Ep 69: Equations of Lines Pt 1

Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:09

And we make the case that mathematizing is not about mimicking steps or rote memorizing facts. It's about thinking and reasoning; about creating and using mental relationships. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keep students for being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?

Kim Montague  00:39

Alright, y'all In this episode, this week, we are going to do some higher math.

Pam Harris  00:45

Dah-ta-dah.

Kim Montague  00:46

Just for you all.

Pam Harris  00:47

Whoo! Alright. And Kim's a little excited. I mean, scared. I mean excited about today. We were kind of talking a little bit about how this episode was gonna go. And Kim was like, "(clears throat) Here we go."

Kim Montague  00:59

This is not my jam. But I'm willing.

Pam Harris  01:01

Yet!

Kim Montague  01:01

But I'm going to learn and grow.

Pam Harris  01:03

Yet! It's not your jam, yet. I mean, you're actually so much further than you think you are on a landscape of learning. Lots of fun, lots of fun, So y'all take a deep breath, get a pencil and paper for this one. You're gonna want to write some stuff down. Just because I'm crazy, and so you should write stuff down. But you might want to look at some things, you might want to pause this episode a little bit after you look at some things. If you're a high school math teacher, okay, maybe not. But maybe. So just be aware. Here we go. Today we're going to talk about writing the equation of a line, given some information. So if I give you a point, and slope, write the equation of a 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 that then we would expect the student to write the equation of a line. 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 a kid 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, find 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 gonna subtract, subtract, and then they're gonna 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 this 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 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 big workshop I did. 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 with 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 twenty... 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'd given them. Their job is to write the equation of a line. 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 be 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, I" whatever, whatever. And she has all these disclaimers about why I should not be looking at her work. And said, "Ok 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, "It just jumped out at me." Ooo I got intrigued. Anytime somebody tells me that their intuition was involved, that thing pinged for them. Mmmm uh-huh. I'm interested, right? I'm thinking about that. And I'm thinking about that a lot right in that moment. So I'm like, "Hey, I'd be really curious what you were just thinking about to find the equation of  a line." She said, "Pam, I just see it." I said, "What do you just see it?" She goes, "Look at them. If you add those together - no, no, if you subtract them, I get one every time. So the equation is just x - y = 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 from one of the x values, you subtract its y value every time I get one. So I just wrote the equation x minus y equals one." I said, "Say that again." Y'all, 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 an x value, and she had one as the y value. She said, "Two minus one is one." And I was like, "Well, that's true." And she goes, "Okay, but look at one of the other values." So if it was six was the x value, and five was the y value. She said, "Look, six minus five is one. So every x minus every one of these y's gives me one. So I just wrote the equation x minus y equals one." To which I said, "Can you do that?" I had never... I, again, I was this 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 Montague  07:36

Yeah. 

Pam Harris  07:36

Can we work on that today?

Kim Montague  07:38

Sure. Do it.

Pam Harris  07:40

I wondered in that moment, could we teach that? Could we teach that strategy of looking at the values, finding a pattern, and then just recording that pattern sort of, generally. Okay, so I'm going to give you - alright listeners, Kim, I'm going to give you guys a set of points, I'm going to give you all a set of points. And then I'm going to pause and I'm going to suggest that you pause and then think to yourself, what are the patterns that you see in that set of points? Do you see any patterns? Specifically, think about if I gave you any x value, could you do something to that x value and tell me what the y would be so that it would follow the same pattern as the four points that I'm going to give you? 

Kim Montague  08:27

Okay.

Pam Harris  08:28

You might even graph the points just to like sort of prove to yourself that they're on the same line. However, caution. If you're a high school math teacher, you might not want to do that. Because if you graph those points, you might not be able to stay out of a rote memorized way of writing the equation of a line. So if you've got some rote memorized way of writing the equation of a line, I'm gonna encourage you to maybe not do that, to see if there's some patterns that you can find. And maybe don't graph them. If you already have a - go ahead if you want to. But maybe if you don't want the surprise spoiled, maybe don't do that. But anybody who doesn't have that down, you don't have a rote memorized procedure down to write the equation of a line. Maybe just like what patterns do you see? Maybe even graph the points and see kind of what they look like. Do they lie on the same line? And what does that mean to you? And then come back. Alright, Kim, here's the first set of points. So I'm going to give you four points. So the first point is zero, negative two. So I would write that as (0,-2), right? 

Kim Montague  09:28

Okay. 

Pam Harris  09:28

So an ordered pair (0,-2). The second point is (1,-1). The third point is  (2,0). And the last point is (3,1). 

