Math is Figure-Out-Able with Pam Harris

Ep 130: The Ratio Table Model, Pt 2

December 13, 2022 Pam Harris Episode 130
Math is Figure-Out-Able with Pam Harris
Ep 130: The Ratio Table Model, Pt 2
Show Notes Transcript

We can't get enough of ratio tables! In this episode Pam and Kim go through another division Problem String, highlighting how they model-represent their thinking on ratio tables. They invite listeners to grab a pen or pencil and practice representing their own thinking on a ratio table.
Talking Points:

  • Noticings from MathStratChat
  • Does context matter?
  • Does a ratio table have to be in sequential order?
  • Does it matter when to erase during the Problem String?
  • Does it matter when to use brackets or arrows?
  • Can color help highlight relationships?
  • How do you introduce a ratio table?
  • A tool to represent thinking that can lead to new thinking
Pam:

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

Kim:

and I'm Kim Montague.

Pam:

And you found a place where math is not about memorizing and 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.

Kim:

So, last week, we started talking about one of our very favorite models of all time, the ratio table.

Pam:

Favorite, favorite. Yep. Yeah, the thinking that we were using.

Kim:

Yeah, or your own. Yeah. And we chatted about the purpose of a ratio table as a model, and how it's useful to help represent thinking as we and students navigate relationships between numbers. In the podcast, we described a little bit about what our ratio tables looked like and shared those to social media. So, this week, as we work through some problems, we'd like to encourage you to grab a pen or a pencil and a piece of paper and practice recording yourself. We would love it if you'd share that to social media as well. And you can record what you did to solve the problems, or record the ways that we use problems to solve.

Pam:

Or, the thinking you used. Yeah. Ya'll, do it with your classes. We'd love to see your board that ended up in doing this Problem String with your students. Or if you have students that are solving problems using ratio tables, and you give them some of these problems, we'd love to see their thinking. Let's start flooding social media with ratio tables. How many of you listening today have found yourself reasoning differently because of ratio tables? So, many people are starting to say things like, "Whoa! I'm looking at that completely differently, now that I'm using ratio tables as a tool." So, let's flood it. So, like she said, grab a pen and pencil, record your thinking, and post it. Let's start showing everybody. Let's give everybody this inroad, this open access to the kind of thinking that mathy people do because we can all be mathy people, now that we know it's a thing,

Kim:

You know, what? I actually am going to say something right here, that when we're looking at the strategies that people are using to solve MathStratChat problems, I find it so fascinating that I gravitate towards the ones where people did their strategy on a ratio table, or with some sort of model, rather than just a paragraph of words. And I don't. You know, I don't know why that is?

Pam:

Oh, I do.

Kim:

I just find it so intriguing to try to make sense of what they were doing. And some people are super thoughtful about the moves that they make, and I just love looking at them.

Pam:

To me, it's so much... What's the right word? Not "easy". It's so much more clear. Like, I don't have to sort of sift through their words. I can just see the relationships.

Kim:

Yeah.

Pam:

Yeah. No, ratio tables are so powerful. It's a powerful,

Kim:

Yeah. Okay, Pam.

Pam:

Hey, before we move on from that. visual way to display the relationships people are using as their reasoning through problems. Yeah, no, absolutely. Yeah.

Kim:

Yeah.

Pam:

So, if you want to get your strategy chosen (unclear) ours, that we highlight in a post that next week, then...Hint, hint...you might want to use a ratio table.

Kim:

Yeah.

Pam:

That's a good way to get it chosen. Alright, cool.

Kim:

That's awesome. Okay, so today, I'm going to give you some problems to solve, right? Alright. And you're going to sketch out on a ratio table how you're thinking, so that we can share...

Pam:

We'll post that.

Kim:

...later what you did. Okay, cool. Alright. So, today. I'm going to give you some problems, and I want you to think about the context of water bottles. If you had 120 water bottles. So, tons of water bottles. And they came in packs of 12. How many packs would you have?

Pam:

Okay, so I just am thinking about packs of 12 and water bottles. And so, I've written. I just spelled "water" with two t's. What's wrong with me? Water. Okay, hang on. We're going to take a picture... No, it'll look terrible. It'll look terrible. Okay, water. I know because I was thinking about bottles. That has two t's. For heaven sakes. Tell me the numbers again. I got stuck on.

