Pam  0:01  
Hey fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  0:09  
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  0:17  
We know that algorithms are amazing human achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:31  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:37  
Ya'll thanks for joining us to make math more figure-out-able. Alright, Kimberly.

Kim  0:44  
Hello, what you got today? 

Pam  0:46  
Should we do some math? 

Kim  0:46  
I would love it. 

Pam  0:46  
I would like to do some math. That is my...

Kim  0:50  
That mean I'm doing math? 

Pam  0:51  
Yeah, that is actually what that means. 

Kim  0:53  
Okay. Kind of figured.

Pam  0:57  
Okay, so we have a member of our coaching group who raised a question the other day and got me thinking. So, Brandon, thank you for raising the question. And got me thinking about the ways that I have done some things. Anyway, I don't want to front load it too much because I kind of want to talk about it afterwards, but.

Kim  1:19  
Okay, 

Pam  1:19  
Brandon Pelter, thanks for getting me thinking. And a couple other people have mentioned some things, so let's do a little bit of math, and then I'll kind of talk about why, how that relates to kind of how they were poking me. Okay, so I've mentioned before that we have a favorite recipe. Oh, in fact, funny. So, I was in Virginia not too long ago. And great group. And a gal, first of all, was like, "Shout out to Kim." She's like, "Where's Kim? Where's your sidekick?" "Oh, yeah, she's not here."

Kim  1:40  
I

Kim  1:41  
really do think you need Kim on a stick. You know like a picture of my head. Like they have at sporting

Kim  1:44  
events. 

Pam  1:45  
I could take a life size. 

Kim  1:48  
Gosh, no. 

Pam  1:48  
Flat. And just carry it around with me. Yeah, no. Kim on a stick. Okay. You know, I could wear a podcast shirt. We have a podcast shirt where it's... 

Kim  1:57  
You could.

Pam  1:57  
...pencil and a pen that would be cute. Anyway, this person or one of the people, said, "Do you really? Do you really have a chocolate chip peanut butter cookie recipe? And I said, "Absolutely!" Now, is it in the ratio that I say maybe or maybe not, but it's close. 

Kim  2:12  
Do you know

Kim  2:13  
I haven't ever had your peanut butter chocolate chip cookies? Just the chocolate chip ones they used to bring to school like forever go I got.

Pam  2:19  
I did take chocolate chip oatmeal chocolate chip cookies to school. I always take those to the school to the teachers before school started. 

Kim  2:26  
Mmhm.

Pam  2:26  
Do you like peanut butter chocolate chip

Pam  2:28  
cookies? 

Kim  2:28  
I mean, they're okay, but I would not turn down your chocolate chip oatmeal ones.

Pam  2:33  
Okay. Well, so next time I make. You know, we have a new family member. 

Kim  2:38  
I don't get it anymore. 

Pam  2:39  
We have a family member that was just diagnosed with celiac, so I'm not doing a lot of baking with flour these days.

Kim  2:42  
My family's not.

Pam  2:42  
So, I could make him and then

Pam  2:42  
you'll eat them all. You'll

Pam  2:44  
take him off my hands.

Kim  2:46  
One hundred percent. I don't even really love sweets. 

Pam  2:54  
Yeah, see, that's

Pam  2:55  
why I wouldn't have like pegged you for you

Pam  2:57  
want cookies. 

Kim  2:57  
Okay, to the recipe. 

Pam  3:00  
Okay.

Pam  3:01  
So, Kim, what if my favorite peanut butter chocolate chip cookie recipe says that I need 2 cups of peanut butter.

Kim  3:08  
Okay.

Pam  3:08  
For every three cups of chocolate chips. Because you should have a lot of chocolate chips in any cookie. 

Kim  3:13  
Yep. 

Pam  3:14  
And a cookie with raisins in it is a muffin. Just to be

Pam  3:17  
clear. 

Kim  3:18  
Oh, you know, I would not be sad about those either. 

Pam  3:21  
Well, I'll

Pam  3:21  
let you eat all the muffins in the world, and I'll eat all the cookies, which means they have chocolate in them. Or I will say. 

Kim  3:26  
Muffins have chocolate sometimes. 

Pam  3:27  
No, then it's dessert. It's not a muffin. And it's like if it has chocolate in it, then it's a cupcake. 

