# Ep 111: The Open Number Line Model

August 02, 2022 Pam Harris Episode 111
Math is Figure-Out-Able with Pam Harris
Ep 111: The Open Number Line Model

The open number line is not just a powerful tool for computation, but it's an amazing model for developing all sorts of mathematical concepts! In this episode Pam and Kim discuss how to help students develop the open number line in a natural and meaningful way.
Talking Points:
What is an open number line vs a closed number line?
Are number lines, closed or open, discrete models and why does it matter?
Why shift to a measurement model vs a discrete model?
What are some common errors students make when we use the traditional way of teaching measurement?
We suggest a rich task by Cathy Fosnot to help students develop an understanding of the open number line through measurement
What can you model on an open number line? Elapsed time, addition and subtraction of whole numbers and integers and more
Is an open number line helpful for higher math concepts?
Check out Cathy Fosnot's amazing rich task:  https://www.heinemann.com/products/e01010.aspx

Pam Harris:

Hey fellow mathematicians, you're listening to the podcast where math is Figure-Out-Able. I'm Pam.

Kim Montague:

And I'm Kim.

Pam Harris:

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But y'all 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.

Kim Montague:

So in this episode, we want to focus on kind of one of our favorite things, the open number line model. You've heard us mention it before. But today, we're going to dial in to how to develop it, how we use it, and why we love it so much.

Pam Harris:

Kim Montague:

I was just gonna say I would bet a lot of third grade teachers know exactly what you're talking about as we start introducing non standard, I mean, sorry, standard measurement measure.

Pam Harris:

Uh huh.

Kim Montague:

Yeah. And especially when you start with something not at the zero. So like when you do activities, where there's a broken ruler or something like that. And it's really helpful to help kids make sense of talking about that span as the measure.

Pam Harris:

Kim Montague:

Yeah.

Pam Harris:

And then says, "What if I had this paper that was 19 cubes long? And what if I then wanted to, what if we were going to all year long create artwork that were two papers together? So what if I had this paper that was 19 cubes long, and I had a different paper that was 21 cubes long?" and she literally writes on the board 19 plus 21. "How long of a strip, a label would we need for that?" And students start to use this burgeoning, this beginning sense of what the open number line is to find those on the open number line and use. Now at this point, it's a bit of a closed number line, because they have all the cubes up there. And she's sort of using the strip where they've put some landmark numbers. It's the beginning of an open number line, and then it transitions over time, to our students are literally using relationships. And as they use those relationships, Hilde is modeling their thinking on an open number line. And then students begin to make the transition for themselves to be able to use that open number line as a tool for them to solve addition, and then later, subtraction problems.

Kim Montague:

Yeah, it's so brilliant. And I think that there were so many bits and pieces that Cathy embedded along the way, and you mentioned a couple of them, but not all of them, right? to

Pam Harris:

For sure.

Kim Montague:

Don't want to steal all the thunder. But I remember you being so excited about them using that strip as an open number line. And as you were describing all the things about it, I you know, I've seen a video multiple times as well. And I was remembering what Hilde looks like, and her working with her students, and like, I was picturing the whole thing again, as you were saying it. And I remember being blown away that second day when she came back and said, "We're gonna put the two papers together." And that was the moment for me where I was like, "Whoa, that is brilliant." Yeah, absolutely.

Pam Harris:

And when you say, "That is brilliant." It's the entire rich task.

Kim Montague:

Yes.

Pam Harris:

That starts with this scenario where are the kids learning about measurement? Absolutely.

Kim Montague:

Which they need, right?

Pam Harris:

Which they need? Yeah. But it gets to where students are not only building the open number line, but they're really building addition.

Kim Montague:

Yeah.

Pam Harris:

And earlier when I was talking about the name of the book, so it's Measuring for the Art Show. And the subtitle is Addition on the Open Number Line.

Kim Montague:

Yeah.

Pam Harris:

Because then at that point, we're all like, with our jaws open, "Oh, my gosh." This is how we get there. This is how we help students transition from that discrete one by one model, to this continuous model that is brilliant for modeling, addition, subtraction. So stinckin cool.

Kim Montague:

We would recommend, right, that at any point, that teachers have an opportunity to develop the open number line in such a way that that would be so much more meaningful to their students, and really help them understand the open number line, much more than just starting work on an open number line.

