Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  0:10  
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  0:18  
Because we know that algorithms are amazing human achievements. But, y'all, they're terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:30  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:37  
Thanks for joining us to make math more figure-out-able. 

Kim  0:40  
Hey, hey.

Pam  0:42  
Kimberly. 

Kim  0:42  
Yeah.

Pam and Kim  0:44  
What's up? 

Kim  0:45  
Somebody said the other day, "Do you go by Kim or Kimberly?" And I was like, "When you say that, I only think of two people who ever call me Kimberly. And my mom did growing up sometimes."

Pam  0:57  
Yeah, I was going to say. You're mom, maybe.

Kim  0:58  
Yeah. And I said, "Pam, does it sometimes." But I think mostly it's response to me saying, "Pamela!"

Pam  1:04  
"Kimberly!" Yeah, I'm not sure I've ever heard anybody call you Kimberly.

Kim  1:09  
Yeah.

Pam  1:09  
Yeah.

Kim  1:10  
Yeah. 

Pam  1:10  
It's a good name. 

Kim  1:11  
It's fine. Yeah, I don't hate it. I don't hate it.

Pam  1:13  
My daughter-in-law's name is Kimberly. I like her a lot. 

Kim  1:15  
Yeah, she's great. 

Pam  1:16  
She is fantastic. Alright.

Kim  1:18  
Okay. Hey, so we've said this before. We freakin' love when you leave a review. Just super fun. It helps people find the podcast, very short and street. Street. Sweet. Wow.

Pam  1:32  
Short and something.

Kim  1:33  
This is going to be a good one.

Pam  1:34  
Thanks for the short and sweet review.

Kim  1:37  
Samantha said... At some point. I don't remember when it was. But she said, "Love, Love, LOVE the pod." Super fun.

Pam  1:45  
All caps on that "LOVE" one.

Kim  1:46  
Yeah.

Pam  1:46  
Yeah, nice.

Kim  1:47  
Thank you, Samantha. That's very sweet.

Pam  1:49  
Cool. Well, we'll take that, absolutely.

Kim  1:50  
But not street, not street. Just very sweet.

Pam  1:52  
Sweet. Sweet, "Love, Love, LOVE the pod." That was a great review. Well left. Well left, Samantha. Nicely done. Alright.

Kim  2:00  
Okay, yeah, we 

Pam and Kim  2:02  
On to a podcast.

Kim  2:02  
Let's do a podcast. Every once in a while, we ask our Journey members, our online coaching group, "Hey, what do you guys want to hear on the podcast? Like, what do you want to hear us talk about?" And Lorna is one of our members who's like such a great learner. And...

Pam  2:17  
Fantastic. 

Kim  2:18  
...she said, "How about some more high school?" So, here you go, Lorna. Just for you. And all the other high school teachers.

Pam  2:25  
Just for you, down under. Lorna's in Australia. 

Kim  2:27  
Yeah.

Pam  2:28  
Yeah, here we go. Let's do some high school. And, honestly, especially the beginning of this Problem String is definitely eighth grade content. So, eighth grade and high school.

Kim  2:37  
Okay, I'll be able to hang. 

Pam  2:39  
Ha! Of course, you will. Math is Figure-Out-Able. Okay, here we go. Kim.

Kim  2:45  
Yeah?

Pam  2:45  
If I were to just give you... I'm going to give you three numbers, and I'm going to ask if you could just sort of brainstorm if there's any connection between the three numbers that you can think of. And if I was doing this visually, I would actually write down. I'm just going to kind of throw a letter in front of them. It kind of helps us keep track of where we are. So, I'm just going to put the letter L. Okay. And then next to that, I'm going to put 2 "comma" 8, equals 3.

Kim  3:11  
So L2 "comma" 8 equals 3. 

Pam  3:13  
Yes. So, is there any relationship you can think of between 2, 8, and 3 that... Yeah. Is there any connection?

Kim  3:21  
Thinking.

Pam  3:21  
Kind of weird.

Kim  3:22  
Thinking, thinking. Yeah, I'm thinking that 2 times 2 times 2 is eight, so that's three 2s multiplied.

Pam  3:35  
Okay, cool. And a way that I could write what you just said would be 2^3 equals 8. So, I could write. In fact, what I wrote in my paper was 2 times 2 times 2. Paper. It's my iPad. Equals.

Kim  3:48  
Oh, are you really on your iPad when you do this?

Pam  3:51  
Oh, every time. Mmhm.

Kim  3:52  
What?!

