Ep 251: Foundations for Multi-Digit Addition & Subtraction Strategies

What kinds of tasks help students build multi-digit addition and subtraction strategies? In this episode Pam and Kim discuss how to sequence tasks to develop the foundational strategies and relationships for multi-digit addition and subtraction: Friendly numbers, place value, adjusting, the commutative property, decomposing

Talking Points:

• Sequencing tasks is more than just moving from one page to the next in the teacher guide
• Sample of three "messy" tasks to:
  • develop number lines
  • make connections between addition and subtraction
  • analyze strategies
  • solidify a strategy

For more sequences, check out the Foundations for Strategies from Hand2mind!: https://www.hand2mind.com/item/foundations-for-strategies-multi-digit-addition-subtraction-small-group-kit

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Transcript

Pam  00:00

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

Kim  00:09

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  00:16

Because, ya'll, algorithms are amazing historic achievements, but they are not good teaching tools because mimicking step-by-step procedures actually traps 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  00:37

We invite you to join us to make math more figure-out-able. You know, I really like how we say the relationship part in there.

Kim  00:44

Yeah.

Pam  00:44

Like that mathing is about relationships with students and mathematical relationships. I like that.

Kim  00:49

Feels know your content, know your kids. 

Pam  00:51

It does, doesn't it?

Kim  00:52

I have a review (unclear).

Pam  00:54

Oh, I love reviews. Yay!

Kim  00:55

I know. 

Pam  00:56

Hey, I go to say, when I met Laura Breaux, who's in our coaching group, our Journey coaching group. I met her in person not too long ago in California. She has been valiantly trying to leave us a review. And I guess the way that... What is it? Google? Or YouTube? Or somebody messed around with podcasts. It's much harder to leave a review. So, ya'll, we get it. We sure appreciate it when you put forth the effort because it helps other people find the podcast. So, thank you. Those of you that are able to put in the time and effort to do the review. It means a ton. We know that it's not trivial. 

Kim  01:27

Yeah, yeah, yeah. This one is from MathInCharlestonSouthCarolina. 

Pam  01:32

Oh, nice. 

Kim  01:33

And they say, beth math...

Pam  01:36

Somebody. You can do it.

Kim  01:39

"Best math podcast for teachers of all levels." And when I saw that, I was like, "Ooh! All levels! Like, some are numerate and some are not yet? Or grade levels? "So, anyway, she says, he says, "Math..." This is going to be a fun episode, Pam.

Pam  01:57

Hey, Kim. How's it going?

Kim  01:58

I can not read today. Oh, holy, now. Okay, 

Pam  02:01

We can do it. We can do it.

Kim  02:02

This person said, "Pam and Kim bring humor and realistic ideas to all math levels. I always take away a new idea and solidify existing math learning. I'm a veteran teacher, but new to math. I taught ELA for years and needed a change. Teaching math and science for me is that change. I was never, quote, good at math, and with time Pam and Kim have given me the confidence to be able to let the numbers dictate how I solve problems. My own number sense has increased with this podcast, and I'm able to help kiddos at all levels. This is the podcast that all math teachers need to listen to." 

Pam  02:35

Aww! 

Kim  02:35

Thank you.

Pam  02:36

I love that. 

Kim  02:37

So sweet. So sweet. 

Pam  02:38

I love that. You know, you can tell that MathInCharlestonSC has actually listened. I'm assuming because this person says, "to let the numbers dictate how I solve the problems. Like, that's a Kim quote, right there. 

Kim  02:50

Welcome to the good side of math.

Pam  02:53

I love it. Alright, cool. Well, thanks for sharing that, Kim.

Kim  02:56

Alright, Pamela. (unclear).

