October 24, 2023
Pam Harris
Episode 175

Math is Figure-Out-Able with Pam Harris

Ep 175: The Most Sophisticated Strategy for Division

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Math is Figure-Out-Able with Pam Harris

Ep 175: The Most Sophisticated Strategy for Division

Oct 24, 2023
Episode 175

Pam Harris

Division is figure-out-able! In this episode Pam and Kim discuss the most sophisticated strategy for division and precursors that students can begin developing in the fourth grade!

Talking Points:

- An Equivalent Ratios Problem String
- Precursor relationships
- Scaling up or Scaling down
- What makes Equivalent Ratios the most sophisticated strategy for division
- 4th grade can begin to reason about division using the "bigger, smaller thing" to reason about division
- 5th grade can begin to reason about equivalence
- Relational thinking

See Ep 64 and 132 for more on equivalent ratios!

Checkout mathisfigureoutable.com/relational-thinking for free exercises to develop the equivalent ratios strategy..

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

Listen on

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Division is figure-out-able! In this episode Pam and Kim discuss the most sophisticated strategy for division and precursors that students can begin developing in the fourth grade!

Talking Points:

- An Equivalent Ratios Problem String
- Precursor relationships
- Scaling up or Scaling down
- What makes Equivalent Ratios the most sophisticated strategy for division
- 4th grade can begin to reason about division using the "bigger, smaller thing" to reason about division
- 5th grade can begin to reason about equivalence
- Relational thinking

See Ep 64 and 132 for more on equivalent ratios!

Checkout mathisfigureoutable.com/relational-thinking for free exercises to develop the equivalent ratios strategy..

Check out our social media

Twitter: @PWHarris

Instagram: Pam Harris_math

Facebook: Pam Harris, author, mathematics education

Linkedin: Pam Harris Consulting LLC

**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 students to think and reason like mathematicians. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keeps students from being the mathematicians they can be. What are you doing in the background?

**Kim **

I'm sorry, I'm sorry.

**Pam **

I'm like, are you walking around?

**Kim **

No, no, no. Well, I just hit my glass of water. And also, I was thinking about what day it is because we're going to be traveling super soon. Like, I don't even know what day you go. But it's almost time for conferences.

**Pam **

Whoa!.

**Kim **

Alright.

**Pam **

Have fun.

**Kim **

Oh, I know. Okay, so hey, we got a review. I don't know. I hate calling them a review when it's somebody just like sharing their story. So, I don't know. It's a share. Anyway, but the title of it was "Long time listener, first time reviewer."

**Pam **

Oh!

**Kim **

Do you leave reviews like Yelp or whatever? I never do.

**Pam **

So, I've left like two reviews in my life.

**Kim **

Yeah.

**Pam **

And I felt really, really strongly about when I did. I guess I should probably try to do that more. Yeah.

**Kim **

Yeah. Okay. Well, anyway. This is from JayYC. And I I don't know. They said, "I've listened to the podcast for a few years now. And just when I think there's no more topics to cover, Pam and Kim keep coming out with more and more great topics."

**Pam **

Oh, nice!

**Kim **

I think there's no shortage of things we'll talk about.

**Pam **

We're not bored of each other yet.

**Kim **

Yeah. So, "One thing I most appreciate about every episode is the very personable and authentic dynamic between Pam and Kim. It feels like you're in a room with your favorite colleagues chatting about math. Just when I think I figured out one or two ways of thinking of a problem, one or both of them will each share a different perspective or way of doing things that I had never even considered. I truly appreciate how these episodes really stretch my own mathematical thinking, gives me pause to think about the intentional ways that I should structure my own instructional practices, as well as ways of lifting up student thinking in mathematics about mathematics. Thank you for making each episode so engaging, inviting, and most of all figure-out-able.

**Pam **

Wow, that was super cool.

**Kim **

(unclear) super, super fun.

**Pam **

Thanks.

