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  00:09

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

Pam  00:20

We know that algorithms are amazing achievements, but they're not good teaching tools because, ya'll, mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop. That was me, by the way, I got trapped. 

Kim  00:35

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

Pam  00:41

We invite you to join us to make math more figure-out-able. Alright, Kimberly.

Kim  00:46

Yes. 

Pam  00:48

You're supposed to say "Pamela".

Kim  00:49

Pamela. I do say that a lot.

Pam  00:52

You do. Yeah. Sometimes it's when you're frustrated with me. "Pamela!"

Kim  00:57

My exasperated tone.

Pam  00:59

I think most often it's when you're like, "Pamela."

Kim  01:01

Yeah.

Pam  01:02

Yeah, love it. 

Kim  01:03

Like, when I greet you somehow.

Pam  01:04

Yeah.

Kim  01:05

I don't see you as much anymore.

Pam  01:07

Yeah, usually you say "hola" 

Pam and Kim  01:08

Hola. 

Pam  01:10

Hola. "Hola, Pamela." Yeah, there you go. Alright, you guys know way too much about Kim and me. There you go. Alright, Kim.

Kim  01:16

Tara Deanne sent us a review.

Pam  01:20

Aww.

Kim  01:20

Yeah. And the title is so cute, "It said I needed this in my life." 

Pam  01:24

Aww!

Kim  01:25

Aww.

Pam  01:25

Thanks, Tara Deanne.

Kim  01:26

Yeah. She says, "I love this podcast, and have already..."

Pam  01:29

Hey, and five stars I'm seeing. 

Kim  01:31

Yeah. 

Pam  01:32

Five. Thanks, Tara. Deanne. Okay, sorry. Keep going. 

Kim  01:34

"I love this podcast and have already felt a change in my math teaching." Aww. "Pam and Kim break it down in real world situations and cover topics K-12. I suggest to anyone just finding the podcast to start at episode 156, literally called 'start here'. Pam and Kim, you're amazing, and you make my Tuesdays brighter."

Pam  01:54

Aww! So, our "start here" worked. That's awesome. Nice! 

Kim  01:57

Yeah.

Pam  01:58

Nice. 

Kim  01:58

"Start here" is a great episode for people who haven't really spent a lot of time in the podcast because it gives them some kind of underpinnings of the Math is Figure-Out-Able movement.

Pam  02:08

Okay, okay. 156. 156, ya'll. Check it out.

Kim  02:11

Yeah. 

Pam  02:12

And maybe that's the one to pass on to your favorite person who you're trying to get to listen to the podcast. Tell them to start at 156. In fact, if you're starting listening to this podcast with today's episode, you might consider backing up a couple of episodes because we are in the fifth of a series that we are doing on exponent relationships. And we're kind of building off of what we've done before. So, highly recommend that you check out the four episodes that have come before this to really get the most out of today's episode with exponent relationships. Kim, we're going to have some fun today. Sit up tall. 

Kim  02:48

Oka12/12/249:45y. 

Pam  02:48

Take a deep breath. Grab your pencil. 

Kim  02:51

Yes, I have it. 

Pam  02:51

I have my pen in my hand right now. Are you ready? 

Kim  02:56

Yep.

Pam  02:56

Okay. Today's Problem String, I'm going to actually tell you what the first problem looks like before I say it. Okay? 

Kim  03:03

Okay. 

Pam  03:03

So, what it looks like is 2, divided by, but the divided by is a fraction bar. So, 2 "fraction bar". Sort of not mathematical, but over. It's going to look like over. So, 2, big fraction bar, and then 2 times 2 times 2. And we just want to simplify this expression. 

Kim  03:21

Okay. 

Pam  03:22

Which really means we kind of want to divide out any common factors. 

Kim  03:27

Okay. 

Pam  03:27

So, "simplify" can often mean different things. Today, it means to divide out any common factors. Alright, so what are you thinking about? Maybe if you could tell us what you're thinking about instead of just the answer, that would be great.