Pam  02:57

Have a

Kim  02:57

Yeah, we are a few episodes in to our conversation about sequencing. And sequencing is so important to actually affect change. And, you know, we do love changing up the types of lessons for so many reasons because you can do different things with different types of lessons. But hey, I want to let you know that we have heard about textbooks and district made curriculum where it is I do a sequence friends, but it is the sameness all the time. Or it is, we do this strategy one day. Turn the page. There's another strategy the next day. Turn the page. Another strategy a next day. And what we're talking about here is not that. We're not talking about turning the page and starting a whole new thing. What we're talking about here is landing for a bit to really develop connections and weaving things kind of all throughout to keep cycling back because we know that's actually how kids develop is to just lob something out, work with it, come back around to it. (unclear).

Pam  02:57

Let's dive in. running thread. Sorry, keep going. 

Kim  03:54

Yeah, no, go ahead. 

Pam  04:05

Running thread of big, major ideas, models, and strategies that we need to have kids keep developing. 

Kim  04:12

Yeah. 

Pam  04:12

And we do that by altering and sequencing different kinds of tasks.

Kim  04:17

Mmhm. 

Pam  04:18

And also, like you said, weaving in and out of like let's start developing the strategy. Let's do some work with it with different types of tasks. Let's start developing another strategy. Let's come back and work on that first strategy. We're not going to pretend that kids can rote memorize a strategy in a day, and that that's worth anything. 

Kim  04:36

Yeah. 

Pam  04:37

It's not about having a set of strategies to solve problems. It's about developing the mathematical relationships that make those strategies natural outcomes. 

Kim  04:49

Yeah, yeah. So, if we were making a list of foundational things for multi-digit addition and subtraction, here would be some of the things on our list. We think it's important to think about friendly numbers. Not just I like 12, so it's friendly, but important landmark numbers. I don't know why I said 12. I actually like 14 better. That's weird that I said 12. 12 is richer, but I like 14. Anyway, so friendly numbers that are landmark, important numbers to our place value system. We also think developing understandings about place value, not place labeling, is super essential.

Pam  05:27

Mmhm. 

Kim  05:28

Doubles are important. The idea, like we talked about with single-digit, the idea of adjusting. So, adding, subtracting, and then adjusting a little bit is crucial. 

Pam  05:40

That's where your friendly numbers come in, right? Like, add a friendly number, so then you adjust, either keep adding to add the rest of it or adjust back because you added too much. 

Kim  05:48

Yeah. 

Pam  05:48

Same with subtraction. Subtract a friendly number, and then keep going if you haven't subtracted enough. Or then if you subtracted too much, then you have to adjust by giving a little bit back. Like, sort of compensating. All those adjustments are based on that very first thing you said, those friendly numbers and place value relationships. 

Kim  06:06

Yeah.

Pam  06:06

Yeah, nice. 

Kim  06:07

We also will continue to develop the commutative property.

Pam  06:10

Mmhm. 

Kim  06:11

Just properties in general are foundational. We also want to develop the idea of partial sums by splitting numbers. So, decomposing. Not just any random way but in important place value ways. So, these are foundational. And I want to just comment that these are add-ons to the foundational things that we talked about in single addition. 

Pam  06:34

Well, actually, can I tweak that (unclear).

Kim  06:35

I just said single addition. I'm struggling today. 

Pam  06:38

You're doing great.

Kim  06:39

Single-digit Addition

Pam  06:41

So, let me say that in another way. We mentioned some foundational, common running threads, major relationships that we want to build in single-digit, addition, subtraction.

Kim  06:52

Yeah. 

Pam  06:52

And really, they just grow up to become the ones that we just mentioned. 

Kim  06:57

Yeah. 

Pam  06:57

So, when you said, "add-on", I'm actually going to request that like  we're not adding on. We're not...

Kim  07:02

Not plus, not plus. 

Pam  07:03

It's not additional to, it's not now you've got more. It's now we're just going to grow up. Like, the relationships mature. They get a bit more complex, more sophisticated.  And that's good news. If we develop those relationships when we're helping kids own their single-digit addition and subtraction facts, then bam. We just continue to develop them. 