**Kim **

So, if you have just started listening, you just wait. You'll soon find out that we mostly have your stuff together. But when we don't, we just go with it. It is what it is. We appreciate the compliment. We love chatting math with all of you too. Okay, so we have tackled the most sophisticated strategies for addition, and subtraction, and multiplication. So, you've guessed it. Today, we're going to talk about division.

**Pam **

Whoa! Let's do it.

**Kim **

Yeah.

**Pam **

Kim, just the other day in our message board. So we have online workshops, and I interact with participants in the message board.

**Kim **

Yeah.

**Pam **

Elfi, who's taking our Building Powerful Division workshop said, "These approaches teach so many different concepts. And at the same time!"

**Kim **

Yeah.

**Pam **

I thought that was a noteworthy comment. First of all, Elfi, well noticed because I think often if we are kind of stuck in that perspective that math is about rote memorizing disconnected sets of facts and mimicking rules and procedures, then, you know, "Learn this. Do this thing," and it's about memorizing a thing, so that then you can add it to another thing.

**Kim **

Yes.

**Pam **

And that's kind of how it looks like you teach all the different concepts.

**Kim **

It's unmanageable.

**Pam **

Yeah. And it's unmanageable because there's so many different ones. And we end up with this kind of mile wide, inch deep kind of thing.

**Kim **

Yep.

**Pam **

Once you really start mathing, you really start really mathing the way mathematicians math, even at young ways, you start to realize that it's so interconnected, and that as you do a Problem String or a routine, there are lots of things that are coming together and making more sense. Which is one reason why you need more than one experience to quote unquote, "Learn a strategy." I don't know if you've noticed, listeners, but I used to say things like "learn a strategy," and I am saying a whole lot more of what I actually mean. Which is, "Develop the relationships, so that a strategy becomes a natural outcome."

**Kim **

Yep.

**Pam **

And it's multiple relationships, so nicely noticed, Elfi. Let's see if we can build the most sophisticated division strategy and learn a whole lot of other things at the same time. Alright, Kim, I got some questions for you. Hey, pretty soonl we're going to turn it around, and you're going to give me questions. But not today.

**Kim **

Oh, that's awesome.

**Pam **

Okay, ready? What is 500 divided by 25?

**Kim **

Oh, that's quarters. Okay, so I'm going to say that there are 2 quarters in $0.50.

**Pam **

Okay.

**Kim **

So, like 50 divided by 25 will be 2, but you're asking 500 divided by 25, so it's going to be 10 times as much. So, 20 quarters.

**Pam **

There's 20 quarters in 500. For anybody who's not familiar with that strategy, you got anything else that we could also reason about?

**Kim **

Yeah. So, I could think of four 25s is 100. So, it's is still quarters. But then, I need five 100s. So, if there's four 25s in 100, then I can scale up by 5 to get the 20 quarters.

**Pam **

Nice. 20 quarters in 500. So, you're saying 500 divided by 25 is 20.

**Kim **

Yep.

**Pam **

Next problem. What is 250 divided by... Oh, you know what? Before I do that... Yeah. Okay, I think we're good. I was trying to decide if I was going to mention something first. 250 divided by 12.5.

**Kim **

I also think this one is 20.

**Pam **

What?!

**Kim **

Yeah. So, I never say this well. I see the connection between 500 divided by 25 and 250 divided by 12.5.

**Pam **

Okay, what connection do you see?

**Kim **

So, this time, I am... How do I explain? This is always really tough for me. So, I have a smaller... Oh, what's the word? Divisor?

**Pam **

Yep.

**Kim **

And it's half as big, and the dividend is also half as big, so my quotient, my answer is going to stay the same. So, I have half as much, but I'm also dividing it by half as much.

**Pam **

So, if you had 20...

**Kim **

There's the same amount of 12.5s in 250, as there are in 25 in 500.

**Pam **

Say that again. There are the same amount of 12.5.

**Kim **

I'm trying to find the words.

