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

Kim  0:10  
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  0:18  
Because we know that algorithms are amazing human achievements. But, y'all, they're 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  0:30  
In this podcast, we hope you teach math-ing, building relationships with your students, and grappling with mathematical relationships. 

Pam  0:37  
Thanks for joining us to make math more figure-out-able. 

Kim  0:41  
Hey, hey.

Pam  0:42  
Kimberly. 

Kim  0:43  
Yeah.

Pam and Kim  0:43  
What's up? 

Kim  0:45  
In this episode, we're taking a look back at some episodes where we chatted about important topics in high school classrooms. We're tackling logarithms in a Problem String you're sure to love, plus sharing when Pam got hooked on thinking about equations of lines, and we've got some discussion about one of the three ways to reason functionally. Here we go.

Pam  1:10  
...Here's another set of numbers. How about 5? So, I'm going to just put that L in front of it again. Just random. 5 "comma" 25 equals 2. Is there any relationship between those numbers? 

Kim  1:20  
Yeah, yeah. 

Pam  1:20  
Okay. What? 

Kim  1:21  
5^2 is 25.

Pam  1:23  
So, 5 to the second power is equal to 25.  Okay. (unclear). 

Kim  1:27  
Yeah.

Pam  1:27  
Yeah. The next one. Again, L3 "comma" 81 equals 4. Are those related in the same way?

Kim  1:36  
In the same way, meaning how the other?

Pam  1:38  
The other two... Like, the other ones were 2 to the third power equals 8. 3 to the... Or sorry. 5 to the second power, I can read, is 25. And then 3 to the...

Kim  1:49  
Yeah, and can I tell you how I...

Pam  1:52  
Yeah. What were you going to? You paused.

Kim  1:53  
Yeah. Yeah, I did. And I was like, "What I know about 81? That's 9 times 9, so that's 3^2 and 3^2, so that would be 3 to the fourth."

Pam  2:01  
Oh, nice. And I might even, as a teacher, write down exactly what you just said. 9 times 9 equals 3^2 times 3^2 equals 3 times 3 times 3 times 3. 

Kim  2:09  
Yeah.

Pam  2:09  
Hey, sure enough, there's four 3s. Yeah. And so, that's like the 81. So, 3 to the fourth equals 81 Okay, cool. Another one. This time... So, we kind of got this sort of set pattern. I would look at those three problems, we kind of have a number, and then the power equals the exponent. Does that make sense? So, like we had 2, 8, 3. And then 5, 25, 2. And then 3, 81, 4. Somebody might say that's a little out of order, but the numbers are all related in a way. Okay, cool. So, this time I'm going to leave one out, and I'm going to ask you to fill in the one that's left out. So, I'm going to write the L down again, and then I'm just going to say 2 "comma" 16. So, what's the equals on that one if it's following the same pattern? And how are you thinking about it? 

Kim  2:57  
Yeah. So, again I thought about 16 as 4 times 4.

Pam  3:03  
Okay. 

Kim  3:03  
And so, I was thinking 2^2 times 2^2, so it would be 2 "comma" 16 equals 4.

Pam  3:11  
2 times. Because 2 to the fourth equals 16.

Kim  3:14  
Yeah.

Pam  3:15  
Cool. So, it's almost like saying 2 raised to some power is 16. What is that power? 4. 

Kim  3:20  
Yeah. 

Pam  3:20  
Maybe I should say 2 raised to some exponent. Sue is always on me about getting these terms right. And she should be. 2 raised to the exponent of 4 is equal to 16. Okay, cool. Or, sorry. Let me say that in the order of the numbers. 2 raised to some power, 2 raised to some exponent is 16, and that exponent is, that's the equals, 4.

Kim  3:40  
Yep.

Pam  3:40  
There, I said it in the right order. Okay, next problem. I leave one of them out again. Just one of them random, and then you can kind of see if you can help. So, this time I'm going to L9 "comma" 81 equals blank.

Kim  3:53  
9 "comma" 81 equals 2.

