# Ep 77: Relational Thinking

December 07, 2021 Pam Harris Episode 77
Math is Figure-Out-Able with Pam Harris
Ep 77: Relational Thinking

We've got another one of our favorite routines for you today! In this episode Pam and Kim demonstrate the Relational Thinking routine, a routine that helps students understand equivalence and become more comfortable with relational strategies. Thanks to the  Cognitively Guided Instruction group for introducing us to this routine!

Resources:

Pam Harris:

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

Kim Montague:

And I'm Kim.

Pam Harris:

And we make the case that mathematizing is not about mimicking steps, or rote memorizing facts, but 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 keeps students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?

Kim Montague:

So you know already that we love problem strings as a really cool routine. And if you were listening last week, you heard us talk about another great routine called As Close As It Gets. There are others that we love so much.

Pam Harris:

Nice. Now, they're not like an infinite number of others. Like there's a few that we really like. And some of them are kind of the ones that we've put our particular bent on, our sort of magic touch to. But we definitely like this one: Relational Thinking. And I'll give the CGI group credit for putting it in their Algebraic Thinking book where they talked about the meaning of the equal sign and got me really thinking about relational thinking problems. And now I use them a lot. And so Kim, you get to be on the hot seat. I'm going to ask you some questions. The point of relational thinking is that you give students an equation, and there's a blank in the equation, and when you give them the equation, so Kim, when I give you this equation today, I do not want you to do a lot of computation. I for sure don't want you to use equation solving strategies. I really want you to use relationships among the numbers to reason about the number that's missing. Okay?

Kim Montague:

Okay. Yep.

Pam Harris:

All right. So the first one: 15 x 18 =30 x __. Let me repeat that. 15 times 18 equals 30 times blank.

Kim Montague:

All right, okay. So I'm thinking that I have twice as many of some things. So 15, twice as many of them would be 30. So then I need half as many as the size of the things. And I don't know that I said that so well. I'm going to let you clean up my language. And I'm going to call this 30 times nine. Because if I have twice as much, twice as many, 30, then the size of those things can only be half as big. So nine.

Pam Harris:

So we can use groups here if I have 15 groups of 18 -

Kim Montague:

Yep.

Pam Harris:

but I'm telling you that's equivalent to twice as many groups, what would have to be the size of the group to have the same number of things? A way to say that. Another way to say that, you and I were both really general, another way to say that

could be:

if I have 15 eightteens, I'm going to need 30 nines to have as much as I want. If I have 15 eightteens, I'm going to need - and I have twice as many... I can't say twice as many eightteens, but I'd need twice as many groups that are half as big. Yep. Okay, cool. So you're filling in the blank with nine. All right, nice, nice. Okay, cool. Next one. Six point one, or six and 1/10, minus 2.9, or two and nine tenths equals blank minus three 6.1 - 2.9 = __ - 3.

Pam Harris:

Totally just do that on my paper.

Kim Montague:

So I've thought about this before. So here's Yep, me too. So if I think about the gap between what I'm thinking. Not about this particular problem. But I those two numbers, I don't really like those. I'm going to know that if I'm going to take away - I want to take away something nicer. So I think about the actual distance actually shift that slightly to the left, like 1/10. between 6.1 and 2.9 on a number line -

Pam Harris:

To the left or to the right?

Kim Montague:

Well, I wrote it like 6.1 minus 2.9. So if I put it on an actual number line then I'm gonna shift to the right. I have the problem written down on my paper.

Pam Harris:

Gotcha.

Kim Montague:

So that's going to be 6.2 minus three, because the gap between 6.2 - 3 is the same distance as the gap between 6.1 and 2.9.

Pam Harris:

Nice. Yeah. So I just wrote on my paper 2.9 on an open number line, and then off to the right, I wrote 6.1. And then I thought about the relationship between that 2.9 and the three and the three is to the right of 2.9. So I knew if I was going to shift that distance up just that tenth, then it would shift 6.1 up, I could say sort of that jump, I kind of have the jump drawn between 2.9 and 6.1. And then I shifted that jump up to three and 6.2. And that's yeah, that looks just like okay, sorry, I had to say out loud what I'd written on my paper. Cool. All right, bring it. Here's the next one. What about 3999 plus blank equals 4000 plus 2587. 3999 + __, plus what, = 4000 + 2587.

Kim Montague:

Okay, so I'm actually looking at the difference between 3999 and 4000.

