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!