Math is Figure-Out-Able with Pam Harris

Ep 112: The Double Open Number Line Model

August 09, 2022 Pam Harris Episode 112
Math is Figure-Out-Able with Pam Harris
Ep 112: The Double Open Number Line Model
Show Notes Transcript

We sure loved discussing the open number line model last week, but have you ever heard of the double open number line? In this episode Pam and Kim discuss how the double open number line can be used to develop Proportional Reasoning and to think and reason about rates.
Talking Points:

  • A Problem String to reason about unit rates and model on a double open number line
  • When could an double open number line be helpful in middle school and high school?
  • When is a ratio table more versatile than a double open number line?
  • What other concepts can a double open number line be helpful for students?
  • The percent bar is a double open number line?

Check out our free download on Double Number Lines: https://mathisfigureoutable.com/double

Pam Harris:

Hey fellow mathematicians, you're listening to the podcast where math is Figure-Out-Able. I'm Pam. And you found a place where math is not about

Kim Montague:

And I'm Kim. memorizing or mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns and reasoning using mathematical relationships. We can mentor mathematicians, as we co-create meaning together. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians, they can be and teachers too, to be clear. So last week, we dove in deeply about one of our favorite models, the open number line, and one of our favorite ways to introduce it. And we also talked about why it's so valuable. If you missed that episode, you should check it out.

Pam Harris:

Absolutely.

Kim Montague:

This week, however, we are so excited to focus on another model that is important, but less discussed. And that is the double open number line.

Pam Harris:

So we did the open number line last week, double open number line, what's that, like double the trouble? Like what? What is the double open number line? And I'm going to be really honest, and say the first 150 times I saw a double open number line, I didn't get it. I was like,

Kim Montague:

Yeah.

Pam Harris:

What is - and I just tuned it out, turn it off. And I'll tell you where I was at that point in my life, I was developing my Additive Reasoning along with my young kids. And I had seen some work, where people were talking about how they could use a double open number line. And I was like, whatever weirdo. It just, it didn't. It didn't land. And in hindsight, I believe it's because I wasn't reasoning proportionally. And so because I wasn't reasoning, proportionally, a proportional model, a double open number line wasn't hitting for me, it wasn't making sense. I wasn't reasoning through proportional reasoning problems at all. And so I would look at, you know, the problem they were solving, and I had so little access to those problems that I couldn't understand what then they were doing on a double open number line. So that, we'll try today to do our best to make it accessible to everyone. But you might consider that if you're still building your proportional reasoning that you might want to dive in and actually do these problems with us and be thinking and reasoning, don't expect it to be just, you know, oh, yeah, of course, that's what it is. It wasn't for me. So if you're like me at all, maybe you're gonna want to sort of co-create it with us as we go today.

Kim Montague:

Cool.

Pam Harris:

So how does a double open number line compare to that normal open number line? A normal open number line, like we talked about last time is different from a closed number line. A closed number line has all of the tick marks, every whole number has a tick mark. An open number line is where we get to choose, we put the tick marks where we want them, we sort of label the numbers that are helpful for us in solving a problem. A double open number line keeps track of corresponding quantities. So I have quantities on the top of the number line and quantities on the bottom of the number line, we still have jumps in between quantities, but I have corresponding quantities being represented. That might sound a little bit like a ratio table. And so maybe at the end of this, we'll compare a little bit ratio tables and double open number lines, because they have some similarities, but they also have some differences. And it might be helpful to, but they both have that similarity, that there's two things corresponding to each other. So to get at what we mean by that, let's actually do a Problem String. So I'm going to ask them questions. Kim, you're gonna tell me what you're thinking about. And then I'm going to, y'all, but we're on a podcast. So hopefully, I'll describe best I can, what I'm drawing what it would look like. So you can sort of see what a double open number line would look like. But this is similar to how we would introduce a double open number line to students. We would do some work. And as they tell us their strategy, we would make it visible. In this particular case, I'm going to make that thinking visible on a double open number line. Well, maybe before I forget, Kim, before we start the Problem String, I will say that a double open number line is an introductory model. See, how do I say this? We have talked in the past a little bit about how the area model is a brilliant model for kind of introducing multiplicative relationships. And helping students really get a sense of this spatial relationships that are happening, sense of size and magnitude and what it means to have this two dimensional measurement thing. But when we actually want to compute with multiplication and division, we don't use the area model and open array, we tend to use a ratio table. That is actually going to be very similar with a double open number line. In my vision for models, the way that we've set up models and modeling, a double open number line is brilliant to introduce thinking proportionally, to get students to develop a sense of size and magnitude and spatial relationships and how things are related. But we don't actually expect students to then use double open number lines to compute.