Kim  3:34  
I want lunch. Can we move on? 

Pam  3:35  
Oh, you're hungry?

Kim  3:37  
I haven't eaten yet. 

Pam  3:38  
Oh my

Pam  3:38  
gosh! It's late in the day for not... Okay, we better. Wow. I want to hand Kim some food.

Kim  3:43  
Cookies! Bring me cookies! Okay, this is going to be a rough episode. I can tell.

Pam  3:49  
Kim, if it takes 2 cups of peanut butter and 3 cups of chocolate chips in my recipe, what if I walk into my pantry and I have 4 cups of peanut butter? How many cups of chocolate chips

Pam  3:59  
do I need? 

Kim  3:59  
You need 6 cups of chocolate chips, yeah. 

Pam  4:01  
Because? 

Kim  4:02  
You have twice as much peanut butter, so you need twice as much chocolate chips.

Pam  4:07  
Okay, cool. What if I only... Twice as many. What if I only had one cup of peanut butter in my pantry, and I need 2 cups for every 3 cups of chocolate chips? How many cups of

Pam  4:17  
chocolate chips? 

Kim  4:18  
1 and a 1/2 cups.

Pam  4:19  
Because? 

Kim  4:20  
I looked back at the 2 to 3.

Pam  4:23  
Mmhm.

Kim  4:23  
And now you have half as much

Kim  4:24  
of each ingredient. 

Pam  4:25  
Of each of them. Yeah, it's half half as much. Cool. What if I am baking for, I don't know, Kim's family.

Kim  4:33  
Mmhm! 

Pam  4:33  
Maybe your whole extended family. And I know that I've got 7 and a 1/2 cups of chocolate chips. So, not peanut butter anymore. Now, I've got 7 and a 1/2 cups of chocolate chips. How many cups of peanut butter would I need in order to make

Pam  4:44  
that recipe with those 7 and 1/2? 

Kim  4:44  
I think you need... Yeah, I think you need

Kim  4:46  
5 cups of peanut butter. 

Pam  4:47  
And you're welcome that I kind of gave you...

Kim  4:47  
That was nice. Appreciate it. 

Pam  4:47  
...the value. Because you had the 4. Or, sorry, you had the 6 cups of chocolate chips. You had a 1 and a 1/2 cups of chocolate chips. Am I right? You just added those together? 

Kim  4:47  
Yeah, yep. 

Pam  4:47  
So, then you added the corresponding 4 cups of peanut butter to 1 cup of peanut butter. And so, it's... So, you're saying 5 cups of peanut butter would need 7 and a 1/2 cups of chocolate chips. Cool. What if I walked in the pantry and I had 10 cups of peanut butter? How many cups of chocolate

Pam  5:13  
chips? 

Kim  5:14  
I need twice as much as when I had 5 cups of peanut butter.

Pam  5:17  
Okay. 

Kim  5:18  
So, 7 and a 1/2 doubled is 15.

Pam  5:20  
Cool.

Pam  5:21  
Got anything else? Because I don't know if you've been writing. If I was doing this in class. 

Kim  5:26  
Oh,

Kim  5:26  
yeah, there's... I had the 1, so I could do 10 times as much. 

Pam  5:30  
So, like

Pam  5:30  
1 cup of peanut butter times 10 would give you the 10 cups? 

Kim  5:32  
Yeah. 

Pam  5:32  
And that 1 cup of peanut butter needed how many cups of chocolate chips? 

Kim  5:33  
1 cup is 1 and a 1/2.

Pam  5:34  
1 and a 1/2. So, 1 and a 1/2 times 10. Ooh, qnd here's an opportunity where now I have some nice place value on the board. 

Kim  5:45  
Mmhm. 

Pam  5:45  
Where a lot of kids are going to do what you did, double the 5 cups  and double the 7 and a 1/2 cups. And so, now I'm really clear that 10 cups of peanut butter is 15 cups of chocolate chips. And now I can high dose kids with the place value pattern of what happens when you multiply by 10 with decimals. I'm not butt cheeking. I'm not moving the decimal, but I'm actually watching that 1.5 shift in the place value system to 15 when I scaled it times 10. And so, does that make sense that we're kind of feeling it two ways? We're like. we're... Not feeling the place value pattern, but we're really clear that I can go from the 5 to 10, so I can double the 7 and a 1/2 to 15. And then I can notice the place value pattern. I can go, "Aha." Okay, it really is true this shifting thing that happens when we multiply by 10. Cool. 