Kim Montague  09:39

Okay.

Pam Harris  09:40

So y'all pause. Let me say it again. (0,-2), (1,-1), (2,0), (3,1). Pause the recording, go see what patterns you can find. Maybe graph the points. Come on back.  Alright, hopefully people paused a little bit. Okay, so Kim, what patterns... What patterns do you see? Just like first thing that comes to you? What patterns do you see?

Kim Montague  10:04

I noticed that the x values increase by one. 

Pam Harris  10:09

Sure enough. 

Kim Montague  10:10

And the y values also increased by one.

Pam Harris  10:15

Sure enough, like they're going up. Now they're different, right? They're not the same. But they're all going up by one. Okay, cool. Did you notice any other patterns?

Kim Montague  10:24

Yes. So I, also looked, I don't know if you say it this way, but kind of between the points. Between the x and y value. And I noticed that - 

Pam Harris  10:35

Like from x to y? 

Kim Montague  10:37

Yeah. So from x to y is a decrease of two.

Pam Harris  10:44

Wait. What do you mean?

Kim Montague  10:45

So I'm looking at the series of points that you just gave us and I started kind of from the bottom up, and I said, how do I get from three to one, and that's subtract two. How do we get from two to zero? That's subtract two. How do we get one to negative one? That's subtract two. And to get from zero to negative two, that's subtract two.

Pam Harris  11:06

So I hear you saying that (3,1) was actually kind of helpful as you looked between the x and y. You said to yourself, "Well, I know that relationship, that's just minus two. I wonder if it works for the other problems?" I kind of heard wonder. 

Kim Montague  11:19

Absolutely.

Pam Harris  11:19

And woo! It did. You're like, yeah, sweet, it did. So I might record your thinking. I might make that visible by saying you took an x value, subtracted two. So I would write down x - 2. And you said every time that equaled y. 

Kim Montague  11:35

Yes.

Pam Harris  11:36

So I would write down x - 2 = y. And you just wrote the equation of a line line that contains those points. (drumroll) I don't know if you could hear the drumroll there.

Kim Montague  11:50

 Okay. Alright.

Pam Harris  11:52

Now, just from noticing a pattern, you were able to like, "Hey, I found a pattern that works for all the points. And so I'm just going to generalize that." I could give you any x value. Like what if I gave you four Kim right now? Now if I gave you four, what would you do? 

Kim Montague  12:04

I would say that the x is four and the y is two. 

Pam Harris  12:09

Because?

Kim Montague  12:10

Because four minus two is two.

Pam Harris  12:13

Yeah, so maybe that's not the best point because we got too many twos in there. So what if I give you five? What if I say the x value is five? What would the y be?

Kim Montague  12:19

The point would be (5,3)

Pam Harris  12:21

Because you're saying x subtract two is always going to give you the y value. There you go. So I could literally give you 1,000,000. 1 million is the x value. And you could say that the y value would be?

Kim Montague  12:32

198,000.

Pam Harris  12:35

Oh.

Kim Montague  12:36

Oh, I did that. I started to write it down. 999,998.

Pam Harris  12:43

There we go. That was fun. 

Kim Montague  12:44

Hilarious.

Pam Harris  12:45

That's what I would typically do. Okay, cool. So you've now found a pattern that works for any x value. And that is what the equation of a line means. It means this is a pattern that if I give you any x value, you can tell me the y value based on this pattern. That's how we describe that set of ordered pairs, that set of points that make up that line. 

Kim Montague  13:03

Okay.

Pam Harris  13:04

So ready for the next one? 

Kim Montague  13:05

Sure. 

Pam Harris  13:06

Alright, here we go. Four points again. 

Kim Montague  13:08

Okay. 

Pam Harris  13:08

And you're gonna look for patterns and try to tell me if I give you any x, what would the y be, Okay, first point, (1, -1). So negative one comma one. Next point, (0,0), (1, -1), (2,-2), Let's say it really fast. (-1,1), (0,0). (1, -1), (2,-2) Alright, everybody. Pause. Pause the recording. Pause the podcast. Go see what patterns you can find. Come back when you're ready.  Alright. Hopefully they paused the recording. I feel funny, because I feel like I should pause longer.

Kim Montague  13:44

Yeah, because I don't have a huge pause. 

Pam Harris  13:47

Oh, that doesn't give you any pause. Does it?

Kim Montague  13:49

That's ok.

Pam Harris  13:50

Kim's like, "Pam! Give me some time here!"

