where you'd make the cuts, where would you draw the line? How would you turn it into a 12 times 4 if a kid said, I doubled in half? And if this is easy for you, put up a yellow smiling face. If this is hard for you, put up the blue face. And then Kelly said she had to leave. Leslie, are you still there? Or was it Leslie that had to leave at 7, 5 your time? Kelly and Leslie, anyone still there? Yep, so sorry, but we have to leave now. Yes, I knew you did. Thanks so much for joining us. Nice to have you. And have fun with many lessons, OK? Go ahead, do some nice work. Can you load up the next video clip for us, Wanda? We're going to go into a classroom. We'll see a teacher notating, doubling and having. All right, so once again, Wanda's put it in the chat box for us. Click on it. Don't forget to close out and give us the green check to let us know you're done. Great. OK, so we saw the doubling and having. We saw the teacher represent really the child, right? So, OK, so here's another string for us to take a look at. It's the focus of the string if you just want to vote on that one. All right, we're going to have a nice discussion here. We've got two D's, two B's and Harry's still thinking. So, let's, let's, so, Pat, why don't you tell us what you're thinking? Oh, Pat doesn't have a mic, right? You can type in, well, we know what you've done because you voted for B, Pat. Let's hear from Susan, do you have a mic? No, Keith, you have a mic, right? And you voted for D. Oh, Pat wants to change your mind. So, OK. So, Keith, let's hear from you. You voted for D. Let's see if, and Pat, you can respond if this is why you decided to change your mind when you hear what Keith has to say. Well, I noticed that A is involved because 170 plus 34 equals 204. So, we're just, we have two parts that we're going to divide by 17, but together they would make, it would be the same as 170 divided by 17 and 30 divided by 17. So, that's involved, but when I'm, when I'm originally thinking about the problem, I'm also thinking about breaking a division problem down into parts as well to do this. So, I'm trying to find friendly numbers to use. So, if you agree with Keith now and you can want to change your, and if you disagree with Keith still, then keep what you voted for. All right. And I agree too. So, I think it's D. I think that both are involved. I think you really described it nicely. Have we convinced you, Harry? Yeah. OK. All right. Let's go to the next one. If a child said, for the last problem, 323 divided by 17. If the child said, I got 19 because I, can you paste Wanda for me down on the bottom right? Would you be able to paste that string from the prior slide? If a child said, I got 19 because I took 17 away from 340. Take a sheet of paper. Try to represent that using an open array. Again, don't do it on the white board. Just see if you can do it on a piece of paper. Yeah. There's the string. So, it's this bottom one down here. If a kid said, I used, I used really 20 times and I took one group away. So, I know it's 19. So, I used 3, if a kid said, I used 340 and then I took 19, I took 17 away. How would you represent that on an open array? So, here's my picture. If it matches your picture, would you put a smiling face up if you were struggling with drawing the array, put up a blue one so I know. Okay, Susan. Was struggling but I can see your picture makes sense. Good. I'm going to also go to this one, a video clip that we'll look at next. And we're going to see a grade, this is a teacher working with a grade 4, 5 combination class. And she's doing the problem that we're just talking about. I'm just going to put up the video clip for us to look at. The teacher represented slightly differently and then I want to have a conversation on how she represents it. All right. So, once again, look at it, close out and give us the green check. Wanda, that's the wrong clip. Yeah. That's the wrong clip. Can people hear us while they're looking at the video? Not your fault. It may have been numbered wrong. I know you had to go back into the CD wrong, a different way to pull these clips off. But that's not the one I need people to look at. So, here's what I'd like everybody to do. The picture that, can everyone hear me? Could you just, if you're listening and not looking at the clip now, great. Okay. Keith, what do, yeah, good. Okay. So, when you go home tonight or tomorrow, you know, when you have a chance, you have some time, the CD wrong that I gave you in the face-to-face session. Actually, if you load it up, notice that up here on the top where this arrow is right here, it says division mini lessons. I'm in the folder and down here is what you're going to click on for your table of contents. I'm in the folder marked division mini lessons. And it is this third video clip right here, this third one, that is, if you run your mouse over it, what you'll see, it says 330. And you'll see the teacher representing in a little bit of a different way. I prefer the representation that you saw me use here on this slide. But I don't think hers is wrong either. So, you might just want to take a look