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  00:17

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  00:38

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

Kim  00:42

Hey! Happy birthday! Well, not really your birthday. But we are recording a little bit early. I know (unclear). We're recording just a little bit early. But when the day the episode comes out, it will be your birthday! 

Pam  00:55

Wow, look at that. The whole world celebrates my birthday. How fun is that? Yeah, there's always a party. 

Kim  01:01

Yeah, it is! Okay, so listen, we've been hearing... I mean, I say that. I know I've been hearing. I know you probably have too. About people loving the higher math that's been happening. (unclear).

Pam  01:11

Yeah, interesting, right? 

Kim  01:12

It's super fun. And I've been hearing from some elementary teachers who have been really excited about trying out some middle school and high school content. And it's been kind of fun, right? 

Pam  01:23

Yeah. Actually, when I was on my trip to Europe, it was fascinating to hear from the number of people who said, "We really like listening to the podcast, even when it's not the content we teach because we're able to mess around, play with, think about that mathematics and realize Math is Figure-Out-Able. Like, stuff that we don't even deal with a lot and maybe never got when we were taking it." Yeah, it's kind of fun to... 

Kim  01:45

Yeah.

Pam  01:46

...figure it out.

Kim  01:46

Back in November, we got a review from Natalie E. from New Jersey, and she said, "I started listening to the podcast while researching ways to support students struggling with dyscalculia. And as I listened to more episodes, I found myself diving deeper into the world of numeracy." Goodness, Kim. "Unlike other podcasts that often seem to focus on selling products rather than providing valuable content, the episodes are full of practical insights. Pam and Kim guide listeners through mathematical concepts slowly and deliberately, helping me reason through problems and make connections in ways I never thought possible. Especially in more advanced math. Thanks to this podcast, I've made incredible progress in my numeracy journey, and now I feel confident helping my middle school son tackle algebra. Thank you for everything you do."

Pam  02:36

Wow! 

Kim  02:37

Isn't that fun? Yeah. 

Pam  02:38

Oh, Natalie E., thank you. (unclear). 

Kim  02:39

We should talk. I got a middle schooler taking an algebra.

Pam  02:42

So, and I'm noticing that the title of this review is "Life Changing".

Kim  02:47

Mmhm. 

Pam  02:47

Aww, that's super cool. Thanks. 

Kim  02:50

Alright, so how about we do a little bit more math today?

Pam  02:54

You know, let's do. When I was in Bulgaria, I was tasked, kind of last minute, just a little bit with doing a Problem String with a group of year 10 students. So, they talk about years and not grades. And so, kind of a little quickly, I came up with one of the Problem Strings I've done before. I didn't write it, you know like... Well, I grabbed a string that I'd done before that we actually have on the website. So, maybe after listening to the podcast if you want to see the modeling, you could actually go check out the Problem String at mathisfigureoutable.com/ps. Don't do it now. Like, do it with us here. Mmhm. But you could actually see it with a group of grade nine students. So, ninth grade students doing it. This is a group of grade 10 students. And yeah, let's have a little bit of fun. You ready? 

Kim  03:40

Yeah. 

Pam  03:40

Okay. So, Kim, pick something that grows on trees that you would want to harvest off those trees to eat. 

Kim  03:50

How about oranges? 

Pam  03:51

Could be oranges, right? Could be... Pick something else, apples, cherries. 

Kim  03:57

Ooh, cherries are yum. 

Pam  03:58

In our in our area, there's a lot of pecans. Do you say pecans or pecans? 

Kim  04:02

Ugh, pecans. 

Pam  04:03

Okay, yeah. There's people here, it's...

Pam and Kim  04:05

Pecans. 

Kim  04:06

Yeah.

Pam  04:07

Yeah.

Kim  04:07

I think it depends on where you are on the states. 

Pam  04:09

Yeah. I think so too. So, lots of... We could picture an orchard full of trees. And you said oranges. We're going to do oranges today. 

Kim  04:17

Okay.

Pam  04:17

Got this orchard full of orange trees. Lots of oranges. And for this particular orchard, we know that it takes 20 workers 6 hours to clear the orchard. So, if we say, "Mark, get set, go," and the 20 workers that we have go out into the orchard, 6 hours later, we'll have the orchard cleared. All the fruit will be picked. And we're happy. In 6 hours. And the 20 workers can go home. 

Kim  04:44

That's a big orchard. Yeah. (unclear).

