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  00:16

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  00:30

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

Pam  00:37

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

Kim  00:42

Hi.

Pam  00:42

Hey.

Both Pam and Kim  00:43

How's it going?

Pam  00:45

Busy, busy, busy. I mean, wait, wait. My life is full of many wonderful things.

Kim  00:50

It is indeed. Hey, I've got a review for us today, and it's super short. 

Pam  00:55

Okay. 

Kim  00:56

It says, "How we teach math has to change." Agreed. 

Pam  00:59

Ooh, bam, yeah. 

Kim  01:00

And it's EtherealLord, EtherealLord. "And it's simple (unclear)..." 

Pam  01:05

We heard from Lord. Okay.

Kim  01:06

Yeah. Oh, and you had a lord in one of your (unclear).

Pam  01:09

I did Lord Math. Lord Math. Sir Math. That was awesome. I'll never forget that.

Kim  01:13

So, "It's simple. Everyone can do more real math than fake math.

Pam  01:18

Ooh!

Kim  01:19

Agreed. 

Pam  01:19

That's a great review. 

Kim  01:20

(unclear) mic drop. 

Pam  01:21

Everyone can do more real math than fake math. Bam! (unclear).

Kim  01:24

Okay, so speaking of real math and fake math, lots of us grew up with a lot of fake math in middle and high school. but the last (unclear).

Pam  01:31

I mean, elementary too. Just to be clear. 

Kim  01:33

I mean, that's true, but... You now have an affinity for elementary teachers. 

Pam  01:39

It's true. 

Kim  01:40

But in the last few weeks, we've been talking about meaning making and sense making about math in older grades because it can be done,

Pam  01:48

Bam! And a few weeks ago, I mentioned that we were going to maybe answer... Was it Nera's question? 

Kim  01:53

Yeah.

Pam  01:54

About systems of equations. And we're not there yet, but we're getting there.

Kim  01:58

We're heading that way. 

Pam  01:58

We are heading that direction. We're actually going to take just a tiny detour today to do a little bit more fun because we can with double open number lines.

Kim  02:06

Okay.

Pam  02:06

But I promise next week we are going to hit systems of equations hard. Okay, so what about, Kim? 

Kim  02:13

Yeah.

Pam  02:13

Inequalities? So, because we've been having fun with equalities or equations and a double open number line. We thought we'd have a little bit of fun with inequalities this week. If you have not listened to the last two episodes, highly recommend that you listen to at least one of them.

Kim  02:32

Yeah.

Pam  02:33

Yeah, that would behoove you to... It would be helpful. Okay. Kim. 

Kim  02:37

Yeah?

Pam  02:38

Here is a lovely Problem String. First problem. If I said to you that x equals 3.

Kim  02:44

Mmhm.

Pam  02:44

You would be bored because you're like, "Okay." But today we're going to put on a number line like we have been. So, I'm going to draw a long number line, and in the middle of it, I'm putting x on the top of a tick mark and 3 on the bottom of that same tick mark. And I'm saying x is in the same location as 3. And the question I have for you is where is 0?

Kim  03:02

0 is to the left of the 3. 

Pam  03:04

Okay, so I'm going to write 0 to the left of 3. I could put 0x is where that is. I'm also going to ask where the opposite of x is?

Kim  03:13

On the other side. So, -3. 

Pam  03:16

Okay, so on the other side of that...

Kim  03:17

Other side of... Yeah, the opposite of the x.

Pam  03:19

Of 0? Mmhm. Yep. So, I've written on the opposite side of 0, the left side of 0, I put -3 on the bottom, and I've put the opposite of x on the top. Cool. So, over there on the left, very far left of the number line, I wrote x equal 3 when we started. I'm also going to write that the opposite of x is equal to the opposite of 3, or -3.

Kim  03:39

Okay.

Pam  03:39

So, we just know both of those things. Cool. What if? New problem. What if x is greater than or equal to 3?

Kim  03:50

Mmhm.

