Pam  0:00  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  0:09  
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  0:16  
And

Pam  0:17  
we're glad you're on that mission with us. Ya'll, we know that algorithms are amazing, historic achievements, but they are terrible teaching tools. Mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than they should be developing and using.

Kim  0:32  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:39  
We invite you to join us to make math more figure-out-able. Hey, Kim. 

Kim  0:44  
Hey, hey. 

Pam  0:44  
Alright, let's record a podcast. What

Pam  0:46  
do you say? 

Kim  0:47  
Okay, sounds great. Hey, this is the shortest and sweetest comment we've ever had, I think. Nico McDaniel said, "I'm hooked! The podcast is fantastic! Pam and Kim challenged me in the most delightful

Kim  1:00  
ways." Isn't that fun? 

Pam  1:01  
Oh, that is delightful. Thank you. 

Kim  1:03  
Pam

Kim  1:03  
challenges me in the most delightful. Oh, wait. Delightful way.

Pam  1:08  
Delightful ways. You're saying, I challenge you in delightful ways? 

Kim  1:11  
Yes.

Pam  1:12  
Well, back at ya. Hey, hey, let's challenge each other

Pam  1:16  
a little bit today.

Kim  1:16  
Okay, sounds good. Hey, we had a great chat last week.

Pam  1:19  
Yeah, so that was kind of fun. We talked all about models, and strategies, and properties for multiplication. Let's do that a little bit with addition. You game? 

Kim  1:26  
Yeah. Yeah, let's do it.

Pam  1:26  
Okay. Alright, so I'm going to start with a problem. (unclear).

Kim  1:26  
Mmhm. Do I need paper?

Pam  1:26  
You know, I don't know if you will for this. But you can keep track of your mental thinking. That's okay. 

Kim  1:27  
Okay. 

Pam  1:27  
Okay, so the problem I'm going to give you is 38 plus 29. 

Kim  1:42  
Okay.

Pam  1:43  
Okay, and just what's your... What strategy pops out at you first?

Kim  1:49  
I'm going to Give and Take. I'm going to move 2 of the 29 over to the 38 to make the new problem 40 plus 27.

Pam  1:59  
And did you do all that in your

Pam  2:00  
head? Did you write anything down? 

Kim  2:02  
I wrote down 40 plus 27. 

Pam  2:02  
That's all you

Pam  2:02  
wrote down, 40 plus 27? Okay, cool. And what is 40 plus 27 just for... 

Kim  2:02  
Oh, I didn't even solve it. 67. 

Pam  2:02  
Okay, so you're saying 38 plus 29 is 67.

Kim  2:08  
Mmhm. 

Pam  2:09  
And you just kind of move some marbles around to make that happen. Cool.

Kim  2:18  
Mmhm. 

Pam  2:18  
So, what I represented for that strategy was I used equations.

Kim  2:23  
Mmhm. 

Pam  2:24  
And I did a horizontal equation. I have 38 plus 27 horizontally. And underneath the 28, I wrote plus 4. That's funny because I think you must have said 40. So, I meant to write plus 2. I fixed it now. And then I wrote under the 29, I write minus 2. And then I put a line under all that, and I wrote... So, underneath the 38 plus 2, I wrote 40. And underneath the 29 minus 2, I wrote plus 27. And then next to that. So, the 40 plus 27 equals 67 next to that. 

Kim  2:53  
Mmhm.

Pam  2:53  
So, this strategy, Give and Take, I would say is the most sophisticated of the addition strategies that we want to develop in kids heads.

Kim  3:02  
Mmhm. 

Pam  3:02  
And I'm going to choose the model of equations to represent that thinking.

Kim  3:08  
Okay.

Pam  3:08  
Cool. What if I forced you to do maybe the least sophisticated strategy we would expect kids to use. If I say that, do you have (unclear). 

Kim  3:18  
Yeah.

Pam  3:19  
Guess what's in my head

Pam  3:20  
there? 