Kim Montague:

Right, right.

Pam Harris:

about that when we get to the how it compares to the ratio table. I'm just making notes. I don't want to forget. Okay, so ready, Kim?

Kim Montague:

Yep.

Pam Harris:

If you were walking, you're walking, you run, I know, you run but can you walk from or if you're walking, if you're walking, maybe you're running actually have to think about these distances I'm about to give you. Maybe you're running actually. So you're moving, you're moving in some way. If you're moving around, but you're moving at a constant rate. So it's a constant, you're not speeding up, you're not slowing down, you're moving at a constant rate. And in 40 feet, sorry, you've moved 40 feet. So you've traveled 40 feet, and you did it in five seconds. So you traveled 40 feet at a constant rate in five seconds. I'm curious how fast were you going? Huh? Like per second? Yep.

Kim Montague:

Okay. So 40 feet in five seconds means that I'm going to go eight feet per second.

Pam Harris:

And how do you know?

Kim Montague:

Because if I've gone five seconds, I wrote down 40 feet in five seconds. And then I divided both of those numbers by five. Because to get from five seconds to one second is divided by five. And so then I also had to divide 40 by five.

Pam Harris:

Cool. And when you divide 40 feet by five seconds, you end up with feet per second, feet divided by seconds, which is a rate and you're saying that you were going at eight feet per that one second?

Kim Montague:

Yes.

Pam Harris:

Nice. If I were to work, if I was working with students, I might say, "Okay, so you said that you traveled 40 feet." And so I would just draw a number line on my paper. And I put zero on the left hand side above the tick mark. And I've put 40 feet on the right hand side. So zero feet on the left, and 40 feet on the right. And I've actually kept those ends. So I'm not going beyond those ends of sort of there they are tick marks on either side. And then on the bottom, I'm going to put seconds. And so on the bottom, right underneath the zero feet, I'm going to put zero seconds. And right underneath the 40 feet, I'm going to put five seconds. And then I'm going to kind of model visually what you just said, because you said, "Well, I've got five seconds. And I know each of those seconds, I went some feet." So it's almost like you took that distance from zero to five and you sort of cut it into five chunks. And y'all I just cut it into five chunks. I'm actually getting kind of good at that Kim.

Kim Montague:

I just did too.

Pam Harris:

To cut a distance into five chunks. That's tricky. That's tricky, in fact, ooh, let's play a little test here. How many tick marks did you draw? How many tick marks in a draw for five seconds? Yeah. So like when you said you cut it into five equal chunks?

Kim Montague:

Yeah.

Pam Harris:

You only drew, so you had a beginning and ending already. But then you just drew four tick marks because you're trying to make five chunks. So that's also something that kids need to mess with a little bit. And it's one thing that we can mess with a little bit on the open number line, definitely when we get to a double open number line, it's the thing that we're going to have to consider. Most students will draw five tick marks and then try, wait what? Or at least many students and then try to figure out why they now have six chunks. And yeah, anyway, okay, so I've got these five chunks. And so I've labeled now on the time sort of underneath the number line where that I have time, I've labeled zero seconds, one second, 2, 3, 4, and then the ending five seconds. And since I just cut the seconds by five, it's exactly what you said, you divide to the five by five, to get one second intervals. Now I'm going to do the same thing to the zero to 40. Oh, and 40 divided by five is eight. So I could put eight above the one second and I could keep going but I'm actually going to choose not to. I could put 16 above the two and so on. And then I was sort of know where I was at every second. I would not take that long with students. The reason I just took so long to say that, I would just do it. I would draw it up on the board. I would fairly quickly talk about what I was doing, and move on. But I took a little bit more time here because we're audio only and I wanted to make sure nobody knew what I was drawing. Second problem. This time Kim, oh, by the way, sorry. Before the second problem. Hey, if you're going 40 feet in five seconds or eight feet per second. What does that feel like to you? Does that does that walking, crawling, running?