Pam  04:47

Okay, there you go. Well, I mean, you know. (unclear).

Kim  04:49

20 workers for 6 hours?

Pam  04:51

Yeah. 

Kim  04:52

Yeah, okay. 

Pam  04:53

Yeah, I don't know how many oranges they're picking, but there you go. Okay. So, if that's true that 20 workers, working at the same rate each work for 6 hours, and they can get the job done. The whole thing is done in 6 hours. What if tomorrow we're ready to clear the orchard, but only 10 workers show up. Oops. Only 10 workers show up. What are we going to need to plan for?

Kim  05:21

Mad workers. They're working longer.

Pam  05:25

Yep. And how long will it take to clear the orchard? 

Kim  05:27

It's going to take them 12 hours. 

Pam  05:29

12 hours. Because... Wait, 12 hours? Hang on. We halved the number of workers, so normally wouldn't we half the number of hours? 

Kim  05:38

No. Half the number of workers means that you need to work twice as long to do the work.

Pam  05:43

So, it's almost too bad that you said that so fast because every time I've done this problem string, whether I've done a lot of ratio table work or not with students, almost to a T, students will say, "Well, if you halve the number of workers, then you halve the number of hours." And so, they say 3. Including, a very quick group of tenth graders that I worked with in Bulgaria. 

Kim  06:03

Do you think it's because they saw you write it? Like, I don't have anything written down, so I'm listening to your story in the context.

Pam  06:09

Yeah, I don't know. But I'll tell you. For the rest of the string, you are going to probably want to write down a table. 

Kim  06:14

Okay.

Pam  06:14

It has workers on one side. Hours on the other. 

Kim  06:17

Okay. 

Pam  06:17

Maybe, maybe. I know I did this with a group of teachers quite a few years ago. And the group of teachers... I asked the question and didn't take any answers and walked around. It was super interesting. There was one teacher who said what you said. If you halve the number of workers, then you double the time. It will take you longer because you have half the number of workers. And this particular teacher was a Special Education teacher. And the teacher sitting next to him laughed and said, "No, no, no, no, no. If you halve 20, then you halve 6. The answer is 3." And I said, "Well, say more about that. Like, why? Why would you if you halve the number of workers, why does it halve the time?" And and she laughed and said, "Because." And I said, "Why do you think it's twice the time?" And the kind of quieter teacher said, "Well, because you have half the number..." And then, interrupted. It was a very uncomfortable situation. The gal interrupted him again and said, "No, no, no, no. It's fine. It's 3." In that moment, I'm not sure I chose the right route to go. I kind of let it go, and I was hoping they would continue to talk about it. But I was smiling a little bit inside because I knew pretty quickly we were going to substantiate his answer and that we were going to let her go, "Oh." Like, maybe instead of just starting to do stuff, it's worth thinking about what's happening. So, if you halve the number of workers, it's going to take  more time. In fact, it's going to take double the time. Okay. So, so far on my paper, I've got 20 workers to 6 hours. And then on that same table underneath that, I've got 10 workers to 12 hours. Cool. What if we had 40 workers show up? Double the number of workers from the 20 show up. What if we have 40 workers show up? How long will it take to clear the... 

Kim  08:04

Well, you just told me it's double number workers. (unclear). 

Pam  08:06

Sorry, sorry, sorry. Well, it's not double from the 10. It was double from the original.

Kim  08:11

Well, you said double from the 20. So, double from the 20. When I had 20 workers, it took 6 hours. 

Pam  08:17

Okay.

Kim  08:17

So, if I have twice as many workers, then I need half as much time as it took the 20 workers. So, instead of 6 is going to be 3.

Pam  08:25

And can you go from the 10 workers and 12 hours?

Kim  08:28

Yeah, so when it was 10 workers, it was 12. So, 4 times as many workers, they're going to need a fourth of the time. 

Pam  08:37

And a fourth of 12 is that same 3 that you just got. Cool. So, 40 workers will take 3 hours Nice. What if I find out that a frost is coming. Oh, no. I don't even know if it freezes where you grow oranges, but whatever. I find out there's some catastrophe about to happen to the orchard, and we've got to clear it, and we have exactly 24 hours to clear it. So, I'm putting now 24 in the hours part of my table. And I know I've got 24 hours to clear the orchard. How many workers do I need to hire?

Kim  09:12

If they're going to work continuously in that 24 hours? 

Pam  09:14

Yep.

Kim  09:14

Then, you need 5 workers. 

