# 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 227: Solving Equations Pt 1

Do we need to memorize a bunch of rules to solve linear equations or can we reason about them? In this episode Pam and Kim use a model to make thinking visible and keep students reasoning!

Talking Points:

- Modeling on a double open number line
- Finding where x lies on a double open number line.
- Grappling with the opposite of x
- Building integer sense as well as building equation solving sense
- Reasoning about an equation versus balancing an equation

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 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 **00:17

We know that algorithms are amazing human 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 **00:31

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

**Pam **00:38

And we invite you to join us to make math more figure-out-able. Hey, Kim,

**Kim **00:43

Hi.

**Pam **00:44

How's it going?

**Kim **00:45

It's good. You know, we knew what our topic was going to be today, and I found something which I think is super fun. The person who asked for this podcast had left a review at one point.

**Pam **00:59

Oh, nice.

**Kim **00:59

So, I think that's super cool, so I found it. This is from InTurk and the title is Middle School Math coach. Which is always a fun job.

**Pam **01:08

Yeah.

**Kim **01:08

And they said, "I love your show. I've been waiting to be inspired, and thanks to you, I'm now inspired to make some great changes to our program. I recommended it to our school district and asked them to consider using your strategies. I'm so bummed I didn't hear of you till last year, but I binged listened to 177 episodes starting last summer." We hear that a lot. That's a lot of Pam and Kim talking. I'm using your strategies in my collaboration meetings, but I have senior staff, and it is hard to convince them to change their practice. I'm close to retirement. I'm determined to convince my staff, if not the whole district, to make math figure-out-able. I think our kids would benefit so much from doing so. You guys are brilliant."

**Pam **01:48

Aww!

**Kim **01:50

That sweet?

**Pam **01:50

Thanks for finding that. Thanks for reading that. Thanks, InTurk. Yeah, that's awesome. Sweet!

**Kim **01:55

So, we'll see how brilliant that I am today because today we are going to live in the land of older grades. Nera asked for some content about substitution and elimination.

**Pam **02:08

Yeah. So, let's talk a little bit about systems of equations and solving systems of equations. But today, we're actually going to back up a little bit and kind of do a little bit of groundwork. So, it's not too, too old. If you're a teacher of younger grades, I hope you're still listening. You'll definitely be able to follow along. I'm going to base a lot of what we do today on some work... Well, based on. That's quite the right way. I'll give credit to Chris Shore and Clothesline Math for some of the ideas.

**Kim **02:35

Yeah.

**Pam **02:35

And kind of getting me started. And then we kind of ran with some ways of making that help us with equations. And we're going to head towards systems of equations. Today (unclear).

**Kim **02:48

Sorry (unclear).

**Pam **02:48

Yeah, go ahead. (unclear).

**Kim **02:49

Did I tell you that I met Chris Shore at NCTM?

**Pam **02:52

No!

**Kim **02:52

Yeah, yeah.

**Pam **02:53

He was there?! Oh!

**Kim **02:55

I'm pretty sure (unclear) game night.

**Pam **02:56

Man, there was so many people that later the two of us meet up, and I'm like, "I just met...." "I just met..." and ah we didn't get a chance to meet everybody. Well, you got to meet Chris Shore. Well, nice.

**Kim **03:04

I did, yeah. And I think Kourtney, our Kourtney, went to a session about Clothesline Math. And I think it was geometry maybe? It might have been him. I'm not really sure.

**Pam **03:14

We'll have to check, yeah. Okay.

**Kim **03:15

Yeah. Yeah, I'm still thinking about NCTM and NCSM. I know it's been a little bit. But I just like... There was so much to see and do. And how in the world did we get to announce so much that we've been working on at the same time? That was a lot of fun. Yeah.

**Pam **03:30

Oh my goodness, yeah. So, Corwin announced my Developing Mathematical Reasoning - Avoiding the Trap of Algorithms book. It's available for pre order, and it'll be shipping around February. We also announced the project that we've been working on with Hand to Mind. We're super excited about Foundations for Strategies. It's a wonderful product out from Hand to Mind. We've been working on our Problem String books. We've got grades three, four, and... No. Two, three, and four are out and shipping. Fifth grades working, coming right along. We're super excited about those. And we also have our Journey membership. And, right now, we here at Math is Figure-Out-Able are busily working on creating our High School Problem Strings workshop. Whoo! That's going to be (unclear).

