Ep 252: Foundations for Multi-Digit Multiplication & Division Strategies

What kinds of tasks help students build multi-digit multiplication and division strategies? In this episode Pam and Kim discuss how to sequence tasks to develop the foundational strategies and relationships for multi-digit multiplication and division: Scaling, Over and Adjust, Partial Products, Factoring

Talking Points:

• Benefits of using diverse task types
• Sample of three "messy" tasks to develop:
• Ratio tables
• Area model
• 2 different kinds of division
• Relationship between multiplication and division
• Practice ideas

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Transcript

Pam  00:00

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

Kim  00:09

And I'm Kim Montague, 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  00:17

We know that algorithms are amazing human achievements, but they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.

Kim  00:30

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

Pam  00:36

Ya'll, we're so glad you're here to help us make math more figure-out-able. 

Kim  00:41

Hi, there. 

Pam  00:42

Kimberly, What's up today? We are on fire. 

Kim  00:48

Yeah.

Pam  00:49

So many things.

Kim  00:50

I know.

Pam  00:50

Cool things that we are doing here at Math is Figure-Out-Able. All the things. Yeah, nice.

Kim  00:54

Okay, so today we're wrapping up a series on foundations and like...

Pam  01:01

How we could build.

Kim  01:02

...sequencing these foundations, yeah. And I'm excited about today because today is about multi-digit multiplication and division, and I know that people are waiting for us to get here. Because, listen, we do MathStratChat on Wednesdays. When you throw out a multiplication problem, there are always so many responses. 

Pam  01:23

Definitely the most popular. 

Kim  01:24

Something about multiplication, yeah.

Pam  01:26

Yeah. Yeah, every once in a while, you and I will say, "Let's do multiplication problem," because we know we'll get a lot of interest, and activity, and engagement.

Kim  01:35

Mmhm. 

Pam  01:37

We encourage everybody to join in on the other ones. But yeah, there's something about multiplication that I think... I don't know. Is it naturally intriguing? It's just kind of inherently like a puzzle. And maybe...

Kim  01:47

Yeah, I don't know.

Pam  01:48

Maybe people own some relationships.

Kim  01:51

Yeah.

Pam  01:51

It kind of makes me wonder if a lot of people were able to do addition and subtraction.

Kim  01:55

Yeah, (unclear). 

Pam  01:56

With the algorithm. Yeah. 

Kim  01:58

Yeah. 

Pam  01:58

But with multiplication, division, they didn't want to sit down and like chug out a bunch of steps, so estimation became kind of a thing. Not like, "Let me teach you how to estimate." But really, like Jo Boaler would say.

Kim  02:02

Life.

Pam  02:05

A Math-ish answer.

Kim  02:13

I wonder. 

Pam and Kim  02:13

Yeah. 

Pam  02:13

And in life, they've come up with some relationships to reason through multiplication. Maybe when they haven't had to for other things. I don't know. Do you think? Is that kind of where you thought I was going to go?

Kim  02:23

(unclear). I do think that sometimes people are more willing to think about strategies for multiplication. Maybe because they did it growing up. Maybe that they did it like at the grocery store. 

Pam and Kim  02:36

Yeah.

Pam  02:37

In life. In life, it sort of demanded. Or at least requested it, and so they went with it. Yeah.

Kim  02:43

Yeah. So, join us.

Pam  02:45

Yeah, join us on MathStratChat. Yeah, check it out. If you just go to the hashtag MathStratChat, all one word, then you can find us on Facebook, Instagram, Twitter at the moment. But we might be heading into some other social media outlets soon. 

Kim  02:59

Yeah. 

Pam  02:59

So, yeah, wherever you do your social media, just look for it. You can lurk. You can just see what other people do. But you can also, we love it when you join in. We especially love it when you comment on each other's strategies. Awesome. 

Kim  03:11

Yeah, yeah. 

Pam  03:12

So, Kim? 

Kim  03:12

Yep.

Pam  03:13

Like you said, we've been talking about sequencing tasks to really help build the foundations for strategies because we do a lot of Problem Strings in workshops. We do a lot of Problem Strings on the podcast. But there are other tasks, and we can sequence those to really help build the foundational ideas, and models, and strategies that can help kids math like mathy people actually math. Today, let's do that with multi-digit multiplication and division. What are some of those important threads that we want to have running through our instruction in grades whose whose mandate it is to develop multiplicative reasoning?