Pam  3:53  
I say every time. It's kind of last year or so. Yeah.

Kim  3:56  
I had no idea. 

Pam  3:57  
Oh, that's funny. Yeah.

Kim  3:59  
I pictured paper. 

Pam  4:00  
No.

Kim  4:00  
Actually, I pictured a little index card for you. 

Pam  4:03  
No. Yeah, I take stock in those index card companies.

Kim  4:08  
Yeah, no kidding.

Pam  4:09  
Though, the other day, I found a stack of index cards that are the... What's the small size? 3 by 5, right? 

Kim  4:14  
Yeah, yeah.

Pam  4:15  
Yeah. So, I write on 4 by 6 now.

Kim  4:18  
Mmhm.

Pam  4:18  
But I had forgotten that there were several years where I did 3 by 5 because my eyes could handle it. But old eyes.

Kim  4:24  
Can we... Okay, side note. I know you're about to give me some problems, but can we side note and just tell everybody that when you would travel, you have this bag. And at one point we were together, and you were like, "Oh, I probably have a Problem String in there," and you reached into your bag, and you pull out this like stack of like...

Pam  4:44  
I have stack of index cards.

Kim  4:46  
...30 index cards.

Pam  4:47  
Oh, try 60. Yeah, try 60.

Kim  4:55  
I mean, yeah, I was trying to be not exaggerating.

Pam  4:58  
Hey, I'm prepared, Kim. That's called prepared.

Kim  4:58  
I mean...

Pam  4:58  
Prepared.

Kim  4:58  
Index cards of just like Problem Strings. And you would like flip through and be like, "Uh, that one's good." 

Pam  4:59  
And they also used to be 3 by 5. My point is now they're all 4 by 6. And the other day, my husband asked me for 5 by 8, and I was like, "Hey, I'm not that old yet."

Kim  5:06  
Okay. Anyway. 2.

Pam  5:09  
2^3 is 8.

Kim  5:09  
Yep, there you go.

Pam  5:10  
Okay, here's another set of numbers. How about 5s? So, I'm going to just put that L in front of it again. Just random. 5 "comma" 25 equals 2. Is there any relationship between those numbers?

Kim  5:22  
Yeah, yeah. Yeah. 

Pam  5:22  
Okay, what?

Kim  5:22  
5^2 is 25.

Pam  5:24  
So, 5^2 is equal to 25. Okay.

Kim  5:27  
Yeah.

Pam  5:28  
Yeah. The next one. Again, L3 "comma" 81 equals 4. Are those related in the same way?

Kim  5:37  
In the same way meaning like...

Pam  5:39  
How the other two. Like, the other ones were 2^3 equals 8. Three to the... Or, sorry. 5^2, I can read, is 25. And then 3 to the... Is 3... 

Kim  5:50  
Yeah, and can I tell you how I...

Pam  5:52  
Yeah, what were you... You paused?

Kim  5:54  
Yeah. Yeah, I did. And I was like, "What do I know about 81?" That's 9 times 9. So, that's 3^2 and 3^2, so that would be 3^4.

Pam  6:02  
Oh, nice. And I might even, as a teacher, write down exactly what you just said. 9 times 9 equals 3^2 times 3^2 equals 3 times 3 times 3 times 3. 

Kim  6:09  
Yeah. 

Pam  6:09  
Hey, sure enough there's four 3s.

Kim  6:11  
Yeah.

Pam  6:12  
And so that's like that 81. So, 3^4 equals 81. Okay, cool. Another one. This time. So, we've kind of got this sort of set pattern. I would look at those three problems. We kind of have a number, and then the power equals the exponent.

Kim  6:30  
Mmhm.

Pam  6:30  
Does that make sense?

Kim  6:31  
Mmhm.

Pam  6:31  
So, like we had 2, 8, 3. And then 5, 25, 2. And then 3, 81, 4. It's all... Somebody might say this a little out of order, but the numbers are all related in a way.

Kim  6:43  
Yeah.

Pam  6:43  
Okay, cool. So, this time, I'm going to leave one out, and I'm going to ask you to fill in the one that's left out. So, I'm going to write the L down again, and then I'm just going to say 2 "comma" 16. So, what's the equals on that one if it's following the same pattern? And how are you thinking about it?

Kim  6:58  
Yeah. So, again, I thought about 16 as 4 times 4. 

Pam  7:03  
Okay.

Kim  7:04  
And so, I was thinking 2^2 times 2^2. So, it would be 2 "comma" 16 equals 4.

