March 12, 2024
Pam Harris
Episode 195

Math is Figure-Out-Able with Pam Harris

Ep 195: Early Addition Important Relationships

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Math is Figure-Out-Able with Pam Harris

Ep 195: Early Addition Important Relationships

Mar 12, 2024
Episode 195

Pam Harris

When should you start developing which relationships? In this episode Pam and Kim discuss the important addition relationships for first grade and how to develop them throughout the year.

Talking Points:

When do I teach what?

- Download The Most Important Numeracy Strategies Free Ebook!
- We want students to understand relationships so that strategies are natural outcomes, not memorized procedures.
- Partners of Ten
- Doubles
- Plus ten
- Modeling is super important

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

Listen on

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When should you start developing which relationships? In this episode Pam and Kim discuss the important addition relationships for first grade and how to develop them throughout the year.

Talking Points:

When do I teach what?

- Download The Most Important Numeracy Strategies Free Ebook!
- We want students to understand relationships so that strategies are natural outcomes, not memorized procedures.
- Partners of Ten
- Doubles
- Plus ten
- Modeling is super important

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

Pam 00:01

Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam.

Kim 00:07

And I'm Kim.

Pam 00:08

And I start, Kim. And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians do. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps can actually keep students from being the mathematicians they can be.

Kim 00:33

We wouldn't want that. (unclear).

Pam 00:36

No.

Kim 00:36

(unclear).

Pam 00:37

No, no, no, no. Hey, Kim.

Kim 00:39

How are you doing?

Pam 00:40

I am a little stretched. We are writing, writing, writing. And it's super exciting all the things that we're getting done, but Whoo! We are...

Kim 00:49

It's some long days.

Pam 00:50

...feeling a little stretched today.

Kim 00:53

Yeah. So, I know you know this, but let's tell everybody that in Journey, your implementation support program, a few months back, one of the things that... Well, one of the things we put in there are videos of Problem Strings. And so, we go film in a live classroom with real kids. Kim's laughing. Yeah. Pam says "real kids", and it makes me crazy!

Pam 01:18

Because I say, "You know, we get video of real kids." And she's like, "What? Not fake kids?" (unclear).

Kim 01:24

They're not scripted.

Pam 01:25

It feels... Yeah, it feels it doesn't feel staged. It's not scripted. Because we actually go in, in a real class. Is that better? Real class? Whatever. Anyway. Yeah.

Kim 01:34

Yes, we go in classrooms, and we film an expert teacher or one of us filming. Anyway, we do video. Then, we put video and kind of a voiceover where we talk about the video and all the extra. Yeah. So, we shared a video of a Problem String with first graders using one of my favorite strategies. It's not my only favorite. But I sure do love it. And a member, at that time, was asking, "Hey, I'm curious what time of the year is this?" And it got me thinking that... And the response that I gave to her were kind of the major relationships that I think first graders need to work with kind of all throughout the year. So, there are some types of things we do first, and maybe towards the middle. But really, these are major relationships that we kind of implement early, and then continue to work on all year long? And so, we've talked about major strategies for each operation before here on the podcast. There's some episodes about that.

Pam 02:32

Well, and can I interrupt you?

Kim 02:33

Yeah.

Pam 02:33

Sorry. So, just to reiterate. What you're saying is, is that this Journey member said like, "When in the year do you do this?" And you pushed back. Go ahead.

Kim 02:43

Yeah. She said, "When in the year is this because I want to know when in my year do I do this thing?" And what I said was... And I'll tell you, it was it was the Over strategy. And the question was, "About when in my year would I do this? Is it like early? Is it late?" And so, the conversation that we had...

Pam 02:59

As if there's a moment, right?

Kim 03:01

Yeah.

Pam 03:01

It's like you do the thing. And you've done the thing. They've got the thing. You're done with that thing. You don't... And we're not saying that that member necessarily had all that in their head.

Kim 03:10

No, no, no.

Pam 03:11

But we hear that a lot because we were taught that way. We were taught that you do it here. You teach you to mastery. You're done. You move on. And so, you push back on that, and you're like, "Actually..."

Kim 03:22

Well, and there are some strategies that we would say are precursors to others. But for this particular conversation, what I shared was that there are some major relationships that I think first graders need to work with. And they might be a little less sequential, like this is, you know, beginning of the year, this is like the second nine or six weeks, and then this is the third. So, we thought that because we've talked about major strategies before and we've talked a little bit about very young counting ideas, we thought we would share what do my kids need to know in first grade? Because we haven't really dialed into a particular grade. So, yeah, let's talk about that today.

