# Math is Figure-Out-Able!

Math teacher educator Pam Harris and her cohost Kim Montague answer the question: If not algorithms, then what? Join them for ~15-30 minutes every Tuesday as they cast their vision for mathematics education and give actionable items to help teachers teach math that is Figure-Out-Able. See www.MathisFigureOutAble.com for more great resources!

## Math is Figure-Out-Able!

# Ep 228: Solving Equations Pt 2

Solving equations is figureoutable! In this episode Pam and Kim do a follow up string to last week's episode to further develop algebraic reasoning.

Talking Points

- Using a double open number line to build proportional reasoning
- Using what you know to find x and explore multiple relationships
- Understanding equations vs mimicking steps
- Simultaneity can be tricky
- Additive vs Multiplicative thinking showing up
- Operator meaning of fractions comes into play

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

**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!