Kim  3:20  
Yeah, I think you're going to do Place Value Partials.

Pam  3:24  
Okay.

Kim  3:24  
Oh, gosh. So, then I wrote down 38 and 29. I'm definitely writing something because I can't hold all this. I wrote 38 and I split it into 30 and 8. 

Pam  3:33  
And when you split it, did you draw like a little line to 30 and a little line to the 8? 

Kim  3:34  
I drew like a little like carrot symbol.

Pam  3:38  
Mmhm, okay. Yeah.

Kim  3:39  
You know what I mean?

Pam  3:40  
Like a little kind of partial triangle, kind of.

Kim  3:43  
Mmhm. 

Pam  3:43  
Okay.

Kim  3:44  
And then I wrote plus 29, and I split that into 20 and 9.

Pam  3:47  
Mmhm. 

Kim  3:48  
And then I drew a line connecting the 30 and 20, and that's 50. 

Pam  3:54  
Mmhm.

Kim  3:54  
And then I drew a line connecting the 8 and 9, and that's 17.

Pam  3:59  
And put those next to each other, so it's like 50 plus 17.

Kim  3:59  
Yeah. And then 50 plus 17 is 67. 

Pam  4:00  
Cool. And

Pam  4:06  
we would call that Splitting by Place Value. And you just described modeling on what we would call a splitting model. Kind of split both of the numbers into place value parts, put them together, then kind of have an equation at the end. 50 plus 17, 67. 

Kim  4:18  
Yep. 

Pam  4:18  
I also could model that. Not better or worse, but I think we would also want this model. I'm going to write 38 on top of 29, kind of like you would expect in a traditional addition algorithm. 

Kim  4:32  
Yep. 

Pam  4:32  
Though I'm not going to... I'm going to model what you did, not the traditional algorithm.

Kim  4:44  
Mmhm. 

Pam  4:44  
So, next to the 38 I'm going to write equals 30 plus 8.

Kim  4:44  
Mmhm. 

Pam  4:44  
Next to the 29, I'm going to write 20 plus 9. And then I'm going to draw a line under those. And under the 30 and 20, I'm going to write 50. And under the 8, 9, I'm going to write 17. And then next to that. So, I've got 50 plus 17. And then next to that, I'll write 67.

Kim  4:49  
Can I just point out really quickly that this is like a little bit of a nudge. This is a little bit of a teacher move. Because when kids are messing around with Place Value Partial...

Pam  5:01  
Partial sums, sums.

Kim  5:04  
There are a lot of kids and teachers who are doing what I did and just like lots of lines, and breaks, and put together, and redraw together. And that can get really messy. And kids don't know what parts they put together. And so, what you just are suggesting is a bit of a nudge, a teacher move, to model the same thinking that I had but in a way that's a little cleaner.

Pam  5:24  
Mmhm. And heading towards equations, which is going to become our model of choice.

Kim  5:28  
Right. 

Pam  5:28  
So, we don't mind a splitting model. It works fine early. But what we're going to kind of nudge away from it. Hey, why do we say nudge and not nudge? Have you ever heard people say nudge? 

Kim  5:39  
Is that a word?

Pam  5:40  
I've totally just heard two people in the last week say, "I'm just gonna nudge student over." And I was like, "Wow, that's a different way to pronounce nudge." 

Kim  5:48  
I have never heard that.

Pam  5:48  
One of them was British, so maybe it's the British pronunciation? Oh, okay. Yeah. I think the other is Canadian. I don't know.

Kim  5:55  
Well, there you go.

Pam  5:56  
Yeah. I mean, they're not British, but they live...  What are they? The Crown? The Commonwealth? Oh, I'm getting myself in political trouble here. Okay, never mind. We like our Canadian friends and our British friends. Alright, so that was kind of the least sophisticated strategy. I actually kind of liked it how you said, "Whoa,

Pam  6:11  
I'm going to have to write this down to keep track of it all."

Kim  6:11  
Yeah, I don't think I've done that in a long time.