Kim Montague:

Eight feet per second?

Pam Harris:

Yeah.

Kim Montague:

I'm gonna fast.

Pam Harris:

It's pretty quick. Right? Okay.

Kim Montague:

Yeah.

Pam Harris:

Cool. Next problem. How about if you're going 20 feet in four seconds? A constant rate, 20 feet four seconds, how fast you're going? Five feet per second. How do you know?

Kim Montague:

The same thing I did from four seconds to one seconds divided by 4 and 20 seconds, divided by four is five seconds.

Pam Harris:

Is five. I mean 20 feet. And you know, 20 divided by four, right? So you know, 20 We need time to be this the same. Okay. divided by four and feet divided by seconds, so you end up with

Kim Montague:

Yeah, I may ask myself later, if I'm glad I did five feet per second. So very quickly, I would draw a double that. But that is what I did. Okay, and then on the top of the open number line. So I've got a line drawn, I've got zero to four seconds on the bottom, I've got zero feet to 20 feet on the open number line, I have zero feet to 20 feet. You said that top. I did draw that about half of the 40 feet that I drew same thing, you divided the 20 by four, or the four by four and before. No, that's not true. I drew the same length. No, that's not true. I drew it. I'm asking myself, if I like what I drew. the 20 by four. So I could divide, when you said you divide I'm gonna tell you what I actually drew, I actually drew the four by four, I could like think about where would those it the same length as the four seconds. Four seconds down first. Yep. But I didn't worry about the fact that the 20 is one seconds be. And I would do that easier than the five half of the 40 feet. So four seconds is right underneath the seconds. I would cut the span that represents four seconds in four seconds ahead in that previous problem, even though I had five seconds, but it put it underneath a four. But I have half for two seconds, and then cut that span in half to get the put 20 feet above that tick mark. one second. That's an easier cut to do, cut in half and then cut in half again, rather than eyeballing the five equal chunks. And then since you divided the 20 by four, I could also, I know that's five and then just stuck the five above that one. But I might ask students, "Hey, once we cut the four seconds in half to get the two seconds, what corresponds to that?" nudging the fact that if I cut the four seconds in half, could I also cut the 20 seconds in half? Sure. And so that would be 10. And once I know that two seconds was 10 feet, could I also then when I cut that in half to get one second, cut the 10 feet and half to get five feet. And I'm just going to kind of bring that up sort of try to draw that out of the student quickly, not belabor it too much, but just kind of like raise it as a possibility. Next question, how about if you moved 24 feet in four seconds? How fast were you moving?

Pam Harris:

Six feet per second. And as you were thinking, I drew the same length of an open number line that I had for the four seconds and I put zero seconds to four seconds. And then you had already said six seconds. So I might at that point say, "How far were you moving?"

Kim Montague:

Six feet per second.

Pam Harris:

That was your speed?

Kim Montague:

Oh, how far am I moving?

Pam Harris:

How far total? Yeah.

Kim Montague:

Twenty-four, 24 feet.

Pam Harris:

And the only reason I asked that was to give me time to write down the zero feet and the 24 feet.

Kim Montague:

Okay.

Pam Harris:

And then I would say, "Hey, did anybody do the thing that we did before? Like the cut and half thing? Did anybody?" And I would just look and I would look for somebody. I would look for a spark, I would look for a glimmer of somebody saying, "Yeah." And then I might say, "Okay, so you cut the four seconds and have to get two seconds. Did you cut the 20?" I would just say just enough, for them to go, "Yeah, that's 12." And then I'd go, "So then did you cut the two seconds in half? That's one second. And then what's half of 12?" So there's another way of getting the six feet per second that Kim had said. So I'm going to notice that right now I have 20 feet in four seconds was five feet per second. And 24 feet in four seconds was six feet per second. Next problem. How about if you went 22 feet in four seconds? How fast were you going? Five and a half feet per second? And I probably should have said maybe pause for a second. That's okay. Because we didn't give listeners a very

Kim Montague:

Sorry. long time to think about that. So my bad. So is that right, listeners? Like 22 feet in four seconds. And Kim, how do you know? How do you know it's five and a half feet per second? Well, I'm, a couple of things that I'm thinking about. I like that you just had me think about half of the four seconds is two seconds. So that would be half of the 22 feet which is 11 feet. And so I've just drawn that I've drawn zero to four, and I cut it in half to for two seconds, zero to four seconds for two seconds and I wrote the 22 feet and then above the two seconds I were 11 feet Okay, keep going. And then half that, again, would be from two seconds to one second. And from 11 feet to five and a half feet.