**Kim **04:14

There's no shortage of work to do.

**Pam **04:16

My goodness. We're so... Yeah. And we're loving it. Lots and lots of really cool things. Okay, so one of the things that we would need to... I think that would be great backup, background, undergirdings, underpinnings... In order to get to systems of equations. So, we're not there yet. Would be a little bit of work with a double open number line. That's where it connects to Clothesline Math. So, a double open number line. To really think about kind of some baseline things. So, Kim.

**Kim **04:47

Yeah.

**Pam **04:48

I have a lovely Problem String to do with you. Do you have a pencil?

**Kim **04:51

I do indeed.

**Pam **04:52

Not a pen?

**Kim **04:53

Never.

**Pam **04:54

Okay. It was so fun. One of the things that was fun at NCSM, NCTM, listeners, is that Kim created... Well, I guess it was Kim and Sue. Right? Sue helped you?

**Kim **05:03

Yeah, always Sue.

**Pam **05:04

Sue's our designer. But they had this really cool idea to create podcast shirts that we could wear while we were there. And it's super cute because there's a pen that... You know like literally a picture of a pen on the shirt that says, "I'm Pam." And then there's a pencil and says, "I'm Kim."

**Kim **05:20

Yeah.

**Pam **05:20

It's just super fun. So, we have a lot of fun wearing those (unclear).

**Kim **05:23

Yeah, people would walk up and say, "Oh, my gosh! I love that shirt! Where did you get it? And I would go, "Uh, I made it. We made it." We made it for ourselves.

**Pam **05:32

Yeah, there are only two in the world right now. Yeah.

**Kim **05:34

Yeah.

**Pam **05:34

That's pretty funny. Okay, so lovely Problem String that we're just going to kind of start just developing some relationships.

**Kim **05:40

Okay.

**Pam **05:40

Listeners, if you have something to write with and something to write on, now would be a good time. If you're driving, totally keep listening and everything, but really try to visualize some things. And I'll try to do as much kind of describing of what I'm drawing as we ask and answer these questions. Alright. So, Kim.

**Kim **05:57

Yep.

**Pam **05:58

If I say to you, if I write down, x = 3. You might be like, "Um... Okay?" Like, there's not a lot to do there. You know, we didn't really usually do something with that when you were in school. But today, we're going to let that mean that x is in the same location as 3.

**Kim **06:17

Okay.

**Pam **06:17

So, I'm actually going to draw a number line on my paper. And just right in the middle of that number line. So, I've got a number line. Right in the middle of that number line, I've put x. I put a tick mark. And above that tick mark, I've written x. And below that tick mark, I've written 3. So, I've got a location, and in that location, I'm saying that's where x is. And x happens to be where 3 also is. So, on the number line, that's where x is. Just so kind of, everybody can kind of picture it. Half of the number lines to the left that. Half a number line to the right of it. Right in the middle. Bam. It's just sitting there. The question that I would ask for this particular problem is, if that's true, where's 0?

**Kim **06:55

You want me to answer?

**Pam **06:56

I do yes.

**Kim **06:57

Okay. So, I'm going to say that 0 is to the left of the 3.

**Pam **07:03

Okay. And how far?

**Kim **07:04

I mean, I just like made some amount. I don't know. And so, you just like, plopped it (unclear). 3. Yeah, it's 3 away.

**Pam **07:12

better be 3 away. Yeah? Right? The 0 better be 3 away from 3. And if I were to ask you... No, that's good enough. Okay, problem number one. Kind of boring. We didn't do a whole lot. We just sort of established 0 goes to the left, and then if you go to the right 3 units, you'd be at 3. And for that particular problem, that's also where x was. Alright, next problem. What if I said for this problem, x is in the same location as -2. So, I've just written down x = -2. It

**Kim **07:38

Okay.

**Pam **07:38

And I'm going to draw a number line, and I'm going to say... And I'm right in the middle again. So, same place it is. This is not in relation to the problem before.

**Kim **07:47

Oh, okay.