Kim  07:14

Yeah. Yeah. 

Pam  07:26

If, on the other hand, we're in kind of a traditional place where we're like, "Okay, kids. Rote memorize these single-digit addition and subtraction facts," we now don't have those relationships to continue to build and develop to mature. We have to start from scratch because all they have is a bunch of rote memorized stories.

Kim  07:43

Mmhm.

Pam  07:44

Or, you know like, pictures in their head. We don't want just a picture of, I don't know, Mrs. Week sat on a chair eating two (unclear). 

Kim  07:51

No! 

Pam  07:52

Okay, but we don't. Those. 

Kim  07:55

Yes.

Pam  07:56

If I just have facts with a rap, or a rhyme, or a story, we didn't have, we haven't built in a kid, we haven't developed those important things that can now grow up into, like you just said, friendly numbers, place value, doubles, adjusting the properties, partial sums. All of that, we could have been have started develop, and now can grow up. 

Kim  08:14

Yeah, thank you for that. That's a really good point because as our kids are getting older, they're able to handle more complex numbers, but the relationships are the relationships. And we want to continue to develop in them how to trust the relationships that they've started to develop. When we develop these foundational things, when kids are really grappling, and then solidifying, and analyzing, and practicing, you're not going to be the teacher that says, "Well, I did this one thing, but they don't know how to do anything with it. Like, I did this one activity." You know, we hear that sometimes from people like, "Hey, I did something."

Pam  08:16

"I taught them the strategies."

Kim  08:17

Yeah, and then, "But they don't use it anywhere," or "They don't. "You know, they aren't applying it outside of this one task." And that's not surprising because kids need to be immersed in opportunities to see these relationships happening, so that then these strategies become natural outcomes.

Pam  09:17

Mmhm. Yeah, "immersed" is a good word. And that's kind of what we mean when we say, "come back to". We cycle back, and come around, and and have a running thread through your tasks because those running threads can be super, super helpful because the kid goes, "Oh, like we did work on that strategy last week a few times, and I'm seeing numbers today that are kind of begging for those relationships." And if some of the students don't notice that, then it can be the teacher to say, "Hmm. Sure wish I knew something about quarters," or "Sure wish I knew something about..." like, whatever it is that we're... "Sure wish I could add a bit too much here."

Kim  09:22

Yeah. Mmhm. 

Pam  09:59

Or, you know like, say we're adding 39. "Sure wish I knew what plus 40 was." 

Kim  10:02

Mmhm.

Pam  10:03

Some kind of a drop to go and the kid could go, "Oh, yeah. Like, we did work on that last week. I could use that relationship." It's not me saying, "Remove a friendly number and adjust." It's me saying, "Hmm, sure wish I could subtract 40 here."

Kim  10:16

It's a scaffold.

Pam  10:17

Yeah, and it's just enough in that moment. But it's an important scaffold that says we've been working on these relationships. Let's see if I can ping that if it's within the zone of your proximal development. If you can see my hands, I'm like creating this oval in the air here. If this is your zone of proximal development, can I throw something out that's just hits the edge of it, so you can grapple with it, and use it, and and bring it in.

Kim  10:42

Mmhm. 

Pam  10:43

Alright, so, Kim, what are some sequences of tasks? What kinds of tasks and what are sequences of tasks that we could do to help build the foundations for multi-digit addition and subtraction? 

Kim  10:56

Yeah. 