Pam  7:12  
Because 2^4 equals 16. 

Kim  7:14  
Yeah.

Pam  7:16  
Cool. So, it's almost like saying 2 raised to some power is 16. What is that power? 4. 

Kim  7:21  
Yeah. 

Pam  7:21  
Maybe I should say 2 raised to some exponent. Sue is always on me about getting these terms right, and she should be. 2 raised to the exponent of 4 is equal to 16. Okay, cool. Or sorry. Let me say that in the order of the numbers. 2 raised to some power. 2 raised to some exponent is 16.

Kim  7:37  
Mmhm.

Pam  7:38  
And that exponent is that's the equals 4. 

Kim  7:40  
Yeah.

Pam  7:41  
There, I said in the right order. Okay. Next problem. I leave one of them out again. Just one of them. Random. And then you can kind of see if you can help. So, this time I'm going to L9 "comma" 81 equals blank.

Kim  7:54  
9 "comma" 81 equals 2.

Pam  8:01  
2. Because?

Kim  8:03  
Because 9^2 is 81.

Pam  8:04  
Cool. So, 9 to some exponent is 81. Oh, that exponent is 2. And we could actually look above where you had said 9 times 9 above...

Kim  8:12  
Yeah.

Pam  8:12  
To kind of connect that one. Okay, cool. Next one. Again, I'm going to leave one out. L5 "comma" 125 equals blank. 

Kim  8:21  
3.

Pam  8:22  
Because?

Kim  8:23  
Because 5 times 5 is 25, times 5 is 125.

Pam  8:28  
Cool. So, nice. Next one. L. And I'm going to do 11 "comma" 11 equals blank. Is that a typo? No, no, no. I think that's what I want. Yeah. 11 "comma" 11. Mmhm.

Kim  8:42  
1. 

Pam  8:42  
You know when I say, "Is that a typo?" I'm kidding, right? That's a teacher move.

Kim  8:47  
Most of the time, yeah.

Pam  8:48  
Most of the time it's a teacher move.

Kim  8:49  
Yeah.

Pam  8:49  
You said it was 1?

Kim  8:50  
Yeah. 

Pam  8:50  
That's a little different than all the other ones. Can you say more about that?
Pretty sure.

Kim  8:58  
Yeah, 11^1 is 11. Like, it's 11 times nothing else.

Pam  9:09  
Any number to the first power is that number.

Kim  9:11  
Yeah.

Pam  9:12  
So, that's seems kind of helpful. Okay, cool. So, mathematicians notice that there is this relationship between numbers, and the exponents, and then that power that if I do the exponent, if I multiply that base times itself, that many bases times itself, then we'll get that power. And they notice this relationship, and they call those exponential relationships. And they started to notice there were times when they wanted to solve for the exponent exactly like you just did where I said, "Hmm. Well, I have 9 and 81. What's the exponent? If I have 5 and 125, what's the exponent? If I have 11 and 11, what's the exponent?" And you were like, "Ah. Well, here's what it would be." And that was like sort of like solving an exponential equation. So, like, for example on the 5, 125 I could have written that as 5 to the x equals 125. But instead, I wrote it as 5 "comma" 125 equals x, or equals blank.

Kim  10:10  
Mmhm.

Pam  10:11  
And so it was kind of this idea that if we have exponential relationships floating around, there are times where we have the base and we have the power, that number that we've multiplied it out to get, but we don't know the exponent. We don't know the missing exponent. And somewhere in history. And I don't actually know a whole lot of the history here. But somewhere in history, somebody said, "Hey, let's call that a logarithm." Now that word is a logarithm not algorithm. So, those are two different things. An algorithm is a series of steps to solve any type of problem in the class of problems. A logarithm is exactly expressing this relationship that we've been talking about. So, I'm going to give you another. That's why I've been using the letter L. It hasn't been random. It was kind of get you ready for what if I were to say that right underneath the problem we just had. L. Where I had 11 "comma" 11 equals 1. This time I'm going to say log. I'm just going to finish the word now. So l-o-g. Then I'm going to write 8. Then I'm going to say of... Oh, little tiny 8 down below. And I'm going to say of 64. So, the 64 is kind of on the same line as log, and the 8's like a little... What do you call that? Subscript. So, log base 8 of 64 or log 64 base 8 is equal to what? So, the numbers are in the same places they were before.

Kim  11:31  
Mmhm.

Pam  11:32  
They should follow the same pattern. So, I'm just going to tell you that we're still asking the question 8 raised to what exponent gives you 64?

Kim  11:41  
2.

