Pam  00:00

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  00:10

And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  00:18

We know that algorithms are amazing human achievements, but they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop. Hello? 

Kim  00:36

I have to tell you in a second what I just thought about. In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:46

We invite you to join us to make math more figure-out-able And to figure out what Kim is thinking in her head when...

Kim  00:52

For whatever reason, when you just said "trap students", I totally pictured like a little mouse trap. Like, a piece of cheese and a little mouse. That's horrible! But I don't want to trap students. 

Pam  01:03

We don't. We don't. We do not.

Kim  01:05

It's bad like that. 

Pam  01:06

We do not want to trap students. That's correct. That is the main point of all the things. (unclear).

Kim  01:11

Listen, we might have to get a different intro because I kind of gloss over a little bit. I try to pay really hard. I try to pay attention. Like, I'm a focused person, but sometimes (unclear).

Pam  01:22

Not seventeen more times? 

Kim  01:23

Yeah. Hey, last week we talked about Integer multiplication. Yay! People been asking about that. 

Pam  01:30

They have been asking for a long time!

Kim  01:32

I know! 

Pam  01:32

Yeah, I hope you liked it. Good, good, good.

Kim  01:34

Okay, so this week, let's tackle integer division.

Pam  01:38

Integer division. Bam! Alright, first thing, if you have not, please go listen to last week's episode because it's going to be super important that you kind of heard how we laid the groundwork for integer multiplication. A thing to know is, I don't know that I would immediately move into integer division. Though, I could. But I would acknowledge that I'm not done with integer multiplication. I would do more integer multiplication strings, kind of like we did with different facts, different numbers. I would, as you do that, try to keep the feeling tone less, "Oh my gosh! You guys are supposed to know these facts!" and more, "Alright, getting good practice with these facts." 

Kim  02:16

That's never helpful. "You should know this." 

Pam  02:19

Yeah.

Kim  02:20

That's not helpful.

Pam  02:20

Yeah, but "Hey, look! Look, we're getting good practice with these facts!" That's a good, maybe a more helpful way, to approach all the things. Okay, so with integer division, it's all about getting better at two things, division and opposite. And when I say "opposite", I mean looking at a negative and being able to think of opposite in a few ways, being able to think of a number line, where if 9 is over here, the -9 is its opposite, which means I'm going to reflect it over 0. It's the same distance over 0. So, I want to use the word "reflection". I want to use the word "opposite". And it's also being able to think about the opposite of a product. Mmhm. So, what does that mean? It means when I see, when I think opposite, I see a negative sign, and when I see a negative sign, I think opposite. And when I see both of those, think about both of those. I see this reflection across 0. So, kind of all of that stuff, kind of intertwined. Alright, so when I say it's about those two things getting better at division and opposite, that also means that there's some division work that could be helpful here. And so, ideally, we would have done some whole number division work, so the kids are actually thinking about the two meanings of division, and they have some strategies for division.

Kim  03:47

Mmhm. 

Pam  03:48

I would much rather have kids take a little bit longer to use a strategy to figure out a division than for them to have a fact rote memorized and spit it out, and then not be able to have relationships for me to use in further math.

Kim  04:03

Right. 

Pam  04:04

Okay, so let's do a Problem String. 

Kim  04:07

Okay.

Pam  04:07

For integer division. Here we go. Alright, Kim, 36 divided by 9. What you got? 

Kim  04:15

4. 

Pam  04:16

And how do you think about that? I mean, that's probably a fact, you know. But like... 

Kim  04:20

Yeah, there are four 9s in 36.

Pam  04:22

Okay, so you're thinking about that pretty quotatively. And if students are not, meaning how many of these are in those, if students are not, then I'm going to nudge that. So, I might say, "Hey, another way of saying this problem is how many 9s are in 36? And kids are like, "Okay, 4." So, I'll just have that. Up till now, I don't really have a whole lot else going on. I've got 36 divided by 9 equals 4.

Kim  04:47

Mmhm. 

Pam  04:47

Let me actually just think for a second. I'm thinking. There is a world where I might have represented that on an open number line, where I've got 0, and I put 36 over to the right, and I say, "So, you're telling me there are four 9s in 36." How many 9s are in that 36? 4 of them. And I might have just quickly drawn 4 jumps.

Kim  05:08

Mmhm. 

Pam  05:08

Maybe put 9s above them. I don't know that I would do that... I don't think I would do the interim landing points.