**Pam **

Yeah. There's the same amount of 12.5 in?

**Kim **

In 250.

**Pam **

In 250. As twice as much...

**Kim **

In twice as much.

**Pam **

In twice as much.

**Kim **

Right, right.

**Pam **

Is that a way to say that?

**Kim **

Yeah, those are equivalent. Right.

**Pam **

Okay. So, if the total is 500, and you're asking yourself, "How many groups would I have if I've got 25 in each group?" You're saying I'd have 20 groups?

**Kim **

Mmhm.

**Pam **

And then, I asked you what if you only had half as much stuff, and you only had half as many groups? Then, you're saying you'd still have the same amount of stuff in each of those groups?

**Kim **

Yep.

**Pam **

Did I say that right?

**Kim **

Yeah.

**Pam **

Can you say that?

**Kim **

If I have half as much stuff and half as many groups, then I have the same amount in each group.

**Pam **

Cool. Can you say that if it was... So, 500. What if I had 25 groups?

**Kim **

Yeah. So, I think that's the way I was thinking about it. Instead of 500, if I have only 250, and I have half as many groups... (unclear) backwards. I'm saying it the same way. Either I have half as many groups and the same amount in each group, or I have half as much in a group and I have the same number of groups.

**Pam **

Because you halve the total as well.

**Kim **

Yeah, yeah.

**Pam **

Yeah. Yeah. Alright, so that's tricky stuff. We might could have done that with smaller numbers to maybe make it more easy. But anyway. Alright, so when you were talking, and you were talking about how many quarters were in 500, I wrote that problem first as 500 "division symbol" 25. But I also wrote it as a 500 "fraction bar" 25. So, like if I was describing what it looks like on the paper, 500 is over 25. That's not mathematical. That's just a description.

**Kim **

Yeah.

**Pam **

So, 500 "fraction bar" 25. It's almost like 500/25. And the answer to that is 20. You were clear the answer to 500 divided by 25, or 500/25 is equivalent to 20.

**Kim **

Mmhm.

**Pam **

So, when you did 250 divided by 12.5, I'm also going to write 250 "fraction bar" 12.5.

**Kim **

Yep.

**Pam **

Another way to say that is 250/12.5. And you're saying that's also equal to 20?

**Kim **

Yes.

**Pam **

So, if they're both equal to 20, can I write 500 divided by 25, 500 "fraction bar" 25, 500/25 is equivalent to 250/12.5.

**Kim **

Mmhm.

**Pam **

So, now I kind of have two fractions next to each other. And when I write those two fractions... Or maybe ratios next to each other. 500/25s and 250/12.5s. I think that might be even easier for people to go, "Oh, yeah. Sure enough. If you..."

**Kim **

Yeah, I think so.

**Pam **

"...divide one in half, then you can divide the other in half." But, teachers, we would suggest that you don't want to go there too fast. You don't (unclear) tell students. Yeah, we would want to do a lot of experiences, so students understand that a division problem can be written as a ratio.

**Kim **

Well, and some students might not have a deep understanding of the two kinds of division.

**Pam **

Yeah.

**Kim **

Right? So, that's the precursor.

**Pam **

Absolutely. Yep. Yeah, so there's lots of precursors to what we're doing today. And we're not going to do any of them. We're diving in. We're saying we've got those precursors. They have happened. And so, if somebody can think about 500 divided by 25, like Kim did, and they have some partitive and quotitive division, and they know we've done some experiences to say I can write division as a fraction ratio or relationship.

**Kim **

Right.

**Pam **

Then, I now can kind of look at those equivalent ratios and say, "Huh. Like, maybe we could think about equivalent ratios to solve division problems."

**Kim **

Yeah.

**Pam **

I wonder if that might influence how you solve a problem like 330 divided by 16.5?

**Kim **

Yes, I just wrote that the way you described. So, 330 "fraction bar" 16.5. And I...

