Ep 181: Integers

Pam  00:00

Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam Harris.

Kim  00:06

And I'm Kim Montague.

Pam  00:08

And you found a place where math is not about memorizing and mimicking, where you're waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can actually mentor students to think and reason like mathematicians did when they were young. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be. Hey, Kim. 

Kim  00:37

Hi. I'm totally going to tell on you.

Pam  00:40

What?! 

Kim  00:42

Yeah.

Pam  00:42

Okay, fine. Bring it on! (unclear).

Kim  00:44

Yeah. So, Pam and I have lots and lots of meetings for different kinds of projects.

Pam  00:50

Oh, gosh. Yes.

Kim  00:51

When it's just the two of us, sometimes we meet on Slack. We huddle. So, no video. 

Pam  00:57

You're going to tell them this? 

Kim  00:58

I totally am. Yeah. And sometimes, you know, kind of back in the day, we just would get on the phone and pace our house and talk. And, you know, sometimes when there's a group of us or when we just want to see each other's faces, we get on Zoom. And so, it's typical when we're going to start a meeting, Pam or I will text each other and say, "How do you want to meet?" And she sent me a text this morning and said, "Do you want to do Slack, or zoom, or text or whatever, phone. 

Pam  01:23

Yeah. 

Kim  01:23

And I wrote back. ZenCaster. Because she had clearly forgotten that we were recording a podcast today.

Both Pam and Kim  01:30

(unclear).

Pam  01:32

Yep, yep, yep. It's true. It's true. To be clear, when do our calendars, we normally will put in the calendar invite what we're doing, and we switched today, right? It was supposed to be a typical meeting, and today we're recording. So. Ha, that's my excuse. 

Kim  01:49

I totally laughed. 

Pam  01:50

Fine.

Kim  01:50

I actually have another story for you. We'll get to some math in a minute. But.

Pam  01:55

This is what happens when you and I spend a week together at NCTM and NCSM, huh?

Kim  01:58

I know, Okay, so I have to tell you. You're going to appreciate this.

Pam  02:02

Okay. 

Kim  02:02

And it's really about why I stand by pencil over pen. So, it's appropriate. No, I'm not even joking. Like (unclear).

Pam  02:10

Alright, bring it on. Because I'm holding a pen in my hand right now.  Oh, no. 

Kim  02:14

Well, you're not going to want to after this. So last night. Or yesterday after school, Cooper came home, and he was super excited because he had been asked to wear his uniform, his Scout uniform, and be in a program and do flags. And he was super excited about it because he loves doing flag type things in the, you know, ceremony or whatever. So, he's super excited. It was kind of like 7:30, 8:30 when he told me. And I said, "Cool, go get your shirt, and bring it to me, and I'll iron it, so that it really looks sharp." And so, he went and got it and he brought it in. And he had just recently gone to a... I don't know. Some other event where he wore the shirt for like 30 minutes. And so, we just took it off and hung it up. He brought me the shirt, and he held it out to me. And I was like, "What is that?" Mmhm. Mmhm. 

Pam  02:26

Did a pen leak?

Kim  02:34

All over this stinking shirt. Big blobs of it. And he looked at me, and his eyes got really big, and he literally started panicking and like getting upset. And I was like, "Bro, like hang on. Let me see what I can do." But I knew. I mean, everywhere. There's like splashes everywhere. And he was like, "Did I do that?" And he had a tiny pen in his pocket. And the pen was off. Like, it it just had leaked everywhere. 

Pam  03:37

Stupid pen.

Kim  03:38

So, anyway, I, you know, did all the things. Google the things. Hairspray, and alcohol, and sanitizer, and blot it, and wash it, and all the things for hours. So, today's podcast is brought to you by three hours of sleep. 

Pam  03:53

Oh no!

Kim  03:54

Mmhm. Mmhm. And thank goodness for friends who bailed me out because that shirt s in the trash. Horrible. Pencils are way better. Stupid pens! Hate them!

Pam  04:05

Kim, that's horrible. I'm so sorry. So, you have a friend that is loaning you a shirt? (unclear).

Kim  04:10

Well, so I actually live in a neighborhood with a couple of our scout moms. And recently, a kid had grown out of one and the other mom is the advancement chair who holds all the patches. And so, they knew that I was struggling and one of them showed up last night with a shirt from one house and patches from the another house. And, you know, I don't know how to sew, so they're all taped on. Whatever hopefully they stay. Horrible.

