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  00:09

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 the mission to improve math teaching. 

Pam  00:16

Because 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. I couldn't make it through that time. I had to take a breath. Oh, well. 

Kim  00:33

You try to hold your breath while you do it? Is that why your out of breath? 

Pam  00:36

I try to take it on one breath. 

Kim  00:37

Okay. 

Pam  00:38

Like, it's just a long sentence for one breath because I'm excited. You know, if I was like saying it not excited, maybe it wouldn't be. 

Kim  00:44

Well, don't do that. 

Pam  00:45

Alright. In this podcast. 

Kim  00:46

In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  00:53

And we invite you to join us to make math more figure-out-able. Hey, Kim. 

Kim  00:58

Hey.

Pam  01:01

Okay, that sounded funny. "Hey."

Kim  01:02

(unclear). Speaking of less excited.

Pam  01:05

How's it going today?

Kim  01:07

Some days are easier than others. (unclear).

Pam  01:09

Isn't that the truth. Yeah. I promise we're going to do some fun things on this podcast. So. 

Kim  01:13

Okay.

Pam  01:14

That will make you happy.

Kim  01:15

Okay. I grabbed a review for us. That's a fun start. 

Pam  01:20

Yeah. 

Kim  01:21

Ally20335. The numbers always get me. I always wonder what they mean. 

Pam  01:26

Thanks. Ally20335 Yeah, yeah. Appreciate it.

Kim  01:30

Ally says, "Thank you. I enjoyed listening to your podcast each week and thinking about how I can encourage my students to be better math thinkers. Thank you for sharing your ideas with all your listeners."

Pam  01:40

Whoop! Whoop! Hey, didn't you tell me the other day that you think thinking and reasoning are not the same? 

Kim  01:48

Did I? (unclear). 

Pam  01:49

I often say thinking and reasoning is kind of the same?

Kim  01:53

Oh! Yeah, I did say that. I think I might have put that in one of your documents. Edit! 

Pam  01:59

You want to... Yeah. Why do you think reasoning? What's the difference? 

Kim  02:04

I haven't fully fleshed this out, but... 

Pam  02:07

Putting you on the spot. Putting you on the spot.

Kim  02:08

Yeah, it's okay. I think reasoning is deeper and involves more. Like, if you're reasoning about something, I think it's like a higher...

Pam  02:21

More complicated?

Kim  02:22

...level. It's more complicated.

Pam  02:24

Complex? (unclear). 

Kim  02:24

Yeah. You thought a lot about math in high school. But you weren't reasoning.

Pam  02:32

Huh. I don't know if I thought a lot about math. I thought a lot during math. 

Kim  02:38

Okay.

Pam  02:40

You know like, if you were to have said to me, "What are you thinking about?" I would never have said math. But I would definitely say I was thinking while I was doing math, but it was always about someone else's thinking.

Kim  02:51

I think reasoning to me just means that you're like, making sense of it in a way that thinking doesn't necessarily mean that.

Pam  02:59

That doesn't have to.

Kim  03:00

Yeah.

Pam  03:01

Huh. Okay. Well, anyway, Ally20335, thanks for listening. And thanks for giving us a review. Ya'll, we said it before, but I'll say it again. If you give us a review, it helps more people find the podcast. And you would not believe the number of people I meet at conferences. 

Kim  03:18

Oh, yeah.

Pam  03:19

(unclear). at NCTM that are like, "Oh, my gosh! We love your podcast" And it's kind of the gateway drug, right? It is the...

Kim  03:24

Right.

Pam  03:25

Can I say that? It's the way to get in. And so, the further we can share the podcast and get it out to more people, then the further we can spread the message.

Kim  03:35

Yeah. 

Pam  03:36

Yeah, it's a good thing. So, we appreciate it. Ya'll, give us a rating. Give us a rating. Give us a review. Whoop! Okay, the last few weeks, we have been doing a sequence of tasks to build exponentiation, to build a sense of exponents and what it means to grow exponentially or to use exponents. Let's do another one today.

Kim  03:56

Okie doke. 

