Ep 70: Equations of Lines Pt 2

Pam Harris  00:02

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

Kim Montague  00:09

And I'm Kim. 

Pam Harris  00:10

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

Kim Montague  00:40

So in the last episode, we did some higher math, right? And we were writing the equation of a line based on patterns. 

Pam Harris  00:48

Yeah, and it was a lot of fun. And you had so much fun, we decided we would do it again. 

Kim Montague  00:53

Yeah.

Pam Harris  00:55

While we were planning for this show, I have to tell you, Kim was like, "Really, you're gonna maybe do it again." I was like, "It was fun. It was great." It was, right? 

Kim Montague  01:03

Super willing. Yep. 

Pam Harris  01:04

So I appreciate your willingness. And it is kind of fun. And it was fun for me to write a brand new string. So this was a little untested, untried and true. No, it's true. It's just a little untested. It's about to be tested and tried. Here we go. Hey, maybe if this one goes really short, we could talk about what I thought about when I wrote the string. 

Kim Montague  01:24

Okay. 

Pam Harris  01:25

It probably won't go short. 

Kim Montague  01:26

It won't go short. Good thought.

Pam Harris  01:29

Alright, here we go. So a Problem String to, all about writing the equation of a line. And we're just going to suggest that we could do this without having a lot of background. Like there doesn't need to be a whole lot of, probably should have started the episode with that. hey, if you don't have a lot of background, hang on, because I think you can think about relationships. However, you might want to have a pencil handy, because that's okay. 

Kim Montague  01:49

I was just about to ask.

Pam Harris  01:50

Yeah, you're probably going to want to write these down. I don't think I would not be able to hold these in my head. I would need to be able to see. Because I'm gonna give you a bunch of points. 

Kim Montague  01:57

Okay.

Pam Harris  01:57

And then ask you, what would the equation of the line that contains these points, okay? 

Kim Montague  02:03

Oh, my goodness, okay. 

Pam Harris  02:04

Alright. So listeners, hit pause. Go find a pencil and a piece of paper and write down these points. So what about the point negative 2.2 comma 2.2. So that's point, negative 2.2 comma 2.2 is a point. Then there's the point one half, comma, negative one half. And the point seven comma, negative seven. So I'm just gonna give you those three points. 

Kim Montague  02:36

Okay. 

Pam Harris  02:37

And I'm wondering, Kim, if you just talk out loud as best you can. What are you thinking, when you see those three points?

Kim Montague  02:44

Um, okay. So the first thing I did, honestly, was I sketched out a coordinate. I don't even know what you call it.

Pam Harris  02:53

Grid, coordinate grid. Axis.

Kim Montague  02:54

Yeah. And I was, I didn't put a whole lot on there. But I was trying to visualize where those points would go. And so I recognize obviously, that the x is negative 2.2. And then, so I knew that it was going to be like in the second quadrant. 

Pam Harris  03:17

That first point. 

Kim Montague  03:18

Yeah. 

Pam Harris  03:18

Okay. 

Kim Montague  03:19

And then the y of the second point is negative and the third point as well, so those were going to be, I knew they were going to be, like, a little further down. 

Pam Harris  03:30

Okay, because the y's were negative. 

Kim Montague  03:33

Yeah. 

Pam Harris  03:34

Did you actually sketch the three points?

Kim Montague  03:36

 I did. I did. Uh, huh. 

Pam Harris  03:37

Does it look like they lie in the same way? 

Kim Montague  03:39

Well, yes. Based on my not probably awesome sketch.  But yes.

Pam Harris  03:46

Feasibly, they could be on the same line? 

Kim Montague  03:47

Yes. They look like they would be.

Pam Harris  03:49

Okay. Okay. 

Kim Montague  03:50

Um, oh, gosh. 

Pam Harris  03:52

Do you notice anything else?

Kim Montague  03:54

So the seven, negative seven, I feel like, obviously, there's going to be a bunch of points in between, but I'm kind of wanting like a few more points in between the half, negative half and the seven, negative seven. 

Pam Harris  04:06

Yeah. 

Kim Montague  04:06

So I'm kind of like, visualizing what some of them would be.

Pam Harris  04:11

Like, what might they be?

Kim Montague  04:15

I'm not sure.

Pam Harris  04:17

Like, if I went over three, about how far down do you think I'd go to stay in that pattern?

Kim Montague  04:25

I feel like you'd go down three as well. Like I'm wondering in my head would (6, -6) and (4, -4) and (2,-2) all be on that line?

Pam Harris  04:37

And it seems like you're saying that every time you have an x value, the y value is going to be that same number but. 

Kim Montague  04:44

Yeah, the opposite. 

Pam Harris  04:46

But negative, but yeah, the opposite sign. So it almost sounds like you're saying that all the y values are going to be equal to the opposite of the x values. 

Kim Montague  04:53

Yeah, yeah. 

Pam Harris  04:55

And so you, I just literally wrote down so then y equals negative x. 

