Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:09

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  00:40

So last week, we had a fantastic Division Challenge for the world, right? So much fun. 

Pam Harris  00:47

Oh, my gosh, that was fun. 

Kim Montague  00:48

So we decided that we're going to tackle a little bit of division today.

Pam Harris  00:52

I mean, we are not done with division. Because let's be clear, there's a lot. Like division is interesting. And it's complicated. And it's complex. Lots of ins and outs, and therefore even more fun, right? Like we can really have -

Kim Montague  01:08

And super Figure-Out-Able.

Pam Harris  01:10

Super, super Figure-Out-Able, not rote-memorizable. Nicely said. Okay. So, you know, we're gonna be a little transparent here and say that sometimes when we do the podcast, we plan it out a little bit more. And I will actually tell Kim the problem that I'm about to give her so she can think through it a little bit. We've done that. We've also done it where we've literally thrown out the problem, and neither of us have done a lot of thinking - uh that's not true. I think I've always thought about it at least a little bit. Or if you've given me the problem, you've thought about it, but the other person hasn't heard it. Today is the cold case. I've created some division stuff. And Kim has not heard these before. So this will be a little fun and exciting. And here we go. Alright.  Totally willing?  Alright, hey, Kim, if we were going on a field trip, and we had 500 students, so it's kind a big, the whole school is going on the field trip. And we decided for the, or maybe it's field day, I don't know, give me a good context. It's Field Day. Because for whatever reason, we need water bottles, and we were taking stock of what we have. And we've got a bunch of water bottles. And we know that right now for those 500 students, we have 250 water bottles, and I'm curious how many water bottles each of those students can have?

Kim Montague  02:22

Okay, cool. So I grabbed my pencil and paper, just in case. But 500 kids, 250 water bottles, we don't have enough water balls for everybody.

Pam Harris  02:33

So how many water bottles is there per kid?

Kim Montague  02:36

One half of a water bottle per kid. 

Pam Harris  02:39

Like that's not very good, right? We can only give them half of a water bottle per kid. So hmm. Okay. So it gets even worse Kim because we don't have enough. But what if I had 1000 students? And still we only had those 250 water bottles? How many water bottles would each kid get?

Kim Montague  02:56

Yeah, so then everybody would just get a fourth of a water bottle. We have twice as many kids, but the water bottles stay the same. So they're gonna get half as much.

Pam Harris  03:08

So it's kind of like before they had half a water bottle each and now they get half as much of that and half of a half is a fourth. Is that a thing? 

Kim Montague  03:16

Yep.

Pam Harris  03:17

Okay, um, let's go back to our 500 students. So we got those 500 students. What if the situation gets even worse, and we only have 125 water bottles. We miscounted. We only have 125 water bottles.

Kim Montague  03:30

Okay, so same number of kids. 

Pam Harris  03:33

500. 

Kim Montague  03:34

500 kids, and when they were 500 kids and 250 water bottles, they got half a water bottle. So now there are half as many water bottles as before. So they get half as much water as they got before. So instead of getting a fourth now they get an eighth. Oh, they got a half before, so sorry. They were going to get a half of a water bottle when they had 250 water bottles. But now you have half as many water bottles so they get half as much water so they get 1/4.

Pam Harris  04:09

Half of the half is a fourth. 

Kim Montague  04:10

Right. 

Pam Harris  04:11

Okay, cool. Okay, let's see. What if I had our same 500 students? And 250 water balls. That was what we sort of started with. 

Kim Montague  04:20

Right. And that was they got a half a water bottle each. 

Pam Harris  04:23

Yeah. What if we had 250 students? And the 125 water bottles? How many? How much?

Kim Montague  04:33

Half a water bottle per student. 

Pam Harris  04:34

How do you know? 

Kim Montague  04:38

Because I know 125 is half of 250. 

Pam Harris  04:42

Yeah. So we had the original 500...

Kim Montague  04:47

Oh, you mean? So we have half as many students, but they get half as much water. So they're still going to get half each.

Pam Harris  04:54

Okay, cool. What if we had 50 students.  Okay. So we had 500 students got 250 water bottles, and then you said 250 students would get 125 water balls. Wait. How much water did they get for -

Kim Montague  05:18

When there were 250 students, and there were 125 water bottles, they each got half a water bottle.

Pam Harris  05:25

Did you say a fourth? 

Kim Montague  05:27

Did I say a fourth? They get a half, because they get the same as when there were 500 students, and 250 water bottles. 

Pam Harris  05:35

How do you know it's the same? 

Kim Montague  05:37

Um, because there's half as many students and half as much water. So what they get is the same.

Pam Harris  05:43

That seems important. 

Kim Montague  05:44

Yeah.

Pam Harris  05:45

Is that always true? If you have half as many students and half as much water, they get the same amount of water? 

Kim Montague  05:51

As the earlier scenario? Yes. 

Pam Harris  05:53

Would that be true? If you doubled the number of students and doubled the water? Would they still get the same amount of water? 

Kim Montague  05:59

Mhmm.