Pam Harris:

Because half of 11 is five feet.

Kim Montague:

Yep.

Pam Harris:

So five and a half feet is now in the corresponding spot as one second.

Kim Montague:

Yeah.

Pam Harris:

Nice. Do you wanna tell us the other thing you were

Kim Montague:

Yeah, well, you already gave me 20 feet per four thinking about? seconds and 24 feet per four seconds. So I was thinking about how 22 feet per four seconds. It's kind of nestled in between the 20 feet and the 24 feet.

Pam Harris:

Like right in the middle.

Kim Montague:

Which was five feet per second and six feet per second. So that would make sense that it would be five and a half feet per second.

Pam Harris:

Right in between. Bam.

Kim Montague:

Yeah.

Pam Harris:

Nice thinking. Alright, cool. Next problem. What if you had traveled 21 feet in four seconds? I'm pausing. Thank you. I was holding my breath. I was like is she going to pause. Sweet, sweet. Okay, so Kim. Okay, so 21 feet in four seconds. How fast are you going?

Kim Montague:

This one is 5.25 feet per second.

Pam Harris:

Five and a quarter feet per second. Okay, and how did you find that this time.

Kim Montague:

Um, so this time, I kind of did the opposite. I looked up, I'm writing in my paper. So I looked above what I'd already been working with. And I found that I had 20 feet per second and 22 feet per second. So again, I just went right between those two answers.

Pam Harris:

Five feet per second, and the five and a half feet per second, you went right in between to, five and a quarter cool. But we could, we could say, "Hey, did anybody cut the four seconds in half?" And so 21 in half, kim, if you don't mind, what is 21 in half? Twenty-one is 10 and a half? So that'd be 10 and a half feet, corresponding to two seconds. And then you could cut 10 and a half feet in half to get the corresponding one second. And that would be five and a quarter.

Kim Montague:

Yeah.

Pam Harris:

Cool. Next problem. How about if I asked you for 23 feet? You've covered 23 feet in four seconds. 23 feet in four seconds. Thanks for the pause.

Kim Montague:

You're welcome. Five and three quarters feet per second.

Pam Harris:

Okay. And what was your strategy this time? I again looked in between, you had given me 22 feet per second and 24 feet per second. And that was- And that totally makes sense. And if I had not. What if you didn't have that? What if I had 23 feet in four seconds cold? Without any of the previous stuff?

Kim Montague:

Yeah, then I would definitely go, halfway would be two seconds in 11 and a half feet.

Pam Harris:

Half of 23 is 11 and a half. And then half it again to get one second. And what did I say 5.75 feet per second? Because half of 11 and a half. I have to think about that a little bit. Half of 11 is five and a half and half and 50. Yeah, that's five point seven five. Yeah, I honestly had to think about that little bit. Last problem, the string, how about 27 feet, if you covered so 27 feet in four seconds? Now, listeners before you go off, I might say to a class. "Can you solve it both ways? Could you think about hey, we've got a bunch of stuff up here. Is there something up here? That's helpful. But what if what if you were cold? What if you know that you hit this problem on the street and didn't have anything up here? What might you do? Go ahead and solve it. Go ahead and solve it anyway, you want, just curious, go. Okay, I got six and three quarters feet per second. Okay. How?

Kim Montague:

So I did the half the feet and half the seconds thing. So I got 13.5 in two seconds.

Pam Harris:

13.5 feet in 2 seconds. In order to cut in half.

Kim Montague:

Feet and two seconds. And then I got six and three quarters feet in one second. And I actually broke the

Pam Harris:

So half the 12 is six, half of one and a half is

Kim Montague:

Yeah. 13.5 feet into 12 feet and one and a half feet. Because- point seven five. Right.