Pam  04:35

Okay. Alright, you got me there. That's a pretty... Okay, there's the pen and pencil argument continues on. Sorry, about that.

Kim  04:43

Alright, we should start. Sorry.

Pam  04:44

Bam! Alright, Kim. Hey, let's have a podcast about math being figure-out-able

Kim  04:49

Well, one more side note. But we got to review. You know, I'm loving the reviews. They make me so happy.

Pam  04:55

That's been really fun to get so many reviews. Ya'll, thank you for sending the reviews. You know, I don't know if you know this. But if you put in a review, and like, or... What do you call it when you rate? Rate the podcast. It actually gets shown to more people. So, we appreciate that because you know our goal here. We don't monetize this podcast, right? There's no. We just do it because we actually want to change the way math is viewed in the world. And so, this will be helpful, you know, if you don't mind giving us a rating and a review. So, all right, Kim, we get a good one today?  Super fun to hear people's stories,

Kim  05:23

too. Yeah. So...

Pam  05:24

It is. Yeah. No, I think he means are getting... Or she, he. "Are getting required to truly understand math." Like, in other words,  in class, we're not getting away with just mimicking we're actually like requiring they actually understand it. And that's amazing. Yeah. Oh, nice. Thanks, Jay Poor. That's super cool.

Kim  05:25

...Jay Poor. The title is "Completely changed my way of thinking", which is perfect. And it says, "I have listened to every episode, and I learn something new every time. Thank you for sharing and making math more figure-out-able. My brain, and subsequently my students brains, are getting required to..." It says, "required to truly understand math." So, maybe "acquired". Amazing. So, thank you.  Alright. Finally, moving on. We are launching into December this week. Can you believe it? (unclear)

Pam  06:11

Yes! It's my favorite time of the year.

Kim  06:13

I know, it's big holidays for you So, we have saved a very big topic to round out the year. 

Pam  06:15

(unclear) Love it. Love it. Oh, yeah! This is going to be fun. 

Kim  06:22

Yeah. It's been asked for. You can search the Math is Figure-Out-Able teacher group. You should join us. But it's come up.

Pam  06:28

You should join us there. Yeah. 

Kim  06:29

A bunch of times.

Pam  06:30

People asking about it all over the place. 

Kim  06:32

Mmhm. I also met some ladies. I forgot to say this. At NCTM. Shout out to Edna and Felicia. And they asked for this as well. And I said, "Oh. I mean, you never know."

Pam  06:41

Alright.

Kim  06:41

So, today, we are short starting a short series on integers.

Pam  06:46

Alright, so Edna, and Felicia, and the Math is Figure-Out-Able teacher group, and anyone else who's ever asked for integer work, let's dive in and do some stuff with integers. Yeah, Kim, I'll also admit that... "Admit". That's not the right word. I will inform. I will tell. I will say that often when I work with middle school teachers, they'll sort of kind of go with me. They're like, "Yeah, yeah, yeah. This is all making sense. Yeah. Yeah. But what about integers?" And I'm like, "What about integers?" And one of the very first times that teachers push back on me. So, this is several years ago. Middle school teachers I was working with. And this was early in my days. I'd kind of had gotten a pretty good handle on whole number numeracy. I hadn't really done a lot of higher math yet. So several years ago, and they're like, "What about integers?" And I thought to myself, "Well, no. Like, you just kind of keep thinking about integers." And they're like, "No, no, no. Everything changes when you get to integers." So, I did some little action research. I wanted to mention to you today. Where I pulled in... So, this is when I was working really heavily with you and the other teachers at my kids elementary school. I was doing a lot of the professional learning K-5 and the whole district. And so, there were kids, who...my kids friends and kids in the neighborhood...who were getting real math every year. And they had real math now for several years. And I grabbed three or four fifth grade kids in the neighborhood, and I said, "Hey!" Like, "I'll give you some cookies or whatever. But, come here, do some math with me." And they were like, "No problem." I mean, we had fun. They knew it was a good thing. And so, I sat them down, and I said, "Hey, you know like, if I just gave you the problem..." Well, first of all, we talked a little bit. I was like, "What is negative 25 mean? And they were like, "Well, you know, it's like on the side of the number line. And it's kind of like you're in debt." They had some decent things to say about what negative numbers mean. And so, I just said something like, "So, like if I had, I don't know, 10 plus negative 25. Like, what does that mean?" And they said, "Well, yeah. It's, you know, you're at 10. And then, you would sort of jump back 25." And I'm actually just putting that on it just because I keep track of my thinking. So, if I'm at 10 and jump back 25, they were like, "Well, you would hit the 0. You know, you got 10, you've hit the 0. Then, you have to go 15 more to get the 25, so now you're back 15. And so, it's negative 15" And I said, "So, like if you've done this?" They're like, "Well, no. We're fifth graders. We haven't done any." And I was like, "Okay, cool." So, like, what if... And I just threw a couple problems at them, and they just kept reasoning the way they had been reasoning about whole numbers. And I found that fascinating. So, first thing I would say to you is, there is some sense in getting kids A, to think and reason with whole numbers, actually think and reason using relationships, using a number line to represent their thinking and it becomes a tool for them to think with. Then, we need to help integers mean something. And so, today, let's spend some time, Kim, on discussing what that means. You know, what do integers mean? How can we do that with kids? So, you up for that? 