Pam  10:57

So, in the last couple episodes, we've talked about single-digit addition and subtraction, single-digit multiplication and division. Today, let's grow up a little bit. Not saying that ornery, but let's get a little more complex and talk about multi-digit. And remember that we've talked about messier kinds of tasks that are are not unstructured. They're not where kids are uncomfortable. It's not about sending kids off to discover math on their own. Kids are completely clear on what they're supposed to be doing. They know  the goal. They know their role and how they're supposed to use the materials that are given to them. What they don't know is you haven't given them steps to mimic. You've given them a problem to solve, and they're clear on the problem. They're clear on how to sort of go about that. We also have a couple other types of tasks that we're going to talk about. And how do you sequence types of tasks like that? So, Kim, here's one of maybe... Should I say my favorite? I think I could say my favorite. Favorite tasks to do with kids who are just beginning to really build the open number line. Which is not just a, "Okay, go do the open number line method." It's not that. It's not about a picture. It's not like making kids draw pictures. It really is about creating mental maps. That mathematicians actually have mental maps in their heads with which they use to see relationships, to feel relationships, and they then they use those relationships to solve problems. So, we want to create literally open number lines in kids' heads. A way to do that is to have kids measure. So, we can have kids measure. And one of the tasks that we've done is we've said, hey, if we want to create these shelves in our classrooms, and we're going to put books on the shelves, and the books are going to kind of be... We're not putting them like in a normal bookshelf where the ends are sticking out. We're putting them... How do I say this? Side by side.

Kim  11:11

Sure.  Mmhm.

Pam  11:53

Like, width to width.

Kim  12:16

Mmhm.

Pam  12:18

Is there another way to say that? Like, you can see the front cover. You can see the front cover, and then it's right next to the front cover of the next book.

Kim  13:09

Yep. 

Pam  13:09

And we're kind of lining them up that way. And if we want to line up, so we can see the front covers and they're lined up right next to each other, perpendicular. Perpendicular? Parallel? They're parallel.

Kim  13:19

Front cover to front cover.

Pam  13:20

Front cover to front cover. And they're lined up on the shelf like that, then how long of a shelf do we need for this particular book? What if we wanted 2 books lined up? How long? If I want 3 books, then how long would we need these shelves? Could we have kids take, I don't know, cubes. We got lots of cubes in the class. We could take kids. They have linking cubes. They could link them together, and they could measure certain books. And then we could say, "Hey, if we're going to put these 3 books together, how long does the shelf need to be. Or these two books together, how long does the shelf need to be?" And then we could say, "Well, let's like sort of represent where those lengths are. And we could put cubes in in a line, and we could do alternating colors of cubes. And, listeners, if you're... Let me give credit, so I don't forget. I want to give credit to the inspiration for this idea to Cathy Fosnot in her Measuring for the Art Show task in her.... What is her... Context for Learning? Yeah. In her Context for Learning program, she's got this....

Kim  14:17

It's on New Perspectives online.  (unclear).

Pam  14:19

Thank you. Thank you. New Perspectives online. In there, she has this fantastic task. And we've tweaked it, made our own context, but it was based on that, so, Cathy, we really appreciate that inspiration.  Yeah, for sure. Could we then look at an alternating set of cubes where we've got sort of 5 one color, 5 of another color, 5 of the first color, 5 of the second color, and maybe we've got 100 of them kind of lined up. And then could we say, "Hey, where is a book that measures, I don't know, 13 cubes long? Where would that be? And then ask kids, if they're looking at these cubes, where is the edge of that 13th cube? And how are they using those fives? And are they putting the fives together to make tens? And how are they... Where would I put a book that is 31 cubes long? And how are they thinking about that? And kids might count by fives. Kids might put fives together and count by tens. Kids might use that we've already marked the 13, which helped us mark a 10, and they might just go from that 10 to the 30. So, different uses of relationships. 

Kim  15:20

Mmhm.