Kim  6:28  
Yeah. 

Pam  6:29  
What if I walk in the pantry and I'd like to make peanut butter chocolate chip cookies, but all I have is 1 cup of chocolate chips. It's all I got. Can I... How many cups of peanut butter do I need to have in order to... You're not happy with that one.

Kim  6:43  
You're pretty sad.

Kim  6:45  
So... 

Pam  6:46  
I'll only

Pam  6:47  
be able to make enough... 

Kim  6:51  
One cookie.

Pam  6:53  
...to bring to Kim.

Kim  6:53  
Two cookies. That's okay. I'm not sad. 

Kim  6:53  
Wait, you want a cup of chocolate chips for one cookie? That's a lot of chocolate chips for one cookie.

Kim  6:53  
I don't know how many in your recipe. 

Pam  6:53  
You don't even eat chocolate chips.

Kim  6:57  
It's been a while. 

Pam  6:59  
Does everybody know?

Pam  7:01  
When we go snack shopping, I go for the chocolate and Kim goes for the... What do you? Gooey? Gummy?

Kim  7:06  
I mean, now, just chips. Like, I love chips. 

Pam  7:09  
Okay,

Pam  7:10  
so that's like a snack. Is that a treat too? Is there a difference between a snack and a treat?

Kim  7:14  
Well, I think you're referring to I used to get like gummy, sour, chewy. I just don't really have any of it anymore.

Pam  7:21  
Really?

Kim  7:21  
Yeah. 

Pam  7:21  
Ever?

Kim  7:23  
No. 

Pam  7:23  
Huh. Okay.

Kim  7:25  
So, I'm looking at my ratio table. 

Pam  7:27  
Is that

Pam  7:28  
because you got healthy or (unclear)?

Kim  7:30  
It's not worth it, I guess. I don't know. I'd rather eat chips, salsa, queso!

Kim  7:36  
Real food! I don't know.

Pam  7:37  
Real food. Alright, alright, alright. Okay, 1 cup of chocolate chips. How many cups of peanut do

Pam  7:40  
we need? 

Kim  7:41  
So, I first looked at the 1 and a 1/2 cups of chocolate chips.

Pam  7:45  
Yeah. 

Kim  7:45  
And I compared that to the 1, and I said, "Okay, 1 and a 1/2 would be three 1/2s of the one that I need.

Pam  7:53  
Like 150%.

Kim  7:55  
Yeah, mmhm.

Pam  7:56  
Okay, okay. 

Kim  7:57  
And so, I thought, well, if I want to get from 1 and a 1/2 to 1, then I only need two-thirds as much.

Pam  8:05  
You need two-thirds of 1 and a 1/2. Two-thirds of 1 and a 1/2

Pam  8:08  
is 1. 

Kim  8:08  
And then I went on the other side.

Pam  8:11  
How do you know that? 

Kim  8:13  
Because there are 3 parts of 1 and a 1/2. A half, a half, a half. 

Pam  8:17  
Okay. 

Kim  8:18  
And for 1, I use 2 of those parts. So, 2 of the 3 parts of 1 and a

Kim  8:23  
1/2. 

Pam  8:23  
It gave you the 1 cup.

Kim  8:25  
Mmhm.

Pam  8:26  
Nice.

Kim  8:26  
So, then on the other side for peanut butter, I need two-thirds as well. So, two thirds of 1 is two-thirds.

Pam  8:33  
Because I gave you 1 cup of chocolate chips, and you're like, "I need two-thirds of it. Two-thirds of... And that's why you said two-thirds of

Pam  8:34  
1. 

Kim  8:34  
Oh, I guess I could have just gone from the 3 to the 1. Divide that by 3. 

Pam  8:43  
Okay, so

Pam  8:45  
you're referring to the original 3.

Kim  8:47  
Yeah.

Pam  8:47  
So, we had 3 cups of chocolate chips. You're like, how could I get from 3 to 1 multiplicatively? You could divide by 3. And then you would scale the 2 cups of peanut butter. You would divide that also by 3. Well, Kim, how do you know what 2 divided by 3 is? 2 cups of peanut butter divided by 3? That seems complicated. 