Pam  03:57

So, first problem, Kim, of today's Problem String is what is 16 divided by 8? Now. tricky because as I write that, I'm actually writing 16/8. So, I'm writing a vertical fraction 16/8. Which is going to kind of throw some kids. Like, I might even write that first. 16/8, and then say 16 divided by 8. 

Kim  04:19

Mmhm.

Pam  04:19

Because 16 divided by 8 is almost too easy. But then I'm not going to like stay here too long. So, 16 divided by 8 is? 

Kim  04:26

2. 

Pam  04:27

2. And I know you know that, but I'm going to go ahead and do a little bit of work to kind of... What's a good word? Substantiate that just a little bit. I might say something to my students like, "Hey, we've been doing a lot of exponent stuff lately that has everything to do with multiplication. And so, if I was going to write this as a bunch of multiplications, might you write that as..." Like, 16. If I were to write that as a bunch of multiplications, that would be what? 2 times? 

Kim  04:58

2 times 2 times 2 times 2.

Pam  05:02

Because that's like 2. That's like 8 times 2. Yeah, mmhm. So, four 2s multiplied together. And then if I was to write the 8, then I would elicit from kids 2 times 2 times 2. So, now I have a bunch of 2s in the numerator and a bunch of 2s in the denominator. And then I could even say, "So, it's almost kind of like we could say that I've got 2, and I'm going to take one of the 2s from the numerator divided by one of the 2s from the denominator." So, I have 2 divided by 2, times 2 divided by 2, times 2 divided by 2. Now, I've got one more 2 in the numerator left over, so I'm just going to write times 2 kind of hanging out to the side.

Kim  05:39

Mmhm.

Pam  05:39

And that's kind of like a bunch of ones times 2, which is sort of 2. And I may even write that 2^1. The last thing that I might do here is say, "So, I know we wrote the 16 as 2 times 2 times 2 times 2. How can we write that with exponents? 

Kim  05:58

Say that again? 2^4.

Pam  06:00

2^4. And then how do I write 8? 2 times 2 times 2 as exponents?

Kim  06:04

2^3. 

Pam  06:05

So, now I have on my paper or my board, 2 to the first equals 2^4 divided by, over, the fraction bar 2^3. Just let that sit there. There it is. Bam. Next problem. How about 64 divided by 8. But again, I'm writing it as a fraction. 64 "fraction bar" 8. Vertical fraction. Okay, So, Kim 64 divided by 8 is? 

Kim  06:31

8. 

Pam  06:31

8. If we were to kind of pull that apart a little bit, what are some ways that we could pull that apart? And, ya'll, I'm going to do this super fast because I'm not trying to bore kids. I just want to have something that we can kind of lean back on, on the board. So, 64. Now, we have some sort of options about how we could write this. I am going to encourage... You know what? I'm actually not. Nope. I'm just going to leave that as 64 divided by 8. Yeah. Third problem. How about 2 times 2 times 2 times 2 times 2 times 2? So, just because some people are listening as they're driving, that was 1, 2, 3, 4, 5... Six 2s. All divided by, fraction bar, 2 times 2 times 2.

Kim  07:20

Mmhm.

Pam  07:21

And what do you get? 

Kim  07:23

8. 

Pam  07:24

How do you know?

Kim  07:26

Because I have six 2s in the numerator multiplied together.

Pam  07:31

Mmhm. 

Kim  07:31

And three 2s in the denominator. So, if I think about it like pairing up what you just said. Pairing up 2 divided by 2. I can do that 3 times. So, that's three 1s. And I'm left with 2 times 2 times 2 in the numerator.

Pam  07:44

Okay, cool. So, it's almost like... And I'm going to write down kind of what you said, but I'm going to say what you said, and then tell you what I write down. It's almost like you said that you had six 2s in the numerator, and I'm going to write down 2^6. And you had three 2s in the denominator. I'm going to write down 2^3. And you're telling me that that's equal to 8, which is also 2^3. So, I've now written down all those 2s divided by the six 2s multiplied together, divided by the 2... Oh, my gosh. The six 2s multiplied together divided by the three 2s multiplied together, equals 8. Equals 2^6 divided by 2^3. Equals 2^3. And it's kind of because you said that three of those 2s were going to divide out. And when they divided out to 1, 2 divided by 2 is 1, then you were left with three of those 2s multiplied together. Which is 2^3. Yeah? Okay.