Pam  15:21

Then we could start to use that sort of those measurements that we've marked on that cube line. And we could say, "Well, what if I've got a book that's 31 long, and I've got a book that's 15 long, how long does the shelf need to be? And we could go to where we'd mark the 31 and then we could say, "Hey, now we have to go 15 more cubes. How are we going to do that? Well, do we know where 31 and..." Now, we let kids sort of decide, do they want to go 31 and 10 cubes? Well, then, now we're at 41. We still have to go 5 more cubes. Hey, where's that going to be? Or could they from that 31 could they talk about, well, I could get up to 40. I know that's 9, but I was supposed to do 15, so I've got to do 6 more. What is 40 and 6? And where would that be? And, again, I'm sort of using that if I'm at 40, I've got that 5 cubes after it flashing at me because then the color changes, right, for the next set of 5? And so, I can use that 5 and one more to just get that extra 6, and then kind of mark it. And, lo and behold, I'm starting to add numbers using an open number line. Well, in that case, it's a bit of a closed. It's kind of a mix close open because I can see all the cubes. But we're marking jumps above that cube line. And we can literally help kids start creating a mental map of being able to use fives to get tens and tens to get other friendly numbers. We just mentioned it. What are some important things to do? Build place value. Build that sense of friendly numbers. Build this mental map of how numbers relate to each other. So, we really like kind of a measurement task to get kids into the measurement model of an open number line, all heading towards adding and subtracting multi-digit numbers. Okay, so one of the kinds of tasks that we've talked about is messy. Let me give you another messy. So, that one's kind of messy because kids are out measuring and then we're putting them together. What if we took I Have, You Need, and we did some more with it? So, listeners, if you have not heard of the super cool Instructional Routine that Kim made up called I Have, You Need, then go check out one of the other podcast episodes. But briefly, we choose a partner. I'm going to choose... Or a total. We're going to choose a total of 100 right now. So, Kim, if I have 70, you need?

Kim  17:30

30.

Pam  17:30

To make 100. Yeah? So, if I have 70, you need 30 to make 100. That's I have, You Need. I can make it more difficult, etc. Where it helps kids really travel the mental path of figuring the partner a lot, so that that path of figuring the partner to the really important number 100. They figure that. So, if I say, I have 67, Kim, what do you need? I have 67, you need? 

Kim  17:54

33.

Pam  17:55

33. We want kids to travel the path of figuring that partner, so that path becomes a well traveled path, and that they recognize partners of 100, partners of 1,000, partners of 10. That's the routine in a snap. If you want lots more detail on it, listen to another episode. But we can take that routine now and we can use it to connect addition and subtraction. So, Kim, when I said...  33 is probably not a great partner. Let me do another one. How about if I have 70... Yeah, 78. No, 77. 

Kim  18:29

Yeah.

Pam  18:29

There we go. 

Kim  18:30

I was going to say, you're going to have the same problem.

Pam  18:32

Am I tired today or what?

Kim  18:32

You are.

Pam  18:33

Okay, 77. You need?

Kim  18:35

23. 

Pam  18:35

23. Okay. So, we're going to use 77 and 23. I'm going to take out my trusty pen, so I can write these numbers down and make sure that I'm actually thinking. What did I say? 77?

Kim  18:45

77 and 23.

Pam  18:47

No, I don't want 77 because then I've got two 7s. Then I could have a 67. No, I don't like that either. 57! I like 57. Kim, if I have 57, what do you need to get 100.

Kim  18:58

43. You should maybe say why are all these rejecting? It's because they're then have a partner that has two of the same digits. You've got some confounding members.

Pam  19:07

I don't want confounding members. If I say 3, I don't want them to look at the 30 or the 3 and not sure which number I'm referring to. So, yeah, I'm going to choose 57 and 43. It's four different digits that we're dealing with. 

Kim  19:19

Yes.

Pam  19:19

Thank you, for letting me do that. Preparation is not. Okay, so if I say, Kim, I have 57. What do you need? It's in effect, I'm asking the question, 57 plus blank equals 100. Would you agree? 

Kim  19:33

Yeah. 

Pam  19:33

So, we could call that a missing addend problem.

Kim  19:35

Yep. 

Pam  19:36

But I could also say how is that related to 100 minus 57?

Kim  19:42

Yeah.

