Pam Harris  00:00

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

Kim Montague  00:06

And I'm Kim.

Pam Harris  00:07

And we answer the question, if not algorithms, then what? Ya'll, we couldn't be more excited because we have just launched our free online workshop all about the Development of Mathematical Reasoning. This completely free mini workshop is based on my most often requested in person workshop. We decided to make it available for free. Don't worry, if you've missed registration this time, head on over to sign up for the waitlist and you'll be notified about it the next time that it runs. So exciting. 

Kim Montague  00:41

Very exciting. So today, we're going to talk about something a little different. Sometimes we talk about math for teaching, and sometimes we talk about just the math itself, right? Today we're going to do some higher math stuff. Parents of high schoolers, this is going to be especially exciting for you, because we're going to help you make sense of some stuff that your kids are doing: functional reasoning. But listen, younger grade teachers don't tune out - Math is Figure-Out-Able. If it gets a little too much, let us know where we lost you, and we can work on that. So, Pam, I'm gonna be real honest with you for a minute here. When you and I talk about the Development of Mathematical Reasoning, like we did last week, from Counting to Additive, Multiplicative, and Proportional Reasoning, I'm with you, 100%. I totally get it. Okay, but here's the deal. When you mentioned Functional Reasoning, I kind of glossed over a little bit. I have to admit, I'm not even sure I know what you mean by that, because I didn't teach high school. And frankly, I know I'm not alone.

Pam Harris  01:43

Yeah, you're totally not alone. And to be honest, right back at you. I'm still working on some Functional Reasoning. And based on the comments we've been getting from listeners, there are plenty of people in the same boat. So let's focus on that today. Now, I know you said that you reason proportionally but if it's okay, let's do some Proportional Reasoning ish a little bit here because in order to discuss Functional Reasoning, I want to take you from what you know, some Proportional Reasoning stuff and connect it to the Functional Reasoning. 

Kim Montague  02:11

Okay. 

Pam Harris  02:11

Yeah?  Alright. So, Proportional Reasoning has everything to do with two things varying in tandem. Both things are varying at the same time. So when you buy more pounds of apples, you pay more. But what about when you start with a ratio of five pounds for $2? That's not as easy as a unit rate of $1 or for one pound. So if it's five pounds for $2, what happens if you just add one more pound? Do you add two more dollars? Well, no, because you can get five pounds for $2. Yeah, so thinking about those two things varying in tandem: five pounds for $2. What if you only want one pound or what if you have $3? Those kind of problems deal with Proportional Reasoning. What if you're trying to build a ramp? And by ratio the law is 1 to 12 or 1 to 20. Those are typical ratios out there. What does that even mean? So that means if it's the ratio of one to 20, that means for one inch of vertical rise, then you have to have 20 inches of horizontal length or run. Those two vary in tandem. So if you've got a certain height that you need to build a ramp for, then you have to consider how long that ramp has to be. Or what if you're trying to get a job done? Say that you know that it takes 20 students 30 minutes to pick up trash for the whole playground. You know that you've got this experience. Kids are all working. It takes 20 students 30 minutes to get that whole job done. But today, you've only got 15 students, how long is it going to take? That's another kind of relationship. You'd want to use Proportional Reasoning. All of these scenarios deal with rates and not just unit rates, but non unit rates, right. It's not just a rate to one or one to something but it's got two numbers varying in tandem, and that is the change from elementary school to middle school. 

Kim Montague  04:05

Yeah. 

Pam Harris  04:06

When you go from unit rate to a non unit rate. Go ahead.

Kim Montague  04:09

So I feel you on the unit rate. Because I feel like if you say there's 32 students on a field trip needing to go in eight vans, we're really focused on four kids to one van, right? We don't really call it the unit rate, we're focusing on the division. Or if we say there's six apples in one basket, we have kids use multiplication to figure out their 30 apples and five baskets is scaling up by five. Is that kind of like what you're talking about?