Kim  09:47

Sure, yeah. 

Pam  09:48

Alright, cool. So, I think you're going to love this first one. I think one of the best things that we can do to help get kids to really understand the meaning of integers is play games. 

Kim  09:59

Yeah. I mean, I love games. 

Pam  10:02

You do love games. That's why I knew you would like that one. I do still... I think you and your family play more games than me and my family. (unclear) 

Kim  10:09

It's because my kids love (unclear). 

Pam  10:10

Well, my kids like them enough. But, yeah. I'm going to start pushing it more. We've played plenty of games. But I hear you, and I'm like, "They play more games." So, I don't know if it's a competitive treak in me or what it is, but we're going to be playing. I now have a couple of daughters-in-law who really like to play games. That helps. Anyway, so games. So, what are... I'll just throw to you. Do you have any games that deal with negative numbers? 

Kim  10:11

Oh, yeah. Actually one of our favorites.

Pam  10:12

I had a feeling. Yeah.

Kim  10:15

Yeah, we play this game called Skyjo. 

Pam  10:29

Okay, tell me about that. 

Kim  10:35

And it's a card game where you lay out three... Is it 3 by 3? 4? 4 by 3 grid of cards. You face them upside down. 

Pam  10:52

Okay. 

Kim  10:53

So, you can't see the numbers. And you basically are exchanging or trading cards based on. 

Pam  10:58

What's on the card. 

Kim  11:00

Sorry, numbers from negative 2 to 12. And you want the lowest score. So, you draw a card. You decide where you want to put it. You can't look at the card first. You're just kind of exchanging, so.

Pam  11:13

You want the lowest score. Sorry, I'm just catching up. Yep. Okay, 

Kim  11:16

It's a bit of a luck of draw thing, where you hope you get the lowest card drawn. But then, you have to like kind of memorize where things are. And you have to (unclear).

Pam  11:26

So, kind of a memory kind of thing. Mmhm. 

Kim  11:28

Well, then there's strategy because if you get three of the same number, then the whole row goes away. So, even if you get three 12s in a row, the score goes away. And so, my kids are super big into taking a chance on that. Anyway, but because there's negatives, and you want to get the lowest score, then when you total your score, you're going below zero a lot of the times.

Pam  11:49

And I wonder when you're totaling your score if you ever say, "Well, I've got a 5 and I've got a negative 5." 

Kim  11:54

Yeah.

Pam  11:55

Yeah. So, you might look for nice combinations that sum out to 0, and then you can kind of go from there. Yeah, that's an excellent example of something that could get kids kind of thinking about negatives. Like, even playing Monopoly, I think. You know that you can't go in the hole, right? You can't go in debt. And what does it mean to be in debt? Oh, go ahead. 

Kim  12:17

Sorry. Yeah. It made me think of a game that we have that you can. Which is interesting. So, we have this game called Payday. It's pretty much younger. But you basically follow a calendar and bills happen to you, and this and that and the other. You get paid. We used to play it younger. But in that, if you can't pay your bills, then you borrow from the bank. And so, then, you keep a total of you're running. You have to pay the bank back plus interest. So, anyway. 