Pam  4:00  
2. Because?

Kim  4:02  
Because 9^2 is 81.

Pam  4:04  
Cool, so 9 to some exponent is 81. Oh, that exponent is 2. And we could actually look above where you had said 9 times 9 above.

Kim  4:11  
Yeah.

Pam  4:12  
To kind of connect that one. Okay, cool. Next one. Again, I'm going to leave one out, L5 "comma" 125 equals blank.

Kim  4:20  
3.

Pam  4:22  
Because? 

Kim  4:23  
Because 5 times 5 is 25 times 5 is 125.

Pam  4:27  
Cool. So, nice. Next one. L. And I'm going to do 11 "comma" 11 equals blank. Is that a typo? No, no, no. I think that's what I want. Yeah, 11 "comma" 11, mmhm.

Kim  4:41  
1.

Pam  4:42  
You know, when I say is that a typo, I'm kidding, right? Teacher move.

Kim  4:46  
Most of the time. Yeah. 

Pam  4:47  
Most of the time it's a teacher move. 

Kim  4:48  
Yeah.

Pam  4:49  
You said it was 1?

Kim  4:50  
Yeah.

Pam  4:51  
That's a little different than all the other ones. Can you say more about that? Pretty sure?

Kim  4:57  
Yeah. 11 to the first power is 11. Like, it's 11 times nothing else.

Pam  5:08  
Any number to the first power is that number. 

Kim  5:10  
Yeah. 

Pam  5:11  
So, that seems kind of helpful. Okay, cool. So, mathematicians notice that there is this relationship between numbers and the exponents, and then that power. That if I do the exponent, if I multiply that base times itself, that many bases times itself, then we'll get that power. And they notice this relationship, and they call those exponential relationships. And they started to notice there were times when they wanted to solve for
the exponent exactly like you just did. Where I said, "Hmm, well, I have 9 and 81, what's the exponent? If I have 5 and 125, what's the exponent? If I have 11 and 11, what's the exponent?" And you were like, "Well, here's what it would be." And that was like, sort of like solving an exponential equation. So, like, for example, on the 5, 125 I could have written that as 5 to the x equals 125. But instead, I wrote it as 5 "comma" 125 equals x or equals blank.

Kim  6:10  
Mmhm.

Pam  6:10  
And so, it was kind of this idea that if we have this x, if we have exponential relationships floating around, there are times where we have the base and we have the power, that number that we've multiplied it out to get, but we don't know the exponent. We don't know the missing exponent. And somewhere in history, and I don't actually know a whole lot of the history here. But somewhere in history, somebody said, "Hey, let's call that a logarithm." Now, that word is a logarithm, not algorithm, so those are two different things. An algorithm is a series of steps to solve any type of problem in the class of problems. A logarithm is exactly expressing this relationship that we've been talking about. So, I'm going to give you another. That's why I've been using the letter L. It hasn't been random. It was kind of get you ready for what if I were to say that right underneath the problem we just had L, where I had 11 "comma" 11 equals 1. This time I'm going to say log. I'm just going to finish the word now. So, l-o-g. Then, I'm going to write 8. Then I'm going to say of 6. Oh, a little tiny 8 down below. And I'm going to say of 64. So, the 64 is kind of on the same line as log, and the 8 is like a little, what do you call it, subscript. So, log base 8 of 64 or log 64 base 8 is equal to what? So, the numbers are in the same places they were before. They should follow the same pattern. So, I'm just going to tell you that we're still asking the question, 8 raised to what exponent gives you 64? 

Kim  7:41  
2.

Pam  7:42  
2. So, you're going to fill in that blank with 2.

Kim  7:44  
Mmhm.

Pam  7:44  
So, I know this is a verbal podcast, so I'm just going to read out loud what we have down now. We have log base 8 of 64 equals 2. And why again, Kim? Can you say why again? 

Kim  7:54  
Because 8 times 8 is 64. 

Pam  7:56  
So, 8^2 is 64. Nice. Okay, so what if I just said, well, then I'm kind of curious about log base 42 of 42?

