Pam  00:01

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're 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.

Kim  00:31

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

Pam  00:37

We invite you to join us to make math more figure-out-able. Hey, Kim, how's it going?

Kim  00:43

Hey, it's good. How are you?

Pam  00:45

Whoo! Hey, it's going fast and furious. 

Kim  00:47

I know! 

Pam  00:48

All the things.

Kim  00:49

All the things. You know, I like to start with a review, and I found one. 

Pam  00:53

Oh, fun. Okay.

Kim  00:54

Listen, I'm going to say though, that they're getting fewer and farther between, and it makes me so sad. We have asked. We have asked people to share them. And I get, like it takes a minute out of your time. It can be super short. But tell us what you love. Tell us what you wish we did differently.

Pam  01:12

And to be clear, it's not just because we want to hear that you guys think we're cool. Which is always fun. But it really is because we're trying to get the podcast out to more and more people, so that more and more people can benefit from knowing real math.

Kim  01:25

Yeah.

Pam  01:25

And all the algorithms work on if you get reviews and more followers. So, if you guys could send us a review, that would be great. 

Kim  01:32

Yeah. 

Pam  01:32

Yeah. Alright. What did you find?

Kim  01:34

This one says, "Wow!" as the title. 

Pam  01:36

Hey! 

Kim  01:37

JMoore456. And this person said, "I absolutely love listening to the podcast. I'm a third grade teacher, and I have been trying for years to teach math differently. This podcast is filled with great ideas and great conversation to help any math teacher or parent. Thank you for what you do."

Pam  01:53

Aww.

Kim  01:54

I love that they said for parents too. 

Pam  01:56

Yeah, nice. 

Kim  01:56

You know, a lot of the conversation we have is like maybe directed in the classroom. But I think if I were a parent... Which I am. But if I were a parent who didn't kind of own some mathematical relationships, this would be hugely helpful.

Pam  02:10

Or maybe even a parent who owns mathematical relationships but would like to have a way to share with their kids. 

Kim  02:15

Yeah. 

Pam  02:15

Could be some way to do it. Yeah.

Kim  02:16

Sure. Speaking of parent.

Pam  02:18

Hmm?

Kim  02:19

You know I like to share stories about my children just because it's like my real life, and...

Pam  02:23

Yeah, love your kids.

Kim  02:24

And I... You know, I am super pro teacher. I want to be the biggest super fan. And, you know, do all the like gift buying and school supply shopping. And, in fact, my high school teacher, or my high school kid, I sent emails to his... Oh, I forgot to buy them, though! Shoot! Oh! 

Pam  02:45

Okay, good reminder, good reminder.

Kim  02:47

I sent... 

Pam  02:48

Finish the sentence. What?

Kim  02:50

I sent an email to all of his teachers and said, "Hey, I know, high school, you don't get supplies and things. What do you need? And one of them replied, and I forgot to buy stuff. It's on my list.

Pam  03:01

(unclear) You have plenty of time. 

Kim  03:02

Oh, man. Okay, anywho.

Pam  03:04

Nice. That's very nice of you to do.

Kim  03:06

Okay.

Pam  03:06

You are pro teacher. I like it. Yeah. 

Kim  03:08

Okay. But, you know, there are some times where I feel like I struggle a little bit because I want to be super supportive, but I do struggle with some of the things that happen in classes that I just can't let go. And I... You know, anyway. So, I...

Pam  03:23

It's hard when you know a better way. 

Kim  03:25

Yeah, it is. It is. Anywho, so I had this paper that, and I sent the picture to you, so we could talk about it. But... 

Pam  03:36

I'm pulling it up. Okay. Yeah.

Kim  03:39

So, the topic is exponents. One of my kids... This has been a while, but one of my kids was wrapping up the year, and all of a sudden this paper came home that had... I think it was division of exponents. And it came out of nowhere. It wasn't something that they were working on. And they came home, and I was like, "Wow, interesting. You know, what have you guys been talking about?" And they said, "Well, not really this, but this paper was one that we were supposed to know how to do. It's just a quick review, like at the last week or two of school." And as I sat down with him, it was clear to me that he didn't have a clue what, how to tackle any of these problems. And it just struck me because when I looked at the paper, there were so many layers for different kinds of problems that it became one of those things where I felt like, oh, this is a situation where if you mimic a rule, then maybe you could solve a bunch of these problems. But if you're trying to make sense of what's happening, it's just a lot all at once. And... 