Pam  09:17

How do you know? 

Kim  09:18

Because when you had 10 workers, that was 12 hours.

Pam  09:22

Okay.

Kim  09:23

And now you have twice as much time. And you can do it with half as many workers.

Pam  09:28

Oh, nice. Nice, I like that. Can you go from any of the other ones? 

Kim  09:32

Yeah, you can do... When you had the 20 workers for 6 hours.

Pam  09:35

Mmhm. 

Kim  09:36

You have 4 times as much time, so you can take fourth the number of workers. So, from 20 to 5.

Pam  09:44

A fourth of the workers can accomplish the same task in 4 times the amount of time.

Kim  09:50

Mmhm. You can go from 40 to 3 if you want. 

Pam  09:52

I liked it when you said that you're going from the time. If you have a fourth of the time, then you're (unclear).

Kim  10:01

Four times as much time. (unclear).

Pam  10:02

Yeah, thank you. Yeah, 4 times much time, you can have a fourth the number of workers. Yeah, that's nice. Cool. What if 15 workers show up? And maybe people think about this before you just hear Kim just like go. But yeah, go ahead. Alright 15. You got something?

Kim  10:20

Yeah. I actually looked at the 10 first. 

Pam  10:23

Okay.

Kim  10:24

I mean... Yeah, I looked at the 10 first, and I wanted... I guess I'll go to the 5. I don't know why the 10. I don't know why I want to do the 10 first.

Pam  10:34

Let's... We could talk about the 10 in a minute. Go ahead and do the 5 if you want to. 

Kim  10:38

Okay, so if I have 5 workers. Yeah, I have 3 times as many workers.

Pam  10:44

Okay. 

Kim  10:44

So, I only need a third of the time, which is 8 hours.

Pam  10:48

8 hours.

Kim  10:49

Mmhm. 

Pam  10:49

So, 5 workers were 24 hours. So, 15 workers, 3 times as many workers, is only going to take a third the amount of time. 

Kim  10:57

Mmhm.

Pam  10:58

Yeah, thanks. I just had to say that out loud, make sure I was... I'm actually writing that down. I'm like, so times 3 divided by 3 when I go. Yeah. So, that scaling, often on a ratio table will write, if I double the amount of packs, then I double the amount of sticks in a pack of gum. And in this case, if we are multiplying by 3, the number of workers, then we're dividing by 3 the amount of time. 

Kim  11:21

Yeah.

Pam  11:22

Yeah.

Kim  11:22

And I think that's what I was thinking about when I wanted to go to the 10. 

Pam  11:26

Yeah.

Kim  11:26

Was the 10 was two-thirds of the time that I had. I had less time. I only had two-thirds. Or, sorry. The workers. I only had 10 workers, So, I had two-thirds of the number of workers. 

Pam  11:38

Okay.

Kim  11:40

So, they're going to be able to do it faster. And so, I needed the 12 to be three 1/2s. You said 15 first, right? I can't remember.

Pam  11:55

Yes.

Kim  11:56

Okay.

Pam  11:56

Yes, I gave you the 15. 

Kim  11:57

So, I had three 1/2s as many workers, so they could do it as two-thirds as much time. 

Pam  12:05

And if you don't mind, I'm going to repeat those numbers. So, from 10 to 15, you're saying multiplicatively to get from 10 to 15, that's 10 times three 1/2s is 15. I might also say that as 10 times 1 and a 1/2. Like, 1 and a half 10s is 15.

Kim  12:21

Mmhm.

Pam  12:22

So, since you scaled the workers by three 1/2s, then you're going to scale the hours by its reciprocal. And you need two-thirds the amount of hours. So, from 12, two-thirds of 12 is going to be 8. 

Kim  12:36

Yeah, it's a little trickier because when...

Pam  12:39

I think it's a lot trickier!

Kim  12:40

Yeah. Well, because I saw the 10, and I think because I know there's two 5s in 10 and three 5s in 15. Right?

Pam  12:47

Mmhm.

Kim  12:47

So, I know there's a two 1/3s or three 1/2s. But because it's also not the same relationship on each side, like going from the (unclear).

Pam  12:56

It's an inverse?

Kim  12:57

Yeah.

Pam  12:57

Inverse relationship?

Kim  12:58

But it gets a little bit twistier to hang on to.

Pam  13:02

Or maybe a lot twistier.

Kim  13:03

Mmhm. 

