Pam  00:00

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

Kim  00:09

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

Pam  00:17

We know that... Oh, for heaven's sakes. Hang on. My monitor is blinking on and off. I don't know why I don't have this memorized. We know that algorithms are amazing historic achievements, but they're not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop. 

Kim  00:37

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

Pam  00:45

We invite you to join us to make math more figure-out-able. Okay, I have to tell you what I did, why my monitor blinked. Because I was trying to roll up my sleeve, and I hit the cord of my headphones, and I think it pulled on the monitor cord, and all of a sudden the monitor was blinking. I was like, "Ah! Are we all going down?!"

Kim  01:05

I... When you were talking, I wasn't really thinking about your monitor anymore, but I do want to know how many times do you think you've said the word "figure-out-able".

Pam  01:13

Haha.

Kim  01:15

I know.

Pam  01:16

Wow. 

Kim  01:17

I was trying to estimate, and I have no idea.

Pam  01:21

At least tens of thousands. (unclear).

Kim  01:22

I mean, yeah.

Pam  01:24

At least. Yeah. A lot.

Kim  01:27

I'm really not loving typing it. Sorry, sorry. It's a great word. Love the meaning. Hate typing it.

Pam  01:34

(unclear). Yeah, yeah. In fact, it's funny because it's super easy to tell people our email addresses, pam@mathisfigureoutable.com. Should I have just given that out to everyone? Hmm. Anyway. But then if I have to type it, people hand it to me, and they're like, "Here, type it." I'm like, Oh gosh. "Math is figure..." And then I always have to think there's an E on there. Yeah, it's kind of funny. Alright, should we dive in? Diving in, diving in. 

Kim  01:57

So, last week, we started some work with solving equations using a double number line as a model, and it really helps to make sense of location and how numbers are related. And as we said last week, we are going to continue that work today. I have my pencil. 

Pam  02:13

(unclear) Let's do it. Let's do it. We thank Chris Shore and his Clothesline Math. If you want to check out clotheslinemath.com. Fantastic work there. So, a double open number line has everything to do with we're going to have relationships that we write on the top, that then are corresponding to relationships on the bottom. And here we go. If you have not listened to last week's episode, may I highly recommend that you do because I'm just going to build from it. So, you could maybe get out of... You could stay with us if you want to, but you might want to go listen to last week's. If I was in a classroom where I had done last week's Problem String with students, today, I would remind them a little bit of what we did the time before. That kind of gives everybody access. Especially if I had students who were absent or students who just need another go at some of what we just did. I would probably do x = -6. So, x is in the same location as -6. And then I would do the opposite of x is in the same location as -2. And so, very much like we did in last week's Problem String, I would put those on a number line, and we would reason about where other things are. Like, if this is where they are, where 0. The opposite of x is here. Where's x? So, check that out if you haven't yet. But then, the third problem would be... So you ready? Kim.

Kim  02:42

Yep. 

Pam  02:46

What if I know that twice x or 2x is in the same location as 12? So, what I wrote on my paper was 2x = 12. But I've also then, next to it, writing a big, long number line. Big, long number line. Covering my whole paper left to right. 

Kim  03:50

Okay. 

Pam  03:51

In the middle of that... And I'm just going to put kind of randomly every time... Not randomly. Purposely. In the middle of that number line, I'm going to put what we know every time. So, every time I'm going to put what we know to start with in the middle. So, I've put a tick mark in the middle of that number line, and I've written 2x on the top, and I've written 12 on the bottom of that tick mark. 

Kim  04:12

Okay.

Pam  04:13

Okay. So, if I know that, is there anything else that I know that you know? What else do you know looking at that?

Kim  04:21

Yeah, if I know 2x is 12, then I know 1x is 6.

Pam  04:28

Okay, okay. So, did you like scoot over some and write 1x equals? 

Kim  04:35

No, I halved it. I thought, "Well, if double the x is 12, then half of it would be 6.

Pam  04:43

Okay, cool. So, I might say, "Let's find that on our number line. And (unclear).

Kim  04:48

I just said half the x. That's not what I meant. Half of the double.

Pam  04:51

Oh, half of the 2x. (unclear).

Kim  04:52

Yeah.

Pam  04:52

Okay, sorry. Maybe a better facilitation might have been me to force where the 0 is first. That might have been smart on my part. 

Kim  05:02

Okay, alright.

Pam  05:03

So, if 2x is in that location of 12, where would 0 be? 

Kim  05:08

Very far to the left.