Kim  03:39

Yeah, so there's a 2 divided by 2 because there's one in the numerator, and I used one of the 2 in the denominator. So, I actually wrote 1 divided by 2^2.

Pam  03:56

That's your sort of answer? (unclear).

Kim  03:59

I wrote two things, but that's the... Yeah, that's one of the things I wrote. 

Pam  04:01

Okay, so I'm going to kind of try to write out what you said. I heard you say that there was one 2 in the numerator and there was a 2 in the denominator. So, I'm going to write 2 divided by 2. But then you had some leftover stuff, so then I'm going to... I wrote the times symbol with the dot. And next to that, I wrote 1 divided by 2 times 2.

Kim  04:22

Mmhm. 

Pam  04:23

So, that first 2 divided by 2 divides to 1. 2 divided by 2 is 1. Times. And then that's where I think you had your 1 divided by 2^2.

Kim  04:32

Mmhm. 

Pam  04:32

And 2^2 is? 

Kim  04:34

4. 

Pam  04:34

4. So, you could write that as 1 divided by 4.

Kim  04:37

Mmhm. 

Pam  04:38

Cool. I'm super curious. Did you ever, as we were doing this, get to 0.25? 

Kim  04:44

No.

Pam  04:46

It's not in my lesson plan. 

Kim  04:48

Yeah (unclear). 

Pam  04:49

And I don't know that I would have thought about it, except I was talking to you. Did you think 25%? (unclear).

Kim  04:55

No, I totally stayed in factions. 

Pam  04:58

Okay, cool. We've got this. 2 divided by 2 times 2 times 2 equaling one-fourth. I might then say, "Could we write the original problem in terms of exponents?"

Kim  05:10

Mmhm.

Pam  05:10

So, that 2, I'm going to write as 2^1. And that 2 times 2 times 2 is 2^3. Okay, so now we've got 2^1 divided by 2^3 equals 1 divided by 2^2. Because you said that earlier. And I'm just going to kind of pull that over, so that kind of at the end of my thing here, I've got 2^1 divided by 2^3 equals 1 divided by 2^2.

Kim  05:37

Mmhm. 

Pam  05:38

And then to make sense of all the things, last week, we talked about how when you're dividing, we really want to kind of think, if I've got a bunch of the same base multiplied together in the numerator and a bunch of the same base multiplied together in the denominator, where do we have the extras? Because they sort of divide each other out. This 2 divides out that 2 to 1. This 2 divides out that 2 to 1. In this case, we just had one 2 divided by another 2. But if you have 2^1 divided by 2^3 where do you have the extras? 

Kim  06:10

In the denominator. 

Pam  06:11

In the denominator. So, today we've got 1 divided by 2^2, and we're kind of... There's the first problem. Alright, next problem. How about 3^2 divided by 3^3?

Kim  06:25

That's 3 times 3 in the numerator, and 3 times 3 times 3 in the denominator. So, then when I divide those out, I have one 3 in the denominator. So, I wrote 1/3. 

Pam  06:41

1/3. And that makes a lot of sense. Sure. With that 1/3 that's 1 divided by 3. Can I put a an exponent of 1 on that 3? So, it's sort of like 1 divided by 3^1? 

Kim  06:55

Sure? 

Pam  06:56

I mean, it's not maybe necessary. But I'm just going to stick it there anyway.

Pam and Kim  07:00

Okay.

Pam  07:01

Because if we have any number just sitting there, it is to the first power. 

Kim  07:04

Okay. 

Pam  07:05

Okay, cool. Next problem. How about 5^5 divided by 5^3. Actually, I meant to do one more thing. Sorry. Can we go back to the 3^2 divided by 3^3? I love how you wrote out the factors. What if you multiplied it out? What if you multiplied 3^2 out? 3^2 is?

Kim  07:30

27. Oh, 3^2? So, it would be 9/27.

Pam  07:35

9/27. And is 9/27 also 1/3? 