Pam  04:50

A lot of things happening simultaneously in a problem to make sense of. 

Kim  04:54

Yeah. And so, just... It was one of those situations where I thought, "Okay, I'm in a pickle here because I want my kids to think and reason and make sense of this, so they don't have to like relearn it next year, or the next year, or the next year. And by "relearn", I mean "re-memorize". 

Pam  05:11

Yeah. Or think that math isn't figure-out-able. 

Pam and Kim  05:13

Right. 

Pam  05:14

You don't want either of those. Yeah. 

Kim  05:15

Right.

Pam  05:15

Yeah.

Kim  05:15

So, I sent you a picture, and I'd love your take on am I overreacting or is this... This felt like a lot of different kinds of problems. Maybe you could describe. The top of the paper says Exponents - Quotient Rule.

Pam  05:28

Yeah, so when I'm looking at this, first of all, there's a lot of problems. And I can tell there's a backside, so it's not just one side of the paper. 

Kim  05:38

Yeah. 

Pam  05:39

And so, it's a lot of numbers raised to powers and numbers raised to powers divided by numbers raised to powers. Or there's some variables as the power. There mostly as the base. There's fractions raised to powers. There's negative exponents, positive exponents. And I can completely hear how a teacher might say, "Well, yeah, yeah, yeah. It's quotient rule. You're dividing powers, you're dividing exponents, and so there's a rule, and you just apply the rule, and then you're good to go." You could do that. Or, you know like, is it figure-out-able? And I can also hear teachers say, "Pam, Pam, Pam. There's way too much to like think about. You know, we give kids some conceptual understanding, then we give them the rule, and we just tell them to go, and then they should be able to solve all this." So, maybe a thing to mention, I think you were telling me, with the grade that your student was in, your kid was in.

Kim  06:30

Mmhm.

Pam  06:31

So, this seems a little early for that. So, typically, negative exponents don't show up until at least Algebra 1, even Algebra 2, second year of algebra. So, this was before that. So, it's kind of a little ironic that they're saying, "So, here. Review. You should be able to know how to do this." I can... Like, we both said, if it's just a rule, I can see how somebody might say, "You know, just go apply this." But wow, there's a lot. There's a lot to unpack. And as we were talking about this episode today, I'm thinking, "What is it like to plan?" Could we plan a sequence of tasks that could actually get kids to these problems. Like, right here, one of them is 7.9^-x, all divided by 7.9. Like, there's a lot happening in that problem because you've got decimals. You've got a negative exponent. You've got an exponent that's a variable. A lot of things are sort of happening. That particular one was find the value of x. There were other ones that were just simplify. I'm just kind of looking. Yeah, there's just...

Kim  06:49

Mmhm.  There's a lot.

Pam  07:47

Yeah, a lot. And, you know, one of the things that we talk about in the development of mathematical reasoning is that as you go out in those hierarchical ovals, right? They contain each other. They build on each other. One of the things that happens as we go further to the right and further... How do I say that? In the ovals that kind of eat each other. Is that more things are happening simultaneously, and so you want to be able to kind of cinch some things, so that that becomes something you can now have a... If you could see my hands, I'm kind of making like a... How do I describe what I'm making? Like, I'm putting my hands together, so it's like. It's like a thing. It's like, now, I've got this as a thing that I own, and so now I don't have to necessarily deal with all of the nuances in that. But I own that thing, so that if I need to go into it to deal with the nuances, I can, but I can also then use that as kind of a thing with other things. I bring those things together, and now those become a thing that now... And I sort of create different schema. But I think the point we're trying to make is that the schema is not just a rule that I've memorized. It's actually a schema that I understand, and I can go into if I need to. Alright, so (unclear).

Kim  08:59

(unclear) say one more thing.

Pam  09:00

Yeah, yeah.