Kim  9:05  
No, that's just two-thirds. 

Pam  9:06  
So, 2 cups divided into 3 portions is two-thirds of a cup? And that is a big idea that we need to help students develop. Not just tell them that division is fractions and fractions are division. But actually like, in this case, like really think about it and reason. And that will help them do what you did, which I think is more complicated, the 1.5... Or sorry. What did you... Yeah, the 1.5 to 1. Thinking about two-thirds of 1.5. I mean, we want kids to be able to do both of those. Yeah, nicely done. So, now we kind of have some relationships up here. We've got 2 to 3, 4 to 6, 1 to 1.5, 5 to 7.5, 10 to 15, two-thirds to 1. And all of these are expressing an equivalent relationship, so we would say that all these are equivalent ratios, and so we could have a lot of proportions here. We can have... This is a proportional situation. That's when we scale one of them, we scale the other by the same. Sort of scaling in tandem. And we started with a non-unit rate of 2 cups to 3 cups of.. 2 cups of chocolate... 2 cups of peanut butter to 3 cups of chocolate chips. I could then ask kids, "Hey, what if we were to graph those as ordered pairs? Would there be a pattern?" Kim, if you were to sketch a graph, could you just maybe? Where would the first point be? Like 2  cups of peanut butter to 3 cups of chocolate chips.

Pam  10:29  
At (2, 3). So, you'd go over 2, up 3, and plot a point. And then over 4, up 6, plot a point. And if you... Oh, let's see. Over 1, up 1 and a 1/2, plot a point. If you were to do that, would there be any pattern with those points? 

Kim  10:43  
Mmhm. They'd all be in a line. 

Pam  10:46  
So, then I would want to ask kids, "What line do all of those points lie on? Is there something I can do to figure out the number of cups of chocolate chips I need given the number of cups of peanut butter?" Like, I just put an x and a y in the table, so right now it's x cups of peanut butter to y cups of chocolate chips. And I'm sort of curious. If I just told you some random number. Like 100. I've got 100 cups of peanut butter. Could you do something to that 100? Like, maybe let me back up a little bit. You just went from cups of peanut butter to cups of peanut butter, and then you scaled in tandem to the cups of chocolate chips. Or you went from cups of chocolate chips to cups of chocolate chips, and you scaled in tandem from cups of peanut butter to cups of peanut butter. Now, I'm... So, if you could see me, I'm vertically kind of moving my hands and these scaling like movements, motions. But now I want you to consider kind of the relationship across the table. I was kind of moving up and down my table because I had a vertical table. Now, I'm asking you to consider the relationship across. If I just give like 100, what could you do to the 100? Like, you could scale from another cups of peanut butter, but I want you to like just between peanut butter and chocolate chips. What are you thinking?

Kim  12:03  
So, I think we had that with the 1 to 1.5. 

Pam  12:05  
Okay, say more.

Kim  12:07  
That you can multiply the x. What are the x's that you give me times 1.5 to get whatever the y is.

Pam  12:20  
So,

Pam  12:21  
let's check that out. If I give you 2 cups of peanut butter, what's 1 and 1/2 to? 1 and a 1/2 to cups?

Pam and Kim  12:27  
That's 3. 

Pam  12:27  
If I gave you... Do we need to talk about that? Maybe not. If I give you 4, what's one and a half 4s? 

Kim  12:34  
6. (unclear).

Pam  12:34  
6. And that's where we have, right? We have 2? So, I'm actually going back up into the ratio table, and I'm examining some of the... And then 1 to 1.5, sure. 7. 7. What's one and a half 7s? Sure enough, that's 7 and a 1/1. What's one and a half...

Kim  13:07  
Wait, wait, wait. One and a half 7s? 

Pam  13:16  
Oh, sorry. 

Kim  13:18  
One and a half 5s.

Pam  12:44  
Thank you. Thank you. I misspoke. One and a half 5s is 7 and a 1/2. What about one and a half 10s? Pause, pause, pause. One and a half 10s is 15. Okay, what's one and a half 2/3. (unclear) two-thirds plus half a two-thirds. That's one-third. And that's 1. So, there's the 2/3 to 1. Cool. So, we've got this other relationship happening. So, we kind of have the within cups of peanut butter and within cups of chocolate chips. That's the within relationship. But we also have between cups of between peanut butter and chocolate chips. We have that within. Or between. We have within relationships, and we have between relationships. Sort of within the x's, within the y's. But we also have between the x and the y. There's a couple different ways of saying that. Cool. 

