Ep 169: Graphing, Integers, Quadrants, Oh My!

Pam  00:01

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

Kim  00:07

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 can be about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor students to think and reason like mathematicians. Not only are algorithms really not helpful in teaching mathematics, but when we rotely repeat steps that actually keep students from being the mathematicians they can be. Alright, Kim, I got the super coolest email the other day that I wanted to share with you.

Kim  00:40

Okay.

Pam  00:41

Little longer than some of the things that we read on, but I think you'll appreciate it. So, from Heather, she said, "First of all, Pam Harris, as a dual credit and AP math instructor/department head/ instructional coach/math lead (unclear) 

Kim  00:58

Yeah!

Pam  00:58

(unclear) district/UIL number sense coach..." Oh, and then in parentheses "(and Academic Coordinator)". 

Kim  01:03

She's busy. 

Pam  01:04

Yeah, wears a few hats. "I have only just recently discovered your existence. And, I mean this as a sincere compliment, I think you are my spirit animal." I thought that was so cute. I was like, "Okay." "A couple of the teachers I've been working with at Comanche elementary are huge fans of yours and have been working to implement your strategies to build a stronger conceptual understanding of number reasoning in their students. Their efforts were impressive to me, and so I began listening to some of your podcasts." Well, thanks for listening, Heather! "So far, I've loved everything I've heard you say. I've been so excited to find someone who believes all the things that I do, but has done the hard work in taking the time to make it concise and present it to others to have an impact beyond your own classroom or your own district." 

Kim  01:48

That's great. That's so great.

Pam  01:50

Yeah. So, Heather, thanks, I appreciate that. And it is exactly kind of what I've tried to do, is like bring all this stuff together that many of us are kind of playing with and thinking about, but putting it in a concise, consistent message. 

Kim  02:02

Yeah. 

Pam  02:02

And then, doing my best to get it out to the world. Yeah. So, if I may, thank you to Jennifer and Anne, who are the two teachers in her district who have been spreading all of the wonderfulness. Thank you so much for A, getting it going in your own brains, and B, getting it going in your own classrooms, and C, spreading it to those that you work with like Heather. So, thank you very much for that. 

Kim  02:24

Yeah, I mean, that's what we're hoping that lots of people do, right? Share the message and talk to their leaders, and their coaches, and their teammates to just keep this movement going. 

Pam  02:35

Yeah, ya'll, let's really change the world. I mean, I'm still a little bitter that I could have actually understood all that stuff I rote memorized. Let's catch kids earlier. Let's catch people as early as we can and help that go. 

Kim  02:47

Yeah. 

Pam  02:48

Hey, so, Kim, working off of last week's episode. Ya'll, if you teach younger students, hang with us today because you're going to be able to enter and do this just like last weeks if you did it with us. At least the first majority of last week's podcast should have made a lot of sense. Same thing with this one. We might go a little algebraic at the end, a little higher. What I mean by that, we might assume a little prior knowledge, but not till the end. So, hang with us. I think you're going to have a good time. Especially if you liked what we did last week. It's kind of fun.

Kim  03:18

Yeah. 

Pam  03:18

Alright, so, Kim, last week you gave me some ages. Sorry, you gave me some famous people, and then I guessed how old they were. 

Kim  03:25

Yep. 

Pam  03:26

This week, we're going to do something similar, but I'm going to give you a location, and you're going to guess the average temperature in that location in July, and the average temperature... Hang with me. Hang with me. 

Kim  03:42

Okay.

Pam  03:42

Maybe I should tell listeners. Kim and I don't really talk about weather. Like, some people are weather people. My husband's a weather guy. I don't know if he's going to listen to this podcast, but he likes to talk about the weather. When we first got together, I was like, "Are we really talking about the weather here?" Because to me, it's like, I don't know. inconsequential. I don't care. Now, he says that's because I grew up in a place that didn't have weather. Which, I mean, maybe. There's not a lot of natural disasters in Boise, Idaho, so you don't really. You don't... I don't know. It has nice four seasons. Things are consistent. Things don't change on a dime really fast. You know like, you're pretty much, if today's today, tomorrow is going to be tomorrow. And it's going to gradually turn into the next season kind of thing. So, maybe that's true. But, anyway, Kim and I don't usually talk about weather. We're going to talk about weather today. Here we go. 