**Pam **07:48

Not in relation to the problem before. I'm just saying, hey, I'm going to put... Everything we know, I'm going to put in the middle.

**Kim **07:53

Okay.

**Pam **07:54

This time we know that in the middle, I'm going to put x. So, tick mark. x above that tick mark. And below that, I'm going to put -2.

**Kim **08:01

Okay.

**Pam **08:01

And now the question I'm going to ask you is where's 0?

**Kim **08:05

Mmm, okay.

**Pam **08:06

Okay.

**Kim **08:06

This time it's going to be to the right.

**Pam **08:09

And how far?

**Kim **08:10

Of the 2. 2 to the right.

**Pam **08:12

And so, if you compare now. I told you not to compare, but if you compared the first problem and the second problem is x closer to... Let's see. Is... Yeah. x and 3 closer to 0 or x and -2 closer to 0?

**Kim **08:25

The -2 is closer to 0. So, hopefully, I did a good job, and it's like two-thirds of the distance of the 0 and 3.

**Pam **08:34

That was very mathy there. Ploping the two-thirds. Nice. Nice. Because it's only 2 units away from 0. It's just 2 units away to the... It's to the left of 0. And 0 is now to the right of where x is.

**Kim **08:45

Mmhm.

**Pam **08:46

Okay, cool. Next problem. No, actually.

**Kim **08:48

Can I ask you a question?

**Pam **08:49

Yeah.

**Kim **08:50

I'm going to derail you for a second.

**Pam **08:52

No, you're good.

**Kim **08:52

Do you always put x on top and the numbers on the bottom?

**Pam **08:55

I do.

**Kim **08:56

Okay. Is there a reason for that?

**Pam **08:58

(unclear) It's just about kind of being consistent for a hot minute. Yeah, and it's going to allow us to do some stuff later.

**Kim **09:03

Okay.

**Pam **09:03

So, I'm not going to mark a kid wrong if they flip it, but I'm going to be consistent because then the problems will kind of fall out. Hey, on that problem. Before I move on. If x is in the same location as -2, do you have some sense of where positive 2 is?

**Kim **09:17

Yeah, it's going to be the same distance between 0 and -2, but that's going to be to the right of 0. So, same span.

**Pam **09:26

Nice. Same span. Same distance. So, there's this thing about distance. And we could actually go up to the first problem where we had x at 3, and I could ask you where -3 is. And where would you put -3? You still have the number line?

**Kim **09:39

Yeah. The first one? I don't have space on my Post-it. I have a (unclear).

**Pam **09:42

You're on a Post-it note? Kimberly!

**Kim **09:44

I know. (unclear).

**Pam **09:44

Kimberly! Do you have a piece of paper somewhere? I can push pause on the recording.

**Kim **09:49

It's okay. It's good. I'll figure it out. So, in the first number line where 0 was, I'm going to go to the left to mark -3. And 0 should fall right between -3 and 3.

**Pam **10:02

Because we've got that same distance of 3 happening. So, we're kind of laying some groundwork for absolute value, and distance, and span, and negatives, and positives. There's also this feel, maybe at this point, that maybe there's this kind of pivot thing happening. Do you feel sort of a pivot around the 0? You kind of have something on the left side and the same distance on the right side. And there's this kind of negative on the left and positive on the right. Super cool. Okay, next question. What if I told you this time that the opposite of x. And I'm going to write that as a negative sign and then x. So, the opposite of x is in the same location as 5. So, on my paper right now, I have written -x or the -x = 5. And I'm going to draw the number line. And just like before, whole new world, I'm going to put right in the middle a tick mark. Middle of my number line, tick mark. And on top, I'm going to write the opposite of x or -x. And on the bottom, I'm going to write 5. Now, Kim, if I was doing this not verbally, not audibly, but I was actually, you could see what I was writing, I would never have said negative. I would have only said "opposite". So, that's...

**Kim **11:13

Okay.

**Pam **11:13

I just want to... The only reason today I'm aware that I said "negative" is because you're having to imagine what I write. And so, if I was just writing, I would have just written the negative symbol, but I would have said "opposite", the opposite, as I wrote negative, and then x. The opposite of x is in the same location as 5. So, just a little thing there, I'm going to... On purpose, I'm using the word "opposite" instead of negative. Cool. So, do you have that down on your paper now? (unclear)

**Kim **11:39

I do (unclear).