Kim  03:57

I think a really important thread is the idea of scaling. 

Pam  04:03

Yes.

Kim  04:03

The idea of scaling groups, different numbers of groups that involves  thinking about relationships that you own. But the idea of scaling is super important.

Pam  04:14

Yeah. So, if I know that 2 of something is this number, then if I double the 2, I can double the other number. And if I scale it times 10, I can scale times 10 the other, the corresponding 

Kim  04:27

Yeah, so (unclear). 

Pam  04:28

If I'm dividing by 10, I can divide by 10. Yeah, go ahead. (unclear).

Kim  04:31

Yeah, so the idea of scaling is big. But there are some specific scalings. Like you just mentioned doubling (unclear).

Pam  04:37

Sure enough, yeah.

Kim  04:38

Times 10 and times 100 are really useful. 

Pam  04:42

Isn't that funny that like we're talking about scaling, and I immediately go to the important scale. 

Kim  04:47

Yeah, yeah. Yeah.

Pam  04:47

Yeah, yeah, yeah. So, Five is Half of Ten is going to be an important relationship. Just like we want to know that you don't have to know your fives if you know your tens with single digit multiplication, super important. Important that you don't have to know your 50s if you know your hundreds.  And we can also use 0.5 in lots of relationships. And we tend to call that all kind of encompassing Five is Half of Ten.

Kim  05:02

Yeah.  Mmhm. 

Pam  05:09

but it's kind of all those relationships.

Kim  05:11

So, I would actually call that a specific strategy that you just mentioned. But the relationship is like the Doubling and the Halving. 

Pam  05:18

Gotcha.

Kim  05:19

The foundational thing that we need to develop is the idea of Doubling and the idea of Halving. 

Pam  05:24

And the idea of making use of 10, 100, 50, 0.5. Yeah. Okay, gotcha. 

Kim  05:29

That's just more specific in my mind, I think.

Pam  05:31

Yeah. Alright, so what's another foundational thread that we would want to make sure we develop? 

Kim  05:36

Well, we have mentioned this in every one of the episodes, which I think is noteworthy about its importance, but the idea of Over and adjust (unclear)...

Pam  05:47

That is super interesting

Kim  05:50

(unclear)

Pam  05:50

Yeah. 

Kim  05:51

(unclear). Yeah. 

Pam  05:51

when you say we've mentioned it in the episodes, we've mentioned it in addition and subtraction, and multiplication and division, whether it's small numbers or now we're growing up into bigger numbers. Super important to realize that we can use a friendly number to go a bit Over and adjust. And

Kim  06:07

It might be why it's my favorite. 

Pam  06:11

Do you know, I remember the day when I realized that that underlying relationship existed in all the operations?

Kim  06:18

Mmhm. 

Pam  06:18

Like that was... Unlike, a lot of the other relationships don't necessarily exist in all four operations.

Kim  06:25

Yeah. 

Pam  06:25

That one does. That's an important one that kind of is overarching. Yeah. 

Kim  06:29

Yeah. 

Pam  06:29

I think on multiplication, it's also super important that we have this idea of partials. So, partial products, if we're dealing with multiplication. Partial quotients, if we're dealing with division. But not just the place value ones that we kind of could turn into another algorithm. It's chunks you know.

Kim  06:46

Mmhm. 

Pam  06:46

How could you use chunks you know, either way, to sort of build up to what you're looking for? That's super important. But then also, completely missing from, I would say, most traditional anything out there is the idea of using factoring to multiply. "Wait, Pam, factoring. Isn't that all about like... That's the inverse. That's the undo." Yeah, but we can actually flexibly factor. We can... Doubling and Halving is really based on using factors. Those are strategies, but they're based on this big idea that we have a feel for the factors. Yeah. In the episode, when we talked about single digit multiplication and division, I talked about taking a product approach. Which really means you look at a product and you dive into all of its factors. So, it's super important that we've got this sense of factoring.