Kim  09:01

Because I certainly don't see every assignment my people bring home. I don't have time for that. But I do see them on occasion. And what troubles me a little bit is with an assignment like this that has so many things happening, so many layers, students sometimes think "Because this doesn't make sense to me, or because I'm not sure about all these simultaneous things that are happening, I'm not good at math. I am having a hard time with this paper, so it's about me." And sometimes, it's the ask. Sometimes, it's not recognizing how many different things are being asked at the same time. So, like I go grab a book that has a paper I copy it. Oh, it says (unclear) at the top, so, you know, maybe it's okay. Anyway, I think sometimes we're giving kids the impression that it's about them and what they are good or not good at. And maybe the ask is not quite appropriate.

Pam  10:02

Meaning, that you might grab a sheet. It might behoove you to actually look deeply at the problems to make sure that the... Yeah. Yeah.

Kim  10:11

Yeah. 

Pam  10:11

Yeah, point taken. So, what would it be like to create a sequence of tasks that can get kids to actually be able to reason through problems like 7^11 divided by 7^-4, and without just a, you know like, "Wait, which rule is that?" and applying some rule. Actually being able to sort of think through that if they need to reason about what's happening, so that then when they deal with, say, an exponential function, they're actually thinking exponentially. Like, they can actually reason exponentially about what's happening. What could that look like? Well, I've actually created a sequence of tasks like that. I did some work with the state of Texas awhile back, and also when we wrote Discovering Advanced Algebra, the third edition. We really thought hard about what would it be like to help kids really think through, build? Maybe what I should say is build their reasoning, so that they can actually reason exponentially. So, Kim, let's start today with a lovely Problem String. 

Kim  11:14

Okie doke. 

Pam  11:15

That maybe you could start. And one of my favorite things to do, Kim...I don't know if I've said this on the podcast...is to think about...I've been doing this more and more lately...think about a task that I could do in high school, but then back it up a little bit and say, "Where, ideally, could I have started?" And then create a Problem String that could start maybe even in third grade, that a third grade teacher could do, Mmhm.  But it could also be a Problem String that the fourth grade teacher could do and take it a little further, and the fifth grade teacher could do and take it a little further, and the algebra teacher could do and take it a little further. 

Kim  11:47

Mmhm.

Pam  11:47

So, let's do one like that today.

Kim  11:50

Okie doke.

Pam  11:50

(unclear). Okay, so if I asked you 8 times 6. I'm going to assume that you can think about that or figure that out but the question I kind of have is what would that look like if I was to draw...Oh, I'm sorry. Yeah, yeah, 8 times 6...if I were to draw an array. So, first of all, what is 8 times 6?

Kim  12:05

48. 

Pam  12:06

And then what would the array look like?

Kim  12:08

It's going to be longer than it is wide. 

Pam  12:11

Okay. 

Kim  12:12

So, it's going to be 8 down, and then I'm going to go 6 to the right.

Pam  12:16

Cool.

Kim  12:16

(unclear).

Pam  12:17

And the area's 48. And I would not spend very much time on that. Then I would move on, and I would say, "Okay, next problem." How about 4 times 12?

Kim  12:23

Okay, it's going to be 48 also. 

Pam  12:25

Okay. How do you know? 

Kim  12:28

How do I know? 

Pam  12:28

Do you just know that one? 

Kim  12:29

Yeah.

Pam  12:30

Okay, is there anything you could use if you didn't just know that one? 

Kim  12:33

Well, I was starting to sketch.

Pam  12:35

Okay, okay. 

Kim  12:36

And I looked at the problem above.

Pam  12:38

Mmhm. 

Kim  12:38

And so, it would be half as long. So, instead of 8, it's only 4. But where it was 6 wide before, now it's going to be 12 wide, so it's twice as long. So, half as long, one dimension. Half as long, twice as wide. 

Pam  12:54

Half as tall. 

Kim  12:55

Half as tall. 

Pam  12:55

Which is twice as wide? 

Pam and Kim  12:56

Yeah.

Pam  12:57

Cool. And so, you didn't lose any area. And we could talk about like cutting that top rectangle kind of where you did. You know like, it was only half as deep, half as tall, and it can move that section over. And we didn't really lose any area, but we created a rectangle that was half as deep and twice as long. Same area. Okay, so I'm going to write a 48 in that rectangle. And then I would probably say, so you're telling me that you could halve the 8, and between the top equation 8 times 6, and the bottom equation, 4 times 12, from the 8 to the 4, I'm going to actually do a sort of scaling loop. Like, almost looks like a big parentheses kind of, arrow.