Pam  05:10

Okay, so very far to the left. I'm going to put a tick mark, and I'm going to put 0. 

Kim  05:13

Okay. 

Pam  05:13

And I've got 0 on the bottom because that's where the numbers are going. So...

Kim  05:16

Yeah. 

Pam  05:16

...what goes on the top where the x's go? Any idea? If 2x is at 12, how many x's are at 0?

Kim  05:26

Oh, 0 x's.

Pam  05:29

0x. Okay, so I'm just going to write above the 0. I'm going to write 0x.

Kim  05:32

Mmhm. 

Pam  05:33

Okay. And then you had said... So, right now, on my paper, I've got sort of 0x on the left hand side. 0 and 2x in the middle at 12.

Kim  05:43

Mmhm. 

Pam  05:44

You said something about 1x?

Kim  05:45

Mmhm. It's right in the middle. Right in the middle between the 0x and the 2x. 

Pam  05:49

And right in the middle of 0 and 12 is? 

Kim  05:52

6. 

Pam  05:52

Okay, cool. So, that's what my number line looks like. 0x is at 0. 1x is at 6. And 2x is at 12. Over there where I had equations on my paper, I've got 2x = 12. Then I would say, "And then we also know that x is at 6." We also could write that 0x is at 0. I don't know how interesting that is.

Kim  06:13

Mmhm.

Pam  06:14

Would that always be true? Will 0x always be at 0? We could think about that a little bit later.

Kim  06:18

Mmhm. 

Pam  06:18

But cool. So, I also might then say, "How did you know where 1x is on the number line?" So, like once we had the 0x at 0 and the 2x at 12, how did you know to put the 1x where you put it? 

Kim  06:33

Like, right in the middle? 

Pam  06:34

Exactly in the middle, yeah. Did you... So, like in the middle, do you mean like you cut it in...

Kim  06:40

In half, yeah. Yeah. 

Pam  06:41

So, then over there where I had the equations, I might say, "Well..." So, on the number line, I might say, "Oh, so you cut. It's like you cut the 2x in half."

Kim  06:49

Mmhm.

Pam  06:49

And over on the equations, I might just write underneath the 2x, "divided by 2". 

Kim  06:55

Mmhm.

Pam  06:55

Underneath, (unclear) divided by 2.  So, like you cut them in half. You cut the 2x in half, and you cut the 12 in half, and that's how you got x = 6. So, what I have written on the equations is actually describing the mental action that we did on the number line, but it will look a lot like when teachers just say, "Well, just divide both sides by 2." 

Kim  06:57

Mmhm. Mmhm.

Pam  07:16

But rather than just divide both sides by 2, we're reasoning about the relationships. Okay, that one wasn't too hard. And that almost might have been too... Let's do a harder one, so it gets a little bit more understandable why I'm putting on the number line.

Kim  07:31

Okay.

Pam  07:31

Because I can hear to somebody saying, "Look, twice the number is 12. Pam, the number's got to be 6." Yes, but I want to kind of establish this distance thing because we're going to use it now. 

Kim  07:40

Yeah. 

Pam  07:41

Okay, so next problem. What if 2x is in the same location as -12? 

Kim  07:46

Mmhm. 

Pam  07:48

So, I'm drawing the number line. Big number line right in the middle. I'm putting 2x on top. Putting -12 on the bottom. I'm going to ask you,this time where's 0? 

Kim  07:56

It's going to be far to the right. 

Pam  07:58

Okay.

Kim  07:59

Because I'm thinking about the -12. And I want 0 to be the right of that. 

Pam  08:03

Okay, how far to the right? You said "far" (unclear).

Kim  08:07

It's... Well, it's the same as the number line above.

Pam  08:12

Okay. And how many units? 

Kim  08:13

Oh, 12.

Pam  08:14

12. Okay. So, it's kind of arbitrary how long were making 12, but okay. So, now we know where 0 is and... 

Kim  08:22

And I wrote 0x above the 0. (unclear). 

Pam  08:25

Okay, okay. Is there anything else you know? 

Kim  08:27

Yeah, so then 1x would be smack in the middle of the 2x and the 0x. 

Pam  08:32

Okay. 

Kim  08:33

And so, that would be -6.