Pam  6:16  
Which it's

Pam  6:16  
part of the reason why we want to nudge kids away from this strategy because it becomes way too cumbersome and way too inefficient soon, quickly. 

Kim  6:26  
Right.

Pam  6:26  
And so, then we want to help them keep one addend whole.

Kim  6:30  
Mmhm. 

Pam  6:30  
So, Kim, if I was to ask you to solve this Adding a Friendly Number, 38 plus 29, what might you do to Add a Friendly Number?

Kim  6:38  
I'm going to start with 38.

Pam  6:40  
Okay.

Kim  6:41  
And I'm going to

Kim  6:43  
add 20. 

Pam  6:44  
Did I totally just hear a pencil? 

Kim  6:46  
Sorry, yeah. 

Pam  6:47  
Well, it feels like you drew a number

Pam  6:49  
line.

Kim  6:49  
I did. I wrote 38, and I drew a harsh line.

Pam  6:53  
I could feel. I could hear it. So, it's interesting that for the two strategies we've had so far, neither of us drew number lines.

Kim  6:59  
Right. 

Pam  7:00  
Because that's a terrible model for either of those strategies that we've used so far. Okay, keep

Pam  7:08  
going. 

Kim  7:08  
So, I made a jump of 20 to get to 58. 

Pam  7:10  
Yep. 

Kim  7:10  
And then what am I adding? 9?

Pam  7:11  
29. 

Kim  7:12  
Hmm. Sadness. Aww. Can I do what I want to do?

Pam  7:15  
Yeah. You have to add 9. I don't care how you do it. 

Pam  7:18  
Okay, well, I

Kim  7:19  
mean, I guess I'll just add 9. I don't love it. 

Pam  7:22  
Did you

Pam  7:23  
think about adding 10 (unclear).

Kim  7:26  
Yes, I want to add 10! 

Pam  7:30  
We'll do that in your head. Okay, so you're at 58, and you're going to add 10.

Kim  7:32  
Plus 9 is 67.

Pam  7:32  
Okay, 67. Cool. So, my number line looks like start at 38, big jump of 20, land on 58, smaller jump of 9. About half the size of the 20. A little smaller than half the size of the 20. And landing on 67. Now, if you want, I could have done add 10, backup 1. We could have. That would be okay. But kind of straight Add a Friendly Number would be a jump of 20. You could also have a kid after that jump of 20. That's the Add a Friendly Number part. Then they could jump 2 to get to 60. And then the leftover 7. That would also work. Okay.

Kim  8:04  
Have we ever said this out loud that we name things based on the first move that students make. So, it's Add a Friendly Number because it's that first friendly jump.

Pam  8:13  
But then... 

Pam and Kim  8:14  
Go ahead. 

Pam  8:15  
Especially in addition and subtraction.

Kim  8:20  
Yeah.

Pam  8:21  
It's a little less so in multiplication and division.

Kim  8:26  
Yeah, yeah, yeah. 

Pam  8:26  
Multiplication and division, it's more what the overarching plan of attack was. But yes. In addition subtraction, it's the first move. I'm so sorry. Please finish your

Pam  8:29  
sentence. 

Kim  8:29  
No, it's okay. Well, I'm actually thinking about what you just said. 

Pam  8:32  
Oh.

Kim  8:33  
But then after that first friendly jump, very often, there's a little bit of Over or a little bit of a Get to a Friendly Number. Like, the next move they make is so based on the numbers that they've been left with...

Pam  8:46  
Mmhm.

Kim  8:46  
...that we often see two strategies in one with Add a Friendly Number. Just want to put that out

Kim  8:52  
there.

Pam  8:53  
And so, we choose to describe the strategy, like you said, by the first move. 

Kim  8:58  
Mmhm. 

Pam  8:58  
Yeah, so since you were adding 20.

Kim  9:01  
First inclination, yeah. 