Pam  11:42  
2. So, you're going to fill in that blank with 2.

Kim  11:44  
Mmhm.

Pam  11:44  
So, I know this is a verbal podcast, so I'm just going to read out loud what we have down now. We have log base 8 of 64 equals 2. 

Kim  11:52  
Okay.

Pam  11:52  
And why again, Kim? Can you say why again? 

Kim  11:53  
Because 8 times 8 is 64.

Pam  11:53  
So, 8^2 is 64. Nice. Okay, so what if I just said "Well, then I'm kind of curious about log base 42 of 42?"

Kim  12:04  
1.

Pam  12:05  
That's just 1 because 42^1 is 42. What if I said, "Log base 10 of 1,000 equals what?"

Kim  12:15  
It's going to be 3.

Pam  12:16  
Because?

Kim  12:16  
Because I was thinking 100 times 10. And 100 is 10 times 10. And then times another 10.

Pam  12:16  
So, 10^3 equals 1,000. Therefore, log base 10 of 1,000 equals 3.

Kim  12:30  
Mmhm.

Pam  12:30  
What if I said... We're almost done. Log base 16 of 4.

Kim  12:38  
That's a half.

Pam  12:40  
Say more.

Kim  12:44  
Because it's 4 times 4 is 16. It's you're dividing it. You said log 16, 4?

Pam and Kim  12:55  
Log base 16.

Pam  12:56  
Of 4. Mmhm.

Kim  12:56  
Yeah, I don't know how to say that. Yeah, I think it's a half.

Pam  13:00  
And let me... Okay, so let's stick... It's a half. And I want to talk about that. And I just want to note that in that moment. Sorry, I probably could have done this later. But in that moment, that was a social thing. When you're like, "I don't know how to say that," that's social. How to say that is social. So, you and I both don't really care that in the moment you're getting that right. Sure, I'm going to help you know that this is how we say that, so that you sort of can communicate with other people. That's a communication thing. It's important. But it's not a math thing. It's a logical mathematical thing. I'm not going to get all tied up about that. So, you do say that log base 16 of 4 or log of 4 base 16. Usually log base 16 of 4. But could be either way. And you're saying it's one-half. And I would wonder if you could put that in terms of an exponent. What's 16 raised to the 1/2 power is 4? Is that what you're saying?

Kim  13:49  
I think it's like -2. Is that right? No.

Pam  13:56  
16^-2

Kim  13:57  
So, it's 1. If it's 16^1 it's 16.

Pam  14:04  
Mmhm.

Kim  14:05  
So, but if it's 2, it's 16 times 16. 

Pam  14:09  
Mmhm.

Kim  14:09  
But if it's -2... Oh, man. I'm trying to decide. In my head right now, I'm trying to decide the difference between one-half and negative 2.

Pam  14:28  
I can hear that. Yeah. And I love that you're going both of those places. Maybe if I... in this moment, I might write. So, we're sort of looking at 16 raised to some power equals 4. When you said 16^1/2. I could also write that as the square root of 16 equals 4. Is that right? 

Kim  14:45  
Yeah.

Pam  14:47  
Okay. I could also write your 16^-2 as 1 divided by 16^2. Is that 4?

Kim  14:55  
Say that again. 16 divided by?

Pam  14:57  
1 divided by 16^2. 

Kim  14:58  
1 divided by 16^2. No. 

Pam  15:01  
Ah. Okay. So, I know why you were playing around with the negative, but that's...

Kim  15:07  
You know what I think I'm struggling with right now is that I see the 16 and the 4. Like, when I'm picturing like exponents...

Pam  15:15  
Mmhm.

Kim  15:16  
...I don't also see the 4. Like, you know, normally when you're thinking about... At least, what I've experienced is thinking about exponents and negative exponents, there's only 1 number there. Like the 16 (unclear).

Pam  15:31  
Oh, it's an expression not an equation?

Kim  15:34  
Yeah.

Pam  15:35  
Mmhm, mmhm.

Kim  15:35  
So, I think I'm like, "Wait. Hang on a second. Like, what number am I focusing on right now?"

Pam  15:40  
Yeah, nice. 

Kim  15:41  
Because I want to say 4^2 because then I want it to be the 16, but I'm like, "Wait, that's not the question you're asking."