Kim  13:33  
Yeah.

Pam  13:34  
So, you just said that we could say the number of cups of chocolate chips. I'm writing y. Could be found. Equals. By having 1.5. I wrote 1.5. Times the number of cups of peanut butter. I'm writing x. 

Kim  13:46  
Mmhm.

Pam  13:46  
So, if I were just to say that with the y's, it would be y equals 1.5x. Now, I could just go throw that in a graphing calculator. Or I could plot it by hand. I threw it a graphing calculator. I'm actually looking at that graph right now. Listeners, if you get a chance, you might just do that. I'm looking at a line. And, sure enough, on that line, it goes through the point (1, 1.5). And it goes through the point (2, 3). And so, we could find all those points on there. But we can also find that unit rate, that rate of change in the line y equals sort of mx. And the m, that slope or that rate, is... It was interesting when you said, "Well, we got the 1." And bam, as soon as we have the 1, we do have that unit rate. So, it's kind of nice that we can see both of those. There's an eighth grade standard in many states that I could draw the triangle from the origin that goes from the origin up to the point (1, 1.5) down to 1. I don't know if you can picture that. So, in other words, the point (1, 1.5). If I draw a triangle from the origin to it and down, then that's going to be similar to the triangle if I go to the point (2, 3) and draw that triangle to the origin. And now, I'm talking about equivalent rates of change based on similar triangles. Which is a lovely standard in many eighth grade classrooms. Kim, it's also interesting to me when I read Susan Lamon's, Teaching Fractions and Ratios for Understanding that then we could look at all of those equivalent ratios that we had. So, we had 2 to 3, 4 to 6, 1 to 1.5. We could look at all those ratios as an equivalence class. Well, equivalence class? That's abstract algebra. That's the first time I ever heard that term. And let's be really clear. When I took abstract algebra as a non-mather as a mimicker. Oh my. I understood so little of that class. Now, that I've been mathing so much more makes sense to me. And when I first saw a bunch of equivalent ratios graphed, and then the fact that they lie on a line, and that line describes the equivalence class, I was like, "That's what an equivalence class is?" And a lot of things came together. And, anyway, so that's sort of brilliant. I would invite everybody to consider that what we've just done is build proportional reasoning by dealing with 2 cups of peanut butter to 3 cups of chocolate chips with these equivalent ratios by scaling in tandem and kind of using those within relationships. And then we're starting to transition to functional reasoning when we start to look at the between x and y relationship and when we write the function to represent that. I'm not going to really call that robust functional reasoning yet, but it's we're definitely heading that way. And so, this is a moment in our sort of standards, our curriculum, our teaching, where we can help bridge from proportional reasoning, having built that proportional reasoning, into functional reasoning. And I don't know about you, but when I was a student, I skipped proportional reasoning. I didn't build any proportional reasoning at all. I cross, multiplied, and divide. And when I cross, multiplied, and divided to solve those proportions, I used an algorithm, so I wasn't even using multiplicative or additive reasoning to do any of it. Well, maybe... No, I was just recalling facts. But here's a way. A way. It's not the only way. That we can actually build from proportional reasoning into baby functional reasoning. And we're looking at the function y equals 1.5x.

Kim  13:47  
You mentioned at the beginning that you maybe you got some pushback that was raising some ideas.

Kim  13:47  
Say more about that.