Pam  13:04

Which then begs the question. If I were to do a Problem String like this, ideally, I would have done direct variation Problem Strings with non unit rates, where we would have had the same kind of thing happening. In fact, you're making me wish we would have done an episode before this, where we might have done something like if I've got 5 slices of pizza for $6.00... Let me actually change those. This is Pam making it up on the fly. If I had 4 slices of pizza for $5.00. But then I wanted to have 10 slices of pizza. So, 4 slices of pizza for $5.00 and I wanted 10 slices of pizza. Is that also that two 1/3s or three 1/2s relationship?

Kim  13:43

I don't know. I didn't write it down what you... You said 4 slices for $5.00. 

Pam  13:46

$5.00. And I want to go from 4 slices to 10. 

Kim  13:49

Okay. 

Pam  13:50

Is 4 to 10 also that that 1 and a 1/2? No, that's 2 and a 1/2. 

Kim  13:54

(unclear) Yeah.

Pam  13:56

So, that's a 2 and a 1/2 relationship, right?

Kim  13:58

Mmhm. 

Pam  13:59

So, then you would want to scale the 5 also times 2 and a 1/2. 

Kim  14:03

Yeah, yeah. 

Pam  14:04

And so we kind of want to do some scaling like that. Another example would be if I'm trying to get from 5 to 6 multiplicatively, I can multiply that by six 1/5s. Then I could multiply whatever the other side of the ratio table is also by six 1/5s. That's... I don't know that we've done a whole lot of that. Maybe we do that sometime on the podcast. But this idea of that there's a multiplicative way to get from any factor, any number to another number, and it has everything to do with fractions. Go ahead. 

Kim  14:37

Well, I appreciate you taking me down the road of doing both at the same time. Inverse and (unclear). 

Pam  14:44

There you go. Yeah. So, we're not only scaling by this nice three 1/2s, but we're also having that inverse relationship. You're welcome, Kim. You're welcome. Because you are up to it. And we are having so much fun. You love being on the podcast with me.

Kim  15:00

I do. 

Pam  15:00

Hey, can I tell you what a lot of kids do at this point? 

Kim  15:03

Sure. 

Pam  15:04

So, some kids will do what you did with the 5 workers and 24 hours to  say that 5 to 15 is times 3. That's 3 times the amount of hours, and so you need a third the number of workers. 24 divided by 3 is 8. Many kids will do that. Other students will say, "Hey, I'm noticing that we've already got 10 workers on the chart. They took 12 hours. And we've got five workers on the chart. They took 24 hours." And then they add the 10 workers plus the 5 workers to get the 15 workers. And if you do that, 10 plus the 5 is 15. Then 12 plus 24, they get 36. And then we get to stand back and say, "You're saying that if we have, say, 3 times the number of workers, it's going to take them more hours?"

Kim  15:51

Mmhm.

Pam  15:51

Like, does that make sense? And this is actually... Like, we are doing this really kind of fast this idea that if you double the workers, it takes half the time. But that often takes some working with students, where they actually have to maybe do a thing. 

Kim  16:07

Yeah.

Pam  16:07

You know like, I actually had students at one point. I was like, "If we were going to erase this board, and it took 3 of us a minute to erase the board. Okay, now you two go sit down. There's only one of us now. How long is it going to take me to erase the board? Less time or more time?" And they were like, "Oh. More time." Like, we almost had to act out the kind of thing to get this inverse relationship a little bit more secure. 

Kim  16:31

Yeah.

Pam  16:32

Another thing that students will do sometimes.

Kim  16:35

Ooh, I know! I know!

Pam  16:35

Oh. Yeah, go. 

Kim  16:36

Well, I was going to ask you, but I wanted us to be a little more streamlined with the Problem String, but we've already derailed a little bit. But do you have kids that say, "Well, it's in between 15 and 20?"

Pam  16:46

Yeah, absolutely.

Kim  16:47

(unclear). Yeah.

Pam  16:48

Yeah. and what is... 

Kim  16:48

That's a trip.

Pam  16:49

We have 20 workers and 10 workers. And their corresponding time is 6 and 12 hours. So, if 15 is in between 20 and 10, then they'll say, "What's in between 6 and 12? And that's 9. Now, we're really close. Now, 8 is the correct hours, but they get 9 hours, and that can definitely be tricky to have them think about. Yeah, nice. Hey, one more question. 

Kim  17:15

Yeah? 