Pam  9:01  
First, "What am I going to do? Attack this? I'm adding 20." Okay, that's Add a Friendly Number." I don't care what you do after it. You added a friendly number. Kim, let's... I'm going to move.... Yeah, I'm going to do this next. So, that was Add a Friendly Number. What about Get to a Friendly Number? Or Add to a Friendly Number for 38 plus 29?

Kim  9:04  
Yeah. So, 38 the next friendly number is 40. So, I'm going to add 2. Also on a number line. Did you hear my? 

Pam  9:05  
I didn't this time. 

Kim  9:05  
Okay, alright. So, 38 plus 2 is 40. And then because I was going to add 29, and I've already added 2 of the 29, I could just make one nice jump of 27 to land on 67.

Pam  9:05  
Because you know 40 and 27 to 67 nice. And your first move, your first inclination was to get to that 40. And you're like, "Ah, I'm going to need 2 to do that." So, on on my paper, underneath the number line that I had drawn for your Add a Friendly Number, I've drawn a number line where the 38s line up and the 67s line up.

Kim  9:05  
Mmhm. 

Pam  9:05  
And I've got a tiny jump of 2 to get to 40.

Kim  9:05  
Mmhm. 

Pam  9:05  
And then a big old jump of 27 to land in that same location of 67.

Kim  9:05  
Mmhm. 

Pam  9:05  
So, again, both of us chose to use a number line to model those two strategies. Last strategy, I'm going to ask you for. How about Add a Friendly Number, Over.

Kim  10:16  
Last and maybe my favorite. 

Pam  10:18  
Which you didn't do

Pam  10:19  
today. I was kind of surprised. I mean, when I said, do anything, and you were Giving and Taking.

Kim  10:22  
I don't know. Maybe

Kim  10:23  
I was feeling... I don't know. (unclear).

Pam  10:25  
It's a very mathematical behavior of you to choose what felt good today.

Kim  10:34  
Well, right now, I feel like an Over strategy would be, I drew a number line, started with 38, made a nice big jump of 30 to get to 68. And then I added 1 too much, so I'm going back 1 to get to 67. 

Pam  10:49  
Nice.

Kim  10:50  
And my 67s do not line up, so don't look at me for modeling. 

Pam  10:53  
Mine do.

Kim  10:55  
I'm sure they do.

Pam  10:55  
My 38s are in the same place. My 67s are in the same place. Which means that nice jump of 30 that you made was longer than any jump I have on my paper so far, right?

Kim  11:03  
Mmhm.

Pam  11:04  
I'm actually on my iPad. I've started writing everything on my iPad. I'm becoming a little iPad addicted. It's kind of handy. Like, I used to carry paper and pencil around with me. (unclear).

Kim  11:08  
Pencil! You got a pencil. Yeah, it's good. 

Pam  11:08  
Well, okay, it was really pen. On the plane. And I would pull out that notebook, and I would write, and whatever. And now, I just brought my iPad and right. The only thing I haven't figured out here. Listeners, somebody can help me. I don't organize my files all that well.

Kim  11:28  
Shocking! 

Pam  11:30  
Hahah! Kim!

Kim  11:32  
Sorry, that just came out.

Pam  11:31  
You're so mean. And correct. I just sort of decided to do it, you know? And I'll just organize later. Hey... 

Kim  11:34  
Somebody will fix that. 

Pam  11:42  
...does that feel like my life? Yeah, sure enough. (unclear).

Kim  11:47  
We're grateful for it. It's good.

Pam  11:48  
Don't look at my desk. Okay, cool. So, now I would actually like to back up a little bit and do each of these strategies with equations.

Kim  11:48  
Yeah.

Pam  11:48  
Because I'd like to talk about properties. Are we good to go there? 

Kim  11:48  
Yeah, that's good. 

Pam  11:49  
Okay, so if you don't mind, I'm going to leave Give and Take till the last.

Kim  11:58  
Mmhm. 