Pam  08:35

It's almost like you're dividing that distance in half, and you're like, "Well, half of 2x is 1x and half of -12..." What is halfway between -12 and 0? -6. Cool. So, then over on the left where I have 2x = -12, underneath that, I'm writing, so we also know then that x is at -6. And I could, if I wanted to, sort of divide both the 2x and the -12 by 2, and I could kind of represent it that way. Nice. Hey, staying on that number line. We don't have to, but we could. I could also ask you where 4x is. I could also ask you where the opposite of x is. There's like lots of other relationships right now that we could start putting on that number line. And I might have wanted to have done some of that maybe in the last Problem String that we did. Or maybe here. Just to kind of... So, it's not always about finding x. It's like finding lots of relationships. And one of them could be finding x. Okay, cool. Next problem. How about if the opposite of 2x, or -2x, is in the same location as 10? So, I've written -2x = 10. I'm also drawing that number line. Right in the middle, I'm putting -2x on the top and 10 on the bottom. Where's 0?

Kim  09:15

Mmhm.  Oh, that's kind of fun. 

Pam  09:50

What's fun? 

Kim  09:50

I've written a bunch of stuff that I would maybe want to use to help me. So, I put -2x is in the same location as 10.

Pam  10:00

Mmhm. 

Kim  10:01

And then I put 0. 

Pam  10:02

And where's 0?

Kim  10:03

0 is to the left of the 10. 

Pam  10:06

Okay.

Kim  10:07

10 spaces. And so, I put... Or 10 whatever. Span. So, I put 0x on top and 0 on the bottom. And at that point I went, "Like, what else do I know?"

Pam  10:16

Yeah.

Kim  10:17

So, I actually wrote -1x is 5. Like, in between what I had there currently. (unclear)

Pam  10:24

(unclear) cut them in half, bam. 

Kim  10:25

Yeah. I don't know that I necessarily needed that for any reason, but I put it on there for funsies. 

Pam  10:30

Okay. Sure, sure. 

Kim  10:32

And then I thought to myself, "Well, I know where..." If I know where the opposite of 2x is...

Pam  10:39

Mmhm.

Kim  10:39

...then I know 2x would be mirrored at the 0. 

Pam  10:45

Reflected over, mmhm.

Kim  10:47

Mmhm. So, I put 2x is -10. 

Pam  10:50

for everybody listening. Right now, on the right hand side, we had -2x was in the same location as 10. So, if you mirror that, you reflect it over the 0, then the opposite of negative 2x is going to be 2x. And the opposite of 10 is going to be -10. So I might over on the equations... We started with -2x = 10. You also said that the opposite of x, or -x, was at 5. So, I've written that down. I've got -2x = 10. -x = 5. And then you said also that 2x is equal to -10. So we know all that right now. Super cool. Alright.  So, let me just

Kim  11:01

But you probably are going to ask me about x. 

Pam  11:29

I am.

Kim  11:29

(unclear) x would be in the middle of 0 and 2x. And then the middle of 0 and -10. So, x is -5.

Pam  11:37

So, x is in the same location as -5. Cool.

Kim  11:40

And what I love about what we're doing here is that, yeah, I'm going to find x, but I'm also exploring so many more relationships at the same time.

Pam  11:49

And actually kind of being clear.

Kim  11:51

I'm tinkering, yeah.

Pam  11:51

Yeah. You're... Oh, tinkering! And, Kim, when I started messing around with double open number lines and solving equations, I found myself doing mental actions that felt very much like the mental actions I do when I reason through numeracy problems.

Kim  12:07

Right.

Pam  12:07

And I found myself realizing that what I had always been doing with solving equations was much more of a "do it, mimic" kind of thing and not a reasoning using relationships kind of thing. And in that moment, I was like, "I'm doing this." Like, "We're going here. We are going here because..." And then the more mathy people that I talk to, they're like, "Well, yeah. That's the kind of stuff that I'm actually doing in my head when I solve equations." And I was like, "What?! Why didn't you ever tell me that?!" Okay, cool. May I just suggest, everybody, that for this problem, -2x = 10, that's what we started with. I now have written -2x = 10, -x = 5, 2x = -10, and x = -5. I've got all of those on equations, and I've got all of those on the number line. I actually think that we kind of have two strategies happening here. So, Kim, your strategy, to me, feels like you started with the opposite of 2x equals 10 and you reflected that over to get 2x equals -10, and then you cut it in half to get x equals -5. Great strategy. You started with a -2x. You reflected it to get a positive 2x. 

Kim  13:12

Mmhm.

Pam  13:12

Then you cut it in half to get x.

Kim  13:13

Mmhm. 

Pam  13:15

But you also could have gone a different direction from your starting point. You want to say what you were going to do? Or what you you could have done?

