Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather.

Kim  0:08  
And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  0:16  
We invite you to consider that algorithms are amazing human achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems should be developing.

Kim  0:30  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:37  
We invite you to join us to make math more figure-out-able. 

Kim  0:41  
Hey, there.

Pam  0:42  
Kim, today we have gotten lots of good feedback that people are loving the podcast episodes where we unpack the facilitation moves for particular Problem Strings.

Kim  0:53  
Yeah.

Pam  0:54  
Rather than just what the numbers are and what they're doing, but actually how it would look like with students. So, today we are going to dive in and do another example of an equivalence structure Problem String.

Kim  1:05  
Mmhm. 

Pam  1:05  
But this time with multiplication. 

Kim  1:07  
Yeah.

Pam  1:07  
Take a deep breath. 

Kim  1:08  
Okay. 

Pam  1:09  
Here we go. Alright, Kim. What is 16 times 18? So, I'm in a class. Oh, I might have started with kids saying, "Hey, today, you're probably going to need paper and pencil. I'm going to give some problems that you're going to have to do some figuring. And remember, mental math, you don't have to keep it all in your head. It's totally legal to keep track of your mental thinking." So, this is not formal notes in their notebook. This is just scratch paper, so kids can keep track of their thinking. 

Kim  1:35  
Yep. 

Pam  1:35  
"Alright, you guys. First problem today. 16 times 18." Now, I'm going to write this problem on the board in the top left hand corner, and I'm going to give myself a lot of space.

Kim  1:45  
Mmhm. 

Pam  1:45  
So, I'm making sure that I've got the whiteboard is cleared. If I got a bunch of stuff markered up, or magneted up on the board, or whatever, I'm I've cleared out my whiteboard.

Kim  1:54  
Move it! 

Pam  1:54  
Move it all. Got a lot of space for this particular string. 

Kim  1:55  
Yeah.

Pam  1:55  
Alright, I'm going to wander around. I'm going to find a kid who uses an Over strategy. So, could (unclear). 

Kim  1:55  
You're going to find Kim. 

Pam  1:57  
I'm going to find at least someone Kim-like. And for today, it's going to be Kim. So, ya'll, pause podcast. Solve 16 times 18. You want to have some sense of the relationships before you kind of hear what we do after this. I probably should have said that before I said finding an Over strategy. Kim, will you just help us with an Over strategy. 

Kim  2:20  
Yeah,

Kim  2:21  
so 16 times 20 is 320.

Pam  2:25  
Okay, as you say that, I'm going to say, "Hmm, what would that look like with an array, you guys? A 16 by 20?" And maybe I'll have somebody say something. But I might just say enough of that to have them kind of look up at the board. "Hmm, what would a 16 by 20 look like? Let's see, if this was 16." I'm going to draw a length. Write 16. "What would 20 be?" Probably, kids are going to say "Longer," and I'll go, "Okay, but not too much longer, right?" So, like I'll kind of draw 16, and then a little bit longer. "So, you guys are saying a 16 by 20." And Kim, you're saying the 16 by 20 is something that is helpful? What's the area of that? 

Kim  2:58  
320. 

Pam  2:59  
Okay, so I'm going to write 320 right smack dab in the middle of that. Keep going.

Kim  3:05  
But I don't really want 16 by 20. I only want 18. 

Pam  3:11  
Okay.

Kim  3:11  
So, I'm going to cut off some of the 20.

Pam  3:11  
Alright, so I'm going to. And you're going to cut off 2, right? 

Kim  3:11  
Yeah. 

Pam  3:11  
So, I'm going to go over to the side. I'm going to cut off a little strip, a vertical strip. I'm going to put a 2 near the top to represent that dimension there. And I'm going to say, okay, so if you cut off that 16 by 2, what's that area?

Kim  3:29  
32.

Pam  3:30  
And I've drawn a little 32 in there. Now, I'm going to erase the 320 I wrote before because that's no longer accurate because it's sitting in the middle of just what's left over. And I'm going to write down what's left over. Oh, 18. So, now I've got an 18 and that kind of cut off piece now on the top. You said the whole area was 320. Below the array that I've now cut into two chunks, I've kind of drawn a carrot, and I've written 320. So, that symbolizes if I put those two pieces together, that we just cut up, that together those are a total of 320. So, the whole array is 320. You've got this little piece that's 32. And, Kim, you're saying that you could use those to help you figure out just the area of the 16 by 18. What is 320 minus 32?