Pam  03:50

So, going to draw the same number line, or not the same number line. I'm going to draw a number line underneath it that correlates with the number line above it. I'm going to put... Let's see. If x is greater than or equal to 3, what's something you know?

Kim  04:05

I know that there are numbers that are... 

Pam  04:09

Can we do the equal to first? Sorry. I left it open. Now, I'm shutting it down. Do you know x equals 3? If x is greater than or equal to 3? 

Kim  04:19

Yeah. 

Pam  04:20

Okay. So, I just wanted... I'm going to start there. I'm going to say, so I know x equals 3. So, just like we had above where we had a tick mark and we had x on the top and 3 on the bottom, I'm going to do that, but I'm also going to... I'm going to kind of color that in because we know where I'm going to put a dot on the tick mark. 

Kim  04:34

Okay.

Pam  04:34

X equals 3. And then, now you can finish your sentence. I'm sorry for interrupting. 

Kim  04:38

So, all the numbers that are to the right, all the numbers bigger than 3 would fit whatever your what x could be. 

Pam  04:49

In that set?

Kim  04:50

Mmhm.

Pam  04:50

Yeah. So, when I wrote x greater than equal to 3. I'm very quickly going to start saying all the numbers every time I see x. I'm going say, so I want all the numbers that are greater than equal to 3. And my number line now has x equal to 3. I have a dot colored in. It's equal to 3. And then I've also sort of colored in the number line kind of darker to represent all of the numbers to the right of 3 are also included in that set. So, I basically have kind of a ray where I've started at 3, and I've colored an array going to the right to represent all the numbers that are greater than equal to 3. Yeah? Okay, cool. Do you think I visually... I said that enough? Yeah, probably. You're like, "Well enough. Pam. Stop." 

Kim  05:36

Well, I drew what you're saying, so...

Pam  05:40

Okay. (unclear).

Kim  05:40

...as you're saying it, picturing it. Yeah.

Pam  05:42

Okay, cool. Hopefully, listeners, hopefully you can picture it as well. Staying with this problem, where's the opposite of x? Well, maybe, let's throw 0 on there. Can you throw 0 on there?

Kim  05:51

Mmhm.

Pam  05:52

Okay, so 0 is kind of what?

Kim  05:54

To the right of x and 3.

Pam  05:57

To the right?

Kim  05:57

I'm sorry, to the left.

Pam  05:58

To the left. (unclear).

Kim  05:59

I'm staring at my paper, and it's on the left.

Pam  06:02

You probably we're at 0, and you look to the right. Yeah, okay. So, we kind of have 0, and then to the right of that, we have this set where x is equal to 3 or greater than. It's all colored in. Okay. Where's the opposite of x? Don't go less than, or greater than, or anything. Just where's the opposite of x?

Kim  06:20

Don't say negative? You want me to tell you the value of x, -x is?

Pam  06:24

(unclear) Yes.

Kim  06:25

Or the opposite of x? -3.

Pam  06:27

Okay, so -x, or the opposite of x, is at -3. So, if we look at where we had kind of drawn the dot, the closed in dot at x and 3, you're reflecting that over the 0, and you're like, "Yeah, so the opposite of x now is at -3." And we could close in that dot. So, where I have x greater than equal to 3, kind of in an equation inequality form, I think we could also say that still x... Or no, sorry. That now the opposite of x is equal to -3. But we're maybe not done with that relationship because now we have to reflect all those numbers. You just reflected 3.

Kim  07:09

Mmhm.

Pam  07:09

But we've got to reflect that whole set.

Kim  07:12

Mmhm.

Pam  07:13

So, if we reflect that whole set, that ray that we colored in is now reflected over. What is that picture on your...

Kim  07:21

it's going to be all the numbers going to the left. So, I drew a circle on negative x is same location as -3. Circle on my tick mark. And I drew an arrow going to the left.