Kim  04:27

Okay.

Pam  04:27

So, I'm going to give you a place. You're going to guess the average temperature in July. I guess, I should have asked you why you were laughing. I'm assuming because (unclear).

Kim  04:33

Well, I mean, it's just hot. All I ever say is it's hot or maybe it's humid. I have no idea what the numbers are. Okay.

Pam  04:38

Because we live in Texas, so it's pretty much hot. 

Kim  04:40

Right now, it's 106. 

Pam  04:41

It's like hot and not so hot is kind of... 

Kim  04:42

Yeah.

Pam  04:43

Yeah. 

Kim  04:43

Okay.

Pam  04:44

Okay, anyway, so I'm going to ask you places. You're going to tell me their July temperature, and their December temperature. Okay? Kind of they're average. I know, I know. Just... It will be fine. We'll do together. Maybe this is how other people feel when we do math, Kim, you know?

Kim  04:57

Maybe. Well, I'm fine with not knowing. It's okay.

Pam  05:01

Alright, Kim, we live near Austin, Texas, and so I'm actually going to ask you how hot? Oh, and this is going to be a little tough. I'm going to actually ask you in Celsius.

Kim  05:10

Oh, I have no idea. Holy cow. Okay.

Pam  05:15

How are we going to do this?

Kim  05:17

Can I tell you in Fahrenheit, and then you tell me in Celsius?

Pam  05:20

Yeah. We might have to do that. I'm not the quickest converter, but yes, we can do that together. Why didn't I think of that before? Alright, so, ya'll, this is sometimes how things go, you know, you plan things. I knew I was going to do it, and I just didn't even think about, I know someone's out there going, "Pam, you should be able to do this mentally." Sure, when you're not live on a podcast. You do it.

Kim  05:38

I don't even know the conversion.

Pam  05:40

The conversion? You don't (unclear)

Kim  05:41

Is that horrible? No, I have no idea.

Pam  05:43

No, but I can't spell "Fahrenheit" right now. Alright, alright. The conversion has everything to do with getting rid of the 32 degrees. You know how 32 degrees in Fahrenheit is equal to 0 in Celsius? 

Kim  05:57

Yes, I do know that.

Pam  05:57

You just sort of get rid of that 32 degrees, and then you multiply by this fraction of five-ninths. Like, yeah, go figure. Who knows (unclear).

Kim  06:03

Okay.

Pam  06:04

Okay.

Kim  06:05

Right, I'm going to tell you Fahrenheit, and let you deal with it.

Pam  06:07

I'll do the conversion. Okay.

Kim  06:08

Okay.

Pam  06:09

So, Kim, what do you think the temperature in Austin in July is about? 

Kim  06:18

98. 

Pam  06:19

You think the average might be about 98. And that would be about 36. And it's actually about 35 degrees. So, right on. Spot on.

Kim  06:29

Pretty close. Alright. Cool.

Pam  06:30

How about in December? What do you think the average in December is in Austin?

Kim  06:35

I'm going to go like 45.

Pam  06:39

You think it might be 45. Which is... I don't know Is that cold for other people? Yeah? So, it's actually not that. So, 45 is equivalent to 7 degrees Celsius. It's actually about 16 degrees Celsius. So, 16 degrees Celsius is 60.

Kim  06:55

Oh, I was thinking like February. Yeah. I always think it should be December because that's like Christmas. 

Pam  07:01

Christmas should be the coldest month. 

Kim  07:03

Yeah, it does. But it's February. February is probably more like 45.

Pam  07:05

I think the end of January, February is more like.

Kim  07:08

Yeah. (unclear) just talking about this. Okay. Or I  was with somebody else. Okay, so it's more like in the 60s? 

Pam  07:16

60s. It's exact 60. You guessed 16, which is 60 degrees. Yeah. Fahrenheit. Okay, so everybody, we're just going to say that Austin, Texas has an average temperature in July of 35, and in December of 16. So, if I were to plot those as ordered pairs, Kim, where would they be? Where would the point... Not "they", but the point 35, 16 where would that be? Just because we can't like do it 

Kim  07:44

Where do you mean where? Like, in the... 

Pam  07:46

Like, just pretend you were graphing it. Talk out loud.

Kim  07:49

I would start at 0, 0. I would go over 35, then I go up 16. So, what do you mean where?