Kim  13:22

(unclear). Tell me what I did. I think I did... I reflected it, and then I cut it in half. 

Pam  13:26

That's the strategy I would say that you actually kind of did, yes.

Kim  13:29

So, I could have cut it in half and then reflected.

Pam  13:31

Bam! And that's actually... I have that all up here on my number line right now because you had started with, "Well, I could do the -x = -5." Then you kind of ignored it for a minute and did the reflection. So, let me just talk through that. So, starting from -2x, you cut it in half to get the opposite of x. So, sorry. -2x. Cut in half to get -1x. And then you reflected it to get x.

Kim  13:55

You know what I'm realizing?

Pam  13:56

Hmm? 

Kim  13:57

When we tell kids, you know, "Just write it down, and then divide by -2," or whatever. We're asking them to do the two things simultaneously. We're asking them to halve it and reflect it.

Pam  14:10

Yeah.

Kim  14:10

And that's... Simultaneous is trickier for kids.

Pam  14:13

And it's not like we don't ever want kids to be able to look at -2x equals 10 and divide by -2. 

Kim  14:18

Yeah, but if we wonder why that's (unclear). 

Pam  14:20

So confusing, so abstract, so... Yeah, yeah.

Kim  14:24

Probably why.

Pam  14:25

And why later, or maybe soon, negatives, and coefficients, and minus negatives, and everything just gets so muddled because it's just all a bunch of rules that we really don't understand or have made sense of. Yeah, not grounded in being able to like go, "Wait. What do I know?" Not what have I mimicked? What have I memorized? But what do I know? And how can I reason from there? Cool.

Kim  14:46

Guess who's doing some math when they get home from school today?

Pam  14:50

Fun. Let me mention one other thing. The fact that we had two strategies here that you could either reflect and then cut it in half, or you could cut it in half and then reflect, that was another moment where I was like, "There strategy!"

Kim  15:04

Yeah.

Pam  15:04

Like, so often solving equations is just, "Do this." You know like, "Get x alone." Like, and it's... But now, all of a sudden, we can talk, "Hey, what do you want to do first?" And now, in the discussing of strategy, we're really building relationships. And that is what we've been doing in numeracy, and so when I saw that here, I was like, "Yes! We are going here. Alright, next problem. What if... Whole new universe. -2x = -8. Or the opposite of 2x equals -8. What I'm putting on my paper right now, right in the middle, is -2x on the top and -8 on the bottom. And, Kim, before you say. (unclear) everybody listening. Where's 0? So, if you could picture -2x on the top. -8 on the bottom. Where would 0 be? And go ahead, Kim. Where's 0? 

Kim  15:47

It's going to the right, distance of 8. 

Pam  15:51

Okay, cool. And if I want to find out where x is, what strategy might you want to?

Kim  15:57

This time, I'm going to halve first. 

Pam  15:59

Okay.

Kim  15:59

I'm going to... So, I put -1x is negative 4. 

Pam  16:03

Okay. 

Kim  16:04

And then I'm going to reflect that and call x, 4.

Pam  16:09

So, over there on the equations, I could write x = 4.

Kim  16:13

Mmhm.

Pam  16:13

I could also write... In fact, honestly, I moved it down because I knew in the middle I was going to write the opposite of 1x equals -4. And then underneath that, I wrote, so x = 4. Listeners, I wonder if you can appreciate the fact that when I had -x = -4, it wasn't about multiplying by -1 or dividing by -1. It was really, well, if I know what the opposite of x is, then I know the opposite of that has got to be x. Like, there's this whole opposite thing happening. I just encourage you to really think about the idea of using opposite and reflecting over that 0 to help you kind of make sense and help your students make sense of integer operations. Cool. I don't know. Often people can do one strategy and they can't really think of the other one. Can you think of the other one, Kim? Or I can just... I can run it. Like, if we start from -2x equals -8, could you reflect first? 

Kim  17:03

Yeah, so 2x would be 8. 

Pam  17:05

Okay.

Kim  17:06

And then half of that x would be 4.

Pam  17:10

Nice, nice. So, two different strategies that we could represent. Alright, super cool. Kim, I got one more kind of thing to do. Next problem. Ready? What if on the next problem one-half of x equals 3.

Kim  17:24

Mmhm. 

Pam  17:25

Okay, so I'm going to put, make a number line right in the middle. I'm going to put 1/2x on the top and 3 on the bottom. And we're going to start with where's 0. 

Kim  17:37

Do I have to?

Pam  17:39

Yes. Yes, you do.