**Pam **

I think I would... Like, described it in the problems before? Yeah, got it. Okay.

**Kim **

Yeah, sorry. Yeah, the way you were just describing how you wrote what I said.

**Pam **

Okay.

**Kim **

So, I have 330 "fraction bar" or over 16.5. And I am making that equivalent to 660 divided by 33. I just felt like that would be an easier problem. And I know 66 divided by 3 is 2. So 660 divided by 33. Is 20.

**Pam **

Interesting.

**Kim **

I'm totally looking at the previous problem. That's hilarious.

**Pam **

Why are you looking at the previous?

**Kim **

Because it's 20, and I didn't see a connection between those.

**Pam **

I'm not sure there is a connection between them. They're equal to 20 for sure. Are you finding one?

**Kim **

Hmm.

**Pam **

I'm not sure I meant there to be one.

**Kim **

Okay. Well, and now I want to play. I want to play.

**Pam **

Alright, I'm going to let you play later.

**Kim **

Okay, that sounds great.

**Pam **

I'm not sure I meant there to be a connection between those.

**Kim **

Well, way to make me think a little bit. Okay. Next problem.

**Pam **

Alright, next problem. 76 divided by 8.

**Kim **

76 divided by 8. Okay, I actually like the divided by 8 because then I can just say that that 76 divided by 8 is equivalent to... What is that? 38 divided by 4? Yeah.

**Pam **

Because?

**Kim **

Half of 70 is 35. And half of 6 is 3. So, then I know that that's 38.

**Pam **

So, you we're halving 76.

**Kim **

Yeah, sorry.

**Pam **

You went straight to telling us how you were halving

**Kim **

Oh, I'm sorry.

**Pam **

So, if you halve 76.

**Kim **

I'm halving both those numbers to make 38 divided by 4. Thanks for slowing me down.

**Pam **

So, it's like you found an equivalent problem.

**Kim **

I forget you can't see my paper. Nobody can see my paper.

**Pam **

That is true. I can't. No one right now can see your paper. Okay, so you have 76 divided by 8 is equivalent to 38 divided by 4.

**Kim **

Yep, mmhm.

**Pam **

Mmhm.

**Kim **

And then, I'm halving those again, and I'm saying it's equivalent to 19 divided by 2.

**Pam **

Okay.

**Kim **

Which is equivalent to 9.5, 9 and a half.

**Pam **

9.5. 19 divided by 2 is 9 and a half?

**Kim **

Yep.

**Pam **

Sweet. So, in the previous problems, I kind of felt like you scaled up both.

**Kim **

Yeah, the last problem is sure did.

**Pam **

The 330 divided by 16.5.

**Kim **

Yep. Scaled up.

**Pam **

You doubled both the numerator and the denominator.

**Kim **

Yeah.

**Pam **

And this time you scaled down. You divided the numerator and the denominator. So, that's interesting. So, you could do either of those to find an equivalent division problem that's easier to solve?

**Kim **

Mmhm.

**Pam **

Cool. How about 36 divided by 144? What?! That's a crazy problem.

**Kim **

a fourth. writing that like a fraction. So, 36 divided by 144. I noticed that they both have a 12 in them. So, 36 is three 12s. And 144 is twelve 12s. So, I'm scaling that down by 12 to get three-twelfths, which I know is I'm

**Pam **

Bam! So, you're saying 36 divided by 144 is one-fourth.

**Kim **

Nice. I like that problem.

**Pam **

Thank you. And if you had seen that problem as 36 inside the house top and 144 outside the house top?

**Kim **

Oh, gosh. Ugh. Do I even know how to do that anymore? I don't know.

**Pam **

I hope not.

**Kim **

Probably not.

**Pam **

Do you agree with me that after all of that work, you would have ended up with 0.25?

**Kim **

Yes, mmhm.