Pam  07:55

Well, so that's where it would be. It would be over 35, up 16. 

Kim  07:57

Okay, okay. 

Pam  07:58

Okay. And so, it's in the first quadrant? Yes, quadrant one. It's in that upper right hand quadrant, and we call that quadrant one. Okay. 

Kim  08:04

Yep. Okay. 

Pam  08:05

Cool. Detroit, Michigan. You got any idea of? Is Detroit, Michigan warmer or colder than Austin in...

Kim  08:12

Colder.

Pam  08:13

In July, in July. Colder? 

Kim  08:15

In July. Well, I think it's going to be colder always than Austin.

Pam  08:17

It is colder. And it's about 7 degrees colder in July. So, it's about 28 degrees in Detroit in July. Do you think it's warmer or colder in December? 

Kim  08:28

Colder.

Pam  08:29

It's way colder. 

Kim  08:31

Yeah.

Pam  08:31

The average temperature for Detroit, Michigan in December is 1 Celsius. So, that's what? 33 degrees-ish. Yeah? So, right around freezing. I mean, so that's not actually. It's not below freezing. 

Kim  08:45

No.

Pam  08:45

To be clear, I lived near Detroit, Michigan for four years. We had a cold day. Well, we had multiple cold days. But the first time we had a cold day, they said, "We're closing school because it's too cold." And I said, "That's a thing?" Like, it got cold in Boise. It went below 0, but it never was negative 40 Fahrenheit. And that was too cold because kids would get frostbite standing outside, waiting for the bus. Okay, so it's kind of cold in Detroit. But in December, it's like you just said, it's not as cold as February. So, it's about 1 degree. Cool. Do you know where Siberia is? Oh, wait. I'm sorry before we go to Siberia, we got to graph that Detroit, Michigan point. So, where's the point 28,1. 28 "comma" 1. 

Kim  09:29

Still in the first quadrant. 

Pam  09:32

And over 28, but only? 

Kim  09:34

Up 1. 

Pam  09:35

Up 1. Cool. If you had listened to... No, maybe I don't want to do that quite yet. Yeah, I think I do. Hey, Kim, where would a point be if the place had the same average temperature in July and in December? Where would that point be? 

Kim  09:56

Would it be at the point of origin? 

Pam  09:59

Well, that would be if it had a 0,0

Kim  10:02

Oh, like all the temperatures it's going to be... It doesn't matter what the temperature is, it's always going to be... 

Pam  10:09

The same.

Kim  10:09

The same. It's going to be on the y equals x line.

Pam  10:13

Yeah, like kind of last week where we talked about where the guesses equal the average or the y's equal the x's. In this case, it's where the July temperature equals the December temperature. 

Kim  10:21

Yeah.

Pam  10:21

We kind of this 45 degree line, that that's kind of where they would all be the same.

Kim  10:25

Yep.

Pam  10:25

So, the reason I brought that up is... So, if that's kind of where. That's the line July equals December temperatures. Where did the two... Where did Austin and Detroit fall? They're both in the first quadrant. How do they compare to that July equal December temperature line? 

Kim  10:42

They're both under the line. 

Pam  10:44

They're both under the line. Because? 

Kim  10:46

Because it's warmer in the summer. 

Pam  10:49

In July. It's warmer in July than it is in December, so we go over farther before we go up a little bit. Over farther because it's warm, only go up a little because it's not as warm in December. Podcast listeners, you're going to (unclear) tell us. We're describing. You know, you can see it. So, like, let us know if you're like, "We did not follow what you were..." You might want to get a pen and pencil. Pen for me. Pencil for Kim. And, you know, kind of like sketch out what we're talking about, so you can kind of visualize. But it's kind of noteworthy. Now, maybe someone's saying, "Pam, all of the points are going to be under that line because it's always warmer in July than it is in December," right? Are they all going to be there? And we'll just maybe raise that question, and then move on. Okay, Kim. How about Siberia? Where is Siberia? Any ideas? 

Kim  11:41

Russia. 

Pam  11:42

Yeah. Yeah. And when I say Siberia, what comes to mind warm or cold? 

Kim  11:48

Cold! Cold, cold, cold. 

Pam  11:50

Cold, cold, cold. Okay, so the average July temperature in Siberia is actually 14 degrees Celsius. 14 degrees. And 14 degrees Celsius for those Fahrenheit-ers. That's about 57 degrees Fahrenheit. So, in July, that's cool. 