Kim Montague  13:53

Well, okay, so can I tell you the first thing that I thought about? 

Pam Harris  13:55

Yes, please.

Kim Montague  13:56

Because I was trying to think what this would look like on a graph. And so I, like the origin, right? 

Pam Harris  14:05

That's nice. Yeah, Zero zero, is in there.

Kim Montague  14:07

But so when I was looking at the first and the third points that you gave me, I was like, Oh, those are opposites. So -

Pam Harris  14:15

Those? So those are adjectives. What's opposites?

Kim Montague  14:17

Sorry, the negative one, one and the one negative one.

Pam Harris  14:22

I just said adjectives and meant pronouns. Oh, okay so the x value is opposite of the y values? Is that what you mean? 

Kim Montague  14:27

Right. Mhmm. 

Pam Harris  14:28

Is that true for all the points? 

Kim Montague  14:30

Mhmm. 

Pam Harris  14:31

Okay, so I might record your thinking that the y value is, equals, the opposite, negative, of the x value. I might write that as y = -x, or the opposite of x. Because you're telling me that the y values always opposite the x value. So if I give you any x value, take the opposite sign, I would get the y value. 

Kim Montague  14:54

Yeah. 

Pam Harris  14:55

And that's the equation of that line. y = -x. Or the opposite of x. No way, right? 

Kim Montague  15:01

Yeah. 

Pam Harris  15:02

And let me just pull back on some prior knowledge that you might have. I think you know what the line y = x looks like, right? It's that 45 degree line. If I were to graph it, that's the line y = x. You and I've done some work that I think you know that parent function. So what if I just said, Kim, that y equals x, I need all the opposites now, of the y values. Every time you had a positive y value, when you went over to an x value, every time you had a positive y there, I need now - don't graph the positive y. At that x value graph the negative y value. I wish you could see my hand because I was like putting a point up in the first quadrant. And I was like, but not that positive y value. And I was like reflecting down into the fourth quadrant to stick it down there. But back in the, let's go in the normal y = x. That's normally in the third quadrant, right? It's down there, that 45 degree angle down there, negative, negative third quadrant. But now those y values shouldn't be negative because I need the opposite of them. I need them to pop up to the second quadrant. I need them to be positive. Now y'all if you haven't heard quadrants for a while and everything, okay, sorry. But hopefully that all of my graphing people that graph more often that made a little bit of sense. That we can sort of talk through why that reflection is happening. Alright, let's do another one. Here we go. I'm gonna give you four points. And (-2,3), (-1,2), (0,1), (1,0).

Kim Montague  16:37

Okay, they're gonna pause.

Pam Harris  16:39

Oh, yeah, sorry. I was pausing. Pause the recording. Alright. So Kim's gonna pause. Oh, so do you want us to really pause longer here and Craig can cut out the - 

Kim Montague  16:50

No, it's okay. 

Pam Harris  16:52

Craig's our editor everybody. He cuts dumb things out. We're not gonna have him cut this out. We're just gonna let Kim think.

Kim Montague  16:58

It's live math. 

Pam Harris  16:59

Kim's like, "Stop talking, Pam so I can think." Okay. What do you notice?

Kim Montague  17:08

All right. So I, you know, I find that now I'm not really necessarily looking from the first point down to the second point, down to the third, down to the fourth, like I did the first time. Because I'm recognizing that you want me to think about how to get from x to y. So that original thing I did has not helped me. So I want to first look from zero to one and I want to call it plus one. But then that's not really helpful for, so x + 1, but that's not really helpful for the other points. 

Pam Harris  17:38

Okay. 

Kim Montague  17:39

I'm looking at -1 to 2. And how can we get from negative one to two? And I'm thinking that the only way to make something work for both of those points is to consider the opposite of x.

Pam Harris  18:01

That's actually interesting. When you said negative one to two, I thought, well, you could add three. 

Kim Montague  18:05

Yeah, I did. 

Pam Harris  18:05

But I'm assuming you did that. And discounted it - 

Kim Montague  18:07

I did.

Pam Harris  18:08

Because it didn't work for any other points. Yeah. So I just wanted to, like, bring that out. So then you said, "For me to get to, get from a negative value to a positive value... That maybe there might be some of the opposite stuff going on." And we could have mentioned that in the problem before, when we had y equals the opposite of x that we did, we had this sort of switchy thing happening between the x's and the y's, we had kind of this negative/positive switch happening. And so there was the opposite of x in the problem before. So in this problem, you also might have some opposite of x going on.