Pam  07:32

An arrow to the left because it's that whole set. It's all those numbers over there. So, how do I describe that set with respect to -x, with respect to the opposite of x? Like, if you look at the opposite of x as a thing, and I want to describe all the numbers that are to the left of it.

Kim  07:54

All the numbers. So, it's going to be all the numbers that are less than or equal to.

Pam  07:58

-3.

Kim  07:59

Mmhm.

Pam  08:00

Yeah. Does that make sense?

Kim  08:01

Mmhm.

Pam  08:02

So, there's a couple things that we know that's true. If we have a set that's where all the numbers x are greater than or equal to 3, we've kind of got that right hand side. And if that's true, then the opposite of x is going to be less than or equal to -3. Both of those things would be true.

Kim  08:24

Mmhm. 

Pam  08:24

Okay, cool. Next problem. Next problem is what if the opposite of x is equal to... You know, I might change this right here, right now. I am going to. What if the opposite of x is equal to 4?

Kim  08:38

Okay.

Pam  08:39

Okay, where's that? 

Kim  08:40

The opposite of x is equal to 4. So, I drew a number line and I put opposite of x in the same location as 4. 

Pam  08:46

Cool. Where's 0.

Kim  08:47

To the left.

Pam  08:50

Okay. 

Kim  08:50

I put 0x and 0 to the left of 4.

Pam  08:54

Cool. And do you have any idea where x is? Oh, is that what you just said?

Kim  08:59

Yeah.

Pam  08:59

Sorry. 

Kim  09:00

I put 0x.

Pam  09:01

Okay, so how about... We know where the opposite of x is. The opposite of x is at 4. Where's x?

Kim  09:06

X would be the same distance on the left side of the 0.

Pam  09:12

So, x is... 

Kim  09:14

-4.

Pam  09:14

...that distance of 4, and so that would be at -4. Okay, so where I have written negative x equals 4, we could also write x is equal to -4.

Kim  09:23

Mmhm.

Pam  09:24

And we kind of use that. We're sort of pivoting over the 0 to do that. Cool. Next problem. What if the opposite of x is greater than or equal to 4? What are you thinking?

Kim  09:36

So, I put a circle on where I have my number line saying -x is 4.

Pam  09:42

Okay.

Kim  09:43

That's where they're equal.

Pam  09:44

That's where they're equal. Okay.

Kim  09:45

Mmhm. And I want the numbers that are greater than 4, so I'm going to the right with my arrow.

Pam  09:53

Cool. So, I'll just describe. -x is in the same location as 4. And then we have a ray going to the right. So, we've described a set where we're equal to or greater than the opposite of x.

Kim  10:06

Mmhm.

Pam  10:06

Okay, cool. So, if the opposite of x is at 4. Just stay with the equal to part here. If the opposite of x is at 4, where's x.

Kim  10:14

If the opposite of x is 4... Didn't we already put x at -4?

Pam  10:20

We put the... Not for this problem. 

Kim  10:23

Oh, I have my number line from... 

Pam  10:25

Oh, you're using the one from before? 

Kim  10:26

(unclear) yeah, sure. Okay, so negative x is 4, so then x would be -4. 

Pam  10:32

Cool. But now we've got to reflect that whole set. Like, we reflected the opposite of x to get x. We reflected 4 to get -4. But now, that whole set has to sort of also be reflected. So... 

Kim  10:44

So, I have a circle on x is -4.

Pam  10:47

Mmhm.

Kim  10:47

And I drew going towards the left with my arrow. 

Pam  10:53

And how would you describe that set with reference to x? 

Kim  10:58

It's all the numbers that are greater than -4. Greater than or equal to -4.

Pam  11:02

Why greater than? 

Kim  11:04

Oh, I lied. Less than or equal to -4. 

Pam  11:06

Less than or equal to -4. Because it's all the numbers left, to the left of -4. 

Kim  11:10

Yeah. 