Kim  05:13

Mmhm. 

Pam  05:14

So, you're telling me there's four 9s in that 36. Okay, cool. What if I ask -36 divided by -9. And I use the same interpretation. How many -9s are in -36.

Kim  05:27

Mmhm. Still 4. 

Pam  05:30

And so, I might go up on that same number line, and I might go, "Where is -36, ya'll?" And I want students to say, "Well, it's got to be the opposite, so it's reflected over the number line." It's going to be the same distance from 0 over there in the negative land. I could have been doing this on a vertical number line. I think I would probably do the first problem on a vertical number line. Maybe the second one on a second set of problems on a vertical one. So, I'm on a horizontal one right now. So, you said also 4. So, I would say, "You're telling me that if I jump back these four 9s, there's four jumps of 9 in this negative 36. But I'm jumping backwards, so those are jumps of -9."

Kim  06:08

Mmhm. 

Pam  06:08

So, how many -9s are in 36? You're saying also 4. And then I'm going to sit. I'm going to do what Ann Roman, a colleague who I have not talked to for it could be 20 years, maybe 15 years. And I called her up the other day. I'm going to quote her in one of my books. And one of the things that she told me once is she's a horse woman. She trains horses. And she said, "When you train a horse, and you have done something, and they've... You've asked them to do something, and they've done it correctly, then you just sit there and you let it soak. This is a moment where I've got on the board two problems, and I've got a number line. Four jumps of 9 to the right, landing on 36. Four jumps of -9 to the left, landing on -36. I got these two problems, and I'm just going to sit, and I'm going to go, "What do you guys see?" What are you thinking about right now?" Now, Kim, I don't know what you're thinking. I'm not sure if you're thinking about what. And I don't know that I'm even going to get anything like really new or fantastic from kids, but I think this is a moment to soak. So, just like. 

Kim  07:13

It's you soaking, gives it a little bit of weight. It gives a little bit that like there's something here to consider.

Pam  07:21

Worth considering.

Kim  07:21

(unclear) is good. Yeah.

Pam  07:22

Yeah. And I would hope that some kid would say, "Well, I'm definitely seeing 36 over there. I'm seeing -36, the opposite. I'm seeing these jumps over there. But those are jumps to the right, so they're positive. These are jumps to the left, so they're negative. And I can see. I can see four of those jumps of -9. Sure enough. 

Kim  07:40

Yeah, either way, it's 4 jumps. Mmhm.

Pam  07:42

Yeah. Notice, kind of like you said in the last episode, there's no rule that I'm putting out here. I'm asking kids to reason how many -9s are there in -36? Cool. So, then I might ask the question, next question, -36 divided by 9. I'm going to pause. Kim, what are you thinking about?

Kim  08:04

I'm thinking about 30...

Pam  08:06

Oh, can I back up? 

Kim  08:07

Sure. 

Pam  08:09

When we were still on -36 divided by -9? A teacher move that I think I would have made is I might have said, "Did anybody use opposite to think about this problem? Did anybody think about 36 divided by 9 and then use opposite like we've just been doing in multiplication?" If I asked that, Kim, does that make any sense to you? 

Kim  08:28

Yeah, if you're thinking about 36 divided by 9 is 4.

Pam  08:33

Mmhm.

Kim  08:34

But you have one negative. So, opposite would be -4.

Pam  08:38

Mmhm.

Kim  08:39

And you have another negative. So, opposite of that would be back to 4.

Pam  08:43

Back to positive 4. 

Kim  08:44

Flip flopping. 

Pam  08:45

Yeah. I mean, you're doing opposite, and you've got sort of two opposites there. -36 divided by -9. You can think about 36 divided by 9. And then you've got opposite that is -4. Opposite that is, like you said, positive 4. So, we have those same two strategies that we've been playing with in multiplication seem to be appearing in division. Hey, that kind of makes sense because they're inverse operations of each other. Alright, so we have those two ideas happening. Now, I give you -36 divided by 9.

Kim  09:13

So, I would think about 36 divided by 9 is 4.

Pam  09:16

Mmhm. 

Kim  09:17

And the opposite of that would be -4. 

Pam  09:19

Cool. Can you think about it the other way? Like, this idea of jumps. How many 9s are a -36? Hmm. 

Kim  09:30

I mean, there's... 

Pam  09:31

That's kind of weird.

Kim  09:32

There's 4, but opposite of it. They're not... There's four 9s and 36.