Kim  17:42

0 is to the left. 

Pam  17:44

Okay. Of 3, right? You're focused on the 3.

Kim  17:46

Yeah. Left of 3, mmhm. 

Pam  17:47

Teachers, this is going to be the thing that might... The reason I'm forcing the 0 is it could be a really help. You don't always have to get 0. I mean, Kim probably wouldn't have to. (unclear).

Kim  17:55

But it is useful to say that your distance is... Like, otherwise, it's just... When I want to go put x, it's not halfway between anything, you know. So, fine. I'll write it down. Yeah. But it gives some relation to, why half? Why is it 1/2x? So, I put my 0 to the left.

Pam  18:21

Okay.

Kim  18:21

And then I put x and 6 to the right. And my 1/2x that I started with is smack in the middle.

Pam  18:29

Because if x is at 6...

Kim  18:30

Yeah.

Pam  18:31

...then half of that distance would be in the middle of 0 and 6..

Kim  18:35

Mmhm.

Pam  18:36

...of 3. Cool. So, it's almost like to put the x there, you almost like had to double that distance from 0.

Kim  18:43

Mmhm.

Pam  18:43

That could be a way of thinking about that. So, on the paper, I could have that one-half times x equals 3. And then I could literally double the 1/2x to get x. And then we doubled that distance of 3, double the 3 to get 6.

Kim  18:55

Mmhm. 

Pam  18:55

Again, the kind of notation that I'm making could look like it's kind of the what we tell kids to do when they're balancing an equation, but really the relationships I'm having in my head are much more about relationships and not just like mimicking a thing to do. Cool.

Kim  19:09

Mmhm.

Pam  19:09

Next problem. What if I know that 1/3 of x is at -2.

Kim  19:14

Mmhm. 

Pam  19:15

So, drawing my number line. Right in the middle, I've got 1/3x on the top. I've got -2 on the bottom. And I am going to force where's 0? 

Kim  19:24

Yeah, and I actually almost started putting it to the left of -2 because I was focused on the third.

Pam  19:29

Ah.

Kim  19:30

So, I put it to the right.

Pam  19:32

Because -2 is to the left of 0. Okay, mmhm. 

Kim  19:36

And so, if 1/3 is a span of 2. 1/3x.

Pam  19:41

Yeah. 

Kim  19:42

Then I know I want x at -6 because I need 3 of those 1/3. So, I'm tripling the 1/3 to get to x. And I'm tripling the -2 to get to -6.

Pam  19:56

Okay, and I believe you. I'm going to slow that down just a little bit because you know what kids, almost every kid I've done this with does? (unclear)

Kim  20:03

They put a 2/3? 

Pam  20:05

They do.

Kim  20:06

Okay. 

Pam  20:06

They look at that 1/3x, and it's 2 away from 0 to the left, and they go, "Well, I know we're two 1/3x is." And so, they go over another 2 units. 

Kim  20:17

Mmhm.

Pam  20:17

They put a tick mark, and they put 2/3x. And then they're like, "Well, if that was... If the 1/3x was at -2, then two 1/3x would be at -4." And then, you know what they write? They write 3/3x above the -6. 

Both Pam and Kim  20:31

And I... Go ahead. 

Pam  20:33

Go ahead, go ahead.

Kim  20:34

I would argue that kids who are multiplicative thinkers with with number probably do this additive approach because it's something new. It's something new they're thinking about.

Pam  20:45

Mmhm. And then they fairly quickly start thinking about, "Oh yeah. I could just like scale up. I need 3 of those 1/3s, so I'm going to scale that." But boy, at first, it's super interesting to let them be a little additive first. And then kind of that thing we started with last episode and a little bit today. What else do you know? And it's okay for them to stick in that 1/3x, so that the 2/3x. Whatever, however many x's they need. Totally legal to do. And so, I think you just found that x was at -6. So, I would make sure that I have that written there. Go ahead.

Kim  21:18

When a kid puts the one-third to start, and then they they step up the two-thirds, three-thirds.

Pam  21:24

Mmhm.

Kim  21:25

In that moment, are then you scaling? Like in your summary of what they did, are you showing scaling marks as a as a nudge? 

Pam  21:33

So, I think it depends on the grade level I'm working with. 

Kim  21:37

Mmhm.

Pam  21:37

So, if it's younger grades, and that means I have a little more time, then I probably wouldn't yet. I would probably kind of leave. I would do the next problem. We would do some more. And then I would look for generalizations about how we got from. "How do we get from 1/2x to x? How did we get from 1/3x to x? How'd we get from 1/4x to x?"