Pam  17:16

I wonder. Kim, you wake up one day and only one worker shows up. Ah! You got to clear the field. How long is it going to take that one worker if takes 20 workers 6 hours to clear the orchard, how long is it going to take 1 worker to clear the orchard? (unclear).

Kim  17:34

That's unfortunate. 

Pam  17:35

Yeah, sad day for that guy. 

Kim  17:37

Yeah, that's going to take 120 hours.

Pam  17:40

Now, that seems really random. Where did that 120 come up? (unclear).

Kim  17:44

Yeah, I went back to the 10, and I said we have...

Pam  17:48

10 what? 10 workers? 

Kim  17:50

Mmhm. 

Pam  17:50

Okay.

Kim  17:51

Yeah, I went to the 10 workers. 

Pam  17:52

Okay. 

Kim  17:52

And I said we have a tenth of as many workers.

Pam  17:56

Ah.

Kim  17:56

Divide by 10 or times... 

Pam  17:58

Nice.

Kim  17:59

...one-tenth And then I'm going to need them to have 10 times as much time to do it. So, 12 times 10.

Pam  18:07

(unclear). Yeah, sorry. I didn't mean interrupt. 

Kim  18:08

That's okay. 12 times 10 is 120. 

Pam  18:10

Okay, that's a great way to do that. Could... I wonder if that just if we were to go from, say, like the 20 workers took 6 hours. The original. If we divide the 20 workers by 20 to get 1 worker, then it's going to take 20 times the amount of time. And 6 times 20 is also 120. 

Kim  18:25

Mmhm.

Pam  18:26

Cool. At that point... Well, actually, maybe I'll do one. Will I do one more? What if we only had 1 hour? How many workers would we need?

Kim  18:38

1 hour? We need 120 workers.

Pam  18:41

How do you know?

Kim  18:43

Because the 1 guy took 120 hours. But if he brings a whole bunch of friends, they can all get it done in an hour. I still... There's 120 hours worth of work to be done.

Pam  18:54

Ah, that's really interesting. You're saying that if it takes 12 hours... 20 workers... Bleh. If it takes 20 workers 6 hours to clear the orchard, there's 120... How did you say that? 120 what? 

Kim  19:05

120 hours worth of work to be done. 

Pam  19:07

Yeah, and sometimes we call those "worker hours". So, in this scenario, it's sort of 120 worker hours of stuff, right? There's like. And we need to get that done, and we can make it happen where we have 20 workers and it take 6. And so, I might, at that point, actually look at the table. And I wish we were visual here, but I've got 20 workers, 6 hours. 20 times 6 is that 120. The next entry we have is 12 workers. Excuse me. 10 workers took 12 hours. 12 times 10 is 120. And the next entry was 40 workers, 3 hours, that's 120 worker hours. The next one's 5 workers, 24 hours, that's 120 worker hours. So, every one of these is sort of ending on this 120 worker hours. So, as I kind of look down the table and we start to note. Sure enough, every one of these workers numbers times every one of these number of hours is 120. I might, below that table, then write it sure looks like the number of workers...x...times...and I'm going to write multiplication....the number of hours...y...is always equal to 120. In this particular scenario. Because we started with the 20 workers taking 6 hours.

Kim  20:20

Mmhm. 

Pam  20:20

So, if that's true, we now have an equation x times y equals 120.

Kim  20:25

Mmhm.

Pam  20:26

That's kind of nice.

Kim  20:27

Mmhm. 

Pam  20:27

And so, if we wanted to solve for one of them, we wanted to say, "Hey, what if I wanted to know how many hours it's going to take me?" Well, that's sort of if y was the number of hours, then y is equal to that 120 worker hours divided by the number of workers. Divided by x. Now, we have a function where y equals 120 divided by x. And we could graph that, and we could look at what happens. The more workers you have, the less time it's going to take. The fewer workers you have, the more time it's going to take. And we get a very nice curve that comes out, showing that non-constant rate of change in that inverse relationship. Kind of cool. 

Kim  21:04

Yeah, that's cool. 

Pam  21:05

Yeah, nice. So, when we've done other ratio table work on the podcast, we did direct variation. Sometimes we call it directly, the situation is directly proportional. Today, we were talking about inverse variation. Or we can say that the situation varies inversely. Or it's inversely proportional. So, that was kind of fun. Let's maybe do some more of that sometime. 

Kim  21:28

Yeah, sounds great. 

Pam  21:29

Whoo! 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!