Pam Harris:

Nice chunks. Nice chunks.

Kim Montague:

Thanks.

Pam Harris:

I like it. Cool. So why would we do a string like this? Well, for many reasons. One, as a high school math teacher, oh, I just realized that for this entire time. I had been leaning away from my microphone. Sorry. I'm just gonna tell our listeners, our editor, who happens to be my son is really frustrated with me right now because I've been quieter for the last few minutes because every time we work problems I lean over on the problem, on the paper that I'm working on. And I just realized I was doing it. Sorry, Greg. So I'm sure, hopefully you didn't hear it podcast listeners, because he does such a good job of like altering the sound. Just okay, couldn't stop not, I couldn't stop saying that. So there you go. Craig. You're welcome to leave this in the podcast, not cut it out, anyway. So why would we, well, as a high school math teacher, I often get students who cannot think about rate. Like, even though they should have spent a lot of middle school time figuring out what rate means. By the time I get to them, if I say something like 40 feet in five seconds they have, they're like, "What?" I'm like, "How fast are you going?" And they're like, "What do you mean?" I'm like, "Well, in a second, how fast were you going?" "I don't know." And then it's not about whether they know what 40 divided by five is. It's literally that they don't know how to conceive how to find that unit rate. But also now you might, middle school teachers up in arms are like, "Pam, of course, my students can." I mean, when you've given them a formula, and you've told them what to do, yeah, they follow up pretty well, in that moment. But as soon as I get them, there's been so many more formulas thrown at them and all things. Really, we need them reasoning about rate, because it's not just can they find the unit rate here, but can they actually reason? So I've had occasions where we've said, "Hey, if you're going 20 feet in four seconds is that faster, or slower than 24 feet in four seconds?" And they're not sure. Or if I've gone 20 feet in four seconds, or I've gone 20 feet in five seconds. And they really have to think about it. Now some students have had some experience with rates. And they're not too bad. But a lot of students hear those kinds of questions as, "I don't know what I don't know what rule is supposed to do. Now, I don't know what numbers are supposed to plug in. Do I add or subtract, multiply, divide?" And you're like, "Aww! Think!" Like it's not any of those things. It's like thinking. And some of you might have been listening to that Problem String and as Kim was explaining, divide 40 feet by, or the five seconds by five and the 40 by five, you're like, "Just divide 40 by five." But she's actually reasoning, she's actually thinking through the problem making sense of what's happening. And that's what we need in students. So A there's a reason to do that Problem String. But B today, we wanted to give you a sense and a feel for why a double open number line. What t could look like. And so we actually have taken notes on today's Problem String. Oh, and we have written up what this looks like. And so you can download what this looks like at mathis FigureOutAble.com/double, for double open number line. So mathisFigureOutAble.com/double, you can download this Problem String or one like it and some extras, some questions to ask and what your board could look like with a double open number lines could look like. Because we realize this was all in the air and all verbal and we want to give you guys a chance to sort of see what it would look like. Why do we use double open number lines with students? It gives them a model on which they can operate. A way for them to think about how can I reason about what's happening with these numbers with the context in the situation, and it can be very, very helpful. Now I mentioned earlier that a double open number line is similar or has some similarities to a ratio table, just like an area model has similarities to multiplication and a ratio table. So similarly to how we want to use an arrow model to build spatial relationships and magnitude, then transition students with multiplication division, then transition students to be using a ratio table as the tool with which to compute. The same thing is true with proportional reasoning and a double open number line. I want to use a double open number line primarily where I as the teacher, I'm the one that's drawing the model. I'm representing what they're thinking with the model. I don't demand back from students very much. Because then I want to transition to having students actually use ratio tables to help them reason through most proportional reasoning problems. So just quickly as an example, if they were reasoning about that very first problem that we did 40 feet in five seconds. I would expect that eventually a student would write down sort of a ratio table looking thing where it would be 40 feet divided by or that division line over five seconds, and then they would create an equivalent ratio. And they would say, "Oh, well, so if I'm going to divide the five seconds by five," and I've literally written down like an arrow five seconds divided by five, that would be one second, then I would also divide the 40 feet by five, and that would give me eight feet. And now I literally have a ratio of eight feet per, over, divided by one second. So we would expect students eventually