This string is intended to build intuition for inverse variation, which can be described as "𝑦 varying inversely to 𝑥. This kind of functional behavior is like the rational parent function 𝑦 = 𝑘/𝑥. If students have prior experience reasoning in ratio tables by scaling in tandem, this Problem String may be challenging. Keep students in context, reasoning about the number of workers it takes to clear the whole field and the time that takes.

If students start to talk about "clearing half the field" or in other ways chopping up the field, remind them that this Problem String is all about clearing the whole field. The number of workers and the time is always based on clearing the whole field, getting the whole job done. 

The first two problems give students a nice factor and product of 20 workers, half (10) and double (40). The next problem, 24 hours, is a multiple of each of the previous number of hours, 6, 12, 3, so students can reason from any/all of them. 

The fourth problem, 15 workers, is most directly figured from the previous 5 workers (times 3). Students may try to "split the difference" because 15 workers is halfway between 20 and 10 workers, so students may reason that the corresponding number of hours is halfway between their corresponding 6 and 12. This "split the difference" strategy only works in linear situations, where the rate of change is constant. The rate of change for inverse variation is not constant. 

The last two problems are given to help nudge students to think about the total number of worker-hours in this situation, which is 120 worker-hours. 

Set the stage for the context. You have an orchard full of fruit. You know that 20 workers can clear the orchard, pick all of the fruit, in 6 hours. If that is true, how long would it take 10 workers?

Ask students to justify their answer. Notice if students are thinking additively or multiplicatively. Encourage students to stay in context. "10 what? If you halve number of hours? What is your 6, the 12? For those numbers, what's happening in the orchard?"

Model the doubling and halving using arrows. Use multiplicative language like, "So if you halve the number of workers, how long would it take?" "If you halve the number of workers, it would take double the amount of time?"

Repeat for the other questions. For the 40 workers, encourage students to look for relationships between both the 20 and 10 to 40 workers. 

When you get to the 24 hours, ask, "What if you had a whole day, 24 hours, how many people would you need? You don't want to get it done any faster. You have all 24 hours. You want to hire just the right number of people.

Before you ask the number of hours it will take for 1 worker, ask students if they notice a number happening all through the table. Wave back and forth from left to right to nudge students to look between the number of workers and the number of hours. Help students notice the 120 "worker-hours". The product of each of the row entries is (number of workers)(number of hours) = number of worker-hours (120). 

If it takes 20 workers 6 hours to clear the orchard, pick all of the fruit, how long would it take 10 workers to clear the orchard?

What if you had a whole day, 24 hours, how many people would you need? You don't want to get it done any faster. You have all 24 hours. You want to hire just the right number of people. How many people would that be?

What is happening in this table? Is this a ratio table? In a ratio table, the ratios 𝑦/𝑥 are all equivalent. What is equivalent in this table? The product 𝑦𝑥? What is that product? Can that help you find the time it would take one person? The number of people you need to clear the orchard in one hour?

Problem Strings are facilitated by the teacher by asking each problem, one at a time. Students work to solve the problem, the teacher calls on individuals to share, and the teacher models (represents) student thinking on the display. Scaling strategies can be represented with arrows. Be sure to keep the arrows consistent. If you are doubling from 10 to 20, the arrow should point from 10 to 20 and note ×2."

If the question demands some think or