Kim  13:34

Mmhm. 

Pam  13:34

And I'm going to say divided by 2, and then I'm going to do that same kind of loop from the 6 to the 12, and I'm going to put times 2. So, I've got 8 divided by 2 is four. 6 times 2 is 12. So, if you divide one factor by 2, multiply one factor by 2, then you get the same area. Cool. Next problem. How about 3 times 16? 

Kim  13:53

Okay. 

Pam  13:54

(unclear) are you laughing about?

Kim  13:55

Sometimes when I don't know like what you want to do or where you're going, and I'm just kind of... I don't want to ruin what you're doing. 

Pam  14:04

And what you're saying is your brain just went fifteen places, and you were like, "Hmm, I wonder which (unclear) go.

Kim  14:09

(unclear).

Pam  14:09

Yeah, mmhm. Okay, alright. 

Kim  14:11

(unclear) That's also 48.

Pam  14:12

Okay, 48. Is there one of the rectangles that we have on our papers right now... We have an 8 by 6 and a 4 by 12. Could one of those be related to the 3 by 16? Or maybe both of them? 

Kim  14:21

Yeah, yeah. 

Pam  14:22

Pick one.

Kim  14:23

The 4 by 12. 

Pam  14:24

Okay, how?

Kim  14:26

3, the length of 3.

Pam  14:29

Mmhm.

Kim  14:30

Is three-fourths of 4.

Pam  14:33

Whoa. Okay. Not where I thought you were going to go.

Kim  14:37

See, yeah. And then 12 is three-fourths of 16.

Pam  14:41

Nice. Okay, so I might, if I was doing this with a third or fourth grade class, go, "Thanks, Kim. That's really cool. Did anybody think of going from the 12 to the 3?" 

Kim  14:50

Yeah. 

Pam  14:51

So, that was a very middle school thing that you just did.

Kim  14:54

Mmhm. 

Pam  14:54

And if I was in third or fourth grade, I wouldn't do it. If I was in middle school, I might not do it as the first strategy either because that's pretty... 

Kim  15:00

Yeah.

Pam  15:01

Pretty intense. 

Kim  15:02

It's funny that you say that because when you were like, "Yeah, let's not go to that one," I went up to the first one. I totally did not even see that (unclear).

Pam  15:10

Let's actually do the first one. I think that's a better choice. Yeah. (unclear). 

Kim  15:12

Okay.

Pam  15:13

 So, from the 8 by 6, how are you getting 3 by 16? 

Kim  15:17

The 3 is half of the 6. And the 16 is double the 8. So, it's like the 8 by 6 was turned. 

Pam  15:25

So, I'm actually going to do that on my paper, and I'm going to create up by the 8 by 6. I'm going to put a 6 by 8. 

Kim  15:30

Okay. 

Pam  15:31

Okay. And then now that I have a 6 by 8?

Kim  15:34

The 6 is double the 3. 

Pam  15:36

So, I'll cut that 6 in half to get 3. 

Kim  15:38

Mmhm. And then 8 is... 16 is twice as much as 8.

Pam  15:43

And then I'll double that 8 to get 16. Now, I have a 3 by 16, and I did it from that 6 by 8. And since the 6 by 8 had a 48, I cut up the 6 by 8.  Didn't lose any area. Then the 3 by 16 is also going to have an area of 48. Cool. We could also look at the relationship between the 3 times 16 and the 4 times 12, and we'll let the readers do that. So, but for today on the podcast, I'm just going to keep going. Next problem. How about 2 times 24?

Kim  16:12

Mmhm. 

Pam  16:13

So, 2 times 24, I'm probably not going to expect you to have used. Like, 3 times 16, often kids are going to actually want to use one of the other problems because they don't know 3 times 16. For 2 times 24, I think a lot of kids will just double 24, and they'll say, "Well, double 24 is 48." I'll say, "What would that look like? And is it related to any..." Is there one up there, Kim? Right now, we have an 8 by 6, a 4 by 12, a 3 by 16. Is there one that's just like super obvious?

Kim  16:39

Yeah, the 4 by 12. 

Pam  16:40

How? 

Kim  16:42

2 is half a 4 and 24 is double 12.