Kim  07:23

Mmhm.  So, kind of the big idea there is that the decomposition of a number. You just mentioned partials, chunks you know, and factoring. Like, those both work towards kids understanding of the way numbers are created.

Pam  07:53

Yes, nice. And when we were in addition, subtraction, we talked about decomposing and composing numbers additively.

Kim  07:59

Mmhm. 

Pam  08:00

Now, we're talking about composing and decomposing numbers multiplicatively. Like, often, when we do the partials using the distributive property, we're kind of breaking one of the factors, or maybe both, into additive chunks, but then multiplying it together.

Kim  08:13

Mmhm. 

Pam  08:13

But when we talk about the factoring part of it, it's really breaking the numbers into multiplicative parts. And that's a whole different idea than I think any of us who were raised to multiply by repeating the steps of the algorithm. It never occurred to most of us to think about factoring. Now, I'm just going to mention, as far as implications for higher math goes, the fundamental theorem of algebra is all about factoring polynomials. So, factoring doesn't go away. Not as a thing to do, but as an inherent component part of what we're talking about. Yeah. So, there's just some of the underlying threads that we would want to make sure we are addressing as we attempt to start building the foundations for multi-digit multiplication and division.

Kim  08:56

Yeah. 

Pam  08:56

Cool. 

Kim  08:57

So, today, we're going to share some sequences of things for multiplication, multi-digit multiplication and division. And you know that we want some messier, messing around kinds of things. We've also talked about developing and analyzing strategies, and we've talked about the idea that some practice is important.

Pam  09:17

Mmhm.

Kim  09:17

(unclear). So, before you share, I want to just raise the idea that we've talked about there are mathematical reasons to have all of those different kinds of tasks.

Pam  09:26

Mmhm. 

Kim  09:26

I also want to raise that for our students, there are reasons to have those different types of tasks. I have one son who is quieter, more introspective. He likes to work alone more. And there are some tasks that are geared maybe more towards that type of student. I also have another student who loves to work together, who's conversational, who, you know, wants to move and busy around. And there are tasks that are more geared towards him. And we want a mixture. So, we have some ideas of different kids shining in different areas. We want to hear them unpack their knowledge. Sometimes solo. Sometimes in partners. Some in a game type setting. Some where it's like deeper, richer work. And so, you know, as listeners are listening to you share some unpacking of sequences today, I wonder if maybe they could be thinking about the types of students that they have in their class.

Pam  10:21

Nice. 

Kim  10:22

And even thinking a lot about specific kids. Like, "Oh, I could see that kid shine in this area, and I might get to understand what their knowledge is in this moment because of that type of task."

Pam  10:32

Hey, Kim, I have a question. I'm not actually sure I know the answer to this. 

Kim  10:35

Okay. 

Pam  10:36

I think we're on the same page that we suggest that we want a mixture of all of these tasks for all kids. 

Kim  10:43

Yes. 

Pam  10:43

In other words, what somebody might...

Kim  10:45

Don't leave my quiet kid to not do any thing with a partner, if that's... 

Pam  10:49

Yeah. 

Kim  10:49

Sorry. Thank you for raising that. Yeah, yeah.

Pam  10:51

Because somebody might have just heard you say, "Oh, okay. So, I'm going to give this kid this kind of task because they'll shine, but I'm not going to have them do this other stuff. 

Kim  10:59

I'm saying that might be their personal preference, and they need all of the tasks.

Pam  11:02

They need experience with all the kinds of tasks. Everybody does, yeah.

Kim  11:07

Yeah, because they're accessing different things that they are using to create a more rich mathematical experience. So, I might have a quiet kid who really doesn't want to talk to a bunch of other people, but that is something for him to develop, and his math will become richer, and his partner's math will become richer when they engage in that.

Pam  11:26

Agreed. And vice versa.

Kim  11:27

Yeah.

Pam  11:28

Everybody needs work in all the areas. 

Kim  11:30

Yeah.

Pam  11:31

Instead of what we've done kind of traditionally, which is really only a reward... What? The kids who know how to do school? Well, the kids who know how to do school and are willing.

Kim  11:41

And are willing.

Pam  11:41

Yeah.