Pam Harris  06:00

Okay, so let's see. If 250 students, 125 water bottles, they each got half a water bottle. What if I had 50 students get 25 water bottles?

Kim Montague  06:15

They're still gonna get half because, and actually, I'm writing on my paper and I wish you could see, this is kind of a thing, but, I wrote 500 students, line 250. So over, positionally over, 500 students, for 250 water bottles, and I actually wrote equivalent or equal-to 250 students with 125 water bottles. And then I wrote 50 students to 25 water bottles, and you found a fifth as many students. And it's a fifth as many water bottles, so that's going to be the same again.

Pam Harris  06:57

So like from the 250, a fifth of 250 is 50. So 50 students. 

Kim Montague  07:01

So I wrote divided by five. 

Pam Harris  07:03

And then from 125 water bottles, a fifth of the 125 water bottles gave you 25 water bottles. So a fifth the number of students get a fifth of the water, they still have are getting the same amount of water per student. Could you do that from the 500? Our original? We had 500 students? 

Kim Montague  07:21

Oh, yeah, that might even be easier. Because you have a 10th as many students and 1/10 as much water. 

Pam Harris  07:29

Because 500 to 50 to 10th, and 250 to 25 is a 10th. Okay, so what if I had only 10 students, and five water bottles?

Kim Montague  07:42

It's still gonna be half of a water bottle each.

Pam Harris  07:44

I mean, to me, when I look at those numbers 10 to five, it sort of screams a half.

Kim Montague  07:50

I would have liked that problem first please.

Pam Harris  07:54

Like if I would have said the ratio of students to water bottles is 10 to five, you could have said, "Oh they're getting a half a water bottle each." At which point you would have said, "Not enough water, Pam, hydrate more." But you could have then sort of worked up and said, "Well, if we had 10 students, and they got five water bottles, then 50 to 25." In fact, when I say that now, you almost could sort of look at some of these and almost see the half screaming. I wonder how many people when you heard these numbers, didn't hear the scale, and you were hearing the half sort of screaming like 500 to 250, 250 to 125, 50 to 25, and 10 to five, sort of that half just kind of screams out the whole time. That's kind of interesting. Why are we doing this? So there's a strategy, that if I can think about scaling the number of students and scaling the number of water bottles with the same scale factor, that I end up with an equivalent ratio of students to water. And that can be helpful when I have other random problems. So for example, Kim, another random problem. What if I asked you 244 divided by 16? Like if I had 244 students, and 16 water bottles? I don't know. That's probably not the best. This is probably not the best context to begin with at all. I probably should have come up with a better context. Sorry, everyone.

Kim Montague  09:22

Yeah.

Pam Harris  09:24

What are you thinking about? Is there something you could do?

Kim Montague  09:25

Well, at first glance, I'm glad that we just talked about the strategy, to be perfectly honest with you, because 244 divided by 16 is kind of yucky. My palms are sweating a little bit. And here's the thing, like, I feel very comfortable taking as much time as I need. And if we need to pause, then we're just going to pause. But, um, there is something to doing math on demand that I want to make sure that I'm explaining my thinking well enough that people can follow. 

Pam Harris  09:57

I mean, you're on a podcast, give a little credit here.

Kim Montague  09:58

I mean, yeah, okay, so alright. So I'm going to think about an equivalent problem that might be nicer. So I'm going to scale this down, divided by two. 

Pam Harris  10:08

Half as many - 

Kim Montague  10:09

Half as many, whatever's so that's going to give me 122 to 8. 122 divided by eight.

Pam Harris  10:18

Okay, because half of 244 is 122. And half of 16 is 8.

Kim Montague  10:22

Um, I think you can do that again. I love that they're both divisible by two again. So I'm going to divide those both by two again, which is nice. I can half those numbers really easily. Oh, I like that. I like that a lot. Because now I have 61 divided by four, so half of 122 was 61. And half of eight was four. So now I've scaled down a couple of times, and I'm left with the problem: 61 divided by four. Does that makes sense?

Pam Harris  10:49

Yep.

Kim Montague  10:50

Okay. And I love that I know. 60 divided by four is 15. And one divided by four is a fourth. So at that point, I kinda changed my strategy a little bit, because I recognize 61 divided by four is 60 divided by four plus one divided by four. And so that gives me 15 and 1/4.

Pam Harris  11:18

Cool. So you're saying you can actually use this strategy of finding Equivalent Ratios to solve a division problem? 

Kim Montague  11:27

Love it.

Pam Harris  11:28

Huh, wow. 

Kim Montague  11:29

Yep. 

Pam Harris  11:30

Hey, I wonder. Is that helpful in any way in a problem, like - hmm, random - how about 432 divided by five? Oh, how are you going to scale those down Kim?

Kim Montague  11:45

Oh, you know what? I don't want to scale down because I don't - I think it's going to lead me to something odd. But what I am going to do is scale up.

Pam Harris  11:54

What? 

Kim Montague  11:55

Yep.