Pam Harris:

Okay.

Kim Montague:

And so I'm thinking about, I'm thinking about moving marbles actually. And I'm thinking about if the left side, sorry, the right side of the equation, the 4000 part. That amount, I want to be equivalent to the amount on the left side of the equation. Right. That's what the equal sign means. And so I'm gonna have 2500 - I'm so distracted. So sorry.

Pam Harris:

It's raining. Just so y'all know.

Kim Montague:

It's thundering.

Pam Harris:

We're recording a very rainy, gray, overcast, loud thunder, lightning. Oh, my goodness. Fun recording day.

Kim Montague:

Fun recording day. Okay. So I -

Pam Harris:

Kim, do you need time or help?

Kim Montague:

No, I shouldn't need either at this moment. I wrote something down. Okay. So I - what did I just write down? I put down 2586, sorry, is what I wrote down.

Pam Harris:

You think it might be 2586. Because?

Kim Montague:

Because I have- did I write the number down? Say the numbers to me again.

Pam Harris:

3999 plus blank equals 4000 plus 2587.

Kim Montague:

Oh, I wrote the wrong number. Say it to me one more time.

Pam Harris:

3999 plus what equals 4000 plus 2587.

Kim Montague:

I wrote the number downk wrong. 2588.

Pam Harris:

Why?

Kim Montague:

Because I have one more marble in the pile on the - I'm looking at the right side of the equation, the 4000. And as I move one marble to the right, from the 3999 to the 4000... So I'm gonna move one marble...

Pam Harris:

You're not liking this one are you?

Kim Montague:

No, I'm not. Why am I so distracted? Okay, I'm thinking. I know that it's 2588. I'm having a hard time verbalizing.

Pam Harris:

I mean, that's huge, right? That's like a real thing. That's like a for sure thing. Once you have that 2588 written down. And you look at the 3999 plus that 2588? What would you -

Kim Montague:

Oh because I'm trying to move marbles across the equations. I want to move a marbel from the 2588 to the 3999 to make it 4000 plus 2588,

Pam Harris:

This is the devil problem for you.

Kim Montague:

Oh, my goodness. Yeah.

Pam Harris:

That's interesting. Because for the moment, you just sort of got caught up a little bit in moving marbles across.

Kim Montague:

Yes, I did.

Pam Harris:

But that's not sort of helpful if you're really just trying to solve the problem 3999 plus 2588. If that's the problem you're trying to solve, then you're really just moving marbles between those two numbers.

Kim Montague:

And all these listeners right now are screaming saying, "this is what it is!". And I love that you asked me, do I need time or help. Because yelling numbers at me. Clearly I'm not writing them down right when you say them the first time, so yelling numbers at me won't be helpful.

Pam Harris:

So at the risk of you, like being done, can I give you one more?

Kim Montague:

Sure, absolutely. I'm in for it.

Pam Harris:

Alright, you can do it. This time the blank is first. So blank divided by 16.

Kim Montague:

Some number divided by 16. Okay.

Pam Harris:

Equals 240 divided by eight.

Kim Montague:

Okay, some number divided by 16. So I'm thinking about how many 16s are in that number, whatever the unknown number is, is equivalent to how many eights are in 240. That's kind of what I'm thinking about.

Pam Harris:

I love that in so many ways. That's not how I thought about it. But how does that help you?

Kim Montague:

So there's some eights in 240. But this time, the group size is twice as big. So I need the amount to be twice as big to maintain the equivalence.

Pam Harris:

To maintain the equivalence. I'm curious, you did that with group size, could you do it with the number in the group? Like if eight was the number in the group?

Kim Montague:

How many eights are in 240?

Pam Harris:

Is that? Yes, yes.

Kim Montague:

How many eights are in 240? Uh huh.

Pam Harris:

Then how do you? How does that -

Kim Montague:

Is that not what I said, How many eights are in 240?

Pam Harris:

Well, is that the number of groups? Or is that the number in the group?

Kim Montague:

How many eights are in 240?

Pam Harris:

That's the number in the group. Yeah. Okay. So you said if there's eight -

Kim Montague:

There's some eights in 240, I think is what I said. Then I need to have that same number of sixteens in some other amount.

Pam Harris:

And that would have to be-

Kim Montague:

480

Pam Harris:

Twice as big. Yeah. So what I was wondering is, if you thought about, I don't know, I don't know if it's possible. If you thought about like there was eight groups in 240.