**Pam **

Does that make sense? So, knowing that we can think about division, as of all those precursor things that we talked about, but also as a ratio that you could then find an equivalent ratio could be really slick to find. Not only slick, but understandable, right? A clever way of doing problems like 36 divided by 144. How about 49 divided by 0.5?

**Kim **

Yep. Oh, I'm going to scale that up to something divided by 1. And so, I'm just doubling 49 and the 0.5. And that's 98 divided by 1.

**Pam **

I will never forget the day when I was working with this equivalent ratio strategy., "Could I turn a division problem into a ratio, find an equal ratio that's easier to solve?" When it occurred to me that anything divided by 0.5, I could scale both the divisor and the dividend, the numerator and the denominator, to something divided by 1. Oh, bam!

**Kim **

Yeah.

**Pam **

So nice. And so much work with the long division algorithm. So much work. Okay, next problem. How about... People are like, right now, trying all sorts of problems divided by 0.5. They're like, "Really?!

**Kim **

Should we pause a little bit. Let them (unclear).

**Pam **

I mean, they can pause the podcast. Nope. We're moving on.

**Kim **

Moving on.

**Pam **

0.88 or 88/100 divided by 0.25. Oh, Kim, decimal division. We've got decimals everywhere. Crazy. There's no way you can think and reason about this one.

**Kim **

Well, I'm going to go with your divided by 1 again. So, I'm going to scale those numbers up by 4, so that I can do 0.88 times 4 and a 0.25 times 4.

**Pam **

That 0.25 times 4 sounds really nice because now you divide it by 1. 0.88 times 4? What are you doing to do that?

**Kim **

I'm going to go with 90 times 4 is 3.60. 3.6. And that's too much by $0.02. $0.02, 4 times. So, it's $0.08 cents too much. So, I'm getting 3.52

**Pam **

3.52. It's kind of like that 360 minus 8 was 352, and then you kind of put the... That's how I did it. And put the place value back in. All divided by 1?

**Kim **

Mmhm. So, 3.52.

**Pam **

So, you're saying that 0.88 divided by 0.25 is 3.52?

**Kim **

Mmhm.

**Pam **

Nice. And if you were to kind of reason about that, are there about 3 and a 1/2 quarters in $0.88?

**Kim **

Yeah.

**Pam **

That's kind of a way to kind of check to make sure our magnitudes made sense. I do not think I would have scale times 4. I think I would have... So, when you decided 0.25 times 4 is 1. Let me just scale the 0.88 times 4. I think I might have doubled, doubled.

**Kim **

Oh, okay. Yeah.

**Pam **

But, I liked your height. I like how you did the times 4. Taking a drink of my tea. Alright, next problem. What is... Are you drinking anything today?

**Kim **

Water, but I'm almost out.

**Pam **

I'm drinking throat coat. Throat coat tea, herbal tea. I'm an herbal tea drinker. Anyway, you don't care. Moving on. How about 1 point... I don't know why I went there. Because I was worried I was taking too long drinking, and my throat is sore today. Last problem of the string. What is 1.6 divided by 0.125?

**Kim **

Okay. So, I know stuff about 0.125. but I'm going to pretend I don't.

**Pam **

Yeah. Can I ask you something really quick first?

**Kim **

Sure.

**Pam **

Could we just estimate? Is this answer going to be less than 1 or more than 1?

**Kim **

Less than 1 or more than 1.

**Pam **

Because it's kind of this ugly fraction with all these decimals, and I'm just kind of curious.

**Kim **

It's going to be significantly more than 1. How do you know? Because you're dividing 1.6 by an eighth. So, it's like how many eighths are in 1.6? It's kind of what I'm thinking about.

**Pam **

So, you know something about 0.125...

**Kim **

Yeah.

**Pam **

...being an eighth. Okay. Alright.

**Kim **

But basically, I mean, another way to think of it is I'm going to scale up quite a bit. What I'd like to do is make it be something nice in the denominator. And I'm going to have to scale up to make that happen. And so my numerator, when I scale it up, it's going to be larger.