Kim  8:03  
1.

Pam  8:04  
That's just 1 because 42 to the first power is 42. What if I said log base 10 of 1,000 equals what?

Kim  8:14  
It's going to be 3.

Pam  8:16  
Because?

Kim  8:18  
Because I was thinking 100 times 10. And 100 is 10 times 10. And then times another 10.

Pam  8:24  
So, 10^3 equals 1,000 Therefore, log base 10 of 1,000 equals 3. What if I said... We're almost done? Log base...

Today, we're going to talk about writing the equation of a line, given some information. So, if I give you a point, a slope. Write the equation of line if I give you two points. Write the equation of a line that contains those two points. I could describe the line. I could say it's vertical, and it goes through this point. Lots of different things that I could say. Or really just a few sets of things that we typically say. Then we would expect a student to write the equation of a line. Okay. Okay. Uncle, right? I cry, Uncle. Of all the things we've talked about, this. This surely, surely we have to have formulas for this, right? Surely we have to have step-by-step procedures. I mean, you can't reason about writing the equation of a line. Or can you?

What would it be like to work with kids' intuition to help them reason and find strategy for writing the equation of a line? Let's do that. So, let me just paint a picture. When I travel around and I look at teachers, classrooms, students all across the nation when I've looked at kids solving equations... Or, sorry. Writing the equation of a line. I see, often, a very particular thing happening. Say kids are finding the slope between two points, finding the rate of change between two points. I often see this. I see, minus minus divide. I see kids finding the rise, the run, and then dividing. Right? They're going to subtract, subtract, and then they're going to divide the rise divided by the run. Rise over run. I see that a lot. And it becomes this very procedural thing to do, where they sort of do that, and then they plug that into a formula, and then they do a bunch of combining like terms and messing around, and then they come up with like y=mx+b. Or maybe the standard form ax+by=c. Like, maybe like all that. And if you guys don't teach higher math right now, you're like, "All the alphabet soup!" Like, all the stuff that's happening. I see a lot of procedures and step-by-step stuff.

One day I was doing a workshop with teachers in Texas. So, I live in Texas. I was doing a lot of work with teachers in Texas. Early in my career, as I was teacher educator, a math teacher educator. And I had already figured out a few ways of helping teachers parse out how we can use intuition to solve or to write the equation of a line, but I had not developed a particular strategy that I want to work on today on the podcast. Yet. I hadn't developed it yet. And so, I handed out some points to the teachers, and I was having them do some things. And we're going to talk actually about that strategy a little bit more next week. But they were just looking at a set of points, and they were supposed to write the equation of a line. And all of a sudden, I was kind of circulating in the room. And we had a huge group. I do not like to do workshops this big. In fact, this might have been the last that big workshop I did because it was put on by the state, and they forced it, and I don't know, I had like 50 people in the room. Just me, 50 in a huge room. It was terrible. So, I'm wandering. I like to do 35. 35 is a great. It's a great in person. It's not bad on Zoom. 35 is pretty good. I don't know why, but that's... Less than 35 can actually not be so. Especially less than 20. Anyway, you don't care. So, I'm doing this big group. I think it was like 75 people in the room. I'm wandering around. Not wandering. I'm circulating, and I'm purposely watching what people are doing with the set of points that I've given them. Their job is to write the equation. This one gal. I wish I remembered her name. She goes, "Don't look over my shoulder." And I was like, "Oh, okay. Alright, do you want to tell me what you're thinking about?" "No!" And I was like, "Okay." She goes, "I'm not math." And then she goes into this whole like, "I'm science. I don't do math." Whatever. And she all these disclaimers about why I should not be looking at her work. And I said, "Okay, but I'm actually curious. You have an equation written there. How'd you do that? Like, were you just following steps and not kind of the way we were talking about? Like, what were you doing?" And she goes, "I just.. It just jumped out at me."