Kim  11:42

Yeah, that.

Pam  11:42

That's kind of the kids who have been rewarded in our system. 

Kim  11:45

Yeah.

Pam  11:46

Let's change it up and have different kinds of tasks to help all kids develop a more well roundedness. 

Kim  11:52

Yeah. Okay.

Pam  11:54

I'm sure there was another way to say that. Okay. So, what are some tasks that we could sequence to help kids build the foundations for multi-digit multiplication, division? Well, I'm going to start with some messy tasks. And to be honest, we don't just say those messy tasks first, because, because. We say that because we like to start with messier tasks.

Kim  12:15

Oh. Whoa, I'm going to push back though because there are times where you do some strategy something first.

Pam  12:21

Oh, that's true. So, maybe what I actually mean to say is there are times... No. When we plan sequences, we tend to plan them around the messy tasks. 

Kim  12:32

Yes.

Pam  12:32

Is that a better way to say that? 

Kim  12:34

Yes, it's centered around it.

Pam  12:35

(unclear). Yeah, centered around that. So, we might say, "Based on this messy task if for it to be successful, are there some things that we need to get bubbling up in class?" Then we might do something to start that messy task?

Kim  12:48

Yeah, yeah. I'm so glad you said that because we have gone in order of messy, and then strategy, and then practice, and I think the people could absolutely think that that's the order. I can actually think of times where I'd have a practice come before a Rich Task. So.

Pam  12:59

Yeah.

Kim  12:59

So, there's not (unclear).

Pam  13:00

And Problem Strings sprinkled throughout, right? 

Kim  13:02

Yeah.

Pam  13:03

But just to be really clear because we're not talking about Problem Strings in this in the last three episodes, doesn't mean that we don't wholeheartedly believe in them. It's just that we spent a lot of time on those. These are what to do with, what to do along with, kinds of tasks to sequence along with Problem Strings.

Kim  13:18

Mmhm. 

Pam  13:18

Okay, cool. So, one of the things that we like to do is have kids dive into a rich, meaty multiplication problem, but in a context. So, we want to start a context. And we could do lots of context. We also like to have specific numbers. We want to have numbers that do decompose nicely, that break apart, chunk nicely. So, we might use numbers that are like $1.25 or $1.50. We might use numbers that are like 125 and 150. And then multiply those by something that's pretty chunkable, like 24, or 36, or 32. So, an example problem might be that I say, "Hey, I'm going to make a bunch of origami. And to fold an origami, I got to have the paper and my supplies to do that, so it's going to take me about $1.25 to make each origami. And I'm going to make, say, I don't know, 24 of them." So, I might just throw that out to kids and say, "How much money do I need in order to get the supplies to make this origami?" So, now I've got something like 125 times 24 or $1.25 times 24. Something like that. And then go, and let them start working on that. And then the teacher, super interactive. If I've got kids that are adding a bunch of $1.25s together, I might intervene and say, "Hey, do you know... Could we group some of these?" I might keep track of those groups on this, I don't know, this table looking thing. So, I might say, "Well, it looks like, you know, you're thinking about $1.25s. And so, if you group those together. 4 of them. How much were 4 of them? Oh, you found that was $5.00? How did you find that? Oh, it was because the four $1.00s and the four quarters. You put that together, you got $5.00." And I might kind of keep track. I might just like that much. Just like 1 to $1.25 and 4 to $5.00. And walk away and just kind of see what they do with that. But I might have other kids that have sort of grouped the $1.25 together, and they've got $2.50s, and I might like record that, "Okay, so 2 of them was $2.50 and walk away."  I might have kids that are doing things like with 10 of them and what that is. So, depending on what kids are... I might have kids gathering the quarters together and gathering the $1.00s together, and we might record those. But I'm going to specifically record all that in ratio tables, nudging kids towards building the idea of a ratio table to keep track of bigger chunks, chunks that you know. Is there anything else you can think of with that messy problem, Kim, that I might add in. It's okay. That's all (unclear).

Kim  15:18

Mmhm. No.