Pam  16:46

So, I could take that 4 by 12, make it not as deep. It's only 2. The 4 becomes 2, and the 12 doubles to get 24. Now I have a 2 by 24. And so, I didn't lose any area. That also has 48. I'm also going to do some of those scaling marks over by the equations where, again, I'm going to have from the 4 to the 2 divided by 2, and from the 12 to the 24 times two. Papers getting a little kind of messy here at this point. The next problem that I'm going to ask you... Well, first of all, I'm going to note that all of these were equivalent to 48. That's interesting. We could just move area around these rectangles. Depending on kind of where I am, I might notice that we did a lot of Halving one dimension and Doubling the other dimension, and that that seemed kind of helpful. But for where we're going to go today... So, we could definitely bring that up. But for where we're going to go today, I'm not going to really hammer that too much. I am going to ask another question. And the next question will be 2 times 2 times 2 times 2 times 3. 

Kim  17:42

Okay, say that again. 

Pam  17:43

So, four 2s, four factors of 2 times 3. 

Kim  17:47

Okay.

Pam  17:48

And I'm just going to ask, "Alright, guys. What's that?" And so, Kim, what's that?

Kim  17:54

It is also 48.

Pam  17:56

Okay, and how'd you do it?

Kim  17:58

I did all the twos first. And then...

Pam  18:04

(unclear). Oh, go ahead. Well, actually, when you said that, I would probably interrupt you, and I would say, "Oh, I'm just going to show that by putting parentheses around the twos." So, I'm going to put parenthesis around the first two on the left parenthesis. And then on the last two, a right parenthesis. So, now on my paper, I have the twos in a set of parentheses times 3.

Kim  18:22

Mmhm. 

Pam  18:22

And that's kind of like, if we look up at the problems that we've done before, that's kind of like the 3 times 16. Just the community property. We've reversed the factors.

Kim  18:31

Mmhm. 

Pam  18:31

Because 2 times 2 times 2 times 2, the four 2s, is 16, times 3. So, it's kind of like you were like, "Hey, I'm just going to figure it out like we did that problem." Cool. Then I'm going to suggest, could you put those four 2s, 2 times 2 times 2 times 2 times 3, could you write that same expression for each of these 48s? So, for example, for the 8 times 6.

Kim  18:54

Mmhm. 

Pam  18:55

If I put 2 times 2 times 2...I'm writing it actually...times 2 times 3, next to that 8 times 6 equals 48, where would you put the parentheses to mean 8 times 6? Instead of like you did. You did all the 4s... Or, excuse me, all the 2s to get 16. Does that make sense? 

Kim  19:11

Yeah. 

Pam  19:12

Okay, how would you do that for the 8 times 6?

Kim  19:14

I would leave the first three 2s out of the parentheses.

Pam  19:18

Okay. 

Kim  19:19

I might put parentheses around the last 2 times 3. 

Pam  19:22

Cool. And I might put two sets of parentheses around the 2s, around the first 2s. Sorry. 2 times 2 times 2. That's 8. 

Kim  19:30

Okay. 

Pam  19:30

And then another set of parentheses around the 2 times 3. Either way.

Kim  19:33

Yeah. 

Pam  19:34

So, now that I have those two sets of parentheses, I can like... There's 8. It's like jumping at me. 2 times 2 times 2. And the 6 is jumping at me 2 times 3. Cool. So, how about the next one? 4 times 12. Where would you put parentheses if you had those 2 times 2 times 2 times 2 times 3. And maybe, listeners, you might pause and think to yourself where would parentheses go to have you sort of scream 4 times 12?

Kim  19:58

The first set of parentheses would be around the first two 2s. 

Pam  20:02

Okay. 2 times 2 (unclear).

Kim  20:03

Yeah. And then the next set of parentheses would be around 2 times 2 times 3.

Pam  20:09

Because 2 times 2 times 3 is? 

Kim  20:11

12.

Pam  20:12

12. Nice. Cool. And then how about the 3 times 16? Oh, we already did that one. How about the 2 times 24?

Kim  20:19

Yeah, I would put the parentheses around everything except for the first 2.

Pam  20:26

Because 2 times 2 times 2.... Wait, sorry. (unclear).

Kim  20:28

2 times 2 times 2. 

Pam  20:30

Times 3 is?

Kim  20:32

24.