Kim  4:15  
288.

Pam  4:16  
That's 288. And so, you're saying the area of a 16 by 18. Now, I've gone over to the problem. 16 by 18 is 288. I've written 288 in both places where I had the 16 by 18 in that cut up part, and it's over there on the problem. Cool. At that point, I might say, "There were some other nice strategies out there. But is everybody really clear that was the area? Everybody agree 16 by 18 is 288? Good." So, I let kids use other strategies. Maybe somebody did, Five is Half of Ten to get ten, five, and one 18s. That would have been a fine strategy. But I'm not using those at this point. I'm just making sure we've got that that relationship up there. Bam. Next problem. "How about if I were to ask you 9 times 32? Go ahead, everybody. Go ahead and solve it any way you want." I'm going to circulate. So, I'm going to walk around, and I'm going to see what kids are doing, and I'm watching for kids who might kind of look up like, "Wait..." So, podcast listeners solve the problem. And then I'm looking at you. I'm looking to see if anybody goes, "Hey..." because once you solve 9 times 32, do you also get 288? So, when I see a kid look up like Kim, who solved it, and they're like, "Hey, that's the same." I'm going to try to catch them and go, "Don't say anything." Like, I'm going to try to make it be, "Yeah, yeah, yeah. I know, I know. But don't..." Like, try not to make a big deal of it. So, I might wink at a kid. I might just kind of nod. Depending on the relationship I have with the kid, whatever will work to kind of keep that under wraps for a minute. Yeah. Does that make sense, Kim?

Kim  5:56  
Yeah. I think at a young grade.

Pam  6:00  
Mmhm. 

Kim  6:00  
I think they might be curious about there being a similar product. But they might be more like curious, not knowing.

Pam  6:11  
Yeah.

Pam  6:13  
Yes, I don't know that they know what they're looking for. They just might have noticed it was the same answer. 

Kim  6:18  
Mmhm.

Pam  6:19  
Okay, so then I might say, "What did you guys get?" And when kids say, "Yeah, we got 288," then I might go, "Huh, that's interesting. What would a 9 by 32 look like?"

Kim  6:31  
Right.

Pam  6:31  
"If we have the 16 by 20 up here on the board, what would a 9... Somebody help me draw 9. Can you use something up here? Because we're going to draw it in relationship." So, Kim, what might kids say to draw 9 by 32 depending on that 16 by 18?

Kim  6:47  
I think they would say as long as the 18 is, it's going to be half as long.

Pam  6:52  
Half the 18. 

Kim  6:53  
Mmhm.

Pam  6:53  
Now, sometimes I've heard kids say "shorter than the 16". 

Kim  6:56  
Correct. 

Pam  6:56  
Yeah. And I might be like, "Okay, yeah. Is there any other?" Like, and I'm going to try to push for half of the 18.

Kim  7:02  
Right. 

Pam  7:02  
In some way to see if I can get that. So, I might say, "Well..." and then put my hand up at the board. "Well, if 18 is this long, then half of that? I'll kind of like use my hands. "So, half of that." And then I'll kind of turn it because now it's going to be the the length or you know, the up and down measurement. So, I'll say, "Okay, so about there." And I'll kind of draw. "Is that about right, everybody?" And they'll say, "No, it needs to be shorter or whatever." I'm going to try to have them kind of fuss with me a little bit. Should be about half. "Is that about half that 18? Okay, that's the 9. And then how about 32? Does that relate to any of those numbers up there?"

Kim  7:33  
Mmhm. 

Pam  7:34  
Which, go ahead, Kim.

Kim  7:35  
Yeah, it's twice as long as the 16. 

Pam  7:39  
As the 16. "Alright, so if that's 16..." I'm going to take my hands, kind of lay them out. That's how long the 16 is. Then I'm going to kind of turn my hands to go, "Okay, then it has to be twice that long." So I'm making a big deal physically that I've just drawn half the 18 and twice the 16.