Pam  15:46

No? We're going to have kids share out what they did and kind of like help focus kids towards, "Hey, we could use this table to kind of help us keep track. Ooh, were there some nice chunks? Oh, nice chunk that you did 4 for $5.00. That was super." A lot of kids are like, "Oh, that would have been nice!" Or, "You gathered the quarters?" And so, we'll have that conversation about, again, nice numbers, and about scaling, and about using chunks that you know. But we also want to develop some intuition and foundation for division, so we want to give kids some experience in thinking about division, so we're going to do a context where kids have groups of things where we sort of would... Well, a whole bunch of loose things, and then saying, "Hey, if I've got a whole bunch of, I don't know, say markers, and I need to find out what colors I've got. Say I've got, you know like, those... Can I say Crayola? Is that? Like, a certain brand of markers. So, I've got a certain brand of markers that in a box gives me 8 different colors. Maybe I want to go find out how many colors. I've got this whole huge pile of markers. Kids dumped them all out. Here's the pile. Like, how many reds do I have? How many? If I have the same number colors in this big pile of, I don't know, say 96 markers, you know how many reds do I have?  How many.. Yeah. And again, I just give them lots of colors. And so, then the kids are kind of thinking about, "Well, if I'm looking for, you know, these 8 colors, how many do I have of each?

Kim  16:59

Yeah. Mmhm.

Pam  17:12

Then they're going to be tempted to kind of deal out those markers into those 8 colors.

Kim  17:17

Mmhm.

Pam  17:17

And then I might say, "Well, alright, great. Now, I know how many colors I've got of each of those markers, but now I got to put them in the box."

Kim  17:23

Mmhm. 

Pam  17:23

"So, there's the box. And each box holds 8 markers. How many boxes do I need to put the markers back in the box?" And really, in a big way, I'm helping kids reason through the two types of division.

Kim  17:34

Mmhm.

Pam  17:34

I want them to really think about how many 8s are in that 96, but I also want them to think about if I've got a group that... How do I? Help me, Kim. How many 8s

Kim  17:45

If you have... Yeah, if you have certain number of colors.

Pam  17:48

Oh, I have 8 colors.

Kim  17:49

Mmhm.

Pam  17:49

Yeah. Certain number of groups. I have 8 groups, 8 colors. How many markers are in each one? Thank you for saving me there. So, I've got both quotative and partitive division kind of happening, and kids are sort of messing around with that. We want them to kind of share what they were doing. Because I'm putting those markers in a box, I can actually stack that box of markers, and I've got an array. 

Kim  18:09

Yeah.

Pam  18:10

I've got an array of those markers. And if I've put the colors in those boxes, in the same order, I've got a different array where I can sort of look at the columns of... Well, not a different array, but a different focus. So, I'm either looking at the rows of boxes stacked up. Or I'm looking at the columns of colors. And I've really got a nice way of kind of looking at that array from two perspectives. Go ahead.

Kim  18:37

Yeah. And I think what's nice here is that the context, you're not just saying like, "Oh, rows. Columns. 8s. 12s." Like, kids can actually see the reds all in a group, the colorings. And so, it gives them... I like that you said perspective. It gives them a way to view the model based on what they just tinkered with in this real life scenario.

Pam  19:01

Nice. So, if we use these two kinds of messier problems, we've got one that kind of develops the ratio table as a model and the marker problem kind of develops the array as a model for multiplication and division. And we're in both of them... Well, specifically in the marker one, we're kind of connecting multiplication and division as well. 

Kim  19:21

Yeah.

Pam  19:21

So, those are some examples of messier. Now, when I say "messy", I don't mean leave kids out to just discover math on their own. It's highly structured. Kids know what they're doing. They're clear on their task. The teacher is clear on the math to bring out.

Kim  19:34

Mmhm. 

Pam  19:34

It's only messier because it's not we're focusing on this one thing. It's messier because there's lots of things that are coming up, bubbling up, and then now we're going to do different kinds of tasks to focus on each one of those things a little bit more. 

Kim  19:47

Yeah.

Pam  19:47

We might cycle back and do something messy after we focused for a little bit. 

Kim  19:51

Mmhm.