at what she's done. All right. Let's go. We're nearly running out of any, out of our time here. So, I'm just curious whether any of you have tried a mini lesson like this and if you've had any struggles with representing. And I want to open it up for conversation. We've only got four minutes left. So, if you have general questions about representing, looks like everyone's been doing many lessons great and not struggling with it. Wow, bravo to you guys. That is really super. I'll tell you, the representing is hard. Yeah, it does take getting used to, Pat. And I'm terrific. Really applaud you guys. All right. There's one last, we're going to come to an end here. There's one last string to take a look at. And just vote what you're thinking this one's about. Well, we really have some disagreements going here. Notice I voted to be. And I see that someone added doubling and halving too. Let's analyze the string together. It starts with, here's the helper, a hundred divided by four. Can you just type in the word helper there for us? Thank you so much, Wanda. Two hundred divided by four. All that's happened there, it's still a helper. Because all that's happened there is that the two hundred is a double of the hundred. So we've doubled. Yep. Now what's happened? We've doubled the divisor. So if you can type in double the divisor. So we're going to end up getting half the answer that we had the first and the second problem, right? But we're going to get the same answer that we got for a hundred divided by four. So a hundred divided by four is 25, but two hundred divided by eight is also 25. And four hundred divided by 16 is also 25. Eight hundred divided by 32 is also 25. Three hundred divided by 12 is also 25. This one, of course, down here, if you can just write the word challenge down here. Because there's no helper for this one. The kids are going to have to make a helper. But if you take a look at what's been going on, they're all equal to 25. Three hundred divided by 12 is a messy problem, wouldn't it? Now that you know that it's going to be equal to a hundred divided by four, wouldn't you want to simplify? Three hundred divided by three is a hundred and twelve divided by three is four. And we're back to the first problem. So this problem and this problem are the same. So really what we've done is we've brought up the idea of simplifying. We tend to use that word with fractions. One might even think how this string really can support the development of reducing or simplifying with fractions as well. So anyone now have a really nice strategy for the last problem if you want to type it in. Nine hundred divided by 18 is a messy problem. But if you simplify it, it might be nice and easy. Yeah, it is proportional reasoning now, Pat. Right. Yep. And a hundred divided by two, isn't that nice? You could divide it by nine. Nine hundred by nine is a hundred. Eighteen by nine is two. And you've got the whole problem done. So I'm going to leave you now with how you can even think about these division strings that one might do. So I'm going to start by helping your fifth graders begin to have a good strong understanding of fractions as well. Here's a clip you can look at again in your leisure at home. It's on your CD-ROM. It's in the folder again marked division many lessons. And I'm going to just end with a look at the upper part of the landscape now. We've looked at how an array can be used as a tool. We've looked at how the open array can be used to represent strategies. We've looked at the associative property for multiplication. We've looked at the commutative property for multiplication. The use of ten times, tripling and thirding, how it generalized to the associative property, etc. We're hoping that by the end of all this string work, year after year after year, kids will have a really strong understanding of number and operation. Any last questions before we close out? All right. Well, I have loved working with you guys. Have fun with many lessons. Yeah. Oh, thank you so much for special day in June. Yes, there's two people from Edmonton coming. Lorelay is coming. And Annie Swinkels is coming. So it would be nice to have Edmonton people there. And I'm glad they're obsessed with these number strings. Yeah, Pat, you know what, applause to all you guys. Just really terrific. Keep the string work going. All right. Talk to you later. Thanks very much, Jackie. As always, we appreciate your involvement and your willingness to communicate with us in this way. And I appreciate your time today, as Pat says. The time has flown by. So thanks very much. My pleasure, Harry. I've enjoyed working with all of you. We will look forward to following up with you in the near future about conversations about what we might do together next year. I know that you'll be doing some work likely with the Southern Alberta Professional Development Consortium. And we'll see how we might be back with that. That would be great, Harry. Actually, I think that Mark has all of the dates and everything down now. So you could check with her. We can piggyback and it would save a lot, too, if that works.