Pam  12:45

Keep track of that debt. Yeah.

Kim  12:47

Hopefully, you don't get too many bills and you live in negative for too long.

Pam  12:51

Yeah, yeah. Let's all not be in debt. That's a good thing. I know, teachers. I know. You're like screaming at your radio right now. Radio? How old am I? Heavens, did I sleep last night? You're listening to your podcast wherever your... Yeah, podcast player or whatever that is. And you're screaming. You're like, "Pam, don't you know we're teachers? We don't make any money." I know. I know. I know. We're both teachers too. So, we get, but. Golly, in fact. Mike... Mike, what's your last name? It'll come to me in a minute. Good friend of mine. Teachers Teaching with Technology. Mike does a really good presentation about debt and helping teachers really think about staying out of debt. Anyway. Mike, sorry I can't remember your last name. Good guy. Anyway, there's another game that we like to play with younger kids, where we don't necessarily have integers involved. But then, you can bring integers in. And it's called, "Close to 100". So, I've heard it called Target 100. Close to 100. I don't know. I can't think of any other names. But there's lots of versions of it out there. Where basically you've got digit cards. So, cards that have the digits on them. And you draw 6 cards, and your goal is to choose 4 of them to turn into two two-digit numbers to sum, to get as close as you can to 100. So, you know, say you get 45 and 67. You add those together? No, that's going to be a little over, and so then. Oh! Well, so when you originally play it, if you go a little over, then your score is how far you went over. So, say you were a sum of 102, then you would have a score of 2. Let's say that your sum was 97, then you would have a score of 3. Well, for young kids, you just sort of play where the score is. No matter if you're over or under, the scores are positive. But pretty soon you have students even super young that will go, "Well wait a minute,. If we're over is that like negative? Like Price is Right?" And so, you know, if your sum is 102, is your score negative 2? Or vice versa. If you're under is that negative? And so, there's this brilliant kind of distance from 100 that comes out, and you can kind of think about it as the absolute value, as just the positive distance. You can also bring in negatives. And then, you've got kids that try to get a little bit of a higher score in one round because they're already in the hole. Like, their score is negative, and so they're trying to get a positive to kind of 0 it out. And all of those kinds of 0 pair ideas are brilliant. That can come out in the game like Close to 100. 

Kim  15:20

Yeah, so good.

Pam  15:21

Alright, cool. So, another thing. So, games. Games are a great way that we can kind of get kids to think about this idea of negatives and positives and how they work together, especially as you're summing scores. Another thing that can be really nice to help kids think about the meaning of integers are versions of Clothesline Math. So, Clothesline Math, we really like. Chris Shore and the Clothesline Math site does a really nice job of talking about different things to do. Well, one thing that you can do is just give kids in order a set of numbers. And so, you can say things like, "Alright, here's the..." You put the clothesline up, and you just say. So, Kim we'll do that. If I had a clothesline, I've kind of drawn a line on my paper, and I were to say, "Hey, Kim, where are you going to put 0?" And you would just choose where you're going to hang this card on that line. Yeah. Where would you put it if the line was kind of hanging there? Describe where you'd throw it. 

Kim  16:12

I put it in the middle, actually. 

Pam  16:14

Yeah. And a lot of kids usually do that. Though, the younger they are they put it on the left hand side. 

Kim  16:18

Sure. Yep. 

Pam  16:19

Right? But, you know, we're talking about integers today, so you're like, "Ah, there might be some things happening on both sides." And then, I might say, "Where 6?" And then, you would?

Kim  16:28

To the right.

Pam  16:28

To the right. And then, you're going to have to decide how far to the right. You might put it all the way to the right. Somewhere. We stuck it somewhere. And then, I'll say, "Where's negative 6?" And then, where would you put negative 6? And that's

Kim  16:39

To the left of the 0. The same distance as 0 to 6. 

Pam  16:40

kind of important. And so, then, we talk about how the distance is the same, but one is on one side of the number line. The other ones on the other side of the number line. And then, I might say, "Where's 10?" And then, you know, if you had put 6 all the way to the right, then you're going to have to adjust 6 because now we have to fit ten on there. So, you'd stick 10 on there. And then, I might say, "Where's negative 10?" But then, I also might start with, "Where's negative 2.5?" 