Pam  11:59  
I want to start with the least sophisticated strategy where we did Place Value Partial Sums. And so, it's kind of like you said, I've got 38 plus 29. So, I've just written down 38 plus 29 equals. And you cut the 30 into 30 plus 8. And then I put a plus in between that. So, I've got a parentheses. Parentheses around 30 plus 8. Plus parentheses around 20 plus 9.

Kim  12:28  
Mmhm. 

Pam  12:28  
And I have a plus sign in between the two sets of parentheses. Does that make sense? 

Kim  12:28  
Yep.

Pam  12:30  
So, I don't want to mistake this for multiplication. We're adding here. 

Kim  12:36  
Yeah. 

Pam  12:36  
Okay, then I'm going to... Then you said, "Hey, I don't want that 20 and that 9 to be associated. I'd rather have the 20 and the 30 be associated." So, you brought that over. Now, I have 30 plus 20 in parentheses, plus eight plus 9 in parentheses.

Kim  12:50  
Mmhm. 

Pam  12:51  
So, you reassociated. You broke up the numbers into place value parts, and then reassociated the tens together and the ones together. A then, you added those together, and then you kept sticking them all together. Yeah.

Kim  13:04  
Mmhm, mmhm.

Pam  13:04  
So, that's the associative property, that Partial Sum strategy is based on the associative property. Yes?

Kim  13:12  
Mmhm, yep. 

Pam  13:12  
Okay, cool. So, then what properties are at work with Add a Friendly Number, Get to a Friendly Number, and Add a Friendly Number Over? So, let's maybe mess with that. So, when you added a friendly number, you started at 38, and you added 30. So, I'm going to write that as 38 plus 29 equals 38 plus parentheses, 30... Sorry, 20 plus 9. And you said to yourself, "I'm going to reassociate. Instead of..." Oh, so 38 plus parentheses, 20 plus 9. Did I say parentheses? I don't remember if I did. So, now I'm going to reassociate and say that's equal to 38 plus 20. And I'm putting that in parentheses. 38 plus 20. Then plus 9. So, I've got parentheses 38 plus 20 parentheses plus 9. Then you add the 38 plus 20 together to get the 58. Then you added the 9 together to get the 67. So, that one's also based on the associative property. Instead of keeping the 29 together, you reassociated the 38 and the 20. Okay.

Kim  14:14  
Mmhm.

Pam  14:15  
Alright, how about the Get to a Friendly Number? So, you started at 38 and you added 2 from the 29. So, I'm going to write 38 plus 29 equals 38 plus parentheses 2 plus 27. Equals. So, now instead of having the parentheses around the 2 plus 27, I'm going to reassociate, and I'm going to have parentheses around 38 plus 2. Plus that leftover 27 outside the parentheses. 

Kim  14:42  
Mmhm.

Pam  14:43  
Now, the 38 plus 2 becomes 40. And then I'm going to add that leftover 27. So, again, based on the associative property.

Kim  14:51  
Mmhm.

Pam  14:51  
Cool. Anybody want to guess what the Over strategy is based on? So, you said that you're going to do 38 plus 29. So, I've just written that down. 38 plus 29. As 38 plus 30. And then you backed up 1. So, I'm going to write that as 38 plus parentheses 30 minus 1. And you said, "I'm going to add that 30 first." So, I'm going to write that now as parentheses 38 plus 30. All subtract 1. 

Kim  15:18  
Mmhm.

Pam  15:20  
38 plus 30 was the 68. Subtract one gave you the 67.

Kim  15:24  
Mmhm. 

Pam  15:24  
So, again...

Kim  15:25  
And before you...

Pam  15:26  
Mmhm. 

Kim  15:26  
Before you move on.

Pam  15:26  
Yeah?

Kim  15:28  
They're all using the associative property. But notice that this slightly more sophisticated strategy is also using subtraction for the first

Kim  15:34  
time.

Pam  15:34  
Mmm, true. Yep. Which is one of the reasons why it's more sophisticated. 

Kim  15:34  
Yeah.