Kim  07:38

Yeah. 

Pam  07:39

Okay, cool. I just wanted to do that as well. Okay, so now we're at 5^5 divided by 5^3. What are you thinking?

Kim  07:49

I think that's 25 because I have two extra 5s multiplied in the numerator.

Pam  07:58

Cool. So, we could write out 5 times 5 times 5 times 5 times 5 in the numerator. Five 5s in the numerator. Divided by three of those 5s multiplied together in the denominator. And then we could line those up. And you're saying that you would have two left over 5s in the numerator?

Kim  08:18

Mmhm.

Pam  08:18

Multiplied together. 

Kim  08:19

Yeah. 

Pam  08:19

So, that's like 5^2. So, on my paper, I have all that written, but I might now kind of pull attention to the 5^5 divided by 5^3 equals 5^2.

Kim  08:32

Mmhm. 

Pam  08:33

And I might say, "Hey, does anybody remember that from last time?"

Kim  08:37

Mmhm. 

Pam  08:37

So I think last time we talked about this idea. Where do I have leftover factors? And that we could sort of write that 5^2 as 5^5-3. 5^5-3 is 5^2. Yeah?

Kim  08:54

Yep. 

Pam  08:54

So, with that in mind, now that we've kind of remembered that from when we've done this before, if we look back at the first problem. So, the very first problem, we had 2 divided by 2^3. Or 2^1 divided by 2^3. Can we also write that as subtraction?

Kim  09:17

Mmhm. (unclear).

Pam  09:17

As subtracting the exponents? Yeah, go ahead.

Kim  09:19

That would be 2^-2.

Pam  09:22

What does that mean? And let me just be really clear what's on my board right now. So, I've gone back to the first problem, and I'm focused on the last part. I have a series of equations. And I'm kind of almost might even circle this last part, where I've got 2^1 divided by, all over, fraction bar, 2^3 equals 1 divided by 2^2. And we landed on that when we did the first problem. Now, I've got equals 2^-2 because we kind of did the 1 minus 3. So, 2^-2. And now, I'm going to step back and I'm going to go, "What?" and let kids just kind of absorb that a little bit. And then ask, "What does 2^-2 mean? What does it mean to have a -2 exponent? And, Kim, by jove, kids will just go, "Well, it means exactly what we have there. It means it's 1 divided by 2^2." Instead of us telling kids, we can actually do a lot of work to say like, "Here it is. Make sense of that." And mathematicians had to do the exact same thing. Mathematicians had to go, "Could that work?" Like, if we say that we can subtract exponents when we're dividing the bases, like bases, then we're going to get negative exponents. And if that's true, will the rest of the math carry through? Can we continue to do things and that will actually work? And sure enough, it does. So, now all of a sudden, we've got some negative exponents to deal with. That's kind of interesting. Let's go to problem two. Problem two, we had 3^2 divided by 3^3, and we ended up with 1/3 or 1 divided by 3^1. Could we also write that as (unclear).

Kim  11:12

3^-1.

Pam  11:13

3^-1? Sure enough. Cool. Next problem. How about 2^3 divided by 2^5 2^3 divided by 2^5. Okay. I'm going to take a drink while you think.

Kim  11:29

I've got 1 divided by 2^2? 

Pam  11:34

Why? 

Kim  11:35

You said 2^3 divided by 2^5? 

Pam  11:37

Mmhm, yeah. And I like your 1 divided by 2^2. But can you tell (unclear).

Kim  11:40

(unclear). Yeah, because there's two extra 2s multiplied together in the bottom.  In the denominator.

Pam and Kim  11:46

Mmhm. 

Kim  11:47

Okay. And so, I'm going to put 1/4, but I'm also writing 2^-2.

Pam  11:55

Bam. And then we would want to let that sit and make sure kids are looking at it and feeling comfortable with it. And at least as comfortable as they can with it.