Kim  7:53  
Mmhm. 

Pam  7:53  
Then I'm going to label that rectangle as a 9

Pam  7:56  
by 32. Go ahead.

Kim  7:57  
Pam, I don't think that you've mentioned that when you're drawing it, the first number for when you represent this array, the first number is going to go down. So, you're saying something about turning your arms. 

Pam  8:07  
Yeah. 

Kim  8:08  
I think that's important to mention.

Pam  8:10  
Yeah, so when I drew the 16 by 20 initially, the 16 represents the length, the number of rows. So it's the up and down measurement. And then when I drew the 20, that represents the number of columns, so that's the left and right measurement. So, when I drew the 9, and I was talking about the how it related to the 18, the 18 is a cross measurement, but the 9 is an up and down measurement, so that's why I was turning it.

Kim  8:35  
Mmhm. 

Pam  8:35  
And then the 16 was an up and down measurement, but the 32 is an across measurement, so I was turning it.

Kim  8:41  
Mmhm. 

Pam  8:41  
Yeah, so now I have... Before we had a 16 by 18. Which was kind of horizontally longer. And now, I have a 9 by 32. Which is really horizontally longer. It's like not 9 by 32. "And you guys just said the area that was 288." So, I have a couple of rectangles on the board. I have some areas up there, and then I might say something like... What would I say next? Let me think for a second. So, I'm thinking about when I would start to try to turn that rectangle a little bit and what I might ask. Kim, I'm stymied in the moment. I'm trying to ask what would I ask kids to get them to help me try to turn that rectangle so that the 9s or the the 16 is next to the (unclear). 

Kim  9:31  
Yeah, I think I might, at this point, say, "Is it coincidence that they're 288? Like, do these feel like they might be related in any way?"

Pam  9:40  
To try to draw out somebody saying, "Yeah, you doubled the..." 

Kim  9:44  
Yeah. I think

Kim  9:46  
if you could get them to say like it's kind of turned. Like, I could see you saying, "Well, like let's turn this one and put it next to

Kim  9:53  
it. 

Pam  9:54  
And so, I'll turn that 9 by 32. And so, I'm going up next to the 16 by 18. 

Kim  10:00  
Mmhm.

Pam  10:01  
And I'm going to say, "So, you guys told me that it was half of that 18." So, now where I had 18 across, now I'm going to draw 0 across. Only half.

Kim  10:08  
Mmhm. 

Pam  10:08  
And wheree had 16 down, now I'm going to draw twice as much. Can we see that 9 by 16 related here? And I might actually take that 16 by 18 and kind of cut it. 

Kim  10:25  
Yeah, draw a dotted line down the middle. 

Pam  10:28  
Draw a dotted line down the middle, and then say, "Ooh, it's almost like this half of this rectangle just kind of moved down and kind of created that 16 or that... I could think. Now, I've got two different rectangles on my paper. Created that 32 that was twice as tall by that 9, which is half as far across. Yeah.

Kim  10:50  
Yeah, this whole conversation, this bit is kind of like, "Let's puzzle about this. Like, how is it possible that they have the same answer? Like, you just told me that one dimension was half as long and the other dimension was twice as long. That's kind of curious. Like, could we put them up next to each other and kind of make them like be equivalent? Because you're saying they have the same product, the same area."

Pam  11:12  
Yeah. And when we do that, you can kind of see that it's almost like we cut the rectangle in half and we stuck half of it next to the other one, and so we didn't lose any area. 

Kim  11:22  
Yeah.

Pam  11:22  
If you cut one, cut the rectangle in half and kind of move it, that has an effect of cutting one dimension in half and doubling the other.

Kim  11:29  
yeah.

Pam  11:29  
"Hey, but we ended up with the same total area. That's interesting." 

Kim  11:33  
Yeah.

Pam  11:33  
"I wonder if that's ever helpful. Because neither of these seemed all that particularly maybe better to solve." Like, I might actually ask, "Were either of these like your favorite to solve?" And then we might have a brief conversation. Did somebody like solving 16 times 18 better than 9 times 32? I could hear somebody saying, "Well, 10 times 32, and just get rid of one 32 was pretty easy to solve. You're like, "Okay, but really? Yeah, whatever." Next problem. So, then what if I were to ask you say something completely random, not related, like 36 times 8. 