Pam 04:03

Excellent. Because there is that super young counting stuff. And then, we've done a lot of work with the multi-digit today. Let's do some work with single digit stuff. And we have an ebook that we've created. We will put the link in the notes, in the show notes, where you can download it. We call it the Major Strategies Ebook. I think. Is what we call it? That's what we call behind the scenes. I don't know what we actually call it.

Kim 04:27

We call it the ebook. It's a... it's a... (unclear).

Pam 04:30

Major Strategies Ebook? Kim's looking it up.

Kim 04:34

Most Important Numeracy Strategies. There you go. There's the official title. But really there's a ton more in there. And...

Pam 04:40

Yeah, there's a lot of information in there. It's jam packed. You are going to want that if you don't already have it. Most of the listeners are like, "Of course, we've already downloaded that, Pam." Well done. Good job. So, we outlined the major relationships that we believe kids need to develop, so that strategies become natural outcomes. There's a hierarchy. And we think that were some of the only work that has been done to develop this hierarchy of strategies. And because if you're not going to rote memorize algorithms, then what? And e put that in this ebook. Yeah.

Kim 05:17

Yeah, and I want to add in real quick. One of the things that we've talked about a lot that you're really dialing in on, and I think doing a really nice job of sharing, is that it's not like "Memorize all these new strategies." It's, "What are the relationships?" Like, what are the actual big ideas of mathematics that if you strengthen those, then these strategies become natural outcomes?

Pam 05:38

Mmhm.

Kim 05:39

And I think you've said that really well, lately. And I just wanted to mention it again because it's less of like, "Now, I have this checklist of strategies."

Pam 05:48

"Kids could memorize before, Pam, only one way. Now, you want them to memorize five?" Nope.

Kim 05:52

Right.

Pam 05:52

I don't want them to rote memorize anything. I want to develop their brains, like you said, so that these relationships ping for them. They have intuition when they see the numbers. They like, "Oh, this is what's coming to my mind. Ooh, so is this. Oh, I have these major relationships that are happening. I'm going to use that one this time to solve the problem." "Actually, I wish I would have used that one. Now, that I've seen what I did it's occurring to me. Oh, I could have just done..." That's mathematical behavior. What pings for you. Use it. Look back. Decide, could I have done something that sort of traveled a little slicker today? Nice. I appreciate that. Thanks. We've gotten more clear. I think we were kind of clear in our heads about what we meant. But then, we would hear people say back, "You know, my kids can't do one way. How do you want them to memorize five ways?" I'm like, "Oh, yeah. It's about memorizing. (unclear)." Yeah.

Kim 06:46

Yeah.

Pam 06:46

Yeah.

Kim 06:47

It's an evolution.

Pam 06:48

Alright. So, yes, we're all getting better at being more clear about all the things. Kim, what are the major relationships that we would want to build in, say, first graders as we look at single digit addition facts?

Kim 07:00

Yeah. So a really, really important one is Partners of 10. Which I think then, you know, that basic understanding of partners also extends to partners of 20, partners of other things. But within each of those other groups of larger partners of 20, partners of 100, there's this basic partners of 10 relationship that we really want kids to own.

Pam 07:23

You know, I remember the first time you said, "When you think about partners of 20," I in my head thought about like... How do I even say this? Kind of like all the partners of 20. And you said it's really 20 plus the partners of 10.

Kim 07:38

Ten and the Partners of 10, yeah.

Pam 07:41

Oh, sorry. Thank you.

Kim 07:43

Mmhm.

Pam 07:43

Wait. Oh, yeah, partners of 20. I was thinking about partners of 30. I don't know why. So, for partners of 20, yes. Wow. It's thinking about 10, and then like... Right, now you have the 10, so now you're in the teens. And so, then you're just using partners of 10. So, let me go partners of 30. If you're thinking about partners of 30... That's probably when I thought of it actually. That's why it was in my head. It was the moment when you said, "Well, then you could work with kids for a short period of time in partners of 30." And I pushed back and said, "No, no, no. Don't do that." And you said, "Yeah. Pam, it's not any random number to get to 30. It's once you have 20 established, then you're really just dealing with partners of 10. If you were dealing with partners of 60, you really want kids to think about I got 50, and then partners of 10."

Kim 08:28

Yeah.

Pam 08:28

That's the work you're doing is to realize that inside these bigger numbers, there's always kind of this big multiple of 10 and just partners of 10.