Kim Montague  18:45

Yeah. And so you know, it's interesting, because I feel like the zeros are actually tripping me up a little bit. If I would have focused on the first two, then it's definitely the opposite of x plus one. Let me confirm. Opposite of x plus one. 

Pam Harris  19:05

You just tried it on the first two problems. 

Kim Montague  19:07

Yep, sure. did. 

Pam Harris  19:07

Yeah. So  like when you look at (-2.3),  did I just interrupt you?

Kim Montague  19:12

Nope, go ahead, I'm good.

Pam Harris  19:13

When you looked at  (-2.3) you said, "Well, if I looked at the opposite of negative two, that's two. Well, to get from two to three, that's just plus one." 

Kim Montague  19:19

Yep. 

Pam Harris  19:20

Let's see if it works on the next. Opposite of negative one is one. One plus one is also... bam! And then you can go check to see if it works on the zero. And often zeros and ones are funky. 

Kim Montague  19:32

Okay, 

Pam Harris  19:33

Often zeros and ones are a little - they behave a little differently because of the identity properties. And so that's interesting. Yeah, those can absolutely trip you up.

Kim Montague  19:42

Yeah, I was drawn to the zero at first, but now I'm-.

Pam Harris  19:45

Sure. So I would record your thinking as, I can get any y value - y equals - the opposite of x, so I would write -x and then add one, plus one. So the equation of the line could be y = -x + 1. Totally cool. If I may share a different strategy for the same problem. 

Kim Montague  20:05

Okay.

Pam Harris  20:05

I wish I could - this is our unnamed gal, our science gal at the workshop. This is the strategy she used. So she looked actually at the zero and the one, even though you found them problematic, she was like, "Okay, if I look at the point (0,1) and I look at the point (1,0), both times, if I add those together, I get one." Like zero plus one is one. 

Kim Montague  20:29

Okay,

Pam Harris  20:29

One plus zero is one. And then she tried that in the other points. So is negative two plus three, one? Yep. How about negative one plus two? One. Yep. So I would record that thinking as take any x value, add it to the y values, and I always get one. So x + y = 1. Now wait a minute, wait a minute, we can't have two equations of lines for the same set of points. Well, Kim came up with y equals the opposite of x plus one. And my science gal would have come up with x plus y equals one. And those are equivalent. Like if you move some variables around, then it's the same equation. One of them is in slope intercept form, and one of them is in standard form. But it's the same. Whoa, whoa, in fact, my science gal was just coming up with equations of lines, in standard form. That's interesting. How many times have we messed with standard form as sort of a last minute thing that we do. It's kind of usually the last thing in the chapter. We don't really do a lot with it except for take the other forms and turn it into standard form. That interesting that we could actually have a strategy that a student uses, to kind of motivate, check it out. This form that we just came up with is called standard form. Alright, y'all. So I'm going to encourage you that there are ways that we can teach strategy, even in lots of higher math things. Now, I know that we did one that's not going to work for all equations of lines. But I would ask you to consider, like, if I give you any set of points, you're not necessarily going to be able to recognize the relationships, like we just did. I obviously gave you sets of points that were fairly easy to recognize the relationships. But that doesn't mean that they don't exist, that that strategy isn't out there. In fact, I would love to hear from you on social media, if you are the secondary teacher who's like, "Oh, actually, I do that all the time. Like I look at the points and first I see if something just jumps out at me. And if it doesn't, then I go ahead and do the you know, whatever else." Well y'all that's a strategy. That's a strategy worth teaching, we can teach that to students. We can teach to students so that they can let their intuition drive which strategy they choose. And if they have a strategy that's more efficient, they can choose that one first before they then go to anything that's less efficient. Not using so much the things that ping for them. We can actually think and reason. You can see patterns, not just doing that, subtract, subtract, divide kind of procedural stuff to find the slope of the line and then put it in a formula and do all that. We don't have to do that. One of the things that I wanted to point out is, as Kim was doing that, notice that she was not just thinking functionally though she was. She was also thinking multiplicatively and additively. That functionally depends -  her being able to see those patterns had a lot to do with the fact that she was already thinking both additively and multiplicatively. We didn't just - she's also thinking proportionally, we just didn't see that here. But because she could think that way in those other ways, she was able to use those reasonings in order to think functionally even more. So just wanted to sort of point out that hierarchy of thought that was kind of included in what we just did. Alright, so if you want to know more about that, pay attention to next week's episode where we will do one more, at least, one more writing equations. So if you want to learn more math, and refine your mathematics teaching so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able!