Kim  12:05  
Mmhm.

Pam  12:06  
Okay, so now I'm going to pause. I'm going to let kids solve 36 times 8 however they do. And I'm really less interested in how they solve it this time, and I'm watching to see. So, podcast listeners, go ahead and solve it. And I'm watching to see if anybody smiles or looks up and like, "Hey, it's happening again." Because 36 times 8 is 288. It's the same product as those first two problems.

Pam  12:31  
So...

Kim  12:32  
Pam, I can see you asking the question, "What would it look like? What would this array look like?"

Pam  12:37  
Do you think I'd ask that before I had them solve it?

Kim  12:37  
I think you would. 

Pam  12:40  
I think I could. I

Pam  12:43  
think it would depend. So, let's talk about what it would depend on. I think if nobody's seeing any connections at this point, I might just go ahead and have them solve it, and then ask them. For sure, I'm going to ask them what the rectangle would look like. I think if a lot of kids are kind of noticing patterns. I might be like, "Hey, what would this one look like?" just to kind of get them thinking about that pattern (unclear). 

Kim  13:06  
Yeah. 

Pam  13:06  
So, I definitely, yes. At some point, I'm definitely going to say, "Hey, what would this 36 by 8 look like? I wonder, could we use something that's already up there to help us think about 36 by 8?" So, podcast listeners, maybe grab a pencil at this point if you haven't already. We've got a 16 by 18 on the board. We've got a 9 by 32 on the board. We also have a 32 by 9 on the board because we sort of drew it next to it to kind of talk about how we cut in half and didn't lose any area. So, if that's what we have on the board so far, is there a relationship between this? Like, what would you use to help us draw the 36 by 8? And I could hear somebody talking about, "Well, it's a little longer than the 32," because now we have a 32 that's going down the board. But I might say, "Okay, it's a little longer. Is there anything that like I could kind of like? Is anything else going on?" And somebody might say, "Well, I see that 18 up there. I think that 18. You've got a 36. I think that 18 is..." I don't know. Will kids recognize 18 as half of 36? Maybe. But maybe they might see. So, it's 36 by 8. Maybe they might see that 8 as half of that 16.

Kim  14:17  
Mmhm. 

Pam  14:18  
Ah, so that seems kind of helpful. So, could I use the 16 that I have to kind of think about half? And so, let's see 36. 36 is going to be double the 18, so I'm going to kind of draw 18, and then another 18. Kim, when I draw that 18 and another 18, I'm going to try to leave a little gap, so  they can kind of almost feel this chunk and that chunk to kind of feel double 18.

Kim  14:39  
Yep. 

Pam  14:39  
So, that's 36. And then that 8 was kind of half of that 16. So, I'm going to kind of put my hand next to the 16 up there and kind of be like, "So, it's only half of that. So, that's a tall skinny rectangle. 36 by 18. And you guys are saying the area that's 288, so I'm going to put the 288 in the middle. And then I'm going to say, "What kind of relationships do you see between the other rectangles that we have up there?" Kim, what's a relationship that you see?

Kim  15:11  
Oh, well, I think some would talk about the other problem. Is that what you mean? 

Pam  15:14  
Yes, yeah. 

Kim  15:15  
Yeah, so some kids will also say... Because they're the same orientation, some might see that the going from 9 to 36. It's 4 times.

Pam  15:24  
Mmm. 

Kim  15:25  
And 32 to 8 is a fourth of that dimension.

Pam  15:29  
So, when they do that, I'm going to say something like, "Like, you're kind of like cutting up the 32 into 4 chunks?" So, I'm going to take that nine by 32 and I'm going to sa, like, "If we kind of..." and I'm going to dotted line down half, and then half those. So, I now have that 9 by 32 cut into 9 by... What is that? Four? 9 by 4, 9 by 4, 9 by 4, 9 by 4. Like,  that 32 is cut into four chunks. So, now I have these four little guys. And I might say, "It's almost like you're saying that I could..." And I probably will off to the side draw that 9 by 32. I probably should have said that. Off to the side, I'll redraw that 9 by 32. I'll cut it into those 4 chunks, and then I'll move... Gal, how do I even describe this, Kim? I'll move each of those chunks below. I'll leave the left hand chunk alone, and I'll move the other 3 chunks underneath it. 