**Pam **11:40

Cool. And so, a question we could just ask. Just quickly. Where's 0?

**Kim **11:44

It's going to be to the left of the 5.

**Pam **11:47

And how do you know?

**Kim **11:49

Because a number line goes from smaller to bigger.

**Pam **11:54

Yeah, but there's a -x on top. So, shouldn't 0 be to the right of the negative x?

**Kim **11:59

No.

**Pam **12:01

Because?

**Kim **12:01

Because the orientation of a number line is numerical, so... (unclear)

**Pam **12:09

Does that mean you're kind of focused on the bottom part of the number line? The numbers?

**Kim **12:12

(unclear) Yeah, I am.

**Pam **12:13

Yeah. So, let's go ahead and throw that down. So, you're saying to the left of 5 is 0, and it would be 5 units over.

**Kim **12:20

Mmhm.

**Pam **12:20

Cool. And that's where 0 is. So, interesting, Kim. I've done this Problem String with a lot of secondary teachers, and there are always super secondary teachers who will get very uncomfortable right now. They're looking at 0 being to the left of -x, and they're like. And they're almost... Here's my guess. I think... And I've heard enough teachers sort of say some things that I'm pretty confident that I have a good guess here. I think they're thinking -x. And they're thinking about what to do. And they're kind of multiplying by negative 1. Or they're... They're doing something. Or they're looking at -x as a negative number.

**Kim **12:59

Yeah, (unclear).

**Pam **12:59

We don't really know. We don't know at this point what -x is because we don't know what x is. If x was a positive number, then -x is negative. But if x is a negative number, then the opposite of a negative number is... So, we just don't know at this point. What we do know is it's in the same location as 5. And you're really focused on the 5, and that helps you place the 0 to the left of 5

**Kim **13:24

(unclear). Yeah.

**Pam **13:24

Cool. Alright, if all of that's true... Or did you want to say something else?

**Kim **13:28

No, I was just going to say, I can hear how somebody would say negative goes on the left side.

**Pam **13:33

Yeah.

**Kim **13:33

Like, I can imagine that that would be a thing that people would have been told or would repeat just to make it "easier"...I'm air quoting "easier"...for kids that negatives go on the left.

**Pam **13:45

Sure. And it's super tricky to look at the minus symbol, the subtraction symbol, and then an x, and think to yourself that could actually represent a positive number. That's tricky. And we need to do some work on that. So, it's one of the reasons why I'm going to be very clear to say the opposite of x.

**Kim **14:04

Yeah, that's helpful.

**Pam **14:05

Yeah. Okay, so if the opposite of x is at 5, and 0 is to the left of that 5 units, where's x? We know where the opposite of x is. Do we know where x is?

**Kim **14:17

Yeah, because the opposite comes in really handy. So, I'm going to go from the same distance from 0 to 5.

**Pam **14:26

Mmhm.

**Kim **14:27

I'm going to go to the left that same distance because if the span from 0 to -x is some amount, then the span from 0 to x is the same amount.

**Pam **14:38

Okay.

**Kim **14:39

To the left.

**Pam **14:40

To the left.

**Kim **14:40

I'm going to call that -5.

**Pam **14:43

And how do you know it's at -5?

**Kim **14:45

Because the span from 0 to 5, or 0 to opposite of x, was 5, so then the span from 0 to x is going to be the same amount in the span from 0 to -5.

**Pam **14:58

And you went left of 0. So, the 5 to the left of 0 is -5.

**Kim **15:03

Mmhm.

**Pam **15:04

And so, if I kind of cover the top of the number line right now and all you can see is negative 5, and then shifted over 0, and then shifted over 5. Well, that feels like number lines that we've seen before.

**Kim **15:15

Right.

**Pam **15:16

-5 is 0. 5 from the left to the right. And on the top now we have x is in the same location as -5. And the opposite of x, that -x, is kind of pivoted over the 0. We sort of have this. if you can see my hand, I'm kind of like reflecting.

**Kim **15:31

Yeah.