Kim  17:06

Right. 

Pam  17:07

And we don't have the positive 2.5. But I might say, "Did anybody use the positive 2.5 to help them think about negative 2.5? Or can you? Once you have one, can you have the other?" That one doesn't last all that long, unless you just start lightening throwing a bunch of rational numbers stuff in there as well. So you know, where's two-thirds? So, they have to think about if 6 is there. You know, they're like, "Well, can we put 1 up? That would help us think about where two-thirds is?" And you can say, "Okay, cool. Where's negative two-thirds?" So, you know, kind of throwing in some. You can't do just positive and negative numbers too, too long. That gets a little boring. So, then you can kind of throw some rescaling in there. You know, like I said, where you had the 6, and you had to shift it because now we have the 10. Or now you have the 6, and then I say two-thirds, and you're like, "Whoa, 6 better be much farther over there if we're going to be able to fit two-thirds in here." And then the negatives and positives. Go ahead. 

Kim  17:57

Yeah, I think it's really important that you do that work, though, because kids are so used to the number digit increasing as you go to the right. And so, you know, sometimes... I've seen plenty of students who when you put negative 6, and then you go to put negative 7. Like, it's a little bit of a mind twist for them to think is it bigger? Is it smaller? Which way is it going compared to the negative 6?

Pam  18:21

And that's a whole conversation about why negative 7 just went to the left of negative 6. Not like, "Doesn't usually go 6, 7?" I'm glad you brought that up. Yeah, nicely done. And sometimes, then, we can draw kind of the result that they have with Clothesline Math on the board and we can say... So, on my paper, right now, I have 0 in the middle. I have negative 6, a distance to the left. I have positive 6, a distance to the right. And I actually have drawn kind of a jump from negative 6 to 0, and I have 6 above it. And then, a jump from 0 to 6, and I have 6 above it. And there's this idea that the distance is the same. And I might kind of even fill that in where I have like negative 5, and I have a smaller jump that's five. And 5 that has a smaller jump that's 5. And we kind of have these sort of rainbow things kind of happening all stemming from 0. And that we can really think about the distance from 0. And so, the negative and positive has everything to do with which side of the number line it's on. And then, like you said, the kind of order things go is a little mind trippy and would give kids experience with that. Clothesline Math can be a really nice way to go. Cool. Some things that come out of that. Kids are talking about absolute value. They're talking about distance. There's this idea of kind of reflection. If I can find the one, then I can kind of reflect it over 0. It's a nice geometry thing that's going to come up with transformations, so we can have this reflection idea coming out. Nice things that can come out with Clothesline Math. Anything else you can think about there, Kim? 

Kim  19:52

No, I think you got it all.

Pam  19:54

Alright, cool. You know, always want to give. Just because I can't think of any more doesn't mean I shouldn't let. You know, I often do that with students, right? I'll say, "Hey, I can think of three ways." 

Kim  20:02

Yeah. I love that you do that. (unclear).

Pam  20:05

Well, thanks, Dr. Harvey Fletcher because he's the one that started it at BYU. He's the one that would say, "I can think of solving that in four ways." And then, he would do it in three. 

Kim  20:13

Yeah.

Pam  20:13

Like, what?! Yeah. 

Kim  20:15

That's great. 