Kim Montague  05:02

Yeah. Yep. 

Pam Harris  05:04

And that would be the equation y equals negative x.

Kim Montague  05:07

So you know when it's interesting, I want that to go through the origin. 

Pam Harris  05:13

Tell, say more about that. 

Kim Montague  05:15

Because based on, like just probably memory, I want y equals x and y equals negative x to go through the origin. Because I want it to say y equals x. And...

Pam Harris  05:32

Plus zero. So you have that b your kind of remembering this b thing about the intercept? And since it's plus zero, you're like, I think the y intercept is zero. 

Kim Montague  05:39

Yeah. 

Pam Harris  05:39

Let's see if (0,0) fits your pattern. Is zero equal to the opposite of zero? 

Kim Montague  05:45

Is zero equal to the opposite? 

Pam Harris  05:47

Like to negative zero? 

Kim Montague  05:48

No. 

Pam Harris  05:50

Yeah, kinda because zero does really have a sign.

Kim Montague  05:52

Well so. Okay.  So yeah, so 0 fits. Zero the same as negative zero? I mean, I guess yeah.

Pam Harris  06:00

Yeah. 

Kim Montague  06:01

All right. All right. 

Pam Harris  06:02

Weird, right? 

Kim Montague  06:03

Yeah, that's a little weird. 

Pam Harris  06:04

All right. That was our first problem in the string. Ready for the second problem string? 

Kim Montague  06:07

Sure. Okay. 

Pam Harris  06:08

Nice. All right. Nicely done.

Kim Montague  06:10

It took up a lot of paper to draw my picture. Okay. 

Pam Harris  06:13

Super job telling us what you're thinking about. That's excellent. Well done. Okay. So here's, I'm going to give you three more points again. So the first point is (-2, -1). 

Kim Montague  06:24

Okay. 

Pam Harris  06:25

(0, 1) and (5, 6). Five comma six. And you're making me want to sketch these, which I hadn't totally done.

Kim Montague  06:38

Yeah, like to sketch them. Okay. So, my line is going to go in the opposite direction. Right? 

Pam Harris  06:45

Okay. 

Kim Montague  06:46

It's a positive line

Pam Harris  06:48

Positive rate or positive slope. Okay. That seems important, okay.

Kim Montague  06:53

I'm looking between the points to be honest. I'm looking less at my graph, and I'm looking between the points to see if I notice anything. 

Pam Harris  07:04

Okay. Do you notice anything? It's a positive slope. That was a good first notice. 

Kim Montague  07:08

Yeah. 

Pam Harris  07:08

I feel like we need Jeopardy music. Doo doo doo doo doo, doo doo doo.

Kim Montague  07:14

No, but I appreciate the time. No. Okay. So I'm noticing that between the x and the y, it's increasing by one. I don't know if I'm saying that right. But from five to six is plus one. And then from zero to one is plus one, from negative two to negative one is plus one.

Pam Harris  07:32

Yeah, that seems important. So I might record what you just said, as I could find the y value. So I just wrote down y equals x one. Okay, there you go.

Kim Montague  07:45

Was that right?

Pam Harris  07:47

I think I might have been talking over but y equals x plus one. The y values were all the x values plus one. 

Kim Montague  07:52

Okay.

Pam Harris  07:53

Yeah, there's the equation of that line. Nice. 

Kim Montague  07:54

Okay.

Pam Harris  07:55

I was a little curious to see since you'd mentioned the y intercept in the last group that we were doing, I wondered, in the last problem, I wondered if you did notice anything about a y intercept in this problem?

Kim Montague  08:04

Oh, because it's plus one. 

Pam Harris  08:07

So do you see that zero, one being the y intercept?

Kim Montague  08:09

 Yes.

Pam Harris  08:10

Don't be disappointed. That's okay. You can just connect.

Kim Montague  08:14

I hear that for a second. Okay, cool. Yeah.

Pam Harris  08:17

Well, there it is. Right? So y equals x plus one, there's the plus one is the b as that y intercept. 

Kim Montague  08:21

Cool, okay. 

Pam Harris  08:22

And the slope is positive, right? Because it's 1x. So all those things you noticed actually played out. Nice. Alright. Let's do the problem number 3, third problem in our string. Okay, this time I'm going to give you four points. So (-1, -3) is the first point; (0, 0); (2, 6); (5, 15). Five comma fifteen. So I'm just gonna repeat those for the people on the radio, radio? Podcast. You just keep working, Kim. So negative one comma negative three; zero comma zero; two comma six; and five comma fifteen.

Kim Montague  08:59

Oh, I'm glad you said that again, because I wrote negative one, three. 

Pam Harris  09:02

I may have said it, sorry. 

Kim Montague  09:03

And I was very confused why it wasn't on the line. 

Pam Harris  09:06

Ah, I love how you're sketching them.