get there but they're, it's not because we told them to do it. It's because their reasoning about how they're finding an equivalent rate that's per one second. So in that way, double the number lines are a little bit limited, because let me say one other limitation. Another limitation of double open number lines is that we're sort of stuck with having to go in order. A ratio table is much more versatile, because we can scale down, scale up, scale down again, scale up again, scale down, down, down, scale up. And double open number line, we have to sort of do things, or we would encourage you if you're doing correctly to stay proportional. So when you've got that zero to four seconds, and you cut it in half, you literally cut it in half, and you put that two seconds in the halfway mark. Again, it's because we're developing that sense of space, and magnitude. And how the, then if we've cut that in half, if we've done that action of cutting the four seconds in half, oh, well, then we would also cut the feet in half. That's a way of kind of helping students develop the sense of ratio, and how those corresponding values multiplicatively, maintain their ratio relationship. Those are their equivalent ratio relationship. And that happens as we keep the distances making sense. But then that also limits the double open number line a little bit. It's not as easy to scale down, scale up and do all of the thinking that we want to do in a more complicated problem. I don't know, Kim, I hope that makes little sense. Because I'm doing it all up in the air, it's kind of hard to do it verbally. Cool. So double open number lines can also be very helpful in things like helping students develop the idea of a common denominator as we can use sort of course lengths to help them. We'll do that probably, I'm just going to mention that today. That in addition and subtraction of fractions, a double open number line can again become a really helpful beginning model for us to help students sort of nudge towards that idea of what they want to use it. If it's helpful to use it. So we'll probably do a whole podcast on that sometime soon. The last thing that I kind of want to end with today is a realization I had not too long ago, that a double open number line is a huge way for us to help students make sense of percents. Now, often, when we do percents, we use what we call a percent bar. But in almost all the ways a percent bar is just a double open number line where we've just let it be, we've let that double open number line, kind of have space in the middle to sort of resemble a download bar. If you think about downloading stuff from the internet, downloading a song on your phone, like however you're, whatever you're downloading, and there's this bar that sort of shows you how much you've downloaded. Well, a percent bar can look like that. But in a huge way, percent bar is just a double open number line where on the one side, today on our double open number line, we had time on the bottom and we had feet on the top. In a percent bar, you have percents on one side and you have then the corresponding values, whatever you're finding the percent of are on the other side. And you just reason the exact same way that we were just reasoning with a double open number line. And so that's kind of a cool connection. And in a huge way, that helps us understand that it's always 100%. That always is what goes on one side of the double open number line, or the percent bar. And then it's that other thing that goes on the other one, and it stays in proportion. That's the brilliance of that model. Cool. Because rates are so important. And we need students to understand this proportionality in rates, and we can use a model like the double open number line to do that, so that when I get them in the high school, they're actually owning the idea of rates. And so that's gonna be important to build in the middle school. But if you are a high school teacher, you could dump in a Problem String like today to help students reason about rates of change, for like, as you're going towards slope, the slope of a line, you could use. So you can use a Problem String like we did today in the middle school to develop rate. You can use it in the high school to quickly grab some development as you then move into the content that you actually teach.

Kim Montague:

Right, and you're meeting them where they are. Right?

Pam Harris:

Exactly.

Kim Montague:

And that's exactly what we have tried to do in Building Powerful Proportional Reasoning. We use Problem Strings to build ratios and rates. And also in your very brand new Building Powerful Linear Functions, which is super exciting. And we show them how to use them quickly right there as they're needed. So if you're a middle school teacher, you should check out Building Powerful Proportional Reasoning. And if you teach eighth grade and Algebra, you need to check out the brand new Building Powerful Linear Functions, because registration is opening very soon.

Pam Harris:

And I will just say, Building Powerful Linear Functions may be my best work. I'm so excited about it, and it's about to open up. We've been working on it for a while. I'm thrilled y'all just wait. Wait until you see it is coming. It's coming. Yeah, very exciting.

Kim Montague:

People have been asking for it and it's almost here.

Pam Harris:

So thank you 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. Let's keep spreading the word that math is Figure-Out-Able.