Kim  12:04

Yeah. 

Pam  12:05

I don't know that that's really cold. That's cool. 

Kim  12:07

I'm not going swimming then.

Pam  12:09

No, but it's not freezing, right? Like I would have... Siberia, I guess I wondered if Siberia was cold all year long. But it's not. Like, there are temperatures that are sort of normal-ish. So, interesting. My brother actually did a mission for our church for two years in Siberia. 

Kim  12:24

Oh, my gosh!

Pam  12:24

I know, right. And when he went, he took a ton of long underwear, because it's Siberia. Well, as soon as July came around, he was sending us messages, "Send me normal underwear!" Like, it was kind of funny. I mean, it was kind of a joke. He could buy underwear. But yeah, he didn't realize that it would be normal. Okay, more than you wanted to know about Siberia. Anyway. Thanks, Brad, for letting me share your underwear story on the podcast. He had a great time for two years. So, in December, how cold do you think it is? Or what's the temperature in December? What's your guess, Kim?

Kim  13:01

Negative 20.

Pam  13:03

So, it's actually negative 14.

Kim  13:05

Oh, I almost said 14! Dang it! 

Pam  13:08

But it's negative 14 Celsius. 

Kim  13:10

Oh. Ooh (unclear).

Pam  13:11

Any guesses what negative 14 Celsius is for Fahrenheit?

Kim  13:15

Negative 40. I'm going to go... Oh, wait. You said five-ninths or nine-fifths?

Pam  13:21

Well, it depends on which direction you're going. 

Kim  13:22

Okay.

Pam  13:23

Negative 14...

Kim  13:24

I'm going to go with negative 30. 

Pam  13:25

So, negative 14 Celsius is actually only 7 degrees Fahrenheit. Isn't that wild? 

Kim  13:30

Oh, okay. 

Pam  13:32

So, 7 degrees Fahrenheit is still cold, right? It's just not below 0. It's definitely below freezing.

Kim  13:38

So, it's 57 in July, for me in Fahrenheit, and 7 in? Huh. 

Pam  13:44

Average temperature in December. So, December, December. Now, 7. I mean, average. That means it's going below 7 to get an average of 7, right? Because it's an average. So, this is a place where we could kind of talk about average. And it's in December, not February. But yeah. Yeah, we've got. Yep. Okay. Okay, how about... How about, how about. So, hang on. I got to think for a second. So, Siberia is interesting because it was 14 degrees. I'm staying in Celsius because it's going to make my graph work better. So, it's 14 degrees in July, and it's negative 14 degrees Celsius in December. Where's that point?

Kim  14:17

14, Negative 14? Is that what you said? 

Pam  14:19

Yep, mmhm.

Kim  14:21

That's in the fourth quadrant.

Pam  14:25

And that's...for all those who don't memorize the quadrants...that's in the right lower quadrant. Lower right. Underneath the one that we were just messing with. So, that's interesting. Like, I don't know, if anybody envisioned that we would have points that would be in the fourth quadrant, in that other. Ah, but Siberia. Sure enough. And what does it mean to be in the fourth quadrant? Let's see. It means something about summer temperatures.

Kim  14:52

It's still warmer in July than it is in December.

Pam  14:55

Yeah. So, warmer in July than in December. And what can you say about temperatures in December if I'm In the fourth quadrant?

Kim  15:02

They're negative. 

Pam  15:03

They're negative. So, they're below freezing for sure. 

Kim  15:06

Yep. 

Pam  15:07

Super cool. Okay, I'm just making a note on my graph. Okay, so how about Santiago, Argentina? Any ideas?

Kim  15:15

Oh! Warm.

Pam  15:19

That feels warm to you? 

Kim  15:20

Yeah. 

Pam  15:21

Okay, it feels warm to you. Well, in July, the average temperature in Santiago, Chile is actually 13 degrees. Not so warm. 

Kim  15:29

Wait, 13 degrees Fahrenheit? 

Pam  15:31

Celsius, sorry. 13 degrees Celsius is 55 degrees Fahrenheit.

Kim  15:38

Well, we were just talking Siberia, so it's warmer. That's in July, though?

Pam  15:41

Oh, warmer. Yeah, that's in July. That's interesting (unclear).