Kim  16:21  
Yeah, so like they're sliding down. 

Pam  16:22  
They're sliding down. And as I move it underneath it, I've now created that 9 by 4, and then underneath it a 9 by 4, and underneath it a 9 by 4, and underneath it a 9 by 4. And hey, golly, that's that 36 by... Oh, not 4, it's 8, Kim. I had four 8s. Oh, good heavens. If you could see my board right now, kids would be like, "Miss, you've done that wrong. I said that I cut the 32 into four groups of 4, but I meant 8. I don't know if I said it. I drew it wrong. So, it was four groups of 8.

Kim  16:48  
Mmm.

Pam  16:51  
So, I've got a 9 by 8, 9 by 8, 9 by 8, 9 by 8, and I've now stacking those on top of each other, and so I've now ended up with a 36 by 8. Which is the 36 by 8 we were looking for. Yeah. So, basically, I've taken that 9 by 32, cutting it into 4 chunks, and then stacked them differently.

Kim  17:11  
Mmhm. 

Pam  17:12  
To create that 36 by 8. Hey, look at that. We can cut one side into 4 and quadruple the other side, and we didn't lose any area.

Kim  17:21  
Yeah, and I think during this time, as you're talking about basically transforming it into a new arrangement, you're also mentioning that you haven't lost any area. 

Pam  17:32  
Yeah, very important.

Kim  17:33  
(unclear). I think the language that you're saying is a lot of like, "We're moving this amount. We're moving this portion of the rectangle down here. But like we haven't changed how much we have. We're just rearranging it into a new rectangle."

Pam  17:49  
Yeah, nice. I think there's a high probability that also some kid will see a relationship between the 16 by 18 and the 36 by 8. Because of that Halving the 16 to get 8 and Doubling the 18 to get 36. So, I'm not going to describe it, but I would also grab one of those rectangles, cut it in half, slide it around to create the other one, and have that conversation again. Like, "Hey, we can Halve one dimension and Double the other dimension, and the area stayed the same." That seems interesting. I might step back at that point and go, "So, if these three problems are equivalent, then we could solve one of them and get the answer to the other two. That seems kind of interesting. Okay. Do you like one of these better than the others? I don't know. They're all kind of similar. Okay." So, at this point, if anybody's familiar with the Doubling and Halving strategy, you're like, "Pam, why did you give them those three problems? None of them are like, super spiffy to solve." Because when we're building a strategy like this, this is a very sophisticated strategy.

Kim  18:50  
Mmhm.

Pam  18:51  
We don't want it to become Double one, Halve the other. You're done. And kids don't know when and why, and they also don't connect it to area. We really want to build area here, and dimension, and the idea of conserving area, and we want it to occur to kids to Double and Halve later because they've actually built the mental connections in their heads. 

Kim  19:11  
And they trust it. They trust that it's a thing.

Pam  19:13  
Yes, because they've literally been shifting these areas around, and they're noticing it happens. So, these three problems aren't really to prove that Doubling and Halving is a great strategy. It's to build the the relationships that we need, so that Doubling and Halving can actually be a strategy that will occur to you with good problems. Cool. So, the next problem that I might ask kids is what is 12 times 24? Now, again, I'm going to let kids just solve that. Do whatever they do. I will probably represent one of their strategies, but I'm going to ask them, "Hey, what would a 12 by 24 look like? Completely unrelated, right?" And, "Oh, wait. Or is it?" So, after they solve 12 times 24, and they get 288. Again, now this, Kim, this is where I might have done your move of saying, "What would the 12 by 24 look like? Let's see, is there any relationships up here? Yeah, we could use?" Kim, is there a favorite one that you... You can't probably see what's on my paper, but is there?

Kim  19:13  
No, but the previous problem of 36 times 8.

Pam  19:14  
Mmhm. 