Pam  20:15

I asked him once he goes, "Well, why would I ever say the number I can think of? Because I would never want to limit your thinking." Yeah. So, thanks, Dr. Harvey Fletcher. I don't think he's still around because he was older when I was his student at BYU. Anyway, moving on. Another super important thing that I'm going to suggest in making meaning out of integers. Well, in fact, maybe I'll just say one other quick thing. In the standards in the state of Texas, I'm clear. And this might be true for other states. I'm just really clear because I did a lot of work with teachers around this in Texas. That it used to be... So, is it still? See, this was a while ago. But it was that sixth grade was the place where it just said, "meaning of integers". And then, it wasn't till seventh grade where the students are supposed to operate on integers. In sixth grade teachers would say to me, "I don't know what to do here." So, for anyone whose standard says, "Help students understand integers," this is what we're doing today in today's episode. And so, one of the most important things that you can do to help understand integers is to use contexts. And I think there's three, maybe three and a half. A fourth. I'll throw a fourth one in there that's okay. To help students think about what integers mean. And here are my favorite four. Temperature, elevation, debt, and American football. American football is the one that's a little like okay. I like to use American football if students are familiar with the game. I don't want to have to spend a lot of time getting them, you know like, what does it mean to have 4 downs because we're going to use the line of scrimmage as kind of our 0 line. So it's very important you're not using the yard lines on the field. It's not where the 0 yard line is, or the 50. You're using the line of scrimmage as kind of the 0. So, if you have to do too much explanation for American football, then it's kind of not worth it. But if you have a lot of middle school boys that you're trying to bring in, and they've got football down, bam. Like, football can be a great way for them to go. They've already thought about a lot of integer stuff with football, and you're just kind of bringing it to the surface, making it more kind of... Concrete is not the right word. Kind of abstracting out ideas that they've kind of been playing with, so that they can like, "Oh, yeah. That makes sense." There are some advantages and disadvantages to those. So, I just mentioned a disadvantage to American football. If kids don't know very much, then don't. You know, maybe. But it's an advantage if they know it. What about debt? Well, an advantages to debt is we would really like kids to know about debt and to not get in it.  Our country's debt's terrible. Financial literacy would be a great thing to bring in. So, that's super good. But debt gets really weird with subtraction because how do you subtract a debt? So, that's kind of weird. Elevation is, I think, phenomenal. And I think we should use it with kids. You might have to do a little bit of description about what it means to be above and below sea level. If kids kind of live at sea level, it's maybe a little bit easier to talk about elevation. I grew up at 4,000 feet above sea level, so it was a little bit weird to talk about the 0. I couldn't even almost kind of picture it unless I'd been to the ocean, you know, where it was kind of at sea level. But I think you could talk about that, and I think you can kind of get that down. So, where you live is maybe a pro or con. And then, the same with temperature. When we lived in Michigan and Idaho, I could talk about above and below 0. Especially Michigan. Oh, my! Kim. 

Kim  22:52

Right.  It's cold. 

Pam  23:49

It was so cold in the winter in Michigan. We were below 0 a lot. So, if you live here in Texas, it's almost weird to talk about below 0 because it's so rare. I don't even know if it ever gets below 0 in Central Texas where we live. It gets below freezing. So, if you're in Celsius, then that makes sense. But it doesn't get below 0. Anyway, so you might have to describe those kinds of contexts to help it make a little bit more sense. But I think these four contexts can be super helpful. You're like, "Pam, what do you do with those contexts?" Well, ya'll, what do I always do with math? Let's do a Problem String. Alright, Kim. 

Kim  24:26

Yep. 

Pam  24:29

Let's talk temperature. If it is 11 degrees. Brr. We could talk a little bit about how cold that is. Be a human, you know. Get a little "Brr, it's cold." 11 degrees. And it raises 2 degrees. How cold is it?

Kim  24:42

Still cold. 

Pam  24:45

Go ahead. 

Kim  24:46

It's increased 2 degrees, so now you're at 13 degrees.

Pam  24:51

So, as Kim says that in a class full of sixth graders, I'm going to say, "So, Kim you're at negative 11. I've just drawn a vertical number line."

Kim  24:57

Wait, wait. Did you say negative 11?

Pam  24:59

I said that negative 11. 

Kim  25:00

Oh, I thought you said 11 degrees.

Pam  25:02

Oh my bad. Negative 11 degrees. 

Kim  25:03

Sorry, that's totally different. 

Pam  25:05

Well, is it? 

Kim  25:06

I mean, if it's negative 11, and it raises 2 degrees, then I'm at negative 9.

Pam  25:10

Ah, okay. I'm not sure I heard you say anything. So, if (unclear) ignoring you. 

Kim  25:15

It's negative 9. 

Pam  25:16

Okay, so as you're saying that, I've drawn a vertical number line. At a tick mark, I've put negative 11. And then, you just said raise 2 degrees, so I've kind of gone up 2 degrees. And where is that? And you're saying it's at negative 9? Yes? 

Kim  25:28

Yep. 

Pam  25:29

In that, I might say to kids, "So, where is 0?" And I might throw a 0 in there, but I might not. Okay, cool. But then, importantly, next to that, I'm going to write... I usually write Problem Strings, the problem, to the left. So, I'm going to write negative 11 plus 2. And you're saying that is negative 9. So, now I have the sentence. negative 11 plus 2 equals negative 9. Next problem. Hey, Kim, what if it's 18 degrees? Still cold. Still cold, but not too bad? 18 degrees, there's a big storm, and the temperature falls 21 degrees. Where are we?