**Pam **15:32

Almost. I'm kind of reflecting that x over the 0 to where the opposite of x is in the same way I'm reflecting that -5 over the 0 to where the 5 is.

**Kim **15:40

Yeah.

**Pam **15:40

Cool.

**Kim **15:41

But that takes... I mean, that takes a little bit of thinking, right? (unclear) Oh, yeah. And you're like, "Wait a second. Did I write something down wrong?"

**Pam **15:47

Yeah, yeah.

**Kim **15:47

But yeah.

**Pam **15:48

It's not trivial. Yeah, not trivial at all. Hey, so the last thing I'm going to write down. If I can just briefly say. I've got the -x = 5. I've got that number line I just described. I'm going to say if the opposite of x, we're agreeing now, is in the same location as 5. Underneath that opposite of x equals 5, I'm going to say we also now know that x...and I'm going to x...is in the same location...but I'm going to write "equals"...as -5. -5. So, I now have two equations written. I have the -x = 5. And underneath that, I have x = -5.

**Kim **16:20

Mmhm.

**Pam **16:20

And to be really clear, I haven't done any multiply by -1. I haven't done like move things to the other side. I've literally reasoned that if I know where the opposite of x is, then I can pivot over the 0 and I can know where x is. So, using the sort of spatial, geometric, kind of number line reasoning. Okay, cool. Next problem. Nothing (unclear)

**Kim **16:42

New Post-it note.

**Pam **16:43

New Post-it note. Nothing to do with the one before it. What if we know this time that the opposite of x is in the same location as -4? So, let's just... Because it's visual, let me just say. As I said that, I wrote -x = -4. But I'm going to say "the opposite of x is in the same location as -4". And I'm going to write... So, I wrote that first. Now, right in the middle of my number lines, I'm going to put in the exact same middle that... My number lines are all lined up, so every time this is happening in the middle, I'm going to put the opposite of x on the top of the tick mark, and I'm going to put -4 on the bottom of the tick mark.

**Kim **17:19

Okay.

**Pam **17:20

Alright, now I'm going to be a little more free here and just ask you what else do you know? If we know that, is there anything else you know that you could just start talking about?

**Kim **17:30

Mmhm. Yeah, so I know that 0 is going to be to the right 4 units. Whatever that span is. Because we had -4. And I have to say, I've done a really nice job with my span of 2, 3, 4, 5. That's not always fantastic. But comparatively, I'm quite proud.

**Pam **17:45

Nice, nice.

**Kim **17:46

So, then, I also know that if the opposite of x is -4, then x is 4. And so, I've kind of mirrored that and made the span between -4 and 0 and 0 and 4 the same.

**Pam **18:00

And you kind of said "mirror". There's that sort of pivot thing. We're kind of reflecting. So, the opposite of x reflected over 0 to x. And you're like, "How far?" Well, it was those 4 units. And so, that -4 reflected over the 0 to 4. And so, on my number line, just so I can be visual a little bit here, I've got from left to right. I've got the opposite of x in the same location as negative 4, then 0, then x is in the same location as 4.

**Kim **18:24

Mmhm. And then, I wrote some stuff at the bottom like you did with the -5. (unclear).

**Pam **18:29

Oh, okay. (unclear)

**Kim **18:30

-x = -4. And x = 4.

**Pam **18:34

And then I might say exactly what you just wrote as if the opposite of x is at -4, then x...or the opposite of the opposite of x....x is at 4. Even though I've got equals written, I might say it sort of as a direction. Cool.

**Kim **18:51

Mmhm.

**Pam **18:51

Any thoughts on that so far?

**Kim **18:53

Mmm.

**Pam **18:55

Making some sense?

**Kim **18:57

Yeah. Oh, yeah. I mean, I don't think I have additional thoughts. But I'm with you.

**Pam **19:03

You're with it, yeah. And it's making some sense. Cool. Alright, nice. Fantastic. So, the next thing that I'm going to ask is what if this time... New number line again. Whole new scenario. Whole new universe. What if we've got this time x subtract 4 is in the same location as 6? And what I've just written is x - 4 = 6. But I'm also going to draw a number line where right in the middle again. Same middle. On the top, I'm putting x - 4.

**Kim **19:33

Mmhm.