Kim 08:37

Yeah. Yeah, and I think people can sometimes here that. You know, when we talk about I Have, You Need. They might hear it as, "Let me step up and do partners of 10, and partners of 20, and partners of 30." And once you get outside of the 10, then it's really what you're saying. The understanding of the larger multiples of 10 and the partnership that you're trying to develop. It's not, "Now let me just step up and do... This week we do partners of 30, and then the next week we do partners of 40."

Pam 08:37

Yeah. Or partners of 35.

Kim 08:38

Yeah.

Pam 08:39

No, no, no.

Kim 08:40

Yeah.

Pam 08:41

Yeah, yeah. Cool.

Kim 08:43

Okay. So, Partners of 10 is one major relationship. Another major relationship is Doubles. You know, kids early on have some intuitive sense about early doubles, but we can do some work to capitalize on that relationship because it extends far beyond just first grade.

Pam 09:30

Yeah, mathematicians play with doubles. And then, once they kind of have a sense of doubling, they play with halving. Doubles and halves are super important. I did not know that as a non real mathy person for very long. And I remember the day when I started realizing doubles are super important. And yeah.

Kim 09:51

Would you call yourself a non... I don't know what you just said. I would say you were a traditional.

Pam 09:57

I said I was a non (unclear) Non real mather. I was not real mathing.

Kim 09:59

Non real mather. Yeah.

Pam 10:03

Not on purpose.

Kim 10:05

Yeah, no, no, no. Yeah, for sure.

Pam 10:07

Yeah. (unclear).

Kim 10:07

And then, the other major relationship is the idea of Plus 10 to help you with things like plus 9 or plus 8. So, a little bit Over. A little bit more than you need. But a lot of it is plus 10 or a multiple of 10, so you're not going 18 to get to 17. Eighteen more to get 17 more. It's really about capitalizing on the 10s or the hundreds.

Pam 10:32

Because there's this nice pattern that you can think about adding 10. Especially in first grade. If I have some single digit number, and I add 10, that's the definition of a teen.

Kim 10:42

Mmhm.

Pam 10:42

So, you know, I've got 6 and I add 10. Bam, I need to know what 6 and 10 is. I need to know what 10 plus a single digit number is. That that's a definition of those teen numbers. Sure. That's super important. And so, we can almost practice that while we kind of try to develop an Over strategy kind of thinking, Like, we can say, "Hey, what's 6 plus 10?" And the kid's like, "I know that. It's 16 because I know what the teens are." And you go, "Cool. What's 6 plus 9? If 6 plus 10 is 16, what's 6 plus 9?" Oh, is one of my favorite questions of all time to ask a kindergarten, first grade kid. That's kind of, you know, right on the cusp of what they're thinking about. Second grade, if they've never really thought about stuff. I can even ask that question of third grade, fourth grade kids. Where, you know like, "Hey, what's 7 plus 10?" "That's 17. I know the teens. You're not going to get me here." "Okay, what's 7 plus 9?" "Well, let me think about that. Ooh, those are related. Hey, that's just 1 less!" And their eyes light up. And it's so exciting.

Kim 11:37

Yeah.

Pam 11:38

Cool.

Kim 11:39

So, when I responded to our Journey member, and I shared those three major relationships, then I think part of the conversation was, "So, I wouldn't say let's do Partners of 10. Let's make sure all the kids own them really well. Now, let's move on to the new relationship." So, if my goal is that I want kids to get funky problems, and they have these major relationships, and they choose which of these relationships that you want to use, I'm going to start pretty early to develop kind of all of them just, you know, with Problem Strings, and with routines, and with activities that we can do, and then highlight those relationships all throughout the year.

Pam 12:20

Yep. And in order to choose, they have to have choice.

Kim 12:24

Yeah, for sure.

Pam 12:25

We want to own them, really develop them, and then when they work with them as they choose. Okay, So, Kim, if I asked you a most missed fact, an often missed fact. If kids are going to miss one, they typically will miss this.

Kim 12:40

Yep.

Pam 12:40

I would love to hear you use those major relationships to solve that particular problem. Okay? Okay. Alright. So, if I were to say... So, teachers, do you agree with me? 7 plus 9. If a kid is going to miss a fact, often they will miss 7 plus 9.

Kim 12:55

Yep.

Pam 12:56

Alright, Kim, how could you use each of those three relationships? Partners of 10, Doubles, and that's sort of Over, plus 10, plus 9 relationship? How could you use those to solve? All of them, each of them to solve 7 plus 9. Go.