Kim Montague  09:09

I am sketching them. Sorry, it's taken a while. Maybe that gives everybody a chance to think.

Pam Harris  09:13

So I mean, it really you're actually doing this live. So that's kind of cool. 

Kim Montague  09:17

100%. Oh, that's gonna, Okay, so I just tried to put five, fifteen and I had no space. And I made my pencil go like way up off the page. Okay. So I think that there's a lot of, this is going to be so wrong. I'm not saying this properly. But there's a lot of gap between the points. Like I'm noticing that there's, like, I don't even know how to describe what I'm what I'm noticing. It's like, increasing three times.

Pam Harris  09:45

Like it feels really steep it sounds like. 

Kim Montague  09:47

Yeah, yeah. 

Pam Harris  09:48

Because I've given you a lot of lines that had a slope of one. 

Kim Montague  09:51

Mm hmm. 

Pam Harris  09:51

And I think you're feeling that this one has a higher rate. 

Kim Montague  09:55

Yep. 

Pam Harris  09:55

Yeah, it's like increasing faster. Okay. Cool. I like it. Something about three? 

Kim Montague  10:00

Mm hmm. 

Pam Harris  10:01

What about three? Like why? That's random? What? Three what?

Kim Montague  10:07

Oh, okay, I see now. So between the points, it's three times, oh, I can do this. So y equals 3x. 

Pam Harris  10:18

Why? 

Kim Montague  10:19

Because for each of those points, the y is three times the x. Then I can see on the graph, that's really nice.

Pam Harris  10:28

That you can see how steep it is? That is over one up three?

Kim Montague  10:31

Yeah, as soon as I couldn't put (5,15) on the graph, I was like, wait, that's way up there.

Pam Harris  10:35

Yeah, that feels really steep. Nice.

Kim Montague  10:38

Yep. 

Pam Harris  10:38

So you could see that like, 15 was three times five and six was three times two. 

Kim Montague  10:42

You know, it's funny because I'm not looking at the, maybe I should be looking at the points before I go sketch. I automatically go sketch. So I'm gonna look at the points for the next one, and see if I can see the relationship there first.

Pam Harris  10:53

Well, I'll just let you do that. Okay.

Kim Montague  10:57

Probably won't be able to see it.

Pam Harris  10:58

Here is the last problem in our string. This is fantastic. Alright, four points again, here we go. Negative four comma negative two. 

Kim Montague  11:05

Okay.

Pam Harris  11:06

Negative two comma negative 1; zero, zero; and five comma 2.5. I'm gonna say those again, for podcast listeners, (-4, -2); (-2, -1); (0, 0); (5, 2 and a half). 

Kim Montague  11:24

Got it. As soon as I looked at the points, that was a poor choice on my part. But you know what? Like, I like the fact that I could see, the first thing I did when I was trying to be comfortable was go to the graph. But when I started looking at the points first on this one, I recognize... 

Pam Harris  11:44

Just the coordinates. 

Kim Montague  11:45

Yes, yes. Sorry, the coordinates. I recognize I was going actually from y to x, and I was like, Oh, it's doubling. Its doubling. But then when I started writing y equals, I realized, oh, wait a minute, it's not doubling it's halving. 

Pam Harris  12:03

Sort of depends on the perspective you look at. Right? 

Kim Montague  12:06

Right. 

Pam Harris  12:06

You could write it in either direction. 

Kim Montague  12:08

Right. 

Pam Harris  12:08

So what did you end up.

Kim Montague  12:11

I ended up with y equals one half x. 

Pam Harris  12:14

Cool, because if I look at any x value, and I multiply it by half, or divided by two, you get the y value. So I wonder we can actually think about the other direction. Like you said, the x value was twice the y value. So you could write x equals two y. 

Kim Montague  12:30

Yep. 

Pam Harris  12:31

Which is an equivalent form of the equation of that line. Well, you just usually write it as y equals, but you could sort of have it either way. 

Kim Montague  12:38

Oh okay. Alright. 

Pam Harris  12:40

Alright, y'all. Could we really look at patterns to write the equation of a line? I think we can. I think that you just heard some really good thinking live. Like, from somebody who hasn't really written the equation of a line for a while? 

Kim Montague  12:56

Oh, gosh, Yes. 

Pam Harris  12:56

Just a few. 

Kim Montague  12:57

Yes. A few decades. 

Pam Harris  12:59

Yeah. And so could we use intuition to help people like, oh, like, what's really happening is it has everything to do with the patterns between the x values and the y values. And we can learn a lot about slopes and, and y intercepts and things before maybe we get a little bit more into other ways. Now, do I want other ways to write the equation of a line? Yes, though, you might be surprised about how much I want those to be based on relationships and connections, and not rote memory of a formula, as well. Alright, that was fun.

Kim Montague  13:31

Turns out writing the equation of a line is figure-out-able.

Pam Harris  13:36

So there you go. Excellent. Well said. So everyone, if you want to learn more mathematics and refine your math 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.