Kim  19:14  
I think is a nice. Like, it's related really easily for a lot of kids.

Pam  19:14  
Cool. Especially, I think, the 8 times 3 to get 24, then they might ask themselves is 36 divided by 3, 12? Sure enough. Cool. So, I could sort of say, "Then, from that 36 by 8, I only need a third of that 36. That's 12. I've drawn that length of 12 and labeled a 12, but now I need three of those 8s. "8". I'm going to pause. "8". I'm going to pause. "8." You know, now, when I say pause, I'm putting a little break in between them as I'm drawing the rectangle. "Okay, so cool. So, I've got a 12 by 24. And you guys are saying that's also 288. I might take a strategy of how they solve that, but I might just draw it, connect it to the 36 by 8 above. Cool. We've got four problems that are equivalent. Whoa! Surely no other ones. Alright, completely new problem that's not related at all. How about 6 times 48?" Let them solve it. I might ask them. "Hey, before you start solving how would we draw this rectangle?" Could we move some area around without losing it?" Well, can we move some area around? "Can we use other rectangles that we have?" So, Kim, do you have a favorite that you want to?" 

Kim  19:14  
Yeah, the one above.

Pam  19:14  
Okay, because? 

Kim  19:14  
It's just Double, Halve. 

Pam  19:14  
So, we have 12 times 24 right above. And you're like, 6 times 8. So, if you cut the 12 in half to get 6. So, as I'm drawing, I'm going to cut that. I have a 12 by 24. I'm going to cut the 12 in half to get 6. And I'm going to double the length of the 24. So, if you guys can picture that rectangle, I now have a rectangle that is half as tall but twice as long. And I could cut that. You know, the 12 by 24, I could kind of dot it, and I could kind of move that little guy over there, and say, "Sure enough. Look, we didn't lose any area. So, you're saying 6 times 48 is also 288." I might smile at some point and go, "Do you think there's any other rectangles that we could draw that are also 288. No, probably not. Surely, surely. Let's give you another problem. Now, this one's got to be different. How about 3 times 96. Hopefully kids are starting to smile at this point. Is there a relationship between that 6 times 48 and the 3 times 96? Now, Kim, I didn't mention. When we did the 6 times 48, we could have related it to several of the other problems.

Kim  22:31  
Right, right.

Pam  22:31  
And at some point, we'll probably want to. Like, for example, the 36 by 8 and the 6 by 48. If I divide 36 by 6, I get 6. And if I times 8 by 6, I get 48. Like, there's lots of nice relationships.

Kim  22:46  
And... 

Pam  22:46  
I could... Yeah, go ahead.

Kim  22:47  
Kids who are tinkering on their own are doing that. On their paper, they're sketching like the relationships to other problems that they are noticing. And that's okay!

Pam  22:59  
Yeah, we could set them loose and let them kind of fly while we're keeping other kids using strategies they know, starting to look for patterns.

Kim  23:07  
Because when you ask, "Is it related to anything else up here?"

Pam  23:11  
Yeah.

Kim  23:11  
And you get a kid to share. Maybe it's related to the problem above, you've opened it to maybe there's more than one. There are more

Kim  23:19  
than one.

Pam  23:19  
Yeah. 

Kim  23:20  
Yeah.

Pam  23:21  
Nice. So, what would a 3 by 96 look like? We already have the 6 by 48 up there, so the 3 by 96 would be half as tall and twice as long. Now, I have this really long rectangle, 3 by 96, and I've written in the center of it, 288, just like I have in all of them. And let's see, at that point, I might say, "Okay, anybody want to guess the next problem?" I might even like look at the 12 by 24, 6 by 48, 3 by 96. "Anybody want to guess the next problem?" I just kind of wonder if anybody might guess the last problem would be 1.5 times... Hmm, what's double 96. 192. 

Kim  24:00  
Mmhm.

Pam  24:02  
Ask them how to draw that rectangle. I will have planned, so that I can either fit this a half by 1.5 by 192 on the board or be really clear, dot, dot, dot, it would have to, you know, carry on over there. 

Kim  24:16  
Yep. 