Pam  09:39

Yeah. You know how in multiplication, we can say, "I've got this many groups, and I've got this many in the group."

Kim  09:48

Mmhm. 

Pam  09:48

So, division can kind of turn around the same way where we've been asking, "How many of the divisor are in the dividend?"

Kim  09:57

Mmhm. 

Pam  09:58

But could we also say... So, this -36 divided by 9, could we say, "I've got 9 groups."

Kim  10:05

Mmhm. 

Pam  10:06

And I want to know... Wait, is it groups? 

Kim  10:07

Nine somethings (unclear). 

Pam  10:09

Nine somethings, and I want to know... Yeah, I'm basically trying to turn it into a missing factor problem. 

Kim  10:16

Yeah. (unclear). 

Pam  10:16

So, 9 times what? Yeah. 9 times what? Nine. What are the things? 9 whats is -36.

Kim  10:23

Yeah.

Pam  10:24

That could be another way of thinking about that. 

Kim  10:25

Mmhm, yeah. 

Pam  10:26

So, then I've got another line where I'm thinking not about jumps of 9. Now, I'm thinking about um...

Kim  10:32

9 jumps. 

Pam  10:33

9 jumps. Thank you. And what would those have to be? Oh, that would have to be 4. And in this case, since I'm landing on -36, they'd have to be jumps of -4. (unclear).

Kim  10:44

And this would come after work you've done as early as third grade with partitive and qualitative division. 

Pam  10:50

Yes.

Kim  10:51

Where they say four groups of 9, nine groups of 4. They're equivalent. 

Pam  10:55

Mmhm, mmhm.

Kim  10:55

Yeah, yeah.

Pam  10:57

So, one thing I've got in my paper that you might not realize is that I've got 9 times blank equals -36.

Kim  11:04

Mmhm. 

Pam  11:04

Kind of underneath and shifted a little bit from the -36 divided by 9 equals blank. I've turned that into a missing factor problem. 

Kim  11:12

Sure. 

Pam  11:13

So, we can kind of think about those jumps that we were thinking about before. 

Kim  11:16

Okay.

Pam  11:16

Cool. Alright, next problem. How about if I were to ask you for 36 divided by -9 in two ways. 

Kim  11:26

Mmm!

Pam  11:27

You want to do your favorite way first or...

Kim  11:29

Yeah. Yeah, the easiest way for me to think about is 36 divided by 9 is 4.

Pam  11:35

Mmhm.

Kim  11:35

And the opposite of that. 

Pam  11:37

Because we got one negative, so the opposite is -4. Okay. That totally makes sense. I might write that. No, I'm not going to do that yet. Never mind, keep going.

Kim  11:46

And then...

Pam  11:47

Can you think of it... Yeah, can you make it a missing (unclear). 

Kim  11:50

Yeah, how many -9s? Oh, that's trickier for me. How many -9s are in 36. No.

Pam  12:01

Yeah, that's weird, right? 

Kim  12:02

That's harder when it's not a -36 divided by 9. 

Pam  12:06

Okay. Can you think about it as -9 times that something? The thing we don't know equals 36?

Kim  12:13

Mmhm. So, -9 times -4 is 36.

Pam  12:19

So, now we're really back to what we were thinking about in multiplication. So, we needed to have done a good job in multiplication to now be sort of able to make sense of that missing addend problem. Or, sorry, missing factor problem. 

Kim  12:31

That might be the toughest one we've done. 

Pam  12:35

Mmhm, mhhm.

Kim  12:36

Yeah.

Pam  12:37

I might, I think, yeah, yeah. The missing factor problem. 

Kim  12:41

Mmhm.

Pam  12:41

So, eventually, I (unclear)...

Kim  12:43

Well, specifically when it's when... 

Pam  12:48

Positive divided by a negative?

Kim  12:50

Yeah.

Pam  12:51

Okay.

Kim  12:51

Because it's hard to picture how many are in that. How many quotatively. How many -9s are in 36. That's tricky to do. But also because you have negative groups of things. Yeah, I think that's the trickiest one. And I think it's fair for teachers to know that some of them are easier for kids to mess with than others.

Pam  13:12

For kids to conceptualize, sure.

Kim  13:13

All problems not created equal. 