Pam  11:11

Okay, so looking back at the inequality we started with, we had -x greater than or equal to 4. And now we've said, if that's true, and we reflect everything, then x is going to be less than or equal to -4. 

Kim  11:24

Mmhm.

Pam  11:25

Yeah? And I'll just point out nowhere in here have we done any multiplying, dividing by -1. Nowhere in here have we talked about flipping the inequality sign.

Kim  11:35

(unclear). Yes. 

Pam  11:36

We're using relationships and connections to think about if we know the number, what's the opposite of that number? Oh, we reflect over 0. If we know a set of numbers, what are the opposites of that whole set? Bam, the whole set reflects over. And I think we didn't do but we could have is I could have chosen some numbers... Like in the problem we just did where we started with the opposite of x is greater than or equal to 4. And I could have chosen like 10 or, I don't know, 13 and a 1/2. Anything that was in that first set that we drew that was from 4 all the way to the right.

Kim  12:09

Yeah. 

Pam  12:09

And I could have said, you know, where is it reflecting? Oh, bam. It's over here. Well, then choose another one. Where's that reflecting? If you can see my hand, I'm sort of... I've got it open, and then I kind of reflect it left to right. So, again, like if it's over here, then reflecting it's over here on the other side of 0. So, that whole set kind of reflects back and forth. Cool. Next problem. How about if the opposite of x is less than -5. 

Kim  12:39

Okay. 

Pam  12:40

So, I've got right in the middle of my number line the opposite of x. And it's in the same location as -5, but I'm putting an open circle...

Kim  12:51

Mmhm

Pam  12:51

...there because it's not equal. It's just less than -5.

Kim  12:54

Mmhm. 

Pam  12:55

So, now when I see that opposite of x, I almost kind of cover it up, and I think all the numbers less than -5. And where is that? 

Kim  13:04

All the numbers less than -5? 

Pam  13:06

Yeah.

Kim  13:07

I'm drawing an arrow to the left.

Pam  13:10

To the left. Okay, cool. So, if that's true that there's a set less than -5, and we've drawn it. We've started at -5 with an open circle, and we've sketched. We've colored in all the the number line to the left. What is also true about positive x? Because we were just dealing with the opposite of x. What about the opposite of it? Like, what about... What's the relationship happening now? If we have a set where all the numbers less than -5 are to the left of that -5, and that's -x, where's positive x?

Kim  13:45

So, positive x is going to be 5, and the set is going to be all the numbers that are greater than 5.

Pam  13:52

It reflects, bam, over that 0. And so, if we know that -x is less than -5, then x is going to be... How would you describe that set? I've got a picture of it where x open circle 5, and that's colored in to the right. How do you describe that set?

Kim  14:06

Greater than 5. 

Pam  14:07

Greater than 5. So, if -x is less than -5, then x is also greater than 5. Reasoning about reflecting those sets back and forth, forth and back. Cool. 

Kim  14:18

Yeah.

Pam  14:18

Alright, you want to try one more?

Kim  14:19

Sure enough. I need more paper. Okay.

Pam  14:23

I'm almost to the bottom of mine. Okay, first one is 3x or 3 times x equals -6.

Kim  14:29

3x is -6. And I'm asking myself questions. 

Pam  14:34

Yeah. What else do you know?

Kim  14:35

I'm saying to myself that I know x is 2. I don't know that that matters,but

Pam  14:41

X? 2?

Kim  14:43

If 3x is 6.

Pam  14:45

-6. 

Kim  14:46

Oh. Haha. Then -3.

Pam  14:48

So, maybe throwing 0 in there would be good.

Kim  14:51

Well, I heard you say 6. That's why I said -6. I mean, 2. So, if 3x is -6, then x would be -2.

Pam  15:01

Okay. Could you tell me where those are on the number line?

Kim  15:05

I put x is -2 to the right of 3x is -6. 

Pam  15:10

Cool.

Kim  15:12

And if you want me to put 0, then I'll put 0. 

Pam  15:14

I kind of do.