Ooh, I got intrigued. Anytime somebody tells me that their intuition was involved, that something pinged for them. Hmm. Mmhm. I'm interested, right? I'm thinking about that, and I'm thinking about that a lot, right, in that moment. So, I, "Hey, I'd be really curious what you were just thinking about to find the equation of line." And she said, "Pam, I just see it." I said, "What do you just see?" She goes, "Look at them. If you add those together... No. No, if you subtract them. I get 1 every time. So, the equation is just x minus y equals 1." I said, "I'm sorry. What?"

She goes, "Go, walk away. I'm not math." Whatever. Blah blah blah. And I said, "No, no, no. What did you just say?" She said... I had given her four points, and she said, "If you look at all of the x values, and you from one of the x values you subtract its y value, every time I get 1, so I just wrote the equation x minus y equals 1." I said, "Say that again." It was so new to me to like look for patterns in the table of values. So, she had four points. She looked at the x values. She looked at the y values. If given one of the x values, when she subtracted the y value. So, like if she had two as the x value and she had 1 as the y value, she said 2 minus 1 is 1. And I was like, "Well, that's true." And she goes, "Okay, but look at one of the other values." So, if it was 6 was the x value and 5 was the y value, she said, "Look, 6 minus 5 is 1. So, every x minus every one of these y's gives me 1, so I just wrote the equation x minus y equals 1." To which I said, "Can you do that?" I had never. I... Again, I was the sort of Z perspective growing up. I was the one who rote memorized all the things and did all the steps. I was hooked. I was fascinated that there might be sets of points, that I could find a pattern, that I could just write down that pattern and have the equation of a line. Kim. 

Kim  14:41  
Yeah? 

Pam  14:42  
Can we work on that today a little? 

Kim  14:44  
Sure. Do it.

Pam  14:45  
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 kind of 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 even a Fruit Roll-Up or something, where I've sort of got something that I can kind of put it in that shape and 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 1, then I go up 1 because 1^2 is 1. But if I go over 2, I go up 4 because 2^2 is 4, and then I plot that point. So, what that means is the further I go to the right, then I'm going to go up that number squared. That continues to happen. It's not a static U.

It's really because U's kind of get vertical, right, 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 the 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's not. 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^2, or 3x^2, or 5x^2. 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 parabola. What's actually happening if I take a parabola like x^2, and I multiply by 3, what actually happens is as I go out, now, when I go over 1, I don't go up 1 anymore. Now, when I go over 1, I go up 3 times that amount. So, I go over 1, I go up 3. If I go over 2, 2^2 is 4. But that 4 times 3, 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 3 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, as I go further to the left, I recognize, "Oh, I'm actually scaling that function by 3. 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 the smaller connotation, but really the y values are getting bigger 3 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^2, and now I scale it by one-half. Well, that kind of... 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 like fatter." But in reality it's much more like an elephant sat on it and squished it. So, if you can sort of picture that kind of u shape before, now it's kind of, oh. It's kind of like... why is it sort of squished? Well, because now when I go over 1, instead of going up 1, now I only go up half as much. So, when I go over 1, It will go up half. Or when I go over 2, normally I would go up 4, right, because 2^2 is 4. But now, I only go up half that much. I only go up 2. 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 up half that amount. That's like a squashed version 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^2 times one-tenth. Whoa, now it's really only going up a tenth 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 in the air. It's really hard to do. I hope you're getting sort of a sense of what I'm talking about. 

Kim  20:25  
Sure, and it sounds like you can almost hear the difference between a student who's thinking functionally based on how they're describing what's happening to the parent function. 

Pam  20:36  
Yeah, absolutely. That's a great way of putting it. Yeah.

Kim  20:39  
Cool. Okay, so we talked about slope a little bit. We talked about transformations. You said there was one more thing that you want to talk about...

9-12 is where we wrap everything together and move into functional reasoning, and there's a lot of ground to cover, but it's so much fun when you can think and reason. Thanks for making math figure-out-able instead of rote memorizable for your students, high school teachers, we know they're better for it. Keep up the great work!