Pam  13:15

Yeah. So, once we sort of... Yes, because if we just give kids rules, then it looks like they're all the same. But if kids are actually trying to think about them, it could be a place that stymies them. We don't want to stymie kids. Let's recognize where kids might get stymied, kind of help them think through it. And also, since we're developing both strategies, then help kind of nudge kids towards, "Can I think about this problem as kind of an opposite problem?" So, if 36 divided by 9 is 4, and I've got that one negative in there, then I'm going to think about as the opposite of 4 or -4. 

Kim  13:52

Yeah.

Pam  13:52

Cool. Okay, so then I'm going to do the same kind of thing with a different set of numbers. So, I might say 24 divided by 8, -4 divided by -8, -24 divided by 8, and 24 divided by -8. So, we're just... And I'm going to model these, we're going to get thoughts on the board, and we're going to bring out both strategies. Okay. Oh, go ahead.

Kim  14:14

When you wrap up this string. This is your first, maybe the first division string.

Pam  14:20

Mmhm.

Kim  14:20

What is your like summary, "Oh, that's interesting," that you're hoping kids say that you kind of wrap up with? I think that's hard for teachers sometimes to come up with what's the like?

Pam  14:35

That's hard for me! (unclear).

Kim  14:36

Yeah. 

Pam  14:37

Yeah, absolutely (unclear). 

Kim  14:38

So, it's the first string, so we're not anchoring the learning. We're moving on. It's very first. 

Pam  14:43

We'd like to give kids multiple experiences.

Kim  14:45

Sure. 

Pam  14:45

So, I think I would be wanting to pull back to everything that we've been learning about multiplication.

Kim  14:51

Mmhm. 

Pam  14:51

And I would want to say something like, "Hey, it seems really helpful that we can reason about how many of these are in those. That's the quotative meaning of division."

Kim  15:02

Mmhm. 

Pam  15:02

But sometimes that gets funky, and so it could be helpful to think about a missing factor problem.

Kim  15:08

Mmhm.

Pam  15:08

But sometimes even that's kind of weird. And so, wow, this idea of thinking about negatives and reflections seems to really be helpful, that we kind of pull that -1 out and kind of think about the problem, and then take the opposite for every negative that's in there. For every... 

Kim  15:23

Yeah. 

Pam  15:25

Yeah, every reflection that's that's going to happen. Yep.

Kim  15:28

And I think that idea of thinking about the fact and then opposite is something that it's kind of tried and true, and it's might be what people end up doing lots of the time. But I think the disconnect is that sometimes we launch in to say, "Hey, guys, just think about this way," and we don't give kids the time or experience to develop that understanding themselves. They don't generalize for themselves. I can just think about the fact and opposite.

Pam  15:57

So, it becomes one more rule to memorize. 

Kim  15:59

Right. 

Pam  16:00

Yeah.

Kim  16:00

Right, right.

Pam  16:00

Yeah. Yeah, and if kids are really trying to make sense of it with a quotative meaning, and you're just telling them to memorize some opposite... Or, well, not even opposite, but just, you know like, negative, negative is positive. I think this is one of the traps of the algorithms is they think to themselves, "Oh, it doesn't make sense to me, therefore I'm not a math person. Bummer." And then either they try to memorize your thing, and they know they're not going to do well because they don't memorize well. Or they just keep trying to make sense of it, and then that bogs them down, and their identity as a math person takes a hit. So, one of the three main traps of algorithms, I think, is the fact that it can trap kids into thinking they're not a math person. When in reality, they're trying to make sense of it more than maybe we might have as we're just (unclear)...

Kim  16:44

Sure.

Pam  16:44

...(unclear) a bunch of rules. Yeah.

Kim  16:45

Yep, yep. 

Pam  16:46

We would be remiss if we did not also talk about the partitive meaning of division. And this is so important. And we need both, so I want to do the work that we just talked about, but I also want to do work with partitive, the partitive meaning of division. So, if quotative is all about how many of the divisors are in the dividend. 36 divided by 9. How many 9s are in 36? Quotative... Excuse me, partitive division is a much more... It's funny because it's harder to do when we get out of whole numbers. But it's much more of a dealing out approach. It's kind of a sharing approach. We know the number of groups, but we don't know the share. We're trying to find the share. Is that right? Did I get that right? We know the number of groups... Sometimes, I don't say that well when I'm... 

Kim  17:32

Yeah, we don't know how much is in each group.