Kim  15:14

To the right of that. 

Pam  15:16

So, a lot of kids will start with 3x and -6, and then they'll think about where 0 is. To the right 6 units. And then they'll go, "Well, if 3x is there, then..." Sometimes they'll even put 2x is at -4, and 1x is it -2.

Kim  15:34

But how would they know? If 3x is -6, how would they know 2x without thinking about (unclear).

Pam  15:40

Well, they... Yeah. Sorry, I didn't mean to say that would be the next thing. So, they...

Kim  15:43

Oh, that's just what they have on the paper. 

Pam  15:44

Yeah.

Kim  15:45

Gotcha, okay. 

Pam  15:46

Yeah, yeah. So, they could have a lot of kind of things on their paper.

Kim  15:49

(unclear) Cool.

Pam  15:49

But so if 3x is -6, x is also -2. Good enough for that problem. Next problem. How about if 3x, this set of 3x, is less than or equal to -6.

Kim  16:04

3x is -6. I'm going to color in a circle there. And you said less than or equal to? 

Pam  16:09

Correct.

Kim  16:10

Okay, so all the numbers to the left of the -6.

Pam  16:13

So, so far, that's what we know. We know that we got the set of numbers to the left of -6. Cool.

Kim  16:19

Mmhm. So, then I could also... Oh, sorry.

Pam  16:21

Go ahead. Go. Go, go, go.

Kim  16:23

So, then that's 3x is greater than or equal to 6.

Pam  16:29

You're going to reflect first. Okay.

Kim  16:30

Yeah. 

Pam  16:31

Oh, so hang on, 3x was less than or equal to -6. Are you saying -3x? The opposite of 3x?

Kim  16:38

Did we say 3x is -6? 

Pam  16:40

3x is less than or equal to -6. Is where we started.

Kim  16:43

Okay. So, the negative 3x...

Pam  16:45

Okay.

Kim  16:46

...is greater than or equal to 6.

Pam  16:49

And how did you get greater than or equal to? The equal to I'm good on because you reflected the 3x to the opposite of 3x. You reflected -6 to 6. How do you know it's greater?

Kim  16:58

Tell me again. 3x. We said 3x is -6. 

Pam  17:01

3x is less than or equal to -6. Yes.

Kim  17:03

Less than or equal to. Mmhm.

Pam  17:05

Mmhm. 

Kim  17:05

So, then I reflected to write -3x is 6. 

Pam  17:10

Okay.

Kim  17:10

But I also reflected the numbers, the set.

Pam  17:14

Mmhm. 

Kim  17:14

So, what did I say? Did I say something wrong?

Pam  17:16

No. I just wanted you to say it again. 

Kim  17:18

Oh.

Pam  17:19

Sorry. I was like, "Wait a second." So, 3...  Neutral response on my part.

Kim  17:23

Yeah, it was. 3x is greater than or equal to 6. 

Pam  17:28

Negative 3.

Kim  17:29

-3x is greater than or equal to 6. 

Pam  17:31

We're audio here, Kim. We got to get this right.

Kim  17:33

I know. I know. I can't even read my paper. -3x is greater than or equal to 6. 

Pam  17:39

Cool. So, just to keep... This is so audio. I hope you guys are grabbing a paper and a pencil, pen and paper. So, I, on the left, I have 3x less than or equal to negative 6. And underneath that, I have the opposite of 3x or -3x greater than or equal to 6. 

Kim  17:58

Mmhm.

Pam  17:58

On the number line, I've got two sets represented. Well, (unclear) two sets? Yeah, kind of. Where I've got 0 in the middle. To the left, I've got a closed circle on 3x and -6, and I've colored to the left of that because it's all those guys less than that. And to the right of 0, I've got the opposite of 3x and 6 in the same location with a closed circle, and the whole set that's been reflected now to the right of that.

Kim  18:23

Mmhm. 