Pam  19:43

And now, it becomes a subtraction problem. And we could think of that the very same question both ways and using an open number line to represent what's happening. We can see that. So, that 57 plus blank equals 100 if I had an open number line from 0 to 100 100, and I put 57, where would it be, Kim, about? 

Kim  20:03

I'm sorry. I was writing on my paper. (unclear).

Pam  20:05

That's okay. I have an open number line. Hey, we're a hot mess today. If the open number line goes from 0 to 100, and I'm going to to put 57 on the number line, where am I drawing?

Kim  20:16

A little to the right of halfway. 

Pam  20:17

Alright, so almost halfway is 50. Or halfway is 50, so a little bit to the right would be 57.

Kim  20:22

Yeah. 

Pam  20:22

Then I'm going to draw a jump from 57 to 100 and put a question mark above it. That's what we're asking for, right? 

Kim  20:28

Yeah. 

Pam  20:28

Well, notice, how that picture. So, let me just draw it again. I've got 0 to 100, open number line. 

Kim  20:33

Yep.

Pam  20:34

57 just past half. 

Kim  20:36

Mmhm.

Pam  20:36

And then I've got a jump to 100. So, I've got this arc from 57 to 100. And above it, I have a question mark.

Kim  20:42

Mmhm.

Pam  20:43

So, 57 plus what is 100. But it could also be 100 minus what is 57.

Kim  20:49

Yep.

Pam  20:50

It could also be 100 minus 57 is what? Lots of nice relationships that are far more than me just saying "turnaround fact", and it's far more than me just saying addition is related to subtraction. Like, we're actually feeling the relationship. And so, we could do some work with I Have, You Need to connect addition and subtraction, right? Those are a bit messier. Another task, sets, category of tasks that we like to sequence with messy tasks are tasks where we analyze strategies.

Kim  21:23

Mmhm. 

Pam  21:23

We like to sequence those with Problem Strings where we really cinch and get better at strategies. So, let's talk about a couple of analyzing strategy types of tasks. So, one of them could be where we pick curated numbers. What do I mean by that? Well, I mean that we could actually choose numbers to give kids problems to solve where we've thought about the numbers, and we're giving kids those numbers in such a way that we say, "Hey, how did you use them?" So, for example, I might say the numbers 38 plus 2 plus 15. And I might say, "Hey, Kim. Which numbers do you want to add first? 

Kim  22:02

Yeah, I'm going to do the 38 and 2.

Pam  22:04

And then we'll sort of, "Hey, why? Ooh, that got us to that nice 40. Now, we're at that friendly 40. And then, whew, it's not too bad to add the 15. But I also might give you numbers like 38 plus 15 plus 20. And I might say, "Well, according to what you just did, you should add the first two numbers, right?" And you might be like, "Ooh, icky."

Kim  22:24

Yeah. 

Pam  22:24

What like 38 plus 15 plus 20. What are you going to do? 

Kim  22:27

I'm going to do 38 and 20.

Pam  22:29

Bam, and then add the 15. So, ooh, there might be times where you Add a Friendly Number, and then tack on the rest. 

Kim  22:35

Yeah. 

Pam  22:35

So, in the first one, 38 plus 2, you got to a friendly number, and then you added the rest. 38 plus 20, you added a friendly number, and then you add on the rest. We can give kids numbers like that. Let me give you one more, Kim. What if we had 38 plus 15 plus 39?

Kim  22:53

I'm writing down, 38 plus 15 plus 39?

Pam  22:55

Yeah. 

Kim  22:56

Oh, can I double 38? (unclear).

Pam  22:58

Yes, but that's not what I was going for.

Kim  23:00

Oh. Okay. Oh, so I probably do 38 and 40.

Pam  23:04

There you go. That's what I was going for. Yeah.

Kim  23:06

Yeah. 

Pam  23:07

So, 38 and 40, instead of 38 and 39, and then you back off 1. And then, now, you'd probably tack on the rest of it. 

Kim  23:13

Yeah. 