Pam  24:16  
"So, if we've halved one dimension and doubled the other, do you think the area would stay the same? That seems really interesting, you guys. Let's keep in mind, area so important. We've been learning about area. Dimension by dimension gives us this sort of square footage that's happening in between, that's happening in the space in the middle. And man, it really seems if we're kind of move this area around, we can reason about the area of rectangles that we don't know yet."

Kim  24:43  
Yeah.

Pam  24:43  
Alright.

Kim  24:44  
I want to point out that the first three problems, maybe the fourth, but the first three problems for sure, kids are solving the problem to find the product.

Pam  24:54  
Mmhm. 

Kim  24:54  
And you're relying on that equivalent product to help make sense of the areas. And then there's a shift. That shift is where students are really not even considering the product as much. At that point, they're like really focused on what's happening to the rectangles.

Pam  25:10  
Nice. 

Kim  25:11  
The 12, 6, 3, 1.5. They're really thinking about what's the transformation

Kim  25:16  
that's happening. 

Pam  25:17  
And the teacher is helping that by not saying, "Okay, everybody go do this problem, find this answer." But like, "What would this rectangle look like? Is it related? How's it related? Can we move some area around?" Yeah, very important additional side effect here is that kids aren't just getting good at multiplication. Notice, that we just did, what, 1, 2, 3, 4, 5, 6, 7 multiplication problems where they're getting instant feedback on their strategies, they're getting instant feedback on multiplication, where sometimes teachers are all like, "Yeah, can't we just teach them to double and have, and then we're done." We're getting so much more out of building these other. Like, we're building area, and dimensions, and conservation of area, and factors, and Doubles, and Halves, and triples, and thirds. And all these relationships, we're building multiplicative reasoning. Cool.

Kim  26:04  
And you

Kim  26:05  
can tell them, "Double one number and half the other." I mean, it's possible. You can. But the result of that is having students say, "I don't know why that would matter. I don't know when or why I

Kim  26:14  
would care to do that." 

Pam  26:16  
And it doesn't occur to them.

Kim  26:17  
(unclear) robbing them of,

Kim  26:19  
yeah, the deep thinking, it occurring to them to use that as a fantastic strategy.

Pam  26:25  
Yeah, and having it just naturally sort of ping for them when it makes sense for the numbers to do that. And also, all the things that we just gained from area. Here's what they're not going to do. Next year, when the teacher says, "Find the area this rectangle," they're not going to go, "Hey, is this where I add up the sides? Or is this where I multiply?" They're going to be like, "Area. Like, I own area." It's like a thing that they own now. Yeah, very cool. Ya'll, if you're interested to know, this Problem String comes from our grade four Numeracy Problem String book that we put out fairly recently. So, if you haven't checked out our grade level numeracy Problem String books, we have kindergarten through fifth grade. They are amazing. They have lessons just like this where you could see the sketch of all these rectangles and how they relate to each other with a sample dialog of what could happen between the teachers and students, additional strings that are echo strings that are like it when your kids need more experience, next step strings that are baby steps between it and the next lesson. There are multiple lessons for this strategy. So, by the time you're done in... This is from fourth grade. So, by the time you're done with the Doubling and Halving strings, the last string is really giving you numbers where kids are using Doubling and Halving to solve the problem, making sense of it still with area, but a lot more of like let's actually put this strategy to use. And by then, it's occurring to them to use the strategy because you've built all this background between them. So, check out our Numeracy Problem String books for each grade. This particular structure of a Problem String is an equivalence structure. Why is an equivalence? Because, ya'll, every one of the problems in this string was equivalent. They all have the same product, which gives us an opportunity to wonder why and make spatial connections, built spatial reasoning as well as multiplicative reasoning in this Problem String. Kim, is there anything else you'd say about this particular one? 

Kim  28:14  
No, I don't think so. 

Pam  28:15  
Maybe we'd end this discussion with kids just saying, "Huh, it really seems interesting that if we can sketch a rectangle for a multiplication problem, it might give us an idea of moving some area around to find a problem that maybe a little bit easier to solve. Maybe." So, don't tie it up too tight in a bow, but we could mention something general like that. Alright, cool. Ya'll, thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Thanks for spreading the word that Math is Figure-Out-Able!