Kim  12:06

Mmhm. 

Pam  12:06

Yeah. Nice. How about... If we were to go back to problem one, one more time. I don't know if this is too crazy because we're on a podcast, but I'm going to try it anyway. Problem one, we had 2^1 divided by 2^3, and we ended up, just a minute ago, with 1 divided by 2^2 or 2^-2. 

Kim  12:28

Mmhm.

Pam  12:29

But we also said that was 1/4.

Kim  12:31

Mmhm. 

Pam  12:33

Could we write that as 4^-1? 4^-1? Mmhm. So, 1/4, 1 divided by 4^1.

Kim  12:51

1 divided by 4^1. What did you ask them to write it like?

Pam  12:56

I'm wondering if we can write that as 4^-1.

Kim  13:00

Yes.

Pam  13:01

That's weird, right? 

Kim  13:02

Yeah.

Pam  13:03

That's a little strange, but kind of cool.

Kim  13:06

Mmhm. 

Pam  13:07

Notice, that this is an opportunity where I could say, "But did anybody actually do the math? Like, did anybody actually do the 2^3?" What is 2^3? 4. Cubed. 2^3. 2^3.

Kim  13:19

Oh, 2^3. 8. 

Pam  13:20

8. And then what is... So, I just wrote 8 in the numerator. And what is 2^5. Let me actually do it. So 2, 8. 8 is 2^3. So, times 2 is 16 is 2^4. So, times 2 is 32? Is that right? 32? 

Kim  13:33

So, 8/32.

Pam  13:34

Yeah. So, is 8/32 also 1/4?

Kim  13:37

Mmhm. 

Pam  13:38

Yeah. Okay, so we've just... We have the ability here to use the numbers to, kind of like play out, is this actually tracking the way that we're thinking about all these guys?

Kim  13:48

Mmhm.

Pam  13:49

Cool. 

Kim  13:49

And this is a little bit like you're attaching the way things are recorded to an experience that they're having.

Pam  13:59

Yeah. Yes, that is the attempt.

Kim  14:01

Yeah. 

Pam  14:01

The attempt is to have the experience with the numbers and what they've been learning with exponents, and then saying, "Alright, with what you're doing, check it out. We can write it this way. When your brain does that, it can look like this. When your brain does that, it can look like this negative exponent. Super weird." We said last time that we could subtract exponents. Yeah, but that made sense because we had more in the numerator than in the denominator, so we just had a bunch left over in the numerator. Well, today we're having more in the denominator, more factors in the denominators of the same base than we have in the numerator, so we can leave them in the denominator. Or, weird, if we can do this subtracting exponent thing, we can do the subtracting exponent thing and get negative exponents. What?! What does that mean? Well, when we have a base in the denominator, we can write it in the numerator with a negative exponent. And that's crazy. That's a little interesting.

Kim  14:57

Mmhm. 

Pam  14:58

Let's see if we can make sense of that in a problem that kids might run into. Like, what's 5^2 times 2^2, all divided by 10^3.

Kim  15:07

Mmm, mmhm. So, in the numerator, you have 5 times 5 times 2 times 2.

Pam  15:16

So, I've just written that down. 5 times 5 times 2 times 2. Divided by?

Kim  15:19

10 times 10 times 10. 

Pam  15:21

Okay.

Kim  15:22

But when you pair up a 5 and 2 in the numerator, you have two 10s in the numerator. 

Pam  15:30

Times each other, mmhm.

Kim  15:31

Mmhm. And then three 10s times each other in the denominator.

Pam  15:35

So, I've just written 10 times 10 in the numerator and 10 times 10 times 10 again in the in in the denominator. Mmhm.

Kim  15:42

So, you have 1/10 or 10^-1. 

Pam  15:47

Or 10^-1. That is a way that we could do that. And that's probably enough for our purposes today. I'm thinking, I'm thinking. The other thing we could do is do the number thing like we've kind of talked about. Number thing. Like, actually multiply things out.