Kim  21:57

Yeah.

Pam  21:58

"Oh, every time we needed two 1/2s, and three 1/3s, and four 1/4s." And I'm like, "Oh, we're..." So, I would sort of build that scaling. 

Kim  22:06

Mmhm.

Pam  22:07

If I was working with older students, hopefully who owned more multiplicative thinking, I'd get there a little faster. 

Kim  22:12

Okay.

Pam  22:12

Yeah. I might do it right here. 

Kim  22:14

Yeah.

Pam  22:14

Yeah, cool. Okay, next problem. What if I know that negative... It's actually the last problem, so you can do it, everybody. Hang in there. What about -1/4x = -5. -1/4x = -5. So, Kim, stop right now. Tell me right now what you've already got written.  Don't answer anything, but just like what did you immediately start doing? I'm just curious. 

Kim  22:37

I put 0. 

Pam  22:38

Okay, cool. Did you also put the negative 1/4x (unclear).

Kim  22:41

Oh yeah. 

Pam  22:41

Yeah.

Kim  22:41

Yeah, I put 1/4x. Drew a line. Put -1/4x in the middle at the top and -5 at the bottom. And then I put 0 and 0. 

Pam  22:49

Okay. 0 and 0x. Mmhm, okay.

Kim  22:52

Mmhm.

Pam  22:53

Alright, what else do you know?

Kim  22:54

And then I reflected first, and I put 1/4x is 5. 

Pam  23:01

Oh, nice move. Nice move. So, 1/4x... 

Kim  23:06

And it's (unclear).

Pam  23:07

...is positive 5. 

Both Pam and Kim  23:09

Mmhm.

Pam  23:09

Because -1/4x was -5.

Both Pam and Kim  23:11

Mmhm.

Pam  23:12

Okay. (unclear).

Kim  23:12

Mmhm. And then if you'll let me, I'm going to scale. 

Pam  23:17

Okay,

Kim  23:18

4 times. So...

Pam  23:20

(unclear) 4 of those one-fourths, mmhm.

Kim  23:21

Yeah, so 4/4x, which is 1x is 20.

Pam  23:25

Is 20. Nice, nice. Could you imagine that somebody might have not reflected first?

Kim  23:32

Mmhm.

Pam  23:32

That they might have gone from that -1/4x in the same location as -5 and found -4/4x?

Kim  23:39

Mmhm. Yeah, that would -20.

Pam  23:42

That would be -20. And now, I've got -x, or the opposite of x, at -20. And reflect that over to x is 20.

Kim  23:49

Mmhm.

Pam  23:50

And by calling on and representing both of those strategies, I think students are getting more clear at lots of things.

Kim  23:58

Mmhm. 

Pam  23:59

Not just getting an answer to solve an equation. You took a breath. What were you going to say?

Kim  24:02

Well, and I'm wondering if when I encountered a problem, would there be times where I'd prefer, based on the numbers, prefer to reflect first and then scale or if I would want to scale first? I'm sure there are times where.

Pam  24:15

Well, we'll throw that out as an open question. Let everybody think about that. So, Kim, what are a couple of things that are happening? I'll just mention. Operator meaning of fractions just came into that last bit. One-fourth of something. One-third of something. So, if you wanted to do a little bit of work before this, you might actually have done some operator meaning of fraction strings to get kids thinking about the operator meaning of fractions. That could be a kind of lead in. But yeah. There you go. Solving equations where the goal isn't just to find x. The goal is actually to build algebraic reasoning along with additive, and multiplicative reasoning, and proportional reasoning because we're on the double open number line. So, lots of things kind of embedded. But building kids mathematizing.

Kim  25:02

Yeah. And it's kind of fun, right?

Pam  25:04

And it's kind of fun. 

Kim  25:05

And as a younger grades teacher who does work on a number line, is super fun to know that a model that we're building like doesn't get left behind.

Pam  25:14

Mmm, nice. Yeah.

Kim  25:15

Middle school teachers hopefully are capitalizing on that work.

Pam  25:20

Yeah.

Kim  25:20

Okay, so earlier Pam shared that you know all of our emails now. So, we would love it if you would email me, kim@mathisfigureautical.com, and you can share a review. You can leave it on the podcast. Tell us what you want to hear about. Ask a question. Tell us if anything that's happened in your class, so that we can share with you.

Pam  25:36

Yeah, and 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 keep spreading the word that Math is Figure-Out-Able!