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 229: Solving Inequalities
Use Left/Right to seek, Home/End to jump to start or end. Hold shift to jump forward or backward.
Do we flip the sign when its negative? Or is it we keep the sign and flip the minus? I'm confused! In this episode Pam and Kim avoid all that confusion and make solving inequalities figureoutable!
Talking Points:
- Open Number Lines for Inequalities
- Using "Reflecting the inequality" vs "flipping the sign"
- Reasoning about balancing the equation
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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!