Kim  15:44

Holy cow. Okay, yeah, I would have said it was warmer than that. Wow. Alright.

Pam  15:47

Yeah. So, then I wonder what you think when I tell you that December temperatures in Argentina are 27 degrees Celsius. So, 27 degrees Celsius is 80 degrees.

Kim  16:00

Oh, dang it! I didn't think about that. Shoot!

Pam  16:03

What just happened? 

Kim  16:04

I forgot about the opposite seasons. 

Pam  16:07

What do you mean opposite seasons? 

Kim  16:09

Yeah, July is more like winter time, and December is more summer time. 

Pam  16:15

Because where's Santiago, Chile? 

Kim  16:17

Yeah, it's below the equator.

Pam  16:19

Ah, bam. Or sorry. Santiago, I said Chile, but I met Argentina this whole time. Though, both of them are below the equator. So, below the equator, the seasons are reversed. Okay, so where does that put the points for places below the equator that kind of have typical-ish temperatures? They're kind of 80 in their summer and 50 in their winter. Which is what we just did. Those are the Fahrenheit, so it's... I'll go backwards.

Kim  16:46

Tell me those numbers again. 

Pam  16:47

Well, let me put them in Celsius because that's what I want you to graph. So, 27 was in their summer, which is the December temperature. And 13 in their winter. So, it's the ordered pair 13 "comma" 27. Does that makes sense? Let me just say that again. the 13 is the temperature in July, but that's their winter. And the 27 is the temperature in December, but that's their summer. So, 13, 27. Where's that point? 

Kim  17:11

It's going to be above the line. 

Pam  17:13

And it's still in the first quadrant, right?

Kim  17:15

Yeah.

Pam  17:15

But now, it's above the line? 

Kim  17:16

Yeah.

Pam  17:16

Now, it's above line. Yeah. Okay, cool. 

Kim  17:18

So, above the line is going to be colder in July than December.

Pam  17:23

And where are places like that in the world?

Kim  17:25

Below the equator, Pam.

Pam  17:27

So, that's kind of interesting. Yeah. No stress, no stress. So, there can be places that we can say, "Hey, they're going to kind of hang in the first quadrant below that equal temperature line, if they're in the northern hemisphere, and they're going to be above that equal temperature line if they're in the southern hemisphere."

Kim  17:45

Yeah.

Pam  17:46

And look at us bringing in geography into math. Whoa! Nice. Okay, Kim, I think this one might be one that you know better than anybody. Nome, Alaska. Now, I know that you didn't live in Nome, Alaska, but I do believe you lived somewhere in Alaska for a hot minute. Yeah?

Kim  18:00

Anchorage, yep. 

Pam  18:01

In Anchorage. There you go. So, how cold do you think it is in July in Nome?

Kim  18:07

I'm going to go like 70. Nome. I'll be like... Yeah, 70. I'll go 70.

Pam  18:20

So, Fahrenheit. So, it's actually 50 Fahrenheit. It's actually 50 Fahrenheit. 

Kim  18:23

Oh, okay.

Pam  18:24

So, a little colder than you thought it was.

Kim  18:25

Yeah.

Pam  18:26

That's about 10 degrees Celsius.

Kim  18:28

Okay.

Pam  18:29

And then, what are you thinking in the winter? Is it cold in the winter in Nome, Alaska? 

Kim  18:32

Oh, yeah. Yeah, yeah.

Pam  18:33

I would think it might be. How cold?

Kim  18:36

I'm going to go with... Oh, with average. People in Nome are listening and they're like, "Kim!"

Pam  18:44

I mean, you were young. You were super young, right? 

Kim  18:46

I was young Yeah. 

Pam  18:47

Yeah. 

Kim  18:48

I'm going to say 20 Fahrenheit.

Pam  18:51

You think it might be 20. 20 degrees. So, below freezing. below freezing, but not super below freezing. 

Kim  18:56

Yeah. 

Pam  18:57

So, it's actually negative 14 Celsius, which is about 7 degrees Fahrenheit. 

Kim  19:02

Okay.

Pam  19:02

So, yeah. Below freezing. A little colder than you thought. Below freezing.

Kim  19:07

I don't remember that so much. Okay, so Anchorage is probably warmer than Nome.

Pam  19:11

Oh, that could be. Sure. Okay, so the Celsius... 