Pam  20:32

24. So, that's kind of cool. And now, we have kind of this, I don't know, sort of way of thinking about prime factorization a little bit. Then I might go back and say, "Hey, did you know that..." And listeners, you probably felt this (unclear), 2 times 2 times 2 times 2. Like, is there a way that we could write that? Mathematicians often seek to be efficient. I mean lazy. I mean efficient. And so, they often seek to write things more efficiently. So, when I had that 2 times 2 times... In the very first problem, 8 times 6. You just said that we could do 2 times 2 times 2 in parentheses.

Kim  21:08

Mmhm. 

Pam  21:08

Mathematicians might write that as 2 raised to the third power.

Kim  21:12

Mmhm. 

Pam  21:13

And then 2 times 3, they would just sort of leave alone. So, I could have 2 times... Sorry, 2^3 times 2 times 3. What about 4 times 12? Mathematicians might write that as 2^2 because it's 2 times 2 for 4. And then they have the leftover 2 times 2 times 3. They might write that as 2^2 times 3. Or I could look at the 3 times 16. Mathematicians might write that as 3 times... And then maybe I'd ask the students. 16 was 2 times 2 times 2 times 2, right? Then, how would we write that, Kim?

Kim  21:52

2^4. 

Pam  21:53

2^4.  And then lastly, for the 2 times 24, how might we write that one?

Kim  22:00

Just what's in parentheses?

Pam  22:03

I would probably write...

Kim  22:04

(unclear) 2 times...

Pam  22:05

Yeah, there you go. Yep. 

Kim  22:07

2^3 times 3. 

Pam  22:09

So, now we have lots of expressions here with exponents in them that all are equivalent to 48, that we've made sense of multiplicatively. And we could sort of say, "Okay, well, let me give you one more." What if I gave you 2^4 times 5?

Kim  22:27

Mmhm.

Pam  22:28

So, 2^4 times 5? What does that mean? And you could just tell me sort of what it would look like with multiplication. You could actually multiply it out and give me the whole number. And I would look for both of those. So, if I just wanted to write that with a whole bunch of twos. What would that look like? 

Kim  22:44

2 times 2 times 2 times 2 times 5.

Pam  22:47

Cool. And then any idea what that is?

Kim  22:52

Mmhm. That's 80. 

Pam  22:53

How do you know? 

Kim  22:54

Because 2 times 2 times 2 is 8. And 2 times 5 is 10. So, it's 8 times 10. 

Pam  23:01

Ah, very nice. So, like instead of doing the 16 and the four 2s to get 16...

Kim  23:08

Mmhm.

Pam  23:08

...times 5, and then having to think about what 16 times 5 is, you grab the 2 times 5 to make a 10. That's very, very slick, very clever. Cool. And then the point of this Problem String is, wow, exponents are multiplicative. Like, we can represent repeated multiplication as an exponent, and we can look at exponentiation. We can look at an expression that is written with an exponent, and we can think multiplication. We can be really clear how those are connected. If I'm a high school teacher, I've also just sort of... If you could see my hands, I'm making this, like... I can't describe what I'm doing today. This shout out motion, like, ping, ping, ping. I often do this when I'm in person. Like, ping. You're like saying to students, "Hey, there's this Doubling and Halving strategy. In case you had never thought about it, you could Double and Halve to solve a multiplication problem. You can represent multiplication as the area of an array, and we can represent repeated multiplication as exponents and you can think about exponents as repeated multiplication. And I think you could do this all in about eight minutes in an Algebra 1 or higher class. 

Kim  24:12

Yeah. And I think that's important that you just said because a lot of times older grade teachers say, "My kids, you know, are struggling with..." blank, and they want to know how to wrap in some of the earlier work into the work they're doing, and this is a really nice example. 

Pam  24:27

And we're not done. Like, we haven't done a ton with exponents. But the question we started with was could we have a series of tasks that could build towards negative exponents and division with exponents? And I think this could be a start. And, hey, Kim, how about next week, let's do another one. 

Kim  24:43

Yeah. (unclear)

Pam  24:44

And maybe we'll do a series here where we'll do a series of tasks that could get kids, and maybe teachers, thinking more and more multiplicatively and understandably about exponents.

Kim  24:56

You'll be writing lessons for people. 

Pam  24:58

Bam. 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. Let's keep spreading the word that Math is Figure-Out-Able!