Kim  26:03

So, I drew a number line also vertical, and I dropped 18 degrees to get to 0. And then, I dropped 3 more degrees to get the negative 21, or the dropping 21. 

Pam  26:16

Okay. so you started at 18, dropped 18 to get to 0? And then, how much more?

Kim  26:20

And then, I dropped 3 more.

Pam  26:22

Why 3 more? 

Kim  26:23

Because I dropped 21 degrees.

Pam  26:25

Because you dropped the 18, now you dropped 3 more. And so now you've dropped your 21. Where did you land when you dropped that 3 more?  At negative 3. So, then, I would write on the board 18. 

Kim  26:32

Negative three. Yep.

Pam  26:38

I have to remember the numbers were. 18 minus 21 is negative 3. 

Kim  26:44

Yep.

Pam  26:44

Cool. Next problem. I'm going to take my glasses off. That's why I'm having a hard time. I can't see. Anybody, don't get old.

Kim  26:50

I put glasses on to see.

Pam  26:51

I know. We're different that way. Okay. So, what about... You've got $20.00. 

Kim  26:57

Yep. 

Pam  26:57

And you told a friend that you would give them $32.00. 

Kim  27:01

Oh, liar, liar. Okay, so I'm going to give them the $20.00 I have because I'm super nice. But, then, I wish I had $12.00 more. I'm in debt. I'm in debt $12.00.

Pam  27:12

You're in debt $12.00. And so, I would now... I'm going to go  a horizontal number line on this one.

Kim  27:16

Oh, I did too! 

Pam  27:17

Okay. I'm going to start at the $20.00. Backup that $20.00 to get to the 0. Backup $12.00 more. Which you've got to go find somewhere. So, you're landing at negative $12.00 because you're in debt 12 degrees. Cool. Next. Hey, let me just say. In the midst of this string, ya'll, I'm doing this in the grade that says "understand integers". I am not pretending right now I'm getting kids good at operations.

Kim  27:38

Right. 

Pam  27:38

I'm really just investigating integers. Where are they? What does it mean when we increase and decrease? Next problem.

Kim  27:44

Well hang on a second. 

Pam  27:45

Oh, yeah. Go ahead.

Kim  27:46

You write the equation on the board. And this is very much reminding me of when even when we work with very young students, when they say, "This is what I did," you and I will "record", quote, unquote, what they did with parentheses and really demonstrating the properties. And people will say, "Oh, they're too young for that.  "Well, not if it's coming from them, and we're just showing them what it could look like. 

Pam  28:09

Yeah, yeah. "When your brain does that, it could look like this." 

Kim  28:11

Yeah. 

Pam  28:11

Absolutely. And to be clear, we're doing it both on a number line, in a very nice visual model, and with equations, so that we're connecting those, the scenario, the context with the model, The open number line with a model of  equations. Yeah, nice. Thanks for bringing that up.

Kim  28:25

Yep. 

Pam  28:26

Next problem. Hey, Kim, you're in debt $5.00. Now, at this point in the Problem String, I'm going to actually start writing the equation. Maybe. You know, it kind of depends. In this podcast, I feel like I would. With kids, I might wait a little bit longer. I might just say the scenario, represent it on a number line, and then write the equation. I'm going to do that for a long time. But I guess I have in my head I'm working with adults. At that point, I would have written. Your in debt $5.00, so I've written minus $5.00, negative $5.00. And then, you spent $15.00 more. So, then, I might write. You spent, add. But then.

Kim  29:01

And you spent. Yeah.

Pam  29:02

Yeah, negative $15.00. So, right now my paper, I've got negative $5.00 plus negative $15.00. Where are you?

Kim  29:08

Negative $20.00.

Pam  29:09

Because?

Kim  29:12

Because I was $5.00 in debt, and then I jumped back minus $15.00 to get to negative $20.00.

Pam  29:19

And when you say jump back, minus $15.00, you kind of make a jump back $15.00 or you subtracted $15.00. Either one. Mmhm. And so, that's negative $20.00. And as you said that, I drew that. Like, literally, I drew as you said. I started at negative $5.00, jumped $15.00, and landed on negative $20.00. Cool. Next problem. How about if you're 5 feet above sea level?

Kim  29:39

Okay.

Pam  29:40

You just written a 6. And you fell 12 feet. Minus 12. 

Kim  29:45

Oh, I don't think I fell. I think I dove.

Pam  29:48

Okay.

Kim  29:49

I hit the water.

Pam  29:50

Because you don't fall. Okay. Okay. Alright.

Kim  29:53

Because if I'm 6 feet above, I'm like on the cliff. A little tiny cliff. Yeah, I'm at negative 6. 

Pam  29:58

You're at negative 6. Because? 

Kim  30:00

Because when I went down 6, then I was at 0, so I hit the water. And then, I went down 6 more to get to negative 6.

Pam  30:09

Cool. And I actually put that on a vertical number line. Started at positive 6. Dropped down 6 to get to 0. Dropped down a negative 6. Landed on negative 6.

Kim  30:19

Does yours have a little person on the edge?

Pam  30:22

It does not. 

Kim  30:24

Mine does. 

Pam  30:25

You're hilarious. Okay, what if you started 11 feet below sea level and you walked up 3 feet?

Kim  30:32

I started at 11, negative 11, and the I went up 3, so now I'm at negative 8.

Pam  30:38

So, I've written negative 11 plus 3 equals negative 8. Yeah?

Kim  30:42

Mmhm. 

Pam  30:42

Okay. Lastly, what if it is American football.

Kim  30:47

Mmhm. 

Pam  30:47

And from the line of scrimmage, the first play you gained 2 yards, but then you were pushed back 7 yards. Where are you with respect to the line of scrimmage?

Kim  30:59

So, if I made 2 yards, then I kind of was at the 0, and I jumped forward 2. So, I'm at the two. But then, I got pushed back 7. So, I got pushed back to the 2. So, I'm back at 0. But I got pushed back 3 more. So I'm at... Oh, wait. You said 7?

Pam  31:17

Mmhm.

Kim  31:17

So, I got pushed back 5 more, and now I'm in negative 5. 

Pam  31:21

Because 5 back from 0 is negative 5. Nice. So, get it together, team, because you're 5 yards behind the line of scrimmage. You got to get. In fact, I wasn't even planning to ask this. But if you're 5 yards behind the line of scrimmage, how far do you need to go to get the next down? I'm testing Kim's knowledge of American football.

Kim  31:41

Oh, do you have to go 15?

Pam  31:42

Why do you think so?

Kim  31:43

Because I know you have to go 10 in a play, but it doesn't count. If you go pushed back. You got to go more.

Pam  31:50

Yeah, kind of. Yeah. So, you got to make up the 5 to get to the 0. And you have to get 10 from that 0. So, it's not 10 in a play, but it's 10 from the zero. So, you have four chances to get the 10 yards. But you're right. Since you were back, you got to make up the extra. Yeah, nice. So, all of that is like negatives, positives, and stuff. Let's just do one more quick one. So, what if I said, "Hey, you got sacked..."You know what? Actually. Yeah, yeah. You got sacked, and you're behind the line of scrimmage 5 yards.

Kim  32:18

Okay. 

Pam  32:18

And then, crud. You got pushed back 3 more. 

Kim  32:21

Gah!

Pam  32:22

I know.  You're at negative

Kim  32:24

So, I'm at negative 5. I got pushed back 3. So, now I'm at negative 8. 

Pam  32:27

8. So, ya'll, all of that are ways. All of those kinds of problems. Noticed we use temperature, we used money and debt, we used elevation, and we use American football from the line of scrimmage. Those are all ways of getting kids just to understand what's happening when you are adding and subtracting negative numbers. But really like also where positive and negative numbers even are. And so, when they do it, when they're actually thinking about what's happening in the context, then we write, we represent their thinking with number lines, and we also represent their thinking with equations. And then, we can get a little bit more crazy. Hey, perhaps in our next episode. Hmm. 

Kim  33:07

Hmm.

Pam  33:08

Alright, so major takeaways from today. Use context. Play with negatives when they come up in real life in games. And represent thinking on number lines and with equations. Ya'll, 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. Let's keep spreading the word that Math is Figure-Out-Able!