Kim  16:09

Mmhm. 

Pam  16:09

Is there a way that you would want to multiply out the 5^2 times 2^2? Do you just kind of want to do it like it said? Or do you mind if I just do it like it said?

Kim  16:17

Yeah, go ahead. 

Pam  16:17

So, I might find a kid who said, "Yeah, 5^2 is 25, and 2^2 is 4." So, in the numerator, I've got 25 times 4. That's just 100. So, I've got 100 in the numerator. And then, 10^3. That's 10 times 10 is 100. Times 10 is 1,000. So, I've got 100 in the numerator and 1,000 in the denominator. 100 divided by 1,000 is that just 1/10? Sure enough. That could be another way of thinking about that. Or like you said, 10^1. One other way that we could, as I'm looking at what we have here. When you said that you had 10 times 10 in the numerator, could we write that as 10^2?

Kim  16:55

Mmhm. 

Pam  16:57

And then, 10 times 10 times 10 in the denominator was 10^3.

Kim  17:02

Mmhm. 

Pam  17:02

So, we've got 10^2 in the numerator, 10^3 in the denominator. There's another way that we're looking at 1 divided by 10, 10^1. Or that 10^2 divided by 10^3, that 2 minus 3, we're looking at 10^-1 in a different way. 

Kim  17:18

Yeah. 

Pam  17:19

So, lots of different ways that we can kind of come at. Sure enough we could. And, ya'll, I really want to impress that nowhere in here have we said, "Oh, and then that's the answer." 

Kim  17:30

Right. 

Pam  17:30

In a big way, what we've done is we found lots of equivalent expressions.

Kim  17:34

Mmhm. 

Pam  17:35

That's a big deal. So, high school teachers, I invite you to consider that we have often said there is a correct answer, when really what we've meant is "write the expression with only positive exponents".

Kim  17:48

Mmhm. 

Pam  17:49

Or "write the expression with nothing in the denominator".

Kim  17:52

Mmhm. 

Pam  17:53

And I think we have to be... I'm going to invite us to be more clear on that. Let's be more clear on what we actually mean because the last thing I think we want is for kids to do the math correctly, get a correct equivalent expression, but because they didn't write it in the form that you meant when you said "simplify" or when you said "do it" or whatever, then they're getting marked wrong.

Kim  18:13

Mmhm! 

Pam  18:13

We never want to mark wrong a correct equivalent expression just because we didn't guess at the one that you meant.

Kim  18:19

Right, right.

Pam  18:20

Yeah. So, let's keep equivalence meaning equivalence. Okay, so I might end this Problem String with a couple more generalizing problems. So, I might say something like, what is 2^a divided by 2^b. And I've written that in that big fraction form. So, 2^a "fraction bar" 2^b. Kim, how are you thinking about that right now? 

Kim  18:38

I think it's the same as it was a string or two ago. It's still 2^a-b.

Pam  18:47

Okay. 

Kim  18:47

But now we've added to our experience that maybe b is bigger.

Pam  18:54

Mmm. That there's more twos in the denominator than in the numerator?

Kim  18:58

Mmhm. 

Pam  18:59

Yeah, sure enough. So, is it kind of... Could we also say that we're looking at a bunch of 2s multiplied together in the numerator, and they're a of them. And they're bunch of 2s multiplied in the denominator. There's b of them. And they're going to... Where they line up and divide out to 1, we're going to have some left over. And we're not sure where, but it's going to be the difference between the 2s in the... How do I say that? The number of 2s in the numerator and the number of 2s in the denominator. Wherever they're left over, they're going to be either a bunch of leftover in the numerator or a bunch left over in the denominator. If they're left over in the denominator, we can also write that as a negative exponent.

Kim  19:37

Mmhm. 

Pam  19:37

Cool. 

Kim  19:37

Yeah. 

Pam  19:38

Then we could have a similar conversation with m^a divided by m^b. So, the same base m^a divided by m^b. And we can have the same conversation about we're going to have a m's multiplied together, divided by b m's multiplied together. And really we're just kind of curious, where do we have the left over m's?

Kim  20:00

Mmhm.

Pam  20:01

And if the leftover m's are all in the... After we divide it out, if the leftover m's are all in the denominator, then we could potentially have... Not potentially. We would have a negative exponent.

Kim  20:10

Mmhm. Yeah. 

Pam  20:11

Okay. And so, then lastly to wrap up, I might say, "Then what is m^-a mean?" I'm kind of switching. And maybe I should have switched variables and done something like n^-p. I don't know. Whatever. 

Kim  20:23

Okay. Yeah. 

Pam  20:24

In fact, let's do. Let's do. We're on a podcast. What is n^-p? Like, what does that mean? 

Kim  20:30

It means that when I was multiplying n's in the numerator and more n's multiplied together in the denominator, I had more in the denominator. However many p times there were.

Pam  20:43

P n's, right? Yeah, that many n's left over?

Kim  20:46

Mmhm. In the denominator.

Pam  20:47

In the denominator. So, I could say, "Yeah, that's like 1 divided by n^p because I've got p of those n's being multiplied in the denominator. So, n^-p equals 1 divided by n^p. Cool. So, here's a Problem String where we're giving... First, starting with numbers, and looking at the relationships, and making sense, finding all these equivalencies, different ways of writing these representations. And ending with a fairly abstract way of thinking about, so what does it mean to have n^-p? What does it mean that we have some base raised to a negative exponent? And bam, we can kind of reason about that, make sense of it, and use it to create equivalent problems. Does it make sense that we can use numbers to help students have experiences, so they can generalize relationships, so that then when they see the generalized relationship, they can attach it to numbers. But not just so that then they can just do rules without thinking about it, but because they've developed a schema they can then use that to help continue. They can use that to do bigger, bigger things. But also get inside that schema when they need to. 

Kim  22:06

Yeah.

Pam  22:06

That would be an ideal place to go.

Kim  22:10

So, this sequence of five tasks. What grade are you thinking?

Pam  22:16

Yeah, so I think you could start the first task that we did in the sequence, so five episodes ago, I think you could do starting in third grade. Definitely in fourth grade and fifth grade. To do the beginning part of it. Fifth grade, especially the part where we get to the prime factorization part. Third, fourth, and fifth grade for Doubling and Halving strategy. After that, I think as soon as you really want to kind of get into exponents and exponentiation and exponent notation.

Kim  22:43

Mmhm. 

Pam  22:43

So, typically that's around eighth grade. That's when you really start hitting exponent notation. Pythagorean theorem comes in, typically in an eighth grade. I don't know all the state standards anymore. But typically it hits around eighth grade. And then from there... Oh, and I'm so glad you asked this. Let me just say. From there... So, in eighth grade, you typically deal with exponents, and (unclear), and solving Pythagorean Theorem problems, and that kind of thing. Square roots. So, that then in an Algebra 1 class, you can do exponent... Excuse me, exponential functions. And my favorite way to do these exponent relationship Problem Strings that we've just done, the sequence of Problem Strings is to do it interspersed while students are learning geometric sequences to begin with into exponential functions, so that I'm actually dealing with exponent relationships, and exponential functions, and exponential data at the same time. That we're sort of building everything. So, you might be like, "Pam, what are you going to do these five Problem Strings in a row?" No, actually, I would... I mean, kind of. But I would intersperse them while I'm also building geometric sequences into exponential functions.

Kim  23:56

Mmhm. 

Pam  23:56

So, for me, this series of tasks could very much be in an Algebra 1 class or in an Algebra 2 class, as we're trying to build exponential functions and grab the exponential relationships as we go. 

Kim  24:10

Yeah.

Pam  24:10

Yeah (unclear).

Kim  24:11

Awesome. 

Pam  24:12

Alright, everybody, thanks 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.