Pam  19:52

So, what are some of those focusing things where we can kind of analyze strategies? And I would combine these analyzing strategy tasks with Problem Strings to then get better at the strategies. So, one thing we could do is we like sort cards every once in a while. It's a different way to kind of change things up. So, what if I put on a card one kind of multiplicative problem like a multiplication problem, or a quotative division problem, or a partitive division problem. So, I've got sort of three kinds of multiplicative problems, multiplication, partitive division, quotative division.

Kim  20:28

Yeah.

Pam  20:29

And I've got several examples of each. So, I hand a group of kids. Hey, group of kids, in this group. I've given you 15 cards. Sort them. Which of these are giving you the two factors and asking you for the product? That's a multiplication problem. Which of these are giving you the total and the number of groups? That's a partitive division. I have to think about that. If I've given you the number of groups, I'm looking for the number in each group. Yeah, that's partitive division. Or I've given you the total and I've given you the number of groups, and you've got... No, I just said that. The number... The other one.

Kim  21:03

(unclear).

Pam  21:03

Then that's a quotative division problem. So, depending on which factor I'm leaving out, whether I'm leaving out the number of groups or I'm leaving out the number in each group, that those are the two types of division.

Kim  21:13

Yeah.

Pam  21:13

Can you see how I really don't care which one's which, as long as I develop both.

Kim  21:17

Yeah.

Pam  21:17

Like, it's mostly important that I develop both kinds. Yeah, go

Kim  21:20

And the task is about making sense. The task is not, "Okay, what's the answer to this problem?" Which everybody does. Everybody does, "Here's a problem. What's the answer?" The task is really about do you understand different situations?

Pam  21:20

ahead. And do you connect that they are multiplicative in nature?

Kim  21:39

Yeah, yeah. 

Pam  21:39

And so, you can solve these using multiplicative relationships.

Kim  21:43

Mmhm. 

Pam  21:43

Yeah, that's where exactly that. Yes. So, we could do some of that. Getting kids kind of connect multiplication and division in these different situations. We can also give kids some context that help them get a feel for strategies. And I'd like to talk about one of them real quick. So, this is... I'm actually super proud of this context. I'm not... I think Kourtney came up with it.

Kim  22:05

Could be, yeah.

Pam  22:06

Where we could say to kids, "Hey, let's go to... I don't know, like..." Not Charlie and the Chocolate Factory because we're not going to exactly copy that. But let's go to a chocolate factory where if you won going to the chocolate factory, you get to create your own chocolate bar. And then the chocolate bars that this particular chocolate factory makes, they come out with 1 by 1 little squares. So, not like a Hershey's Bar, where it's like a rectangle that you break off. But it's square. Square little pieces. You put the square pieces together, and you have a chocolate bar. You might recognize like a Ritter Sport. It's a yummy, German chocolate. Those have those 1 by 1 squares in a Ritter Sport chocolate bar. But unlike a Ritter Sport, they could be rectangular. They're not just... A Ritter Sport is a square chocolate bar. These can be any rectangular size. And the kid who wins the prize gets to go choose the size of candy bar they're going to create. So, could you picture a kid who says, "Alright, bam, I want to create an 8 by 12, and I'm going to make it with nuts, and I'm going to make it with raisins." Yuck! No raisins in chocolate. They don't belong together. But a kid could like create an 8 by 12. And then the task is the kid gets to say, "Well, if I want an 8 by 12..." They have to tell. Part of the order form is they have to tell the factory how many squares of chocolate that will be. So, then they have to actually figure out what 8 times 12 is. But it's all in terms of squares of chocolate. They have to sketch out what the chocolate bar looks like. It's brilliant. The brilliant part of it, though, really is... So, maybe I'll say that's great. But the brilliant part of it really is that then we have an order form that we give kids. So, we give kids, "Hey, here's an order form that's been filled out." And the kid asked for, I don't know, say a 17 by 40. Why not? So, the kid said, "Here, I want a 17 by 40 chocolate bar, and I want these ingredients in it." I want it to be dark chocolate because dark chocolate is the best. That gives my age away, huh? It's only old people that like dark chocolate? Anyway.

Kim  22:37

Mmhm. No! 