Pam  15:34  
Yeah, absolutely. And also... It's one of the reasons. Another reason that it's more sophisticated is you kind of have to think ahead a little. You have to say to yourself, "I'm actually going to add something that's not here."

Kim  15:50  
Mmhm. 

Pam  15:50  
Something too big, and then I'm going to have to adjust from there. Not part of what I'm supposed to add and keep going, but I'm going to actually like imagine something that's not here and deal with it. And that requires a little bit more wherewithal, a little more pre-planning, a little more kind of rising above the problem.

Kim  16:08  
Mmhm. 

Pam  16:09  
And to do that, we need kids to have lots of experiences with these other kind of more simultaneous strategies.

Kim  16:16  
Yeah.

Pam  16:16  
So, that they get confident and comfortable with those. While we also get them confident and comfortable rounding. And we bring those two together, and now all of a sudden, Over strategy kind of can start to make sense. Cool. So, the last one that we haven't talked about properties for is the first one that you did, Give and Take. So, I'm going to write that one as you said that you can think about 38 plus 29, and then you gave 2 to the 38 because you got it from the 29.

Kim  16:43  
Mmhm. 

Pam  16:43  
So, there's a couple different ways I could write this. I'm going to choose today to write it as 38 plus parentheses 2 plus 27. So, 38 plus the quantity 2 plus 27. And then I'm going to reassociate that 2. And I'm going to put parentheses 38 plus 2. Close the parentheses. Plus that leftover 27. And now you end up with 40 plus 27. So, again, based on the associative property because you started with 38, and you had 2 plus 27, and you reassociated to get the 38 plus 2 plus the leftover 27.

Kim  17:23  
Hmm.

Pam  17:24  
You thinking about that one?

Kim  17:26  
I am because when you think about the properties, that is exactly how you wrote the reassociation for Get to a Friendly Number.

Pam  17:35  
Mmhm, sure enough. 

Kim  17:36  
And I

Kim  17:37  
think part of what makes Give and Take a little bit maybe... Not too challenging, but up the ante for kids is that both of the numbers are changing simultaneously.

Pam  17:51  
Mmhm. 

Kim  17:51  
And so, I'm wondering about what you think about recording 28 plus 2 plus 29 minus 2.

Pam  18:00  
You said 28. 38, right? 38? 

Kim  18:02  
Yeah. Sorry, 38 plus 2. 

Pam  18:03  
Say it again. 38 plus 2. 

Kim  18:05  
So,

Kim  18:05  
I wrote 38 plus 29. And beneath that, I wrote 38 plus 2 in parentheses. Plus 29 minus 2 in parentheses. 

Pam  18:14  
Mmhm.

Kim  18:15  
I think I've seen you write that several times.

Pam  18:17  
Sure, yeah. And I might write that as 38 plus 2 minus 2 plus 29. Group the... So, like it's almost like 38 plus 0 plus 29. And the 2 in the zero becomes 2 minus 2. Am

Pam  18:37  
I making sense? 

Kim  18:38  
Yeah, yeah. (unclear) Because...

Pam  18:38  
And then I would do

Pam  18:38  
your parentheses the way that you did it. 38 plus 2 and the 29 minus 2. When I started doing the Give and Take, I was like, "There's a few ways we could do this."

Kim  18:38  
Yeah, sure, sure. 

Pam  18:38  
But what I hear...

Kim  18:38  
And I think it depends on the age of your students.

Pam  18:45  
Well, and

Pam  18:47  
what I hear you saying is you want to represent the simultaneity that kind of happens when you're actually Giving and Taking. 

Kim  18:59  
Yeah.

Pam  19:00  
Sure. And I think that's why we like to use the equation model where it's 38 plus 2, 29 minus 2, either horizontally or vertically, to kind of represent that sort of, "Hey, if I give 2 to this, what will that do? Can I get it from the other one? Yes, let's do it." And it's kind of a simultaneous decision that you make. 