Kim 13:07

Okay. Well, this is a good problem because it is a most missed. So, I'm going to actually write, so I don't forget something. But I'm thinking if I'm using Partners of 10, then I could think about 7 plus 3, which gets me to 10. But then, I also have 6 left. So, I take the 3 from the 9. So, 7 plus 3 plus 6 gets me the 16.

Pam 13:08

And that's using the Partners of 10.

Kim 13:13

Yeah, but I can also use the commutative property and turn that around. And I don't actually have to turn it around, but I could also say the 9 plus 1 plus 6 would also get me 16. So, I'm either making 7 and 3 be the partner or I'm making 9 and 1 be the partner. But in either case, I have a leftover 6.

Pam 13:33

Nice, nice. And that's using Partners of 10. Cool.

Kim 13:36

Yep. Okay, so then I also am thinking about Doubles.

Pam 13:45

Mmhm.

Kim 13:46

And I think maybe one of the first things that kids do is think about the Under Doubles. So, an Under Double would be 7 and 7, and then you have a leftover 2 from the 8, and so also 16. But also we want kids to develop this idea that I could use the Double Over. So, 9 plus 9 would be 18, and then I have 2 too much. So, I would subtract 2 to get to 16.

Pam 14:34

Nice.

Kim 14:35

That's another double.

Pam 14:37

I was talking with a slightly older kid the other day.

Kim 14:39

Mmhm.

Pam 14:39

And I said, "Could use doubles to help you think about that." And the slightly older kid said, "Well, if I know 7 and 7 is 14 and 9 and 9 is 18, then 8 and 8 has to be in the middle of that. That's 16."

Kim 14:51

I love that.

Pam 14:52

It's for a different problem, but it just kind of reminded me because I was writing as you spoke, and so I have on my paper 7 plus 7 is 14 and 9 plus 9 is 18. So, it's cool to see that kid sort of, "If I don't know that one in the middle, I know it's in the middle. Yeah. Anyway.

Kim 15:04

That's super cool.

Pam 15:05

Yeah.

Kim 15:06

And then, the last one is kind of this Over strategy or using 10. So, I could say 7 plus 10 is 17, but I didn't really have 10. I only had 9. So, I'm going to backup 1 to get to 16.

Pam 15:19

Add a bit too much. You're supposed to add 9. Add a bit too much, add 10. Backup that 1 because you added too much.

Kim 15:27

Mmhm.

Pam 15:27

Nice.

Kim 15:28

And you could theoretically do 9 plus 10, and then backup because you only need to do 7.

Pam 15:36

Bleh.

Kim 15:37

It's not horrible. It's just the problem is set up so nicely to have the 9 be the second addend. So, (unclear).

Pam 15:42

Yeah, yeah. But I think what you're saying is as as these relationships grow up, as the numbers make... Yeah, as the numbers in relationships grow up, you might have a problem that you're adding something like 97. And you could say, "Well, I can I'm going to add 100 and backup 3." Oh, well, in that case, that's not too bad.

Kim 15:58

Yeah.

Pam 15:58

Yeah. Cool.

Kim 16:00

Cool.

Pam 16:00

Alright. So, 3...

Kim 16:01

Can I give you one?

Pam 16:02

Yeah, yeah.

Kim 16:04

You always have me do math. Okay, you gave me 9 plus 7. But I'm going to give you what I think is the very most missed.

Pam 16:12

Okay, okay.

Kim 16:13

And I'm going to give you 8 plus 7. But maybe the listeners can pause and think if they can come up with those ways to use those relationships first.

Pam 16:24

So, for 8 plus 7, listeners, can you think of a way to use Partners of 10? Can you think of a way to use Doubles? And can you think of a way to use Over, like this idea of plus 10 and backup?

Kim 16:36

And if these relationships are not new to you, I wonder if you could think about how you'd represent them with young students.

Pam 16:42

Oh, nice.

Both Pam and Kim 16:43

Yeah.

Pam 16:44

In fact, you might be interested to know that on my paper, as you were talking, I have 2 number lines and a set of equations.

Kim 16:51

I only have equations, but I wasn't trying to model for anybody but me right here, so.

Pam 17:00

Well, so I would be curious what you model as I talk through 8 plus 7. Alright, so listeners, if you're... Pause, right now, if you're going to think about it and model, and then come on back. Okay, 8 plus 7. Alright, Kim. For the relationships about partners of 10, I'm going to go ahead and say I'm going to get 8 to 10. That takes me 2, so 8 plus 2 is 10. But I was supposed to add 7. I've already added 2, so I have 5 leftover. Ten and 5 is 15.