Pam  15:47  
Nice, nice. That's really well said, yeah. And it is super weird to have fractional exponents. Those are weird. So, we would then do more work on fractional exponents. But nicely done. I like where you went. It's cool. So, then, I might ask kids to kind of step back and look at everything that we've got on the board so far, and say, "So, it sounds like you're saying that if I've got a log base a..." And I would just write down log, and then a little a. Log, little subscript a of c equals b. If those are sort of... I'm just going to put a, b, and c in there. 

Kim  16:18  
Yeah. 

Pam  16:18  
Then what do we know is happening? How could I write that as an exponential equation?

Kim  16:26  
a^b is c.

Pam  16:28  
a^b equals c, yeah. And we've got kids introduced to a logarithm, how it's connected to exponents and exponential relationships, and they know that what we're looking for is the exponent.

Kim  16:40  
Yeah. 

Pam  16:40  
And that's what you were looking for the whole time.

Kim  16:42  
Yeah, I like. I like a lot. I like how you were starting with numbers and relationships that they've probably can see some sort of, you know. Especially the 2 "comma" 8, 3. You know, at first they might be like, "What are you talking about?"

Pam  16:58  
Sure.

Kim  16:58  
You know, but then throughout the next several I was like, "Okay, I know what you're asking." And I think that's important. Like, I have to know what you're asking. And I think that's a tricky thing where, you know, sometimes kids can mess with relationships, but they have to understand what the question is asking. Like, I'm picturing in my mind right now. We're filming not too long ago where, doing a Problem String, where the kids were asked a question in a way that they probably have never been asked a fraction question before, and there's a little bit of, like, "Wait, what?" And it probably doesn't speak to their understanding or their knowledge. And it's more about like...

Pam  17:33  
Never heard. I've never seen that question before...

Kim  17:35  
Yeah, yeah. 

Pam  17:36  
Conceived of it yet.

Kim  17:37  
Yeah, so I'm wondering if, you know... I don't know how much time we have. But I'm wondering like when you get to a problem like log base 16 of 4. Is that how you say it?

Pam and Kim  17:50  
Log base 16 of 4.

Pam  17:52  
Mmhm.

Kim  17:52  
Where you go, "Hang on a second, kid. Like, you've messed with negative exponents before. Like, what's the problem? Like, what's the..." Like, I'm wondering if this with logarithms if this ever collides with teachers ever wonder why kids struggle with that particular thing if they've also understood exponents.

Pam  18:15  
Mmhm, mmhm.

Kim  18:15  
Like, fractional exponents or negative exponents. Is that a thing? Is that a thing where the new type of question makes kids go, "Hang on a second." Like, "I'm questioning what I thought I knew."

Pam  18:27  
Absolutely. Yes. And that doesn't mean that we didn't do the first part well. Yeah. It does mean let's juxtapose these. Let's put them up next to each other to compare and contrast. And which one means which? 

Kim  18:41  
Yeah.

Pam  18:42  
Absolutely, yeah.

Kim  18:42  
I love that you asked me is that also the square root of 16? Is that 1 divided by 16^2? Instead of being like, "No, it's this one, Kim." I mean, right? That's important, you know, because I was able to connect stuff. So, anyway. I like.

Pam  18:56  
Super cool. It's interesting to me that I remember learning logarithms for the first and probably second, third, and fourth, fifth time where my teacher said a logarithm is an exponent. And I said, "What?" And never. I didn't have... Then, we just dove into the rules.

Kim  19:11  
Yeah.

Pam  19:11  
And performed a bunch of rules. And I did the rules with the best of them.

Kim  19:15  
Sure.

Pam  19:15  
But I never. It wasn't, honestly, until my kids were growing up and they were messing around with exponential relationships. And I was like, "Oh, my gosh. It is an exponent."

Kim  19:24  
Yeah.

Pam  19:24  
Like, I was so ingrained in the... Anyway, I think this is a nice way of saying we've got these two numbers. How are they related? They are related with this exponent.

Kim  19:32  
Mmhm.

Pam  19:33  
2 and 8? How are they related? They are related with that third exponent. 3 and 81? How are they related? Ah, because I can have that 3^4 is 81. 

Kim  19:42  
Yeah.

Pam  19:43  
I think that's nice that these numbers are related in a way that that is the exponent.

Kim  19:48  
Mmhm.

Pam  19:49  
Super helpful for me and everybody. Okay, cool. 

Kim  19:51  
Very cool.

Pam  19:51  
Y'all, this isn't the only intro to logarithms that I've got. I think there's a really nice way to do that graphically as well. I think I'd want to have multiple approaches to sort of thinking about logarithms. But here's one way that you could help students understand exponential and logarithmic relationships. Bam. Alright, y'all 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!