Kim  19:20  
Mmhm.

Pam  19:21  
That then you're like, "Bam, I'm going to do Give and Take." Which is  sophisticated, requires all that stuff that we just talked about with the Over strategy. Can I introduce a number that I don't have here by grabbing it from the other one? And in addition, it requires simultaneous thought.

Kim  19:39  
Right.

Pam  19:39  
Like, what will happen to the other number if I grab what I need for this one?

Kim  19:43  
Mmhm.

Pam  19:43  
And so, yeah. It's more sophisticated.

Kim  19:43  
Yeah.

Pam  19:43  
And we would definitely say it's based on the associative property. Cool. Hey, one thing that we kind of didn't talk about at all. And so, just for kind of completeness sake. Somebody's probably going to email us and ping us on social media. The communitive property is also involved in several of the strategies that we did last week with multiplication and this week with addition and subtraction. We didn't do subtraction. We just with addition this week.

Kim  20:10  
Mmhm. 

Pam  20:10  
We kind of assume that we're going to develop commutative pretty early.

Kim  20:15  
Mmhm.

Pam  20:15  
And then we used it in a lot of what we did last week and this week. So...

Kim  20:19  
Yeah, you gave us a problem that was already starting with a bigger number. But if you had given me 29 plus 38.

Pam  20:27  
You might have. We might have used the communitive property. 

Kim  20:29  
Yeah, sure.

Pam  20:30  
Yeah. And a couple of the times, I thought about things like switching the order of the of the addends. And so, yes, the communitive property is involved. But the strategies are less based on the commutative property. The main strategy. So, we need to build the communitive property. The strategies are really based on for addition. They're based on the associative property.

Kim  20:52  
Yeah.

Pam  20:52  
Yeah. Which is super cool. So, Kim, why did you and I only draw number lines and equations?

Kim  20:57  
Hmm.

Pam  20:58  
Because...

Kim  20:58  
Well, yeah.

Pam  20:59  
We've used other models. We've used number racks for early addition and subtraction. But we don't actually like them for multi-digit addition and subtraction.

Kim  21:08  
Right. 

Pam  21:08  
Too much counting by ones. It's okay. I've seen the multiple like the 100 rack with 10. Sorry, the 100 rack with 100 beads, where there's 10 rows of 10. 

Kim  21:18  
Yeah. 

Pam  21:19  
I'm kind of okay with doing a little bit of work. Can you hear the hesitation here? 

Kim  21:23  
Yeah.

Pam  21:24  
If you already have one. Don't go buy one. But if you already have one, I'm kind of okay saying things like, "Where's 67? And how do you know?" But I really want you to not count 6 groups of 10s. Like, I really want you to nudge towards can you find the 50 to get to the 60? And then can you find the 7 using the 10 structure? The 5 and 10 structure. So, I'm kind of okay if you do a little relationship building with that 100 bead number rack. But really, by the time we get to double-digit, multi-digit addition... 

Kim  21:52  
Yeah.

Pam  21:53  
...it's number lines and it's equations.

Kim  21:55  
Yeah, for sure. 

Pam  21:56  
Yeah. So, that's why, in our Problem String books, and in the Developing Mathematical Reasoning - Avoiding the Trap of Algorithms books, and in our Hand to Mind - Foundations for Strategies kits, you'll see that we emphasize open number lines and equations for all these strategies.

Kim  22:13  
And the same caveat that I gave last week. Again, these are not that students need to know the names, and the definitions, and describe all the properties. It's really important that you know them, and that you can use them. You can describe. When a student describes their thinking, you can choose the model that best represents what's happening. You know, if they're Giving and Taking, you're going to represent that on an equation rather on number line. But number lines are fair game for all the others. If you'd like to see what we call the major strategies, you want to grab the major strategies ebook you can find that at mathisfigureoutable.com/big to get that big download.

Pam  22:53  
Nice. Check it 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. And thanks for spreading the word that Math is Figure-Out-Able!