Pam  23:57

Okay, so 17 by 40. And you can see on the order form that there's a place where you fill out the number of squares. But then, you also see on the order form where there's a stamp on there that says, "Adjust". So, for whatever reason, the chocolate factory has come back and said, "Nope. Can't have your 17 by 40. But you can have a 17 by 39." So, Kim, what is that suggesting to you? If you see this order form that clearly says 17 by 40. Fill in the total squares. Stamp, "Adjust". You actually only get to have a 17 by 39. What are you thinking? 

Kim  24:35

Yeah, that's that's really facilitating the opportunity to use an Over strategy.

Pam  24:40

Bam! 

Kim  24:41

Yeah. 

Pam  24:41

Like, the kid is just like. It's almost like begging you to find 17 by 40. Just find 17 by 4. Scale that up by 10. That's one way. And then to get 17 by 39, you got chop off a row. Or column. Whichever one. 17 by... That would be a column, right?

Kim  24:57

Mmhm. 

Pam  24:58

Yeah. Chop off a column. Chop that sucker off. And bam, get rid of one 17, and you have solved 17 times 39. So, we can put kids in situations and contexts where the strategies are like flashing at you. We're like, "Oh, use me. Use the 17 by 40 to help you," in brilliant ways that kids can be just, again, they can kind of notice these patterns. We're high dosing them with the patterns.

Kim  25:23

Yeah.

Pam  25:24

And they can analyze those patterns, and then get better at them in Problem Strings.

Kim  25:30

Yeah.

Pam  25:30

Bring Problem Strings in to get better. So, in previous episodes, we've talked about also looking at smudge problems. We love that. Giving kids ratio tables. Giving kids open arrays where we smudged out part of the work, had them fill in the work, and then analyze what the kids were doing. We also, as we build those strategies, using the smudge problems and Problem Strings to get better at them. Then, help clarify those relationships, make anchor charts that list those relationships. What about for practice with multi-digit multiplication, division? Well, very similar to what we suggested with the multi-digit addition and subtraction, we can put a strategy in a model, on a card but smudge out some of the work.

Kim  26:11

Yeah. 

Pam  26:11

So, let's say that I was doing that 39 times 17. Could I put an Over strategy on a ratio table? And on a different card, put that Over strategy on an array? And on a different card, put that Over strategy using equations? Then, put the same problem but using a different strategy on an array, and on a ratio table, and on equations. And again, smudging out part of the work, having kids fill in the work, then analyzing what was the kid doing for this strategy on this model? Oh, bam. Let me like put some words to that. Sort that out a little bit. Get better at the idea. What are the relationships? What are the models building those, the major foundations for strategies to then become natural outcomes. So, ya'll, you can go take these kind of hints that we've given you, these principles, these big ideas that you can say, "Wow, we really want to give different types of tasks, and we want to sequence them, and we want to bring in Problem Strings and change up the way that we're having kids work together and how they're solving problems as they're analyzing strategies to build the foundations for strategies." If you would like a ready made version, where it's easy to use, easy to read, then check out our Hand2mind Foundations for Strategies, multi-digit multiplication and division, where we walk you through sequence set of lessons to build foundations for those strategies. 

Kim  26:12

Yeah,

Speaker 1  26:14

and I hope that for the last couple of episodes what you've taken away is that when you look at your own resources that you've found, that you've been given, that you can start to think about if this is something that I'm trying to develop, if this is a foundational idea for multiplication, division or for addition, subtraction that I'm trying to build in with my kids, how could you tweak sequences that are in your resource? How can you add things like Problem Strings, or other games, or build it around some messy task? What can you take away that you find is just repetitive, that it's not reaching the foundation that you're trying to reach? What can you cycle back to? So, we hope we've given you some things to think about as you mess with the tasks that you have at your disposal. But you, like Pam said, can always get your ready made resource from our product with Hand2mind.

Pam  28:29

Hand to Mind, yeah. Hand2mind, thanks for giving us the opportunity to really dive in and create sequences... 

Kim  28:35

So fun. 

Pam  28:36

...based on things that we've been thinking about and working with for a long time. We really appreciate the good, good, good people that we worked with at Hand2mind. We're super proud of how that's turned out. Ya'll, thank you 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. Ya'll, thanks for being on the journey with us for spreading the word that Math is Figure-Out-Able.