Pam  17:34

Yeah, we're trying to find that share. Okay, cool. So, ideally, we would have done work with partitive division with whole numbers, so that students would see things like our 36 divided by 9. So, that first problem that we did in the string before. Let's just think about it again. But this time when I write 36 divided by 9, I'm going to write 36 "division symbol" 9. So, that's the line with the two dots. And then I'm going to write equals 36, "ratio" 9, "over" 9. I don't... I hate the word "over" because it's positional, but since we're audio here only, and you can't see what I'm writing, I'm writing the fraction 36/9. But it's really the ratio 36 to 9. And so, I'm thinking about this division problem kind of as a ratio or a fraction. So, I've got really 36/9 on the board now. And kids might go, "Oh, well, I can simplify that fraction." And I might go, "Okay." And that's exactly the kind of thinking that we want to connect. We want... Now, again, ideally, you would have done some work, so that kids are already connecting division with ratios and the equivalent ratio strategy in whole numbers. We could put a note to that in the show notes. We've done equivalent ratio division strategies with whole numbers. I want to be able to be thinking about that here, so I'm thinking about like 36/9. And then I can simplify that. That's not crazy. We're just going to say that's 4. Cool. Then I'm going to say, "So, then what's -36 divided by -9?" And I'm, again, going to write that as a ratio. So, now I've got -36 over -9. I'm only saying "over" here because we're audio, and you can't see what I'm writing. It's the ratio. And then I'm going to say, "Guys, what do we do when we simplify a fraction like that?" Well, now I've clearly got 36 divided by 9 with these two -1s. We could factor out those -1s. So, it's almost like I've got -1 divided by -1.

Kim  19:30

Mmhm.

Pam  19:30

Times 36 divided by 9. 

Kim  19:33

Mmhm.

Pam  19:34

And there's our opposites that we've been talking about before. 36 divided by 9, we know is 4. I can go. I can do it a couple... I think about it a couple of different ways. Once I get 36 divided by 9 is 4, then I can take the one -1. And the opposite of 4 is -4. And the other -1, the opposite of that -4, is back to 4.

Kim  19:54

Mmhm. 

Pam  19:55

But I can also just look at that -1 divided by -1. Bam! Anything divided by itself is 1.

Kim  20:00

Mmhm. 

Pam  20:00

And so, I've just got 1 times 36 divided by 9. Or 1 times 4 is 4. A couple different ways to think about that. I didn't really ask you how you thought about that. I kind of talked through that. Sorry. 

Kim  20:08

No, it's fine! Totally fine! 

Pam  20:10

Let me do the string where I just tell you what I'm thinking. 

Kim  20:12

No, it's fine. 

Pam  20:12

Okay. So, next problem. What if I were to ask you -36 divided by 9? But I'm going to write it as -36/9. What are you thinking about? 

Kim  20:22

I'm going to factor out the -1.

Pam  20:24

Mmhm.

Kim  20:25

And write 36 divided by 9.

Pam  20:27

Mmhm. 

Kim  20:27

Is 4.

Pam  20:29

Yep. 

Kim  20:29

And then I'm going to multiply that by -1, which is -4. 

Pam  20:32

And multiply by -1 we could also say is the opposite. And the opposite of 4 is -4. I just want to keep bringing that language in with kids. Cool. And, ya'll, I can keep going with that exact same string that we just did. So, now I would have the next problem, which was a -36. Where am I? No, 36 divided by -9, but I'm going to write it like a ratio. So, 36 over -9. And I'm going to have kids think about that one. And then I'm going to give the next set of numbers, 24 divided by 8. -24. Well, so let me say it this way. 24/8s.

Kim  21:09

Mmhm.

Pam  21:10

-24... No, I don't know how to say it that way. -24 divided by -8 as a ratio. So, -24 over -8. And every time we're asking kids, "Like, can we factor out these -1s? And what does it mean? How many opposites are we doing here? I'm going to keep bringing in the word "opposite". We've got -1s divided by each other or just one -1. It's just one opposite. And all of that is coming together to just connect with all the work that we've done with integer multiplication.

Kim  21:41

Right.

Pam  21:41

And it all just works. 

Kim  21:43

Yeah, yeah, yeah. So, teachers, do a good job with multiplication, and then with division, both meanings, and the definition of integers being opposite.

Pam  21:53

And you've got kids actually understanding integers, and then you can move on and use that integer, those integer operations, in solving problems. Ya'll, thanks for tuning in. Thanks for tuning in when it's a little bit of higher math. I hope you guys enjoy the higher math when we try to talk it out verbally. But we sure appreciate you trying to improve your mathematics and for trying to teach 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!