Kim  08:38

And I particularly like how you... In the first problem, when you said it was divided out, then you you called that a 1. Because I think a lot of people who try to be... Like, try to make sense of these, just cross them all out and just say they they like cancel each other.

Pam  08:57

Yeah, yeah. That wonderful word "cancel". In my upcoming book, Developing Mathematical Reasoning - Avoiding the Trap of Algorithms, I write a bit about the word "cancel" and how many different meanings "cancel" can have. Many of them which aren't actually mathematical. And yeah, so it can be super helpful to like what is actually happening? And thanks, we're actually dividing. A number divided by itself is 1. Since all this is multiplication, multiplying by 1. Yeah, cool. So, the last thing in this problem is that you said that this was equivalent to 8. We also had the problem before equivalent to 8. And usually at some point somebody's going to say, "Hey, that's also equivalent to the 2 times 2 times 2 times 2 times 2 times 2. Those six 2s multiplied together, that's 64. And so, I just wrote down equals 64. Divided by 2 times 2 times 2. which is also 8. So, that 64 divided by 8 is 8. Sure enough, played out when we wrote all the 2s out, multiplied together, and we divided them out. We could think about lots of different relationships that are happening. Cool. Alright, next problem. How about 5^5 divided by 5^3? What are you thinking about? 

Kim  10:19

I'm thinking about in the numerator, there are five 5s multiplied. And in the denominator, there's three 5s multiplied. So, there's more 5s multiplied in the numerator. 2 more of them. And so, 5 times 5 is 25.

Pam  10:34

Cool. So, I actually, as you were talking, wrote out those five 5s multiplied together, divided by those three 5s multiplied together. And then you said there were more 5s in the numerator?

Kim  10:46

Mmhm. 

Pam  10:46

So, like if we lined up those 5s. And I might even take the time again to like 5 divided by 5, times 5 divided by 5, times 5 divided by 5. There's 3 of them. We've now, there's no more in the in the denominator, so we've got 5 times 5 left over. And that's what you're saying is 25. Cool. And then can we write that 25 in exponents? 

Kim  11:08

Yeah, 5^2. 

Pam  11:10

5^2. So, now we have on the board, 5^5 divided by 5^3, equals the five 5s divided by the three 5s, times each other, equals 25, equals 5^2. So, 5^5 divided by 5^3 is 5^2. And I might at this point say, "Well, what kind of like..." Makes sense of what... Sometimes kids will just go, "Well, you just subtract." Makes sense of, are you just subtracting that exponent of 5 and the exponent of 3 to get an exponent of 2. Ooh, like, how do you make sense of that?

Kim  11:50

Are you asking me?

Pam  11:51

I am, yeah. Sorry. Yes. 

Kim  11:53

So it's really about you have that many less multiplied together. So, if you have five 5s multiplied and three 5s multiplied, you have three less in the denominator than you do in the numerator.

Pam  12:13

Wait, say that again. 2 less?

Kim  12:15

Yes, two less. Sorry. (unclear). 

Pam  12:18

Yeah.

Kim  12:18

Yeah. So, you're like basically dividing out the ones that pair up that you have the same in the numerator and the denominator. So, you can really just focus on how many additional 5s you have multiplied.

Pam  12:34

Yeah, how many are sort of left over. And since it's all being multiplied, you still got those factors kind of left over. Cool. So, if we did something kind of general like 2^a divided by 2^b, how could we kind of think about what that means? And I would let kids kind of talk about that a little bit. But then, Kim, what are you envisioning if you have 2^a, divided by or fraction bar. Kind of over the way it looks, but we're dividing. 2^b. Like, what's happening? 2^a. What does that mean? I feel like we did that a couple strings ago. 

Kim  13:13

Yeah, it's a number of 2s. 