Kim  19:15

I am worse at this than you are at ages for sure.

Pam  19:19

I mean, if you would have been asking me these questions. Geography might be the the only thing I'm worse than guessing celebrity ages, might be geography. Okay. So, I could tell stories about. My husband is a geographer. That's like part of what he does. He loves geography. So, I'm abysmal. And we work well together. It's good. Okay, so the values in Celsius for Alaska for Nome, Alaska were 10 and 14. So, where's the point 10, 14? 

Kim  19:47

Also above the line. 

Pam  19:48

Also above the line. So, that's interesting. No, wait, wait, wait. Above the line. I should have said 10, negative 14. That would help.

Kim  19:55

Oh, okay. Then yeah, it's going to be in fourth quadrant. 

Pam  19:58

So, that negative kind of made a difference there.

Kim  20:00

Yep. 

Pam  20:02

So, that's in the fourth quadrant. It's below the x axis. And so what is that? What can we start generalizing about points below the x axis? Siberia was there. Nome, Alaska was there. Was there any? 

Kim  20:16

They're super cold. 

Pam  20:17

Super cold. And also northern hemisphere. Northern Hemisphere, super cold. Cool. So, I wonder right now if anybody is starting to think about other quadrants. Like, I wonder if we've got points right now that we focused on that were in the first quadrant. And we started to make a differentiation between whether they were in the Northern Hemisphere, the southern hemisphere, whether its seasons were flipped. We've also got some points still in the northern hemisphere, but they're in the fourth quadrant because they're super cold in the winter. I wonder if anybody's starting to think about points that could be in the second quadrant or the third quadrant? Like, where are places like that in the world? That might be an interesting thing to think about. Alright, here's your next location, Kim. The Greenland ice cap. Where's Greenland?

Kim  21:06

Super north and freezing all the time. (unclear) it gets warm there at all.

Pam  21:11

Well, I'll tell you. In July, the temperature is negative 11 Celsius. And in December, the temperature is... Ready? Negative 47 Celsius. So, that's the coldest one we've said so far. Just for those of you who are Fahrenheit bound, negative 47 is negative 52 Fahrenheit.

Kim  21:29

Good gravy! 

Pam  21:30

That's really cold. 

Kim  21:31

Really cold.

Pam  21:32

That is super, super, super, super, super cold. Where's that point, Kim, if you were to graph that one? 

Kim  21:37

That's in the third quadrant, for sure.

Pam  21:39

Negative 11, negative 47. So, somewhere that is cold all year long. And, in fact, not just cold, but below freezing all year long. The Greenland ice cap, right? It was negative in July, negative 11. Negative 47. Didn't get above freezing at least in those two months. Is in the third quarter. Alright, so we found one in the third quadrant. Bam! I don't know if it's noteworthy that it's in the northern hemisphere. I don't know. How about Antarctica?

Kim  22:10

Horribly, horribly, horribly cold. Don't go there.

Pam  22:14

Do you think it's colder than Greenland? I'm kind of curious.

Kim  22:18

I think it depends when of the year you're talking about.

Pam  22:22

Ah. Say more about that. 

Kim  22:24

So, I think it's colder than Greenland in the summer. Well, what we would call summer. In July. I'm going to go with the month. I think it's colder in Antarctica in July, but I think it's warmer in December because one's in northern hemisphere and one's in the southern hemisphere.

Pam  22:43

(unclear) Ah, there you go. Okay, so we flip the hemisphere, so you're flipping the two. And sure enough, Antarctica is...

Kim  22:48

So, then that's the second. 

Pam  22:51

Well, maybe. Antarctica is negative 57 Celsius in July, and negative 18 in December. 

Kim  22:59

Oh, no. It's still in the third quadrant.

Pam  23:01

So, I'll just say those temperatures again. Negative 57 in July and negative 18 in December. So, super, super cold. In fact, that is the coldest temperature we've had so far. Negative 57 Celsius. And negative 57 Celsius, just for those of you who are curious, is negative 70 Fahrenheit. Super cold. So, Antarctica in July is really, really, really cold. But in December, it's still really cold. Right, like what would a place have to... Because you were thinking second quadrant? Kim, I'm stretching you here I think, but what would have to happen to a location for it to live in the second quadrant? In the upper left hand quadrant, upper left hand quadrant. This is the second quadrant. What would have to be true of the numbers? Or maybe the...