Pam Harris  11:55

You mean if I have twice as many water bottles,  for twice as many students, they still get the same amount of water?

Kim Montague  12:00

Yep. And so I'm going to scale up, I'm going to double the number of water bottles and double the number of students.

Pam Harris  12:07

Random, okay.

Kim Montague  12:09

Then that gives me 864 divided by 10, which is 86 and four tenths.

Pam Harris  12:21

Or 86.4. Because you know, this thing about divided by 10. Because division by 10 is so brilliant. And you have a lot of numeracy and you recognize that division by five, if you scale it up to division by 10, bam! Like anything divided by 10, you can just sort of like think about that pattern of multiplying and dividing by 10 in our number system. And  864 divided by 10 becomes 86.4. I totally just repeated what you did, sorry. 

Kim Montague  12:46

No, it's great. 

Pam Harris  12:47

Made me happy to cement in my own head. I wonder if you could do something similar, listeners, if you were to think about - oh, and this whole time, we should have been saying as soon as we give the problem, pause the podcast, solve it on your own. Retroactively: when you hear a problem, pause the podcast, solve it on your own, and then listen to what Kim's doing. So do that on this last problem. How about 61 divided by 2.5 or 61 divided by two and a half? Or 61 divided by two and five tenths? How would you solve that problem? So pause, pause, pause. Alright, Kim, what do you got?

Kim Montague  13:26

Okay, I'm going to scale up again, because scaling down when it's divided by 2.5 doesn't feel like it's going to be super helpful. So I'm going to double both of those numbers and create the problem 122 divided by five. 

Pam Harris  13:45

Because double 61 is 122. Double two and a half is five. 

Kim Montague  13:50

And then I don't love that. But I'm going to scale up again. I'm gonna double 122 to get 244 and double five to get 10. So now I have created the problem 244 divided by 10, which is 24.4.

Pam Harris  14:12

Bam!

Kim Montague  14:12

I like. Thanks for giving me problems that I like. 

Pam Harris  14:17

You're welcome. 

Kim Montague  14:18

Which really, they're not. Those problems, like, they're not, it's not like you gave me super nice ones. It's just that we have a good strategy that make them.

Pam Harris  14:30

Yeah, so if we parse that out a little bit, we've developed the strategy that if I have a division problem, I can think about that as a ratio of the dividend to the divisor. Do I do  that right? Yes. Of the dividend to the divisor. I think about that ratio. And then I can find equivalent ratios that might be a problem easier to solve. And often it's not but when it is, then I might as well use it. I'll tell a real brief, brief story. Y'all, I was a high school math teacher, I was an expert in teaching with graphing calculators, I was doing a workshop, I remember the day I was in San Antonio, Texas. And I was working with a group of teachers. And we had a rational function. So we had like variables on the top of the fraction or the rational expression and variables in the denominator. And I remember, I had this graphical way of looking at it, we're going to do this really cool stuff on the graphing calculator. And one of the participants said, "Well, yeah, but you could just scale both of them. And then the denominator will be so much easier to work with." And it was kind of when we were actually figuring out a particular point. And so we had numbers at that point, not just variables. And I remember looking at that person going, "Whatever weirdo. That's like, really - that is this weird trick that you just pulled out of a hat. And I don't do tricks. I just do, you know, I do mathematics, not tricks." And it's because I didn't have any numeracy. But I remember in the moment going, "I have no idea what you just did." Y'all, let's free up the world to know that these relationships exist. Once we know they exist, then we can play with them, we can experience them. And we can build relationships so that when we reach a division problem, we have choices. Like all of these problems, we could have thought about quotatively, we could have thought about how many of the divisors were in the dividends. But instead, kind of like what came when you solve 61 divided by four, you kind of thought about how many fours are in 60. And how many fours are in one, to sort of think about 61 divided by four, that's a quotative strategy. But we can also think about it partatively, where again, you think about it as a ratio, and then you can find equivalent ratios to solve the easier problem once you found an equivalent ratio. Bam! Like those are important relationships that we need in division so that we can all think and reason more multiplicatively about division, which leads then into all sorts of success with fractions, decimals, and percents, all those rational number things, which then leads to more success in higher math. And I love it. That's like one of my favorite things to do ever. Thanks for playing live math with Pam on the podcast today. Kim, that was fantastic. If y'all like what you just heard, if you're interested to know more about division, we just created the Building Powerful Division workshop and it has launched and we are thrilled to offer to the world where you can learn not only how to divide yourself, like we just did, but how to teach powerful division to your students. Yea verily, even powerful polynomial division. So we have a little bit of an extra bit in the workshop for those of you that teach higher math that want to know how all of this applies to thinking and reasoning about polynomial division. Not just division with whole numbers and decimals and fractions and all the things so if you want to check that out, MathisFigureOutAble.com, more and more in depth content workshops coming out, day by day, not really but at least semester by semester as we try to get more workshops out K-12 for more teachers to learn to teach more and more Real Math. So if you want to learn more mathematics and refine your math teaching so you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able!