**Pam **19:33

And on the bottom, I'm putting 6.

**Kim **19:35

Mmhm.

**Pam **19:36

If we know that x - 4 is in the same location as 6, can we figure out where x is?

**Kim **19:42

Yeah. So, I'm looking at the x - 4, and I'm thinking that's where it landed once you subtracted 4, so I'm going to imagine going to the right a span of 4. Like before you subtracted the 4.

**Pam **19:55

Okay.

**Kim **19:56

And I'm going to call that 10 because 6 plus that 4 is 10.

**Pam **20:01

And so, you're saying x is in the same location as 10?

**Kim **20:04

Yep.

**Pam **20:05

Because if you then take x and you subtract 4, you'll be at x minus 4.

**Kim **20:10

Yep.

**Pam **20:11

And if you take 10 and you subtract 4, you'll be at 6.

**Kim **20:14

Mmhm.

**Pam **20:14

So, on my number line, if I could show you what I've written so far. First I had the x - 4 in the same location as 6. And then I wrote your x and in the same location as 10. But then when I said "from x". I said, "From x, if you subtract 4..." I drew a jump from x to x minus 4, and I wrote minus 4 above it. So, now I've got x with a minus 4 and a jump lands on x - 4. And then below that, I've got 10, and I put a jump minus four to get to the 6. So, my number line now looks like... Oh, this is so hard to do visually. Maybe I don't need to describe it again. No, maybe I will. So, I've got... From the left to the right. x - 4 in the same location as 6. And x in the same location is 10. And between those, I've got these jumps of minus 4. So, then I would say... I'd look back at the equations, and I'd say, "So, we started with the equation of x - 4 = 6. And you're saying that if x - 4 is in the same location as 6, then x is in the same location as 10." And I would write below that x = 10. And in between, I would say, "And what did you do?" It's almost like you said to yourself to get from x to x - 4. Excuse me. Let me go the other direction. To get from x - 4 to x, it's almost like you sort of go 4 to the right. So, it's almost like I could write + 4 next to the x - 4. And I could write + 4 next to the 6 because if I go to the right from where we started, we land on x = 10.

**Kim **21:47

Yeah. You know what I love about... Sorry, go ahead (unclear).

**Pam **21:50

Let me say one more. Yeah, if you can. Don't forget what you're going to say. So, what you see on my equations is actually very similar to what you might see in a balancing the equation sort of perspective.

**Kim **22:01

Yeah.

**Pam **22:02

Where you like, "Add 4 to both sides." But I'm going to suggest that that is more of a "do it" kind of mathematics perspective versus what we've just done is relationally think if I know where x - 4 is, then I can reason where x is. x is going to have to be 4 greater than that because I subtracted 4 to it, 4 from it to get to 6. Alright, what do you got?

**Kim **22:26

I was just going to say that this feels a little bit like a slider to me. Like, I can picture sliding the x - 4 and the 6 kind of up and down the number line. And in my head, I'm like, "Where else would I want to put something?" Like, what would other values be? And how would that affect x? Like, I'm kind of in my head like, "Where would x - 10 be? Where would..."

**Pam **22:45

Oh, nice.

**Kim **22:46

So, it just like opens up like that.... It's like a model that opens up other opportunities, rather than just answer the one, "Where's x?"

**Pam **22:53

Oh, I love, love, love that you said that. And let me just say to algebra teachers. I'm not actually suggesting here that there's magic on this double open number line. I am suggesting that this is actually a mental map that many mathy people actually sort of have in their heads. They might not be drawing a double open number line. But here's what they're not thinking. They're not thinking... When they see x - 4 = 6, they're not like, "I better add 4 to each side." I mean, I can give a second grader this problem and say, "What minus 4 is 6?" And that second grader can go, "What minus 4 is 6? 10." Like, they're actually... And that's kind of what we're thinking about. Like, if we're thinking about that x - 4 in the same location as 6, we're like, "Well, then I had to have something bigger in order to subtract the 4 from." That's the kind of thinking that the second grader is doing. And I want to engender that as we get more complicated with equations. Alright, so one more. One more for this. What if I were to ask you something like x + 4 = -5.

**Kim **23:47

Mmhm.