Kim  23:48

Well, I don't even know what I put on the x and y to be perfectly honest with you. I'm working with a tiny post-it note. (unclear).

Pam  23:54

So, x is average July temperatures and y is average December.

Kim  23:58

So, in July it would have to be negative, and in December it would have to be positive.

Pam  24:05

Okay, let's keep talking. So, if July is negative, that means it's cold in July.

Kim  24:09

So, it's cold in July, and it's warm in December. So, it's got to be in the southern hemisphere.

Pam  24:14

Yeah, because the temperatures or the seasons are flipped. 

Both Pam and Kim  24:18

Yeah. 

Kim  24:19

Well, flipped for us.

Pam  24:21

(unclear) Flipped for us. Thank you. Yeah. Ethnocentric there. Kim, I have to tell you, I spent a long time. Now, I didn't do this recently. I did this quite a while ago when I was gathering this data. But it took me quite a while to find anywhere in the world, where it would be cold enough in July to be negative and warm enough in December to be positive. And the only place I found was a place called Butlers Gorge, Australia. Whoa! Hey, Aussies, there we go. And it was so close. If we round up, it wouldn't work. So, I'm not going to round. So, negative 0.4 degrees Celsius in July. Negative 0.4. So, we're just back from the 0. And then, 5 degrees Celsius in December. So, just above. And we are barely in the second quadrant. It was the only place I could find. Now, I didn't... Hey, so podcast listeners, anybody want a free T-shirt? Somebody find me another place that's in the second quadrant. First person to ping me. Where do I want you to ping me? Ping me... (unclear).

Kim  25:26

They can email me.

Pam  25:27

Nah, lets do social media. 

Kim  25:30

Oh. Okay. 

Pam  25:30

Yeah. So, Facebook, Instagram, or Twitter. First one that... We'll look at the date, the timestamp. Anybody that can find a second quadrant location other than Butler's Gorge, Australia. Yeah. We'll send you a Math is Figure-Out-Able t-shirt. That would be kind of fun. Alright, so, Kim, I think this is a cool task to get kids doing a lot of things. But let me tell you one... Oh, we're kind of running a little long here, so maybe I'll do this a little short. But we just described locations in the world that would be in the first quadrant. And not only in the first quadrant, but that would be above and below kind of an equal temperature line, that y equals x line. We also describe places in the world that could be in the second, third, and fourth quadrant. One thing we didn't do...but, algebra teachers, this could be fun...is described between the... Oh, I can do it. I can read my own writing. Was it Australia and Greenland that we're in the third quadrant? I think so. Between... No. It was Australia. Yeah, Australia and Greenland. Yeah. Between Australia and Greenland in the third quadrant, which of those were above that equal temperature line, and what does that mean for their location. So, there's something going on in the third quadrant, just like there was in the first quadrant with that equal temperature line. Because that equal temperature line goes to the third quadrant as well, right? Like, y equals x is not just in the first quadrant. It's also in the third quadrant. So, there's something going on there. And then, could we get a little general and say, "Hey, what was true about..." And I wish you could see me write. But what was true about all of the July...and I'm writing the word July...temperatures that were greater than...and I'm writing the greater than symbol...the December temperatures? Or in other words, what was true when all of the x's were greater than the y's? And we're starting to graph inequalities. An inequality where y is less than x, or y is greater than x. So, I've literally written an inequality. And we could talk about locations in the world where it's true. What's true with temperatures when all those x's are greater than y's or the x's are less than the y's? And we can actually shade in half planes, making sense of that inequality with a context, with locations in geography. Bam! 

Kim  27:43

Yeah. 

Pam  27:44

And I'm a little particularly proud. So, I made up this task based on Gail Burrell’s really super cool age guessing task. And I kind of like it. I think there's some really nice higher math that we can get to that is actually figure-out-able.

Kim  28:00

Yeah. Well, this is super fun for me. I love when we get an opportunity to have conversations about stuff that may have been... I almost said "shoved at me". But I'm going to... Yeah. Shoved at me without a lot of meaning and not a lot of context. And I can't wait to do this one with my kids too. 

Pam  28:18

Woah!

Kim  28:18

Everyone is telling you listeners that higher math is rote memorizable, and we are telling you that it's all figure-out-able

Pam  28:26

Ya'll, so 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!