**Pam **

Okay. Than the denominator?

**Kim **

Mmhm.

**Pam **

Yep. Alright. As it started out that way, right? The numerator is larger than the denominator.

**Kim **

Yeah.

**Pam **

Okay, keep going.

**Kim **

I know that it's times 8, but I'm feeling like maybe I'm just going to double.

**Pam **

Okay.

**Kim **

So, I'm going to call that 3.2 divided by 0.25. And then, I'm going to double it again to get 6.4 divided by 0.5. And then, I'm going to double again to get 12.8 divided by 1. So, my answer is 12.8.

**Pam **

Bam! Isn't it cool when you get to a place where the question is the answer. What is 12.8 divided by 1? I don't if the question is the answer, but it's so like, "Pa dum. We're done." Yeah?

**Kim **

Mmhm.

**Pam **

So, we call this the equivalent ratio strategy.

**Kim **

Yep.

**Pam **

Can you turn a division problem into a ratio, and then find the equivalent ratios that are easier to solve? And we would suggest that this is the most sophisticated division strategy that we're going to try to help students develop the relationships to make this strategy become a natural outcome. Alright, Kim, here's your quiz. What what is one way, one thing that could ping? Why are we calling this? What's a thing that's true about the equivalent ratio strategy that makes it a most sophisticated strategy?

**Kim **

There is some anticipatory thinking. Right, you have to think about what's going to happen with maybe both of the numbers to decide if this is a strategy that makes sense to be using for these numbers.

**Pam **

Which also means that you are considering things simultaneously. You're, "Alright, if I turn this into a ratio, does it make sense to scale the numerator and the denominator to something that will..." So, I'm simultaneously considering the kind of strategy I'm using, both the numerator and the denominator. And the key to all of these sophisticated strategies is that they're also equivalence strategies. That you're also considering, "Can I make an equivalent problem that is actually easier to solve?" And so, then, also developing equivalence at the same time. So, three major kind of hallmarks that we're suggesting are, are you considering things simultaneously? Are you using anticipatory thinking? And is it an equivalence strategy? And when those three things come together, we think it's a pretty sophisticated strategy.

**Kim **

Yep. Yep.

**Pam **

Nice. So, how do we model this one? This is maybe... How do I say this? We could say how we modeled it today. And also, then I want to back up on how we would actually develop the strategy.

**Kim **

Yeah. I think you maybe described a little bit of a what was on your paper. And mine was the same. So, after the first problem when I was thinking about quarters, then from that point on, I represented with equations, but in a fraction bar...

**Pam **

Notation

**Kim **

...kind of way.

**Both Pam and Kim **

Yeah, mmhm.

**Pam **

Yeah. So, if you look at my paper, you would see a lot of fractions with equal signs in between.

**Kim **

Yep.

**Pam **

A lot of equations with a lot of fractions with equal signs. Is that a way to describe that?

**Kim **

Mmhm.

**Pam **

Yeah. I think the only time that I kind of did something slightly different is when I wrote the long division housetop when I was talking about what it would look like. But everywhere else, I've just got...

**Kim **

Put a big x on that, man.

**Pam **

Well, except we can write division using that symbol. It doesn't mean that then we have to follow steps. It just means now divide. And oh, hey, how about if we divide actually thinking and reasoning?

**Kim **

I think that's a really good point. Because actually, somebody recently asked at at what point do we... Gosh. It might have been Kristy in Journey. I think maybe she said something like, "Hey, at what point? My teachers are going to ask about division notation." And we had a chat about all the different notations there are. There are several, and kids need to see them all. And that doesn't dictate the way that they solve it.