Pam Harris  04:40

Yeah, totally. So in a big way, in third, fourth and fifth grade, when you guys are dealing with multiplication and division, often you have rates like you just said. Like you'll have, I think you said eight kids or four kids in a van. That's four kids per van, that's a rate for kids per van. But you also had a rate where you had 32 students per eight vans, but you don't really think of it that way typically in elementary school. We're just sort of focused on, like you said, the multiplication or the division. So then when we move up to middle school, we're still focusing on multiplication and division. But with these non unit rates, then it's a bit more complicated. So like five pounds for $2, 4 slices of pizza for $5. If you look at the tag on a bulk food order, you know, when you go down the bulk food aisle and stick a bunch of food in the bag, you know, don't touch that stuff, right? Use the scoop. And then when you look at the tag, after you've weighed it, you might have gotten 6.2 pounds for, I don't know, $10 and 20 cents, that's a non unit rate. 6.2 pounds for $10 and 20 cents. Maybe you drove 420 miles with a tank that holds like 16 gallons, how far could you drive if you only had five gallons of fuel? Those are all two things varying at the same time. That's complicated because there's a bunch of stuff happening simultaneously. So let me tell you a quick story. We actually have a family in our church and he rides a scooter to get along. And in order for him to use that scooter effectively, he needs to be able to get in and out of his house. And right now there's no ramp for his house. So some guys in our church are getting together. They're going to build a ramp. And so in order to do that, they had to look up the necessary dimensions for that ramp in order for them to make sure that they had enough for that horizontal run for that rise. Because the rise is a given when we know how tall it is from the driveway up to the doorstep. And so they have to figure out now how they're going to build a long enough ramp for it to be accessible for his scooter. So that's all dealing with Proportional Reasoning. Those are Proportional Reasoning things.

Kim Montague  06:40

Yes. Okay. I'm with you on that. Right. I did a ton of talking about Proportional Reasoning with my middle schooler this year. It feels like that's kind of in the land that he was living. And we had a lot of fun talking about that. Um, but pretty sure soon he's going to get into content that I have not reasoned through yet.

Pam Harris  06:59

Right? Because when you were in high school, I think we've talked about before a little bit, about that was a point in your life where, yeah, boys came along and you were like, yeah, math was fun. Now I'm just gonna kind of memorize it. Spit it out and you got other things to do. Sure. So totally know you're not alone there.

Kim Montague  07:15

So here's what I'm gonna do. I'm gonna sit back and just take some notes while you school me a little bit. You said there were three things that you're going to talk about today.

Pam Harris  07:23

Yeah, so today I'm going to talk about three ways of reasoning functionally. I'm going to talk about slope, specifically rate to change. We're gonna talk about transformations. And then lastly, we'll talk about rational functions. 

Kim Montague  07:35

Okay, so you're going to talk about slope first. Let's go there.

Pam Harris  07:39

Okay, cool. So, slope often is sort of pictured as kind of this physical thing and I want it to be a little bit less of, literally, we talked about ramps earlier, literally less about the slope of like the grade that you're driving on a road, or the slope of stairs or even the slope of that ramp, though we could, but the more general case of that is the rate of change. It's how something is changing over time. And that's sort of the more general case. So interestingly, often we talk about slope as one number. And this is not a good thing. So when I was learning slope, and I was beginning to teach high school math, at first, I would say, "Ah, you know, that line has a slope of two or that line has a slope of a five." And we would talk about which one was steeper and everything. But I'll tell you a quick story. So, Kim, you and I both know Garland Linkenhoger. She's a good friend of ours and an excellent mathematics teacher educator. A long time ago, she and I were working together, and this is early in the day while I was still building my numeracy, and she was complaining about a meeting that she had just been with some national people. And one of the mathematicians or at least was supposed to be the mathematics expert on the panel said something about the rate of change of a line being like two or three or something. And she, she was like, "Ah! You can't talk about slope as one number. That's got to be three to one, if the slope of that line is three, no, it's three to one." And it was kind of funny at that moment, because I definitely used one number to describe the slope of a line before. And, I kind of smiled at her and I said, "You're making kind of a big deal or this." She said, "It's a huge deal! We must understand rate as a, like I said earlier as those two things vary in tandem. So it's not a rate of three, it's a rate of three to one." And that's going to help kids more than when we say, "Oh, this, this slope is one half." One half, that's kind of like one half to one. But it's also one to two. And so slope, if we've got a really good sense of this proportional reasoning, it will just lean really well into this idea that slope is not, we shouldn't represent slope by one number, but we should also represent it as a rate as a ratio of two things to each other.

Kim Montague  09:56

So I can see how my Proportional Reasoning will help me better understand slope when I consider that it's not just one number. 

Pam Harris  10:04

Yeah, nice. Cool. 

Kim Montague  10:05

Cool. Okay, so you said three things. Tell us about number two.