Kim 17:25

Okay.

Pam 17:26

But I could also turn that around, if I needed to or had to, and I could start on the 7. And I could say well, to get to 10 from 7, that's 3. But I was supposed to add 8. I've already added 3, that's 5 leftover. And 10 and 5 is also 15. That would be two different ways to use Partners of 10 to add 8 plus 7. Yeah?

Kim 17:45

And this is not a modeling conversation, but if you are representing because you already use these relationships, your 10 and your 15 should be the same place on your two separate number lines.

Pam 17:54

And your 7. So, if I look at my paper right now...

Kim 17:58

Yep.

Pam 17:59

...the 7 on the second number line is to the left of the 8. But everything else lines up. The 10s are in vertical lineup, and the 15s are in vertical lineup. The jumps of 5 are the same size.

Kim 18:10

Yep.

Pam 18:11

The jump of 2 and 3 are different sizes.

Kim 18:13

Yep.

Pam 18:14

Yeah. All important. By the way, Kim, that's important for teachers to get right.

Kim 18:18

Yes.

Pam 18:19

It's not important to enforce on students.

Kim 18:22

Yeah. And you're not going to be perfect right away, right?

Pam 18:25

Teacher's aren't going to be perfect. Just do your best. Yep, do you're best.

Kim 18:28

Get better and better.

Pam 18:29

In fact, one of the ways that we suggest teachers get better at making their jumps proportional, so they sort of fit the size is to actually talk out loud while you're drawing them. That's going to be helpful for kids. If you say, "Okay, so 8 and 2 to 10. You said you had 5 leftover? Okay, 5 is much bigger than 2. So I'm going to make this jump bigger. Alright, 15." And then, if you do the one below it, you can say, "Well, if I'm going to start on 7, that's back here, that's 1 back from 8." So, I'm kind of pointing to the left a little bit. "I'm going to start on 7, when I jumped 3, that better be bigger than that jump of 2." Like, you literally talk out loud about what you could be thinking about. And if you miss something, because you're talking out loud, you might actually have a student say, "Hey, shouldn't that jump be longer?" And bam! You are building their proportional reasoning at the same time (unclear).

Kim 19:12

(unclear). That's a favorable...

Pam 19:14

Outcome.

Kim 19:15

...comment from your kid.

Pam 19:16

Yes. Yeah, it's. nice. Alright, so 8 plus 7 we did Partners of 10. What's the next one? Doubles?

Kim 19:21

Doubles.

Pam 19:22

Okay. So, if I'm thinking about 8 plus 7. That's almost too easy with doubles. But I could think about the Under double, like you said, 7 plus 7. If 7 plus 7 is 14, then 8 plus 7 is 1 more than that, 15. (unclear).

Kim 19:34

Oh, and I... I'm sorry. As I was over on one side of my paper. But when you went to doubles, I totally went back to where I was writing. Because it would be a really fun conversation to find the doubles within my problem and the doubles within your problem and have a conversation with kids about how you needed plus 1 and I need to plus 2.

Pam 19:52

Oh.

Kim 19:53

And, you know how that's related to the 15 and 16.

Pam 19:57

Because my numbers are only one off and your numbers... Your 7 plus 9 were 2 off. My 8 plus 7 is only 1 off. Oh, nice. Okay. Cool. So, I'll finish the double by saying I can think about 8 plus 7 as 8 plus 8, which is 16. But it's only 8 plus 7, which is 1 less. So, 8 plus 7 would be 15.

Kim 19:59

Yep.

Pam 20:03

Cool. Okay. How about... Ooh, Kim, don't let me forget, I want to talk about teachers needing to know all of the strategies.

Kim 20:24

Okay.

Pam 20:25

Yeah. Okay. Last one. The idea of Over. So, I've got 8 plus 7. Typically, if I was going to do Over would kind of use the commutative property first. So, I'm actually going to do that because that's where my brain is going. So, I'm thinking about 7 plus 8 as 7 plus 10. That's 17. But I've added a bit too much. I only needed to add 8. I added 10, so it's 2 too much. So, I backup 2 from 17, and that's 15.

Kim 20:53

Mmhm.