Pam  13:16

So, it's like 2 times 2 times 2, "dot, dot, dot" times 2. And there's a of those, a 2s. Mmhm. Divided by. 

Kim  13:23

Mmhm. And then b number of 2s in the denominator.

Pam  13:27

And so, when I'm going to write those b number of 2s, I'm actually going to line them up a little bit. So, when I have the 2 times 2 times 2, "dot, dot, dot" times 2, a of those in the numerator, underneath each of those, I'm writing 2 times 2 times 2, "dot, dot, dot". And then we don't really know how many are in the denominator. And so, a bunch of them look like they're going to divide to 1s, but I'm going to have some left over. And how many am I going to have left over? Well, the difference between a and b.

Kim  13:59

Mmhm. 

Pam  14:00

And now, here's where I wish we had done some subtraction work. 

Kim  14:04

I know. 

Pam  14:05

Kids could think about that it's that's then 2^a-b. And at this point, we could kind of look back at the problems that we had done today, and we could see where we had 2^4 divided by 2^3. And 4 minus 3 is 1. And the equivalent expression that we had was 2^1. We could look at 2^6 divided by 2^3, and that simplified to 2^3. And 6 minus 3 was 3. But it's really not about saying, "Alright, so therefore we've given you a little bit of conceptual understanding. Go memorize the rule." I really want kids to look at 2^a divided by 2^b. And rather than think about 2^a-b, I really want them to think about what's the difference? Like, where do I have more 2s? Do I have more 2s in the numerator? Do I have more twos in the denominator? And how many? And then, that's where they are. The rest of them divided to 1. And we're sort of left with those leftover 2s. 

Kim  14:59

Yeah. 

Pam  15:00

Cool. Could we do the same thing with like x^a divided by x^b, and just do everything we just did, but with x's instead of 2s? So, we're not going to spend too much time on that. Could we apply that to something like 3^3 times 5, all divided by... So, I've drawn the big fraction bar. 3^3 times 5 all divided by 3^2. Alright, Kim. What are you thinking about? 

Kim  15:24

I'm thinking in the numerator, I have 3 times 5. So, three 3s multiplied in the numerator and two 3s multiplied in the denominator leaves me with one 3 in the numerator by 5. 

Pam  15:41

And that 5 is still hanging around.

Kim  15:43

Mmhm. 

Pam  15:43

So, I've just written down 3 times 3 times 3 times 5, all divided by 3 times 3. And then, depending on the age of kids that I'm working with, and maybe individual kids, I may even write that again as 3 divided by 3, times 3 divided by 3, times... Let me just... Now, I'm out. I don't have any more 3s in the denominator, so I'm left with 3 times 5.

Kim  16:08

Mmhm. 

Pam  16:08

And bam, 3 times 5 is 15, and we've simplified that expression. 

Kim  16:12

Yeah. 

Pam  16:13

And then I might look back and go, "Huh. It looks like it's sort of playing out. 3^3 in the numerator and 3^2 in the denominator. I had 3 in the numerator. 2 in the denominator. That leaves me just with 1 in the numerator. Oh, times 5. Yeah. So, like 3. And I could write that as 3^1 times 5 with that 3 minus 2 to be that first. That's... I wish you could see me point. That 3^3 and 3^2. The power of 3 minus the power of 2.

Kim  16:48

Mmhm. 

Pam  16:48

I could rewrite that 15 as 3^1 times 5. If I had to. But I really want what's happening in kids' heads to be less about subtraction and more about how many did I have in the numerator of this factor? And how many of that factor did I have in the denominator? Therefore, how many do I have left? 

Kim  17:05

Right. 

Pam  17:06

Yeah, cool. Alright, one more. Just for grins. How about 2^3 times 3^2, all divided by 6.

Kim  17:15

I like this one. Then, I have 2^2 times 3 in the numerator because the 6 in the denominator is a 2 times 3. 

Pam  17:27

Okay. 

Kim  17:28

So, then I'm left with 2 times 3, times 3 times 2 in the numerator. Divided by 3 times 2.

Pam  17:36

Whoa! I did not follow that. 

Kim  17:38

2^3.