Pam Harris  10:09

So number two is all about transformations. It used to be back in the day when I took high school math, we didn't even talk about transformations. So I really applaud the fact that now we've got transformations helping us understand how we can think about parent functions and how those transformations just affect those parent functions. And we can think about the graphs of the functions as they change. However, if we're not really thinking functionally, one way we might look at transformations, is we might look at a typical parent function like a quadratic function. For those of you that haven't looked at quadratic functions for a while, that's the one that looks kind of like a U, we might call it a parabola. So that parabola or that quadratic function kinda looks like a U. If I'm not thinking functionally, then I'm actually thinking of that as a static shape like a U. Like I might think of it like a pipe cleaner that I've sort of put in the shape of a U or, or even a fruit roll up or something where I've sort of got something that I kind of put it in that shape and it sort of sits there. But a parabola or quadratic function isn't a static shape. It's actually the relationship between two variables. It has everything to do with when I go over a certain distance on a coordinate axis. When I go over that certain distance, I go up that distance squared. So if I've gone over one, then I go up one, because one squared is one. But if I go over two, I go up four, because two squared is four. And then I plot that point. So what that means is, the further I go to the right, that I'm going to go up that number squared, that continues to happen. It's not a static U. Because U's kind of get vertical pretty quick. But a parabola never gets totally vertical. A parabola is always getting higher the square as I go over to the right that number. But it also then is to the left. As I go over to the left, then I square that number, and that number squared is positive. And so then I go up that and that's why we sort of get this kind of U shape. But it doesn't turn vertical in the end like a U does, it continues to grow as we go either to the right or to the left. So a first way is to understand the parent functions correctly, that they're not these static shapes that we can just sort of move around. They are in fact relationships between two variables. Well, then we talk about transformations on that parabola, on that shape, which isn't a static shape, right?It's a relationship between two variables. But the transformations of those is we might multiply the whole function by a number. Here's a way that you can know if you've kind of have a limited view of functions, functional understanding, is if you think about when I do something like 2x squared, 3x squared, 5x squared, if you think about that as the parabola getting skinnier, that's a bit of a limited view. That's an incorrect, it's not a functional view of what's happening to the problem. What's actually happening if I take a parabola like x squared, and I multiply it by three, what actually happens is as I go out, now when I go over one, I don't go up one anymore. Now when I go over one, I go up three times that amount. So I go over a over one, I got three, if I go over to two squared is four, but that's four times three, that's 12, I'm going up 12. So when I go over, instead of going up the same amount I was before, I'm going up three times that amount. That might look skinnier if I'm thinking real static shape. But if I'm really thinking about what's happening to the points as I go further to the right or as I go further to the left, I recognize, oh, I'm actually scaling that function by three. Everything's three times as tall as it is. So the function feels very much bigger than it did before, not skinnier. Skinnier, has kind of a smaller connotation, but really the y values are getting bigger three times as fast. That has a feel to it of much bigger. So you might notice if students are talking about transformations, especially when we're dilating transformations, we're scaling them. If they talk about that function getting skinnier when in reality, it's getting bigger, faster. That's a way that you can tell they're not thinking functionally. But a way you can tell they're thinking functionally if they're really thinking about how that scale factor is affecting all those y values. For example, if I take that same x squared, and now I scaled by one half, well, if I have kind of a limited view, I might think about it as fatter. I might look at that parabola, oh, it's kind of looks like fatter. But in reality, it's much more like an elephant sat on it and squished it. So if you could sort of picture that kind of U shape before now it's kind of squished. Why is it sort of squished? Well, because now when I go over one, instead of going up one, now I only go up half as much. So when I go over one, I'll go up half. Or when I go over two normally I would go four right? Because two squared is four, but now I only got half that much. I only go up two. And so whereas I used to when I would go over, I would go up a certain amount now every time I go over I only go half that amount. Wow, that's like a squashed version of the of the function. And so if I have this sort of sense of it got wider. That's not correct at all. In fact, it's gotten squashed. It's only going up half as much as it was before. Or say if I have x squared times one 10th. Whoa, now it's really only going up a 10th of the way that it was every time. And so viewing transformations more as the parent functions are these infinite sets of points. And when I transform them, I really have to think about what's happening, sort of point by point, and then accurately describing what's going on. So I know I'm doing this all on the air. It's really hard to do. I do hope you're getting sort of a sense of what I'm talking about here. 

Kim Montague  15:47

Sure, and it sounds like you can almost hear the difference between the student who's thinking functionally based on how they're describing what's happening to the parent function.

Pam Harris  15:58

Yeah, absolutely. It's a great way to putting it. Yeah.

Kim Montague  16:01

Okay, so we talked about slope a little bit. You talked about transformations. You said there was one more thing that you want to talk about.