Kim Montague:

Hmm, I see what you're saying.

Pam Harris:

But now I've made 16 -

Kim Montague:

The group size has doubled.

Pam Harris:

No, no, the number of groups. I've got eight groups in 240. So if I make 16 groups, what total number would give me the same number of people in those groups? I don't know if I just said that right.

Kim Montague:

I feel like we're saying the same thing. I was thinking about how many eights are in 240. So I need that same number of sixteenths in 480. And I think the opposite, or the other way of thinking about that would be, I have eight of some size in 240. And I need the same amount. But now I'm thinking about the size of sixteens.

Pam Harris:

Yep, that works for me. So what if I also thought about it as equivalent ratios. So if I could picture that, as - I said the symbol divided by, but if I put sort of a blank divided by like over, I'm just saying so how it looks like on my paper, like out of 16 and then I had 240 out of eight, then I can kind of think about equivalent ratios. And to get from the eight to 16, I could double and to get me to 240 I would double the dividend.

Kim Montague:

Yeah. I need to get myself to record division more in that way. That's a thing I'm going to work on. Because I do like that.

Pam Harris:

I like it. I like it. So another way that I'm thinking about it that I hadn't thought about until right now, is what if I was thinking about dimension and area. So it's almost like I'm saying I've got a dimension of 16, I just drew like a side length of 16. And I don't know what the area of the rectangle is. And that has to have the same side length. The other missing side length has to be the same as if I had a side length of eight, and an area of 240.

Kim Montague:

And you can even consider that - I just sketched out what you were saying - you can even consider that without knowing the other dimension. I just have a kind of like a line of one dimension, that's just kind of doesn't end. And I have eight and 240 and 16 and some amount. And I can tell that because the eight doubles to the 16, then whatever the area is, is going to have to double as well.

Pam Harris:

It's gonna have to double. It's almost like you take the eight by whatever is 240. And then you literally, like copy that rectangle and put it next to itself to get the 16 by double the area. And it's all based on the fact that we have to keep that unknown dimension the same.

Kim Montague:

it's a really nice routine for cementing equivalence.

Pam Harris:

Yeah, because the equivalent sign, the equals sign, means equivalence, it doesn't mean do it. All too often we get students who see an equal sign, and they're like, Oh, now I must do the thing. Whatever comes before it, they sort of do. So like, for example, that first problem 15 x 18 = 30 x what. They might say, No it doesn't. Like 15 times 18 is not 30. And to which we want to say no, no, like the equal sign means equivalence, you've got to figure out what goes in that blank to make what's on this side of the equal sign equivalent to what's on the other side of the equal sign. And that can help sort of build this idea of what the equal sign means. I think there's another thing that this particular routine that we call Relational Thinking, I think there's another really nice thing that it can do, which is help sort of cement the equivalence strategies. So for each of the four major operations, there's a strategy that is based on

equivalence. It's based on:

how can I find an equivalent problem that's easier to solve? So for example, if I was in addition, a give and take compensation strategy is: oh I've got this nasty problem, kind of like the one you did 3099 plus 2588, you're like, yuck, how could I solve that easier? Well, I'm just gonna move the marble over and make it 4000 plus 2587. That's an easier equivalent problem. So you just solve the easier one. Similarly, for subtraction, the constant difference strategy, which we actually used to solve 6.1 minus 2.9, we used that, we thought about the difference between the two numbers and we shifted it up, we kept the distance constant to find an easier problem to solve. We're going to subtract three instead of subtracting 2.9. For multiplication an equivalence strategy could be doubling and halving. So when you double and half, you're creating an equivalent problem, like 15 times 18, is equivalent to double a little bit, double the 15 to 30, then I've got to halve the 18 to nine. I'm finding an equivalent problem that's easier to solve. And then for division, it's the equivalent ratio strategy. Can I turn that division problem into a ratio, find an equivalent ratio that's easier to solve and then solve that problem? So this routine, Relational Thinking, can be really helpful to sort of cement those equivalent strategies. And we like that. So if you're interested to see where we've got these, Kim, tell us about where they can find Relational Thinking instructional routines.

Kim Montague:

Absolutely, you can go to mathisFigureOutAble.com, and check out the 'Learn Now' section. If you go down to 'Learn Now', then you will find the Instructional Routines Hub that has all kinds of relational thinking. You can click on each one, find out what people have said about them on social media. You can also find the link in the show notes that's bitly/instructrout.

Pam Harris: