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 a mission to improve math teaching.

Pam  00:22

We know that algorithms are amazing historic 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.

Kim  00:36

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

Pam  00:43

And we invite you to join us to make math more figure-out-able

Kim  00:48

Hey there, speaking of grappling with mathematical relationships.

Pam  00:52

Yeah? 

Kim  00:52

I pulled a paper that I had from last week. 

Pam  00:55

Okay. 

Kim  00:56

Just to take notes on... I knew we were doing a string today.

Pam  00:59

Mmhm. 

Kim  01:00

And I have two things circled on my paper. That when we did the last problem, which was 2^4 times 2^2. I called it 2^6. And kind of at the end you said, "And I'm going to also pull out that that's 4^3. 

Pam  01:14

Okay.

Kim  01:15

For whatever reason, I circled those equivalencies. And I've been.. 

Pam  01:19

The 2^6 and the 4^3?

Kim  01:21

Yeah. And I've been tinkering with it a little bit. Because at first I thought like there's this nice double the base, halve the exponent. And I was just trying to figure...

Pam  01:31

Wait, wait, wait, wait, wait, wait. What? Double the base.

Kim  01:35

Mmhm.

Pam  01:35

Halve the exponent. (unclear).

Kim  01:37

Yeah, that was true there. But I mean, honestly, I'm not sure if that's always true. I just have been tinkering for a second while I'm waiting for you. And at first I thought, well, maybe the base and the exponent has to be even. But then 3^4 is equivalent to 6^2, so in that case, the base is odd. So, anyway.

Pam  02:02

Hey, but 3^2 is not equivalent to 6^1. Is that the pattern? Double the base, halve the exponent.

Kim  02:11

I'm wondering when that would be true? 

Pam  02:14

Oh. 

Kim  02:14

I don't think it's always.

Pam  02:15

So, you're not suggesting it's always true?

Kim  02:15

No, no, no. I'm just wondering like.

Pam  02:17

Because I just found a non-example. 

Kim  02:19

Yeah. I'm wondering like when. Is it just a fluke that that was true? Or is there some sort of (unclear).

Pam  02:26

Okay, I have an idea, but will you let me think about it? 

Kim  02:28

Yeah.

Pam  02:29

Okay.

Kim  02:30

Alright, on to today. Sorry, to derail us.

Pam  02:33

So, this is fun because this is one of the things that we do, right, is that I will come at a lot of things from kind of a traditional background of having taught higher math, and then you'll come in and go, "Well, what about this?" And I'll be like, "Um... I've never thought about that. Let's think about that together." 

Kim  02:50

Oh, and I just... Sorry. I just realized what I wrote and said, and they're not equivalent.

Pam  02:55

Which ones?

Kim  02:59

3^4 and 6^2.

Pam  03:01

Oh, and I wasn't listening because I was doing my own thing. 

Kim  03:03

I know.

Pam  03:05

No, those are not equivalent. Not even close. Which is super interesting, right? Because like 3^4, you might look at that and go, "Well, 3. That's so small. Raise to the fourth, that's not going to be very big. 6, that's a really big number. So, you square it." No, no. Like 3^4 is much bigger than 6^2. Like, tons. Anyway, okay, okay. So, funny story that kind of follows along with what we were just talking about. I was doing some filming for some wonderful work that we were doing. And we were in a wonderful teacher's classroom. Abby Sanchez has been an amazing thought partner over time. Really have enjoyed working with her. Several years ago, we were doing some filming, and in the middle of a Problem String, Kim looked at me, and she just kind of shook her head. And I was like, "What?" So, Kim was there as kind of a pedagogy check. Because at that point you had done less higher math. Your kids were not even old enough that you hadn't done higher math, and you hadn't really done it since high school. So, you weren't really there for the math. You were there for the pedagogy. And you kind of just shook your head, and I was like, "What?" So, Abby got done doing that particular Problem String that I had written, and you handed me a little piece of paper. I think it was a post-it note. And you were like, "This would be a better string." I mean, I think you said it maybe more judiciously than that. 

Kim  04:29

Really? Really, did I? 

Pam  04:31

Your point was like, "Here's a better..." And I was like, "What?" And she goes... She. You. You said something like, "Like, the way you're doing it is fine, but this would get at those relationships a little bit better. And for those of you that have followed our work over time, if you've taken one of our Building Powerful Mathematics workshops, gotten any of our free downloads, you'll notice that we now give you what we affectionately call the "half sheet", and that half sheet is a Problem String on the left. Well, so it's split into two, and on the top there's a Problem String on the left. And next to it are kind of some hints or some things to think about, some important things not to forget while you're doing each of those problems. And then on the bottom is what your board could look like when you've finished the Problem String. Well, that day was the birth of those half sheets. Which, for a while, we called them "Kim's notes" because, Kim, you handed me that little sheet of paper, and on it, it had the problems on the left, and then it had little things that you would think about, not to forget, and ways to say, and things to emphasis to the right of each of those problems. And we talked about it with Abby, and then in the next class period, she did it your way. It was much better. Went much better. And the reason I'm telling that story today is it was actually this Problem String is the one that you rewrote. So, you haven't looked at it for years. 

Kim  05:55

I remember that happening. I remember "Kim's notes".

Pam  05:58

Mmhm. 

Kim  05:58

I don't remember what string it was. 

Pam  06:00

Alright, so I'm just going to throw this string at you.

Kim  06:02

Okay. 

Pam  06:03

We're going to do it together. 

Kim  06:04

Hopefully, I can do it.

Pam  06:06

So, listeners, if you've been listening the last couple episodes, we threw out that it could be possible that we could do a sequence of tasks that could help kids really reason exponentially. Exponents are multiplicative at their heart. This is multiplicative reasoning. And we're heading towards exponential functional reasoning. Which would be in the functional reasoning. But we got to reason exponentially first. Well, I say "first". That it's included in. It's within. And so, what's a series of sequence of tasks? So, we did one two weeks ago. We did one last week. Let's do another one that could continue to get kids reasoning exponentially. I think you could do this particular string in eighth grade. I think it would be totally appropriate in an eighth grade class. Maybe a little bit earlier. But I think most people's standards, it would appear in about eighth grade. Okay, so first problem. What is 5^3? 5 raised to the third power. While you're thinking, I'm going to take a drink. 

Kim  07:04

Okay, I'm going to call that 5 times 5 times 5. Which is 125.

Pam  07:10

And how do you know that 5 times 5 times 5 is 125?

Kim  07:14

Because 5 times 5 is 25 and 25 times 5 is 125.

Pam  07:20

Okay, so this is what I have written on the board right now. I have 5^3 equals 5 times 5 times 5. Which is what you said. Equals 25 times 5. Equals 125. Does that track? Yep. Cool. Next problem. What is 5^2 times 5? 

Kim  07:41

Also 125. 

Pam  07:43

And we wouldn't spend a lot of time here, but I would ask kids, and a lot of kids would say, "Pam, it's just 25 times 5. We did that above. That's 125." And so, now I've written 5^2 times 5, equals 25 times 5, equals 125. And then I might say, "You're saying that we had just done 25 times 5 above, and so you know that that's 125? Well, what did we also say in the first problem? 5^3." So, now I've got on the board, 5^2 times 5, equals... And then a few things. But also equals 5^3. Cool. I might throw above that 5. 5^2 times 5. I might throw 1 above there. Just to like. Okay, so it's like 5^2 times 5^1 is 5^3. Cool. Next problem. How about 5^2 times 5^2? What do you got for that one?

Kim  08:40

That is 25 times 25. 

Pam  08:43

So, I've written that down. 25 times 25.

Kim  08:45

Ugh, do I know that? I know... Actually, I know 5^3 which is 125. Times 5 is 625.

Pam  08:57

So, I wrote down 25 times 25, equals 5^3 times 5, equals 125 times 5. And then you know that's 625? 

Kim  09:06

Yeah. 

Pam  09:07

And does that feel like 25^2?

Kim  09:09

I thought 25 squared was 625, but I don't have them all memorized so.

Pam  09:14

But you just kind of played it out. You're just like.

Kim  09:16

Yeah.

Pam  09:16

Sure enough. Sure enough. Indeed, it is. Cool. 

Kim  09:19

Yep. 

Pam  09:19

Alright. The last thing that I might write down is how many 5s did you just multiply? 

Kim  09:26

4 of them.

Pam  09:27

So, then I'm going to write equals 5^4. And then I might back up and say, so 5^2 times 5^1 was 5^3. And 5^2 times 5^2 is 5^4. So, if I'm multiplying the number of 5s, I can put that as the little exponent, and I could say the base is 5 and the exponent is 4. That means I'm multiplying four 5s together. Okay, cool. Next. Actually, before I'm done with that problem, I might say something like. It looks like in 5^2 times 5^1, we could write that 5^3 as 5^2+1. 

Kim  10:08

Mmhm.

Pam  10:09

And in the 5^2 times 5^2 equals 5^4, we could write that as 5^2+2. Cool next problem. How about if the next one is 2^3 times 3^2. 2^3 times 3^2.

Kim  10:29

2^3 is 2 times 2 times 2. Which is 8. 

Pam  10:34

Okay.

Kim  10:34

And 3^2 is 3 times 3. Which is 9. So, 8 times 9 is 72.

Pam  10:40

So, I've just written on my paper 2^3 times 3^2 equals 8 times 9 equals 72. I'm asking myself if I should have written out the 2 times 2 times 2 times 3 times 3. But I think I might not yet. Even though you clearly said it. I think I might not yet. Because I think I might say, "Based on what we kind of did above where we had 5^4 equals 5^2+2 and 5^3 equals 5^2+1. Can we write 2^3 times 3^2 as 6^5. And I put a question mark above that. So, I wrote equals "question mark" 6^5. Because  above we just said we could add stuff. And so, can we add that 2 and 3. Those two bases. And that 3 and the 2. The two exponents. Can we add those? Because you just said... We kind of said above that we maybe we could add them, so I wonder if we could add them here. And is 2^3 times 3^2, 6^5. 

Kim  11:42

It is not. Do you want more than that? 

Pam  11:48

Yeah, maybe.

Kim  11:49

Because when you are adding the exponents, you're saying you have that many of the base. But you change the base. When you multiply 2 times 3, you change what the base was. 

Pam  12:03

So, this might be the moment where I would have somebody say, "Yeah, 2^3 is 2 times 2 times 2. And 3^2 is 3 times 3." So, if I'm looking at that 2 times 2 times 2 times 3 times 3, there is a 6 there, but there's not 5 of them, right? Like, 6^5, no. So, then I would say not equivalent to 6^5, not even close. Is that okay?

Kim  12:29

Yep. 

Pam  12:29

Alright, cool. Next problem. How about 2^2 times 3... Sorry, 2... I said 2^2. I wrote 2^3. 2^2 times 3^2. Okay, so now the exponents are the same. So, maybe that means we can do something cool now.

Kim  12:50

2^2 times what? 3^2?

Pam  12:54

Mmhm. 

Kim  12:56

2^2 is 4. And 3^2 is 9. So, 4 times 9 is 36. 

Pam  13:05

Okay. And then I might ask everybody, "Is everybody clear on that? We're all good?" Cool. I wonder. Before we said that you had to have something the same. So, I wonder if this time we could say is that equivalent to 6^4? Like, maybe now we can add stuff?

Kim  13:24

Also, no. Because the bases... You change the base again. So.

Pam  13:30

So, it's not the exponents that have to be the same? (unclear). 

Kim  13:33

You still have 4 of something, but it's different somethings. 

Pam  13:36

Ah. So, like if we wrote it out what it means, it would be like 2 times 2 times 3 times 3. And we don't have a base of 6?

Kim  13:47

Right.

Pam  13:48

So, not equal to 6^4. Alright. Rude. Okay. Next problem. Kim, how about 2^2 times 2^3? 2^2 times 2^3.

Kim  14:02

That is 4 times 8. Which is 32.

Pam  14:08

Okay. Is that equal to something?

Kim  14:13

Mmhm. That is the same as 2^5?

Pam  14:16

How do you know?

Kim  14:18

Because I have two 2s multiplied and three 2s multiplied, and that's five 2s. 

Pam  14:26

So, I wrote underneath... Sorry. 2^3 times 2... Sorry. Wow! 2^2 times 2^3 equals 4 times 8, is what you said. Underneath the 4, I wrote 2 times 2. And underneath the 8, I wrote 2 times 2 times 2. There's your five 2s. "And, class, how have we said we're going to write five 2s? Ah, that's 2^5."

Kim  14:50

Mmhm. 

Pam  14:52

Nice. Okay. Could we also write that as 2^2+3?

Kim  14:57

Mmhm. 

Pam  14:59

So, when can we sort of add exponents? Because it looks like we just came up with some things that we could do and couldn't do. When can we add exponents?

Kim  15:10

Only when the bases are the same. 

Pam  15:13

And how do you know? Like, is that a rule you've memorized?

Kim  15:17

No. Because when the bases aren't the same... Like, in the previous two problems, if we multiplied them together, then we got a different base. Here, we're just saying we have two 2s and three more 2s multiplied. It's the same thing, just... It kind of reminds me of like in a younger grade like repeated addition can become multiplication, but only if you're adding the same number over and over. 

Pam  15:45

Oh, that's very nice. Yeah. So, if you're adding the same number over and over, you can write that as multiplication. If you're multiplying the same number over and over, you can write that as an exponent or exponentiation. Nice generalization. Cool. So, next problem. What if I said 2^a times 2^b? Could we write that in an equivalent expression?

Kim  16:12

Mmhm. That would be x^a+b. 

Pam  16:16

So, 2. 2^a and 2^b.

Kim  16:20

(unclear). Did you say x? 2?

Pam  16:21

I said 2. You just went really general on me.

Kim  16:24

2^a+b. 

Pam  16:27

And what does 2^a+b mean?

Kim  16:30

It means two... a 2s and b 2s. The a number of 2s multiplied, and the b number of 2s is how many 2s we're multiplying. 

Pam  16:42

We're just multiplying a ton of 2s.

Kim  16:44

Mmhm.

Pam  16:45

And we first multiplied a of them, and we multiplied b of them, and we're multiplying them all together. 

Kim  16:51

Mmhm. 

Pam  16:51

2^a plus b. Nice.

Kim  16:52

Mmhm. 

Pam  16:53

So, I might write that as 2 times 2 times 2, and then with... How do I say this? With like a bracket underneath? A brace underneath it? So, it's like I'm encompassing 2 times 2 times 2. Oh, let me say that better. 2 times 2 times 2 "dot, dot, dot" times 2. So, like, and then I put the brace underneath it to say that's a of them. A of those 2s. Times. And then some more 2s times each other "dot, dot, dot" times 2. And a brace underneath those to say there's b of those. So, how many total 2s are multiplying together? A plus b? That's how many I have multiplied together. Be a way of doing that. Cool.

Kim  17:29

Mmhm. 

Pam  17:30

And then I might say, "So what if I had..." Now, I'm going to get even more general. x^a times x^b?

Kim  17:39

Ah, here we go. x^a+b.

Pam  17:42

x^a+b. Because we've got a x's times b x's. We got a whole bunch of x's times each other. How many? How many times each other? 

Pam and Kim  17:50

a plus b. 

Pam  17:52

Yeah, nice. Cool. How would that help you think about something like 3, a^4, b. So, 3 times a^4 times b. All times 5, a^2, b^3.

Kim  18:12

So, I have four a's and two more a's. 

Pam  18:16

Multiplied together? 

Kim  18:17

Multiplied together. So, I put a^6.

Pam  18:19

Okay.

Kim  18:20

And I have b times b^3, so I have b^4.

Pam  18:27

Okay.

Kim  18:28

And then 3 times 5 is 15, so I have 15 times a to the 15, a^6, b^4.

Pam  18:35

And how did you know b times b^3 was b^4?

Kim  18:41

Because I have four b's being multiplied together. 

Pam  18:44

It's all being multiplied together. 

Kim  18:45

Mmhm.

Pam  18:45

So, one way that I might do this with students is... Well, first of all. I'd throw it out and let them do what they do. But I'm going to have some students that are going to be like, "What? I don't even know what's happening." We could have somebody share that in that 3, a^4, b, you got a bunch of stuff multiplied together, and we could write it all out. We've got 3, a times a times a times a times b. And we're going to multiply that by 5, a^2, b^3. So, I could write that as 5, a times a times b times b times b. Now, I don't want to do this too many times.

Kim  19:18

Right. 

Pam  19:18

But what I want to engender in kids heads is that when they see b^3, they're thinking b times b times b.

Kim  19:26

Right. 

Pam  19:26

Like, they're like, "Okay, that's b. I've got three b's multiplied together."And when we've got multiplication, we can use the commutative property. Then we could do kind of what Kim just did, and we can move things around and put all the numbers together. Multiply them,the 3 times 5. We can put all the a's together. How many a's did we have multiplying together? 6. a^6. And then put all the b's together. How many b's did we... And with that sense of what's actually happening, bam, we have kids that are clearly adding exponents, but they're doing it because they are reasoning about what's happening. 

Kim  19:59

Mmhm. 

Pam  20:00

Because math is reason-about-able. Alright, Kim, do you think it's possible that we could do a sequence of tasks like we've done? And now, somebody's going to say, "Pam, that took you three tasks to get kids to one of the exponent rules." Then, I'm going to say, "No, it took me three tasks to get kids to reason about exponentiation, to like own what's happening." And I'm going to suggest that maybe not every kid in the room owns it yet. But I'm doing enough things to keep kids challenged. Now, you're like, "How are you keeping kids challenged? Well, you heard Kim start today's podcast thinking about some generalizations. Because we're actually understanding what's happening... 

Kim  20:42

Yeah.

Pam  20:43

...kids like Kim are off thinking about generalizations, while they're getting good practice. Notice, I'm giving kids problems. They're getting instant feedback on how they're thinking about these exponent problems.

Kim  20:55

Mmhm. 

Pam  20:56

We're getting quite a bit of practice done in just a few minutes. And we're building understanding, not only for quote, unquote "exponent rules". Which I'm going to call exponent relationships. But towards exponential functions. Like, we are building exponential growth. We're feeling that it's multiplication is happening here. It's not... What a lot of kids, I think, come out of when we give a couple of rules is they're like, "Oh, okay. So, for exponents, you add, you subtract, and it doesn't make any sense, but you just like... Okay, got it. Like, these weird rules, and I don't know which ones to do, but I'll just memorize." Or we can actually build all this groundwork that really is going to come out and pay dividends. Now, because they actually understand what's going on. They're confident what they're doing. And later when we need to build on. 

Kim  21:41

Yeah.

Pam  21:42

Yeah? Alright, I got a couple more for you. Kim. 

Kim  21:45

Okay, I can't wait.

Pam  21:46

We'll do a couple more in the next couple episodes. Sounds great. Ya'll, 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. And keep spreading the word that Math is Figure-Out-Able!