Pam  18:23

Nice. Alright, do we have time for one more? I think we do. Let's do one more quick. Okay, 

Kim  18:27

Paper, paper, paper. Okay.

Pam  18:28

Paper, paper, paper. Okay. What if this time we've got the opposite of 3x is less than or equal to -6? Ooh, now I kind of wish I had that... I had flipped papers, and I wish I was looking at my other one. Maybe. The opposite of 3x is less than or equal to -6. What do you got?

Kim  18:47

Okay, so I wrote -3x and 6, the same spot.

Pam  18:51

-6. 

Kim  18:54

I did write that. -3x and -6 are the same (unclear). 

Pam  18:59

Well done. Well done. 

Kim  19:01

Jeez. 

Pam  19:01

Sorry.

Kim  19:02

And you said, it's all the numbers less than that, less than or equal to?

Pam  19:07

It's less than or equal to -6. 

Kim  19:09

Okay, so I colored in a circle, and I drew my arrow going to the left.

Pam  19:14

Okay, cool. 

Kim  19:16

Alright. 

Pam  19:16

Well, I'd love to know how x relates to numbers if -3x is less than or equal to -6.

Kim  19:23

Okay, so I'm going to put 0. 

Pam  19:25

Okay. (unclear).

Kim  19:26

For funsies, and then I'm also going to... It's to the right (unclear).

Pam  19:30

Of -6. That makes sense. Okay.

Kim  19:32

And then I don't know. I want to play. I don't know what you're actually wanting me to do yet, but I put 3x and 6.

Pam  19:40

Because you reflected everything. Okay, mmhm. 

Kim  19:43

So, I know that 3x is going to be greater than or equal to 6. 

Pam  19:49

Nice. And I've colored that in, and I've written the equation. 3x is greater than or equal to 6. And I've colored in the 3x and 6 location, and then everything to the right. Okay. 

Kim  20:00

Yep.

Pam  20:00

Nice. 

Kim  20:02

What do you want to know?

Pam  20:03

X.

Kim  20:04

Oh, okay.

Pam  20:04

You told us 3x, and now I want x. 

Kim  20:06

Okay. So, x is going to be where 2 is. It's equal to 2 or greater than 2. X is greater than or equal to 2.

Pam  20:17

Nice. Nice, nice, nice. We could have found out the relationship from -2x  Yep. And that would have been everything less than or equal to -2, and then reflected that set. 

Kim  20:28

Wait, wait, wait. You said -2x?

Pam  20:30

I meant (unclear).

Kim  20:31

-x. 

Pam  20:32

Sorry, you're right. -x and -2. 

Kim  20:35

Yep.

Pam  20:36

And then reflected that to get the x and the 2. 

Kim  20:38

Yep. 

Pam  20:38

Either way. Nice. Kim.

Kim  20:41

Yeah?

Pam  20:42

Inequalities are figure-out-able.

Kim  20:45

They are. And so not figure-out-able for kids if they only get taught flip some things, do some things. But.

Pam  20:53

One other thing I'll mention just really, really fast. There is the move that you could make, teachers, that if you start off with -3x less than or equal to -6 where you do the balance thing, if you must, and add 3x to both sides, and add 6 to both sides. And if you do that, you'll end up with 6 less than or equal to 3x. And again, you don't have to worry about this bit about multiplying and dividing by -1 and flipping the sign or reversing the inequality. By the way, I don't say "flip the sign". I say "reverse the inequality". If you must say it. But I don't know that that ever should become a thing to do. I think it should always be something that we're reasoning about. "Well, yeah. If I reflect this whole set, then I've reflected it. Then, yes. Now, the new set is going to be the other relationship because I've reflected over 0.

Kim  21:45

Mmhm. 

Pam  21:46

Yeah. Hey, Kim, that was fun. Next week, ya'll, tune in because we're going to do some systems of equation magic, and you're going to love it. Math magic. Magic Math. Yeah, whatever. Alright, 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. Let's keep spreading the word that Math is Figure-Out-Able!