Pam Harris:

And it's about measurement. And so wherever you are, if you have older students and you're like, "Pam, we're not going to..." then hang back, lean on, emphasize the idea that you're measuring. In fact, at one point when they're talking about where to put the mark Hilde says, "If we wanted a paper that was 66 cubes long, where would we put the mark?" And you just see kids like light bulbs go off. And they're like, "Well, you would have to put it at the edge of the cube, it would have to, you know, or otherwise, you'd have 65 and a half cubes long, it wouldn't go to the whole length of the paper." And so that idea helps students really understand that it's that span, it's not the, I'm gonna put them tick mark at the edge of the cube. So that I get that entire length of the cube are all 66 cubes. It really helps students like, own this idea of measurement, which means, but which we need in order to use an open number line for lots of different things. So Kim, what are some of our favorite things that once we've developed this open number line as a model as a tool for thinking, what are some of our favorite things that we'd like to use it with?

Kim Montague:

For sure, one of my favorite things is for elapsed time, right? Three-five grade level is, you know kids are still wrapping their heads around time. And so elapsed time in third grade, I remember being a little bit of a struggle. And then -

Pam Harris:

A little bit?

Kim Montague:

Oh, my gosh, you know, you could put a start and end time or start and it's been this much time or an end time and how much was it previous to that. We could stick those amounts on a number line. And kids had had experience with addition, subtraction of whole numbers on an open number line, and it just made so much more sense to them.

Pam Harris:

And to be clear, you're actually putting the 24 hour clock on a number line. So it's not a traditional typical open number line. But we're literally, I guess I shouldn't say you're putting a clock on the number line, but you're representing times.

Kim Montague:

Right.

Pam Harris:

On the number line. Yeah, so you just like stick them down there. And then you can figure out what's in between them. And all of a sudden, bam, you've got elapsed time. Brilliant.

Kim Montague:

Yeah.

Pam Harris:

It's also very helpful for subtraction because we want to consider subtraction as both difference and removal. And you can model that so well on the open number line, not to mention the connection between addition and subtraction. We can do integer addition, subtraction, or even integer multiplication work as we think about something like five times negative six as five jumps have six in towards negative six. So we're going to the left on an open number line. Ssort of jump of negative six is a jump to the left and we have five of those jumps. Where would we land? Oh, we would land on negative 30. All the way, like I mentioned earlier to where it literally becomes a coordinate axis and all the time and higher math, we choose where to put the tick marks, I'm thinking of calculus problems where we have a function and we're trying to find the area underneath the curve, or we're trying to rotate a section of a curve and find the volume of that of the thing that we're creating. What am I trying to say? The surface area of the surface or the volume of the solid, there's the word I wanted. Volume in the solid that we create as we rotate things. Every time we do that, we only put tick marks where it's helpful, where it's handy. And that is the definition of an open number line. We're just putting two of them together, we have the x axis and the Y axis. And then we wanted to go 3d. We get the Z axis. I mean all the things that really is the precursor that all too often we just start the moment that we start graphing points, but we could start it in this way as a model. It is a measurement model started early to help students make that transition and really learn measurement at the same time learning addition, it's so so cool.

Kim Montague:

It's for everyone. Right? So you just mentioned-

Pam Harris:

Absolute.

Kim Montague:

I mentioned elementary, you mentioned middle school in high school. The open number line is for all Right?

Pam Harris:

Yes.

Kim Montague:

So if you are actually interested to learn more about how to help students learn, counting and counting strategies, and then transition to use an open number line, we have a super cool workshop called Building Addition for Young Learners. And this workshop is for teachers of young students, and for leaders pre K through second grade.

Pam Harris:

You will love it. So we do a lot of work in that workshop with young kids and you get to like notice video and all. It is an amazing thing. And we only open up registration for workshops three times a year, and that's coming soon. So if you're listening to this podcast when it first drops, that registration is opening soon.So hop on our email list so you'll know when that's going to happen. Keep listening to the podcast. We'll let you know when registration opens, but be planning that that's happening soon. 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 mathisFigureOu-Able.com. Let's keep spreading the word that math is Figure-Out-Able