**Pam **

Nicely said. Yeah, nicely said. And since this... Oh, wait. So, sorry. Let me... I was going to move on to something else. But if I was developing this with students...and I think we can start developing this in fourth grade, and then continue in fifth grade...a thing that we would do in fourth grade is what Kim and I affectionately call the "bigger, smaller thing". So, if I had something like 32 divided by 8, I might think about if I had $32.00 to share among 8 kids, we would discuss that. And then, I might say, "Well, then how does that relate to something like 16 divided by 8?" And, Kim, if we were talking. There's not an equal sign in between those. But if I said $32.00 divide among 8 kids, now we only have $16 divided by 8 kids? How would you reason about how much money each kid would get?

**Kim **

They're each going to get half as much because you have half as much total.

**Pam **

Yeah, the total was halved, and you have the same number of kids, so they each get half as much. But what if I said from the 32 divided by 8, I've got still that $32.00, and now you're going to divide it among 16 kids.

**Kim **

Poor kids.

**Pam **

How do you reason that one?

**Kim **

They're going to get half as much.

**Pam **

Because?

**Kim **

Because there's twice as many kids.

**Pam **

Yeah, and the total stayed the same, right? So, that needs to happen in fourth grade. There needs to be this conversation about this bigger, smaller and how things relate, so that then in fifth grade, we can do things like, If I start with those $32.00 divided among 8 kids, and then I doubled the money to $64 and I doubled the kids to 16, could you use 32 divided by 8 to help you think about 64 divided by 16? Or even double that again to 128 divided by 32. That's a little bit more of a... Like, kids would have to think about that problem. But if they can say, "Well, if I had $128.00 shared among 32 kids? Well, if I halved the money and halved the kids... Ooh, I could do that again. Halved the money halved the kids. I end up with 32 divided by 8." Bam. That's a problem kids should be able to... Well, they might be able to think about 64 divided by 16 as well. And they could also then think about 16 divided by 4, right? Like, if they can keep finding equivalent ratios. So, we would really then want to develop that in fifth grade. But also, I'll just mention, I would also do some work with area models...

**Kim **

Yeah.

**Pam **

...to talk about what happens when the area stays the same and the dimension halves. What happens when the area haves and the dimension halves? And I'll just sort of raise that, and we could talk about that more some other time. Cool. Since this is an equivalence strategy. It's even called. It's in the name. Its equivalent ratio. Since it's an equivalence strategy, we can also do Relational Thinking with this strategy to help sharpen kids thinking about equivalent ratios. So, Kim, use Relational Thinking to solve this problem. It looks like 21 "fraction bar" 3.5. So, 21/3.5. I don't know how to say, but 21 divided by 3.5, but in a fraction ratio notation. Is equivalent to 42 divided by blank. 42 "fraction bar" and the blank is in the denominator of that fraction. Okay?

**Kim **

Yep. So, instead of 21 divided by 3.5, I'm going to scale that up because I want to figure out the relationship between 21 and 42. And it's doubled. So, 3.5 doubles... I just wrote 6. It's 7.

**Pam **

So, 21 divided by 3.5 is equivalent to 42 divided by 7.

**Kim **

Yep.

**Pam **

Which is also equivalent to what? Your 6, right?

**Kim **

6.

**Pam **

Yeah, your 6 was screaming at you.

**Kim **

(unclear) Yep.

**Pam **

So much so that your hand just took over and wrote 6. Yeah, cool.

**Kim **

Yep. So, hey, ya'll, check out the Relational Thinking problems that you can take back to your classroom and use right away at mathisfigureoutable.com/relational-thinking. It will be in the show notes. They're fantastic. And while you're there, you know, peruse around. There's a bunch of great stuff in that Instructional Routines Hub. All there free for you.

**Pam **

Absolutely. Yeah.

**Kim **

Okay. Also, we would highly recommend that if you want to learn more about the Equivalent Ratio strategy. We've actually talked about this before on some other podcasts. That's episode number 64 and 132. There's some fantastic nuggets there that you can learn more about this particular strategy.

**Pam **

Super cool. Alright, ya'll, 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. Thanks for helping spread the word that Math is Figure-Out-Able!

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