Kim:

120 water bottles in packs of 12.

Pam:

Packs of 12. So, I'm thinking about 12 water bottles

Kim:

Mmhmm. in a pack. Mmhmm.

Pam:

And then, I'm going to scale that to 120 water bottles. So, I'm going to go ahead, and... I wrote 1 pack to 12 water bottles. So, times 10. So, that would be... 10 packs is 120 water bottles, so that 120 divided by 12 is 10.

Kim:

10 packs. Okay.

Pam:

10 packs. Yeah.

Kim:

Cool. What if I asked you for 240 wattle bottle. Wattle. Wattle bottle.

Pam:

Hey, I can't spell it, and you can't say. That's excellent. So...

Kim:

240 water bottles.

Pam:

So, 240, I put in the water bottle column. And I'm thinking that's double 120. And so, double 10 packs is 20 packs. And so, I've just written times 2 from the 120. And times 2 from the 10 to the 20.

Kim:

Okay, cool. What if you had 360 water bottles?

Pam:

360 water bottles. So, I can think of two ways to do this. I already have the 120 and the 240 water bottles. If I add those together, that's 360. And I was kind of actually using sort of the 12 and the 24

Kim:

Mmhmm. to get 36. Like, twelve 10s. Is 120. Oh, it's a 12 packs actually, 12. I can think of it as twelve 10s, but I'm thinking about 12 packs as well. So, 12 packs plus 24 packs is 36 packs. Wait, is that right? Wait, no. 10 packs (unclears).

Pam:

No, that's 10. There's not 10 water bottles in a pack. There's 12 water bottles.

Kim:

Yeah.

Pam:

That's where I got messed up. So, I do have to think about it as twelve 10s. But anyway, so thinking about the 120 and the 240 to get 360 water bottles would mean that I would have 10 plus 20 packs, which would be 30 packs. But the other way I was thinking about it was to get from 12 to 36, I could scale times 3. And so, that would be 3 packs. And then, I could scale up from the 36 to the 360 times 10.

Kim:

Yeah.

Pam:

So, that would just be another way that I could. And then, that would be 3 times 10 would be 30.

Kim:

Nice.

Pam:

Okay.

Kim:

Okay.

Pam:

And let me tell you, I've been writing off to the side, so that problem that you gave me was 360 divided by 12. And I'm saying that it's 30. So, I'm actually recording some division problems off to the side.

Kim:

Oh okay. Yeah, you know, sometimes I do that, sometimes I don't. I don't know if there's a rhyme or reason to that. Alright. Next problem.

Pam:

Maybe there should be... I mean...

Kim:

They're should be probably, right? Alright, what if you had 1,200 water bottles?

Pam:

So, 1,200, I stuck in the water bottle column. And I could scale from the 12 times 100. So, that would be 100. I'd scale the 1 times 100. And that would be 100 packs is 1,200 water bottles.

Kim:

1,188 water bottles. How many packs? Nice.

Pam:

1,188. So, I've got that in the water bottle column, and. Sorry, I'm just actually writing the division problem I forgot to Okay. write down. Okay, so 1,188 water bottles. What do I know? Well,

Kim:

Alright. What about 600 water bottles? How many packs? it's really close to the 1,200, so I'm asking myself how close. And hey, it's just 12 water bottles, which is just 1 pack is 12 less water bottles than the 1,200. And so, it's 1 pack less than the 100. So, 99 packs. In other words, 1,188 divided by 12. Is 99 packs.

Pam:

So, I put 600 water bottles in the water bottle column. And I'm looking. What else do I know? So, I can think about getting from 12 to 600. I could think about going from 120 to 600. The 360 to 600 does not make me happy, so... Oh, unless I added it to the 240. That would work. So, one thing I could consider would be adding the 240 to the 360 to get 600. Therefore, I'd add the 20 plus 30 to get 50. But I actually don't love that so much. I actually then kept looking. And I looked at the 1,200. So, 1,200 water bottles was 100 packs. If I divide the 1,200 in two, I would get the 600 water bottles. So, divide the 100 and 2 to get 50. But now that I'm doing that... No, that's probably what I would do.

Kim:

Yeah,

Pam:

Yeah, that's probably what I would do.

Kim:

Earlier, you said something about the...

Pam:

No, I do have one other. Can I just (unclear).

Kim:

Yeah, go ahead.

Pam:

I'm thinking about. I didn't used to know this, but I do now. I'm thinking about 12 to 60 because (unclear).

Kim:

Yeah.

Pam:

(unclear) 600.

Kim:

Yep.

Pam:

So, if I think about 12 to 60, I did not used to know that 12 times 5 was 60.

Kim:

Mmhmm.

Pam:

But I'm kind of confirming it in my head right now. Because I've just already figured out that 12 to 600 was 50. So, then, 12 to 60 would be 5, but I could have started there. I could have said 12 times 5 is 60, so then 12 times 50 would be 600.

Kim:

Nice, and that's what I was just about to ask you about. Very nice. What if I asked you about 576 water bottles, but this time they come in packs of 24.

Pam:

Say the number again? 576.

Kim:

576 water bottles, but this time they come in packs of 24. divided by 24.

Pam:

576

Kim:

Mmhmm.

Pam:

So, I'm going to think about 576 divided by 12, and then I'm going to reason about how those are related. I think. I think that's what I want to do.

Kim:

Okay.

Pam:

Because I've already been dividing by 12 in this whole thing. The packs had 12 water bottles. So, I'm asking myself how far 576 is from that 600 we just had. And if I have 76, you need 24. So, 576 is 24, water bottles away from 600, which is 2 packs. So, that would be 48 packs. So, 576 divided by 12 is 48. But you just said, now you want to have the packs have 24 water bottles in it. Because you're asking me to think about 576 divided by 24. So, if the packs have 12 water bottles, and there's 48 packs. Now, I have packs that are twice as big, I'm going to have half as many. So, I think it's 24.

Kim:

Nice.

Pam:

Oh, and 24^2 is 576.

Kim:

Nice. So, talk to me a little bit about your ratio table when the number of water bottles in a pack changed. Did you write it on the same ratio table?

Pam:

I did not.

Kim:

Yeah, did you... You left the... It's almost like you had to jump off to another ratio table because the size of the packs changed.

Pam:

Mmhmm.

Kim:

Yeah. And so, then, if you kept it on the same ratio table, then that wouldn't be a ratio table, because then they wouldn't be equivalent. They wouldn't be proportional.

Pam:

Yeah, because like right now, I have 1 pack to 12 is equivalent to... In fact, if I just may. So, we mentioned this last week in last week's episode that a ratio table is a table where all of the ratios are equivalent. Let me just list those. So, in this particular case, the 1 to 12 is equivalent to 10 to 120 is equivalent to 12 to 240, equivalent to 30 to 360, which is equivalent to 100 to 1,200, which is also equivalent to 99 to 1,188. And I could keep going, but all of those ratios, in all of them, there was always 1 pack to 12 water bottles. Until at the very end, you threw a wrench in it. Thanks a lot for that. And you said changing the size of the pack. And so, now, I've got 1 pack to 24. That's no longer equivalent to everything else that I have in the ratio table, so I had to get out of that ratio table to be able to think about that. But it was a super nice problem because I can use what I had there to think about, I'm actually a little curious to go. Like, what if I had just gotten that problem naked? Like, cold. Like, without the beginning problem? So, I might have been tempted to say 1 pack has 24. So, I have a new ratio table. 1 pack to 24 water bottles. And then, I'm asking myself, "How do I get from that 24 to that 576?" And I'm thinking to myself. So, I'm going to talk out loud, even though I would have normally sort of done this kind of quickly. Times 10 would be 240. That's not enough. Times 2. 24 times 2 is 480. That gets me kind of close. So, times 2 was 48. So, times 20 would be 480. And then, I'm kind of asking myself how far apart they are. And 480 to 500 is 20, plus 76 is 96. So, then I'm thinking, "What do I know about 96?" Ooh, bam! I already had the 2 packs to 48 water bottles, and 96 is twice 48, so that must be 4 packs. And so, then, I can think about the 480 plus 96 is the 576. So, 20 packs, plus 4 packs, is 24. So, that's a different way of getting 24.

Kim:

Yeah. I love strings like this where kids can think about relationships that they can make use of along the way. So, you know, you use the 120 to help you with the 240. Or the 1,200 to help you with the 600. But I love when there's kind of a clunker at the end, where you are asking, "Are there any relationships that you could use? Or can you use the ideas that we've been tinkering with?" I thought for sure that you were going to say that you were going to go back to the 1,188 divided by 12. Because in the very last problem, you're asking for half as many water bottles, and the size of the pack is twice as much. So, maybe we don't want to say quite yet. I mean, you might want to, but I just want to throw out there that there's a relationship there that might be nice to

Pam:

you want to think about 576 divided by 24?

Kim:

Yeah.

Pam:

Based on what we did from 1,188 divided by 12?

Kim:

Yeah.

Pam:

Ah, that's a... I wonder how we could ask that in a MathStratChat? I might have to think about that.

Kim:

Okay.

Pam:

Like, how? That will be interesting, right? How does this problem relate to that problem?

Kim:

Yeah.

Pam:

Yeah, cool. So, let's talk about a few things that maybe are noteworthy when we use ratio tables for division, which is kind of what we just did. When you gave me the total number of water bottles in a pack, and then, you said, "So, what if we have like a lot of water bottles? How many packs?" So,

Kim:

Mmhmm. you're giving me number of water bottles, and you're asking for the number of packs. You're giving me the total amount. We know the number in a group, and you're asking me for how many groups.

Pam:

And so, that's one of the meanings of division. And so, these were division problems. And we're using a ratio table to sort of think and reason using multiplicative relationships. What are some of the things that we think about when we use ratio tables to solve division problems? One thing you might have noticed that we just did is we did it in context.

Kim:

Yeah.

Pam:

And Kim and I actually had discussed, "Do we want to do this in context?" And I said, "Yeah. Because, like, if we have anybody listening that is not adept yet, with using ratio tables, then doing it in context should be very helpful." You can start to reason about what's legal to do in the ratio table. Another thing you might have noticed in both last week's episode and this one is that the tables don't have to be in sequential order. So, if you looked at my table right now, it's got... I'm just going to read the number of packs. 1 pack, 10, 20, 30, 3 100, 99, 50, 5, 48. Like, it's definitely not an order, and you don't have to go in order. You just have to keep the ratios equivalent. When I did the 50 packs. When you gave me 1,200 water bottles, and then you gave me 600 water bottles. So, we had 100 packs, and then. I was able to cut that in half. The first time that students do that, often they'll say, "Well, half the number of water bottles. So, half the number of packs." And so, on my ratio table, you'll see that I wrote "divided by 2". And I think "divided by 2" is often a way that we talk about halving something. Cut it in half, we divide it by two. But eventually, we're

Kim:

Mmhmm. going to also want to say things like times one-half, and make those kind of an equivalent way of thinking about that. And so, as a teacher, I'm going to be kind of purposeful about doing "divided by 2" first, and then start gradually introducing multiply by one-half to represent that, and then kind of switch up between those. What are some other things that you think about when you are thinking about ratio tables? And with multiplication and division? Well, as I was saying last week, that the purposeful time to erase really matters. So, you know, I bet your ratio table is a little bit messy.

Pam:

It is because I'm writing in pen.

Kim:

Oh, well, that's the problem. So, you know, I think sometimes you have to be really thoughtful about when you want to erase, when it matters to keep the relationship connection that's on the ratio table there. Are kids going to come back and use that relationship again? Or is it one that you've moved on from for a short bit that you can erase and just kind of clean things up a little bit. We tend to write sometimes with arrows and sometimes with brackets. So, when we're scaling times 10, or divided by 10, we

Pam:

Mmhmm. tend to use arrows. And when we're adding or subtracting, we tend to use brackets. Just for making sense in my mind, I don't know that there's a mathematical reason to do that. You can say if you think there is. No, Yeah. It totally started when I was writing Building Powerful Numeracy. The editor asked me very nicely. Thanks, Catherine. She's like, "Might you want to use different symbols, so that the reader at a glance can tell when you're multiplying?"

Kim:

Yeah.

Pam:

Like you said "scaling". Or when you're, like, adding packs together, adding the number of water bottles in 2 packs plus the number of water bottles in 8 packs. Those we tend to use brackets. And it's kind of a visual way of just helping the reader, or whoever is discussing, looking at the model kind of just to be clear on which relationships we're using. I don't know that that's all that important. And we certainly don't demand it of students.

Kim:

Yeah.

Pam:

Right? We let students... As long as students are kind of clearly notating what they're doing times 2 versus plus this 1, plus that, then we're not very picky about how students do that. But with teachers, I think it can kind of be helpful. And you'll recognize what I was doing in the books, and on the website, and anytime you see me sort of represent things that are ratio table, then you... Yeah, you can kind of recognize that.

Kim:

I think also one other thing is that we often will see a little bit of work off to the side. So, last week, we described that we were doing some addition or subtraction within the problem. And that will be kind of like a side note. Not in the midst of the ratio table, but just to kind of a little bit off to the side. Often, for me, it's smaller. I don't know why. Like, "Side note, I'm doing this addition and subtraction also." Just to continue the thinking on paper or on the board, rather than try to hold stuff in my head.

Pam:

Yeah, totally. One other last thing I'll mention is I've seen some teachers do a decent job of using color. That as they're... The problems are kind of represented with one color. You know like, the 1 to 12, or the 10 to 120 are all written in one color. But as they scale, or they add the packs together, that's in a different color. It kind of helps it stand out. And then when you're, like we said, when you're done with those relationships, you kind of erase it. And so, you end up with your final board, final display is really just the numbers in the ratio table. (unclear) the scaling marks, the brackets, and everything. That stuff kind of goes away. Maybe even your open number line where you're doing the off to the side addition, subtraction. That what's important is the equivalent ratios, that sort of what stands out at the end of the ratio table. Yeah, that's going to help.

Kim:

One last thing I want to mention this week as we're talking about ratio tables is sometimes people will say to us, "My kids aren't familiar with ratio tables. How do I introduce them? How do I get started?"

Pam:

"How do I teach?"

Kim:

Yeah, "How do I teach ratio tables?" And I think the answer to that may be unsatisfying, maybe to some. Because I'm just going to say, you just start. So, we would recommend...

Pam:

Or, maybe scary to some.

Kim:

Right, okay. That could be true.

Pam:

Intimidating, maybe?

Kim:

Yeah. So, we recommend that you just start. You choose a Problem String that builds a ratio table. Or, just ask kids what they're thinking about. And there's no formal, "Today, boys and girls, I'm going to teach you how to write on a ratio table." We just start pulling thinking out of kids and represent it and say, "Hey, I'm going to represent your thinking this way. When you do that in your head, this is a way that we can represent that on paper to make sense to all of us, so we know what's going on in your head."

Pam:

And the student can also say, "Oh, when my brain does that, it could be like that. Huh? Okay. Alright. When my brain does that, it could look like that. Okay" Yeah. Nice. So, I ran across a quote lately. I'm

Kim:

Yeah. actually reading a fascinating book about visionaries and some other things. And in the book, Rocket Fuel, they quote Buckminster Fuller. Who, I didn't really know who that was, but he was an American architect, system theorist, writer, designer, inventor, philosopher, and futurist, which I don't know what that means, either. But sounds like an all around cool guy. He invented the geodesic dome. So, anyway, random. But he said a quote that I find kind of interesting that sort of plays here. He said, "If you want to teach people a new way of thinking, don't bother trying to teach them. Instead, give them a tool, the use of which will lead to new ways of thinking." Nice.

Pam:

May I suggest that a ratio table is that kind of tool. That if we want to teach multiplicative reasoning, really kind of a simplified version of proportional reasoning, and we want to get people really reasoning multiplicatively and reasoning proportionally, then it's not about, "Okay..." Like you just said. It's not about, "Alright, today, we're going to learn the ratio table, and Step One, Step Two. And it's got to look like this." Instead, give them a tool. Give them the ratio table, represent students' thinking. The use of which will lead to new ways of thinking. Teachers, this could be a tool that could lead to your new ways of thinking, that then you can represent. (unclear). Not sequentially, but as you do that, students can then be led to new ways of thinking, as they use the ratio table as a tool to think with, to actually reason with. Super, super cool. Alright, ya'll, thanks 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!