Pam  17:40

Mmhm. 

Kim  17:40

Times 3^2 in the numerator.

Pam  17:43

Mmhm. 

Kim  17:43

And in the denominator, I rewrote 2 times 3 instead of 6.

Pam  17:47

Yes. 

Kim  17:48

So, then I'm left with 2^2 times 3 when I divide out the 2 times 3.

Pam  17:56

Okay, so when I wrote down the 2^3 times 3^2 divided by 2 times 3, I think you're you're looking at the 2... I'm going to actually write out 2^3 divided by 2, times 3^2 divided by 3. And the 2^2 divided by 2, you're saying is 2 ^2. And the 3^2 divided by 3 is 3^1. And so, you were left with 2^2 times 3.

Kim  18:22

Mmhm. 

Pam  18:22

Which is 4 times 3. Which we could write as 12. Cool. Yeah. I like it. Nice. There's lots of 2s and 3s happening in that one, so it can be really helpful to kind of write everything out and help kids kind of find which 2 and which 3. I can imagine the listeners right now, were like, "I don't even know which 2 and 3 you meant. So, (unclear) write that one out. Have some fun with it. Yeah?

Kim  18:42

I feel like that's a little bit of a tricky thing with exponents. That in order to keep the amount small, you live in the land of 2s and 3s and maybe some 5s. 

Pam  18:53

Mmm mmhm. 

Kim  18:54

And it's, you know, you could do other numbers, but then like your intuition kind of goes off rails a little bit if you're living in something to the seventeenth. 

Pam  19:02

Well, and, Kim, let me... I actually really wanted to go here next, so I'm super glad you brought that up. You might have noticed, listeners, that in the last few weeks and in this string, like Kim said, we use these smaller numbers because that intuition is going to be helpful. And for another reason. I'm actually looking, as I give kids problems like 5^5 divided by 5^3 or the 16 divided by 8, I'm actually looking for kids to solve that problem using the numbers.

Kim  19:34

Mmhm, yeah. 

Pam  19:34

That's what Kim means by intuition, right? Like, use the numbers and get something. And then go to a more abstract like, "Okay, I've dealt with the numbers. I'm super confident that with the numbers, this is what it means. Now, let me think in terms of exponents. Ah, yes." So, that's a helpful thing. If we give kids 7^2 divided by... Or 7^5 divided by 7^2, and we're like, "Is that 7^3?" Figuring 7^5, the only way they're going to do that is probably on a calculator. I mean, that's a huge number. So, then it's it's going to have to be this abstracted thing because they're not actually able to just reason quickly about the numbers themselves. But when I give you something like 2^6 divided by 2^3, you can actually figure 2^6, and figure 2^3, and do that division before you just start, quote, unquote, "slashing and burning" and dividing common factors out. So, that's super purposeful that in these Problem Strings, we're giving kids numbers that they can reason about, then we push it. So, in this Problem String, we did push 5^5 divided by 5^3 and didn't actually expect you to do 5^5. That was a moment where I pushed you further to go, "Yeah, we're not even  figuring that one out. We're going to use what we've been thinking about to go from there." So, it's a little bit of a push pull and an interplay between giving problems you can figure out. "Yep, sure enough. Oh, and that does follow the pattern." Now, let me push you a little bit ones that you really don't want to go figure out, and let's use the the generalization here. 

Kim  21:10

Yeah. 

Pam  21:12

So, Kim, I think at the end of this Problem String, we could have something like x^a divided by x^b. I shouldn't even say "end" because we did it kind of two-thirds of the way through. But x^a divided by x^b is equivalent to that same base, since we had the same base x^a-b. And we could walk away with a relationship where kids are thinking about the difference. Where did I have more x's? In the numerator? The denominator? And bam, we're kind of feeling it. Today's string only had more leftover factors in the numerator. 

Kim  21:48

Yeah. (unclear).

Pam  21:49

Hey, everybody, check out next week's podcast. Wonder what we might do there. Ya'll, thanks for joining us on the podcast and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visi mathisfigureoutable.com. Let's keep spreading the word that Math is Figure-Out-Able!