Pam  16:21  
Yeah, I appreciate that. Yeah. So, in some of the work that I've done talking about proportional reasoning, I have focused on the between, the within relationships. So, scaling in the table, I have, it's vertical. So, scaling from 2 cups of peanut butter to 4 cups of peanut butter, therefore then scaling the 3 cups of chocolate chips to 6 cups of chocolate chips. And really helping build the proportional reasoning... Help me. Within. I have to think about which one within the peanut butter, and then scaling in tandem within the chocolate chips. I've really done a lot of focusing on that work. And the pushback I got was, "When are you going to work on that between relationship and write the function for that proportional relation? And I kind of had to grapple a little bit with why I have done less work with that in a proportional reasoning workshop. And what I had to grapple with is it proportional reasoning? Or is it more functional reasoning? And so, where I've landed? It's the bridge. It is absolutely proportional reasoning. And when we reason with functions, often we're going to reason proportionally. Definitely not always. But that's a part of, right? It's nested within. It's built on proportional reasoning. And I think when we write the equation that represents that proportional relationship, that's bridging from proportional reasoning to functional reasoning. Somebody might say, "But it's a function." I'm like, "I know, but it's like  proportional." Somebody might say, "No, it's proportional reasoning." I'm like, "I know. And it's bridging towards functional

Pam  18:48  
reasoning."

Kim  18:48  
It feels like laying the groundwork along the way and maybe not calling it out specifically like this is the work that we're doing right now, but asking the questions a little ahead of time, or wondering, and noticing, and being curious about those relationships just means that it's not brand new. It's not starting from scratch. When you're doing the work on a ratio table, you're getting kids to make sense of the relationships... Between? Within? Going down or across the ratio

Kim  19:16  
table.

Pam  19:16  
Within. 

Pam and Kim  19:17  
Within.

Pam  19:17  
Because it's within each kind of...

Kim  19:19  
Okay. 

Pam  19:19  
Yeah.

Kim  19:20  
But then when you can just turn your attention towards let's look at this a slightly different way, it doesn't feel brand new. It just feels like you're asking more questions. And I think when we talk about, even with the younger grade, talking about multiplication, division, sometimes we will work with multiplication problems on a ratio table, and then we say, "What about division? What if I ask you this division question? And because they've already made sense of a ratio table and how to make use of a ratio table as a tool, then you ask them a division question, they're like, "I can tackle that." So, I feel like this is kind of in the same way. That you're making relationships more visible, and they're thinking proportionally, leading into some functional reasoning.

Pam  20:00  
Yeah, and let me just make one other point that I really wish we would make early, so that then later this was kind of built. I brought up earlier that I have the point 1 to 1.5 on that line, and I also have the point three to (3,2)... (2,3). On the line. We wrote the equation as y equals 1.5x, but we could also write that equation as y equals 3 divided by 2x.

Kim  20:23  
Mmhm. 

Pam  20:23  
We could also write that equation as y equals 6 divided by 4x.

Kim  20:27  
Right. 

Pam  20:27  
And maybe I would do the 3 divided by 2, since it's sort of quote, unquote simplified. But it gives us different information. Like the 3 divided by 2 gives us information that is maybe equally as valuable as the 1.5 to 1. And it also keeps... Gah, how do I say what I'm thinking? It keeps the notion alive that slope is a ratio.

Kim  20:53  
Mmhm. 

Pam  20:54  
So, when we say the slope of that line is 1.5 or the slope of the line is four, I hear Garland Lincolnhogar in my head right now, saying, "Slope is a ratio." So, if you're going to say the slope is 1.5, you got to say 1.5 to 1. Or if it's... But then if the slope is 3 to 2, then you can say it's you know, three 1/2s, you can say it's 3 to 2. Like, by writing the rate of change as a ratio, you keep alive that it is this ratio of the change in y to the change in x. It's not... Slope isn't one number. It's a ratio. Does that make sense?

Kim  21:27  
Yeah, absolutely. I can tell you that when my youngest was graphing, he loved when the slope was in a fraction form because he's like, "It tells me exactly where to plot that." And so, you know, we had to have conversations about when it says 1.5, that is also a ratio that you can graph. But that felt more to him like a thing to do, not easier graphing

Pam  21:51  
Not reasoning about the ratio of the rise to the run. 

Kim  21:54  
Yeah.

Pam  21:55  
(unclear). Oh, that's interesting. I could see that. I could see that in him. I love the fact that he understands so well. That's cool. Alright, so everybody who pushed back on me a little bit about not going into that bit of functional reasoning with proportional reasoning, you're right, and so am I, and it's all connected. But it was good work, and we'll do more of that. We'll keep doing more of that. Nicely done. Alright, ya'll thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. And if you're a leader, school leader, we would love to talk to you about how we work with schools and districts. And let's keep spreading the word that Math is Figure-Out-Able!