Pam  23:14

So, now we're having a conversation about adding a bit too much and adjusting back. 

Kim  23:17

Yeah. Hey, I'm going to jump in here for a second because this is an example of helping kids become judicious strategy pickers. So, if you've developed some strategies with your kids, it is perfectly okay to lob out specific numbers that encourages them, that nudges them, that... Like, you're like laying the groundwork for them to make choices. Yeah.

Pam  23:50

It's like, we're flashing patterns in front of them. 

Kim  23:52

Yes. 

Pam  23:52

We're high dosing patterns.

Kim  23:54

Yes.

Pam  23:54

And saying, "Notice, this pattern." 

Kim  23:57

It's the scaffold again. 

Pam  23:58

Yeah. And we're very purposeful about it. 

Kim  24:01

Yeah.

Pam  24:01

I can't tell you the number of times I've had conversations lately where people say, "Pam, what you're suggesting is terrible." They say, "When you suggest that we don't teach towards algorithms, you're suggesting that we just let kids loose, and we're going to be mean, and not tell them how to do math, and we're going to expect them to go out in a field by themselves and discover the algorithm." And I say not what we're suggesting at all. We're actually very purposeful and focusing kids by flashing, high-dosing these patterns at them. It's just what we're high dosing at them are the major patterns. It's the major relationships that lead to the major strategies that mathy people actually use. We're just helping them actually math the way mathy people actually math. Yeah, nice. I'm glad you brought that up. So, one thing we could do is use curated numbers to have kids realize what they're doing kind of naturally.

Kim  24:51

Mmhm. 

Pam  24:51

But we're picking the numbers on purpose, so that they kind of do those things, and then we can point out what they did and go, "Huh. Let's put some words to that." 

Kim  24:59

Well, I'm going to... Sorry, I'm going to interrupt you one more time because we... In these sequences, we choose numbers to strengthen the muscle of looking at numbers because then the natural outcome is that they tend to want to look at numbers. They're not going to always have the perfect numbers for the rest of their life. They're going to look for the relationships that they have developed and have gotten stronger in.

Pam  25:22

Nice. So, one other thing I want to mention is we've talked in the last couple episodes about smudge problems. 

Kim  25:27

Yeah.

Pam  25:28

Where basically we put kids work down. We smudge out part of the work. And then we have kids fill that in and then analyze what the kid was doing on the model that they just filled in. One of the things that we could do, specifically here. I just talked about using I Have, You Need to connect addition and subtraction. But in those smudge problems, we can also have kids notice that kids could answer a subtraction problem by finding the difference between numbers.

Kim  25:53

Mmhm. 

Pam  25:53

We can like put it in front of them. We can say Kim was solving the subtraction problem, 78 minus 39, and and fill in the smudges. And when they look, the kid looks at the number line, it starts at 39 on the left and goes up to 78 on the right. And then we've got some jumps in between, and they have to fill in the sort of. Like, we smudged out where they jumped, and we smudged out where they landed. And so, the kid fills that in. And then we're like, "Wait. The problem was 78 minus 39. Why is 39 on the number line?" So, we can actually put in front of them that the kid was solving a subtraction problem using distance, using the difference between the numbers. And then right underneath that, we can have a kid removing the 39 and landing on the answer. Again, filling in the smudges, and then analyzing that. The big idea is that we're analyzing strategies and going, "Huh. Why would a kid do that? That's interesting. Want to work on that? Let's do a Problem String to make that, to get that strategy actually pretty good. So, we sort of put it in front of kids, high dose, flash them these patterns, and then do Problem Strings to help them get better at them. 

Kim  26:48

Mmhm.  Yeah.