Pam 20:54

Now, I'll do the way it was given to me. 8 plus 7. So, I can think about starting on the 8, adding 10, that's 18. But from 8, I was only supposed to add 7, not 10, so that's 3 less, and three less than 18 is 15. And on my paper right now, I have two number lines. I have the 7 is the furthest number to the left on the first number line. And right below it on the second problem, when I started with the 8, that 8 is to the right a little bit. The jump of 10s are the same length. And so, when I jumped 10 from seven to get to 17. When I jumped from 8, 10 to get to 18, the jumps are the same size. But because I started farther to the right, the 17 is one place and the 18 is just to the right of it. But when I backed up 2 on the first problem, backed up 3 on the second problem, they both line up. And the 15s. I'll be honest, are not quite in line. But they should be. On my paper, they're not quite. I was getting to the edge of my paper, and I was kind of running out on. 18 didn't go quite as far as I wish it.

Kim 22:01

Excellent.

Pam 22:01

I wish it did. Cool. So, teachers, no matter where you are, you can sort of practice figuring out, can I use all these relationships? Have you built the relationships? If you have, then you can practice representing them. How are you modeling those and everything in between?

Kim 22:15

Hey, you wanted to talk about teachers knowing all the strategies.

Pam 22:18

Thank you. I appreciate that. It is most important for teachers to know all these strategies, so that you can clearly and expertly help your students build them. It's also important for students to build these, but as long as they got most of them, we're pretty happy.

Kim 22:36

Yeah.

Pam 22:36

Like, our goal is to always develop all the relationships because we just want to keep developing all the math. But it's most important for teachers to have them all because you want to be able to. When a student says, "I don't know 8 plus 7," you want to be able to say, "Well, what do you know?" And then, you have in your repertoire things to say, "Well, do you want to get to 10? Oh, okay, that might be helpful. Do you know a double? Do you know, 8 and 8? Do you know, 7 and 7?" Or you might say, "Do you want to add a little too much, and then adjust?" Like, you need to know those, so that when a kid is stuck, and you say, "Well, what do you know?" And they're like, "I don't know. Well, 8, 9, 10, 11..." And they start counting by ones, you're like, "You can count by ones, but what else? Do you know..." And then, you can kind of lob out these major relationships, so that then they could go, "Oh, yeah. I do know that. I can work on that. I can build from there." So, it's super important, teachers, that you don't just decide, "Well, I got one. I got a favorite. I'm good enough."

Kim 23:33

Yeah.

Pam 23:34

You need to know them all, so you can prompt them (unclear).

Kim 23:36

Well, and you and I feel so strongly about knowing your content, knowing your kids. And if you're sitting down with a student, and you're not sure what they know. The more that you know your content, the more you can kind of poke into their brain and find out where you can meet them.

Pam 23:50

Yeah, absolutely. One of the things that we did in our Problem String books is think really hard about how to develop each of these major relationships, how many Problems Strings to give teachers as examples of building them, how to tweak the Problem String from string to string, how to up the ante in between strings, so that once you've done some work, they can get a little bit more complicated, more complex, so kids continue to build their thinking. We really thought long and hard about that. It's been super fun to work not only on these younger relationships, but then to also, in the same book, once we've done Problem Strings with single-digit facts like we've just done, to then have those strategies grow up into relationships for bigger numbers. So, like, Kim, Partners of 10, that grows up what we're doing double-digit numbers, to...

Kim 24:43

Partners of 100 and Get to a Friendly Number. If we know partners, then we can get to a friendly number really nicely.

Pam 24:49

Yeah.

Kim 24:50

Well,

Pam 24:50

So, if I'm adding 48 and something, I can think about, "Oh, well 48 is almost 50," and I'm kind of thinking about that partner of 10 in that 40. And that 10... Oh, yeah, I can just use that 8 to get to... Ooh, the 50. To add whatever. I can get to that friendly number. So, Partners of 10 kind of grow up to Get to a Friendly Number, using partners. Doubles, kind of can grow up to bigger doubles, and then the plus 10 grows up into add anything friendly that's too big. and really the idea of Overing crosses operations as well. So, once kids make sense of the idea of using more than you need, that's just not (unclear) (unclear). Yeah, yeah. You can Over subtract and adjust. You can multiply by something too big, and then adjust. You can divide by something too big, and then adjust. Oh, actually, usually you... It's the dividend that's too big. Anyway. So, these small relationships are not only important, so kids have access to these single-digit facts, but they grow up and become the major relationships that kids use to solve bigger numbers.

Kim 25:57

Yep.

Pam 25:58

Totally cool. Kim, this was fun. Thanks.

Kim 26:00

Yeah.

Pam 26:01

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