Pam Harris  16:09

You bet. And so y'all, this is gonna be the one that's gonna be the hardest for me to describe not using visuals, but I'm gonna do my best. So a rational function is something that is a thing divided by a thing. So like a rational number is like three fourths, three divided by four. Sometimes you might use positional language to describe what it looks like three over four. But that doesn't, that's not really mathematical, but we might have x squared plus one over x cubed minus one. That's an example of a rational function, what it would sort of look like if I wrote it. But you'll notice that if I have that division bar, x squared plus one divided by x cubed minus one, that division bar really is like a way of talking about ratios. So I could also talk about rational functions as x squared plus one, the ratio of x squared plus one to the ratio of x cubed minus one. So if I'm thinking functionally, I'm not thinking over, I'm thinking about the ratio of those two polynomials. So a rational function is the ratio of two polynomials. So you can tell if a student or teacher's thinking functionally about how they describe the way they're thinking about rational functions. So, for example, if a student says x squared plus one over x cubed minus one, that's kind of thinking, like, I'm just describing what the x's look like. I'm not really thinking about the fact that it's a ratio of those two polynomials. But also, it's a ratio of two polynomials. So now I'm going to bring in what I know functionally about polynomials. If I know functionally that polynomials have these long run behaviors, then I know something about the long run behavior of x squared. And I know something about the long run behavior of x cubed. Then I could use my Proportional Reasoning to bring in the ratio and I can think about the ratio of those long run behaviors. So that's why I have to have Proportional Reasoning to know that I'm dealing with a ratio. And then I have to Functional Reasoning to know that I'm dealing with longer behaviors of polynomials. And then I think about the ratio of those long run behaviors to think about the long run behavior of a rational function. Alright, so yeah, that probably was a little bit much for anybody who hasn't thought about polynomials and long run behaviors and functional -

Kim Montague  18:16

Well, that's -

Pam Harris  18:18

For you high school teachers out there, I hope you can sort of picture if you know something about the long run behaviors of polynomials, then you should be able to use what you know about ratios to think about the ratios of those long run behaviors to inform the long run behavior of the rational. Let me finish with one more thing. Also, those polynomials have short run behaviors. What's happening in the short run, like right around zero usually. Well, I can use what I know about the short run behavior of those polynomials to also help me think about the short run behavior of the rational function. And you know that rational functions have this funky behavior. Kim, you probably remember from high school, where we have these asymptotes. And things are like, going like they were sort of sucking up to these vertical asymptotes. All of that vertical asymptote, and the removable discontinuity behavior that's happening in a rational function is all based on the short run behavior of the polynomials. Because that's the short run behavior of the rational function. So I'm not gonna go on too much more about that. But that is a way that you can sort of understand what I'm talking about with functional understanding, Functional Reasoning, in high school math.

Kim Montague  19:27

You know, and I'm sitting here thinking about how it's really clear to me that the stronger of a proportional reasoner you are, the better you're going to be able to make sense of Functional Reasoning because you're able to relate so much of it to that type of reasoning. I have written so many notes, and frankly, I wish that I had you for my high school math teacher. Because, while they were really nice and, and explained well, they, I don't know, were tapping into what I'd made sense of already. So listen, I'm gonna ask you to expect some phone calls when my kids get to high school, because we're gonna need some more talking.

Pam Harris  20:08

Absolutely, I'm there for you. And listeners lest you think that those phone calls that Kim's talking about are going to be like whole tutoring sessions. Actually, often when Kim will call and say, "Hey, my kid's working on something." I can just sort of say, "Oh, that connects to this thing that you know." And bam, they're off and running. So it's not like I have to sit down and do you know, teach the whole semester or anything. Because both Kim and her son, Luke, are so well versed in all of the kinds of reasoning up to where they are, I can just sort of connect it back to the way they've been thinking and reasoning about something, then they can, you know, fly with sort of a new topic. Okay, ya'll, I know that that was really up  in the air. We didn't have any visuals to sort of help with all that. So we'd love some feedback on that. Let us know where we, I sort of lost you and maybe where I can do some blog posts or maybe a video or a blog where we can do a lot more visualizing. And we can use some visual models so that we can make that thinking more visible. Don't worry if you didn't pick up on all that stuff. We believe that math concepts are developed. And so really if, how do I say this? Since math concepts are developed, then I don't think I should be able to just tell you some stuff over the radio and that you should own it like deeply and completely. If you're coming from, maybe you're still an additive reasoner, then yeah, maybe I gave you some sort of sense of what Functional Reason means. But in reality, we would want you to build your Additive Reasoning so that then you can build your Multiplicative Reasoning so they then you can build your Proportional Reasoning and then come back and listen to this podcast, this episode one more time and then the Functional Reasoning stuff should make a whole lot more sense. Because Math is Figure-Out-Able, not just telling. It's not just that I can tell you, unzip your head and pour stuff in. We actually believe Math is Figure-Out-Able. That we have to figure it out. We have to dive in and experience it using relationships and connections. It's not enough to just tell. Okay, so today we talked about Functional Reasoning on the Development of Mathematical Reason.

Kim Montague  22:10

Absolutely. And if you want to learn more check out Pam's website mathisFigureOutAble.com. We would love it if you would join us on Wednesdays on your favorite social media for Math Strat Chat. And if you like the podcast and would like to give us a review, that would be so fantastic.

Pam Harris  22:27

Alright, so if you're interested to learn more math and you want to help students develop as mathematicians, then the Math is Figure-Out-Able podcast is for you. Because Math is Figure-Out-Able!