Pam  26:55

Give those to kids and let the kids fill in the smudges. Then, what do they have in common? I'm going to let kids look at those cards and say, "Find the same strategy. Find the same model," nd let them actually sort and get just more clarity around the different strategies and the different models, what the different strategies can look like on the different models, while they're practicing the strategies by filling in Those smudges and analyzing those strategies. And then we could give them a new problem and let them go at that new problem, different models, different strategies, sorting all that out, making sense of it. That would be a way that kids can sort of practice those strategies. Often, Kim, teachers will say, "I've taught them the strategies, and they just still do an algorithm. Or they still are splitting and doing partial sums. Or they're still like typing in their calculator." Whatever. They won't use the strategies for whatever reason. Well, this is a way... Well, let me, let me say one more thing. "I try to force them to use the strategies, and, Pam, they won't do it." So, here's a way to... Not force. That's not a polite word or a polite thing to do. But to kind of insist that kids are diving into the strategies because their task right now is to fill in the smudges and analyze what they're seeing. That's the task. 

Kim  26:55

Yeah, Yeah.

Pam  26:55

As they get better at those strategies, then we can make anchor charts that sort of solidify those strategies. Now, we have a few of the major strategies on an anchor chart, and then we can not only clarify the meanings by putting, having the kids co create those anchor charts with us. Now, kids can reference those anchor charts when they're in their work. They're doing stuff, and we could go, "Huh. I sure wish I had some strategies I could think about to solve that problem." Kids could look at the anchor chart, and they could be like, "Oh, that's right. Like, I have those relationships I could use." Alright, so cool. We've talked about some messier problems we could do. We've talked about some analyzing strategy problems we could do. How do we practice that multi-digit addition and subtraction? Well, let me give you one way. I'm not going to say this is the only way, but here's a way that we kind of like. Could I put some smudge problems on a card for... I'm going to pick a problem, a specific problem. Maybe... I'm not going to tell you what the problem is. But let's pick an addition problem. So, say it's a two-digit plus two-digit addition problem. Could I solve that addition problem using one strategy on an open number line? Smudge out some of the work. Now, I'm going to on a different card put that same exact problem with the same model on an open number line, but I'm going to put a different strategy. Smudge out some of the work. Could I have a third card, same model, but with a different strategy, same problem. So, basically, now I've got the major strategies with on the same model. Now, I'm going to do the same thing with the major strategies on a different model.  it's intriguing. It's not what's the answer. It is...  Exactly.

Kim  29:46

...what's going on here?

Pam  29:55

if it's just getting the answer, kids will just get the answer any way they want.  Yeah,

Kim  29:59

Mmhm.

Pam  30:00

But if it's analyzing the strategies by filling in smudges, and then looking at it, now I'm intrigued. Now, it's a puzzle to solve. Yeah, thanks. Go ahead. 

Kim  30:07

I like about this particular practice as well is as they're doing this intriguing work, kids are developing a better sense of the idea of strategies and models, and so it's working on the idea that you have kids say, "I did a number line". Because laid out in front of you for one problem are all of these different strategies and different models. I also recently spoke to a leader on a campus who said, "How do you know when you see a problem, which model or which strategy you want to use?" And so, we got to have a conversation about the type of problem and the strategies. And I feel like the sense was that sometimes you do these models. Yeah, and sometimes you do these models. What And that we should know right off the bat which one's the best.  Right. 

Pam  30:19

Yeah.

Kim  30:34

Yeah. 

Pam  30:36

Well, and in reality, we need experience to say, "Ooh, based on these numbers, what strategy feels great and what model would is flowing for me today?" And if I own the models and own the strategies, then those choices become kind of natural. 

Kim  31:11

Yeah.

Pam  31:12

Yeah. That's nice. I like it. Ya'll, you can create tasks like this, and you can sequence them with Problem Strings to help your students build the major relationships that lead to the major strategies for addition and subtraction. If you'd like, readymade tasks that can help you do exactly this to build the foundations for strategies, check out our Hand2Mind product Foundations for Strategies, Multi-Digit Addition and Subtraction for easy to read lessons that can help you exactly do that. 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. Let's keep spreading the word that Math is Figure-Out-Able!