**Pam **23:52

So, x... I'm drawing my number line. In the middle, I'm putting a tick mark. I'm putting x + 4. And then underneath that, I'm putting -5. So, x + 4 is in the same location as negative 5. So, what else do you know?

**Kim **24:06

Oh. I mean, I think you're going to want me to say x, but that's the part that I love is I know lots of things based on what you just said of like all different things. But I think you want me to think about x.

**Pam **24:15

I do.

**Kim **24:16

So...

**Pam **24:17

But I wouldn't mind if you threw something out like where's x plus 5? In fact, let's just do that really fast. Where is x plus 5?

**Kim **24:23

It's going to be to the right 1. Which is? -4. Sorry, I was trying to write, but I put my x on my number line, and then I was trying to erase and write it above. Fail, fail. Sorry,

**Pam **24:38

sorry, sorry. Yeah.

**Both Pam and Kim **24:38

Okay.

**Pam **24:39

Okay. That was great. Okay, so go ahead and find x. Yeah.

**Kim **24:42

Okay, so before you added the 4. That's kind of how I'm thinking about it. Where would x be? And so, before I added 4, x is going to be to the left of x + 4.

**Pam **24:52

Okay.

**Kim **24:52

And so before I added 4, it would be negative 9.

**Pam **24:57

So, when you said before you added 4, x would be to the left 4, I went left and I put x on the top. But I also drew a bubble of 4. And then, now, I could literally ask kids, "Well, you're trying to find what's 4?" And so, now, I've drawn a bubble or a jump below the number line of 4. What is to the left of -5? 4 units. And what is that, Kim? You said it already. -9. So, here's what we didn't do. We didn't, "Alright, you're going to add or subtract 4 from both sides, and now you've got -5 - -4. And is this the Keep, Change, Flip? Nope. Keep, Change, Change? Nope." Like, I can just ignore all of that silliness. And instead, you can just reason like where did x have to be? Well, bam. Then I got to go 4 units to the left also of -5. We're keeping kids in sense making and meaning. They have a feel for what's going on underneath. Oh, go ahead.

**Kim **25:48

And because they're thinking about a number line, whether on paper or in their head, it makes sense that you're going to the left, it's going to become more negative. Like, directionally that that makes sense.

**Pam **26:00

And at the same time, we're building the sense of integers.

**Kim **26:03

Mmhm.

**Pam **26:03

So, I could actually say, are we... Ask the question. Are we building equation solving sense or are we building integer sense? And the answer is yes. Like, we're building both of it. We don't expect kids to have this down as we go. We're actually building it. So, on my paper, I have that number line I just described. But I also have what we started with. x + 4 = -5. And I have that jump of 4 on there, so I could say, "Well, what did we do? We went back 4." So, underneath the x + 4, I could write - 4. Underneath the -5, I could write - 4. And I could record what we did on the number line underneath that as, so now x is in the same location as -9. So, what you see with the equations, you could maybe not even tell whether I was kind of doing the balancing stuff or I'm reasoning through using the relationships. It's going to kind of look... Not the number line, but the equations kind of look the same. Cool. Alright, listeners, here's your chance to go have some fun on your own. Try this one reasoning through it. What if I gave you the next problem? x - 4 = -10. Or x minus 4 is in the same location as -10. Go for that. Especially those of you who are like, "Negative!" Ya'll, put that on a number line and reason about where x was before you subtracted 4, and I bet you'll be able to find out where x actually was to begin with. Yeah?

**Kim **27:27

Yeah, super fun.

**Pam **27:28

Alright, Kim, so this is not about kids... It's not about a new procedure, and now I'm going to have to teach kids how to draw number lines. Nope. It is about using a model to make thinking visible, make the relationships, so that we can point at them and discuss them. And these are the relationships that mathy people actually use. They have a mental model that they're working from, and now we all can as well. We can help the rest of it develop those two. Alright, I know we haven't gotten to systems of equations yet, but we wanted to lay some groundwork for really understanding equations and messing with variables and equations. By the way, I'm not suggesting this is the only thing we would ever do, but it's kind of a thing that we're going to do to get this going. Ya'll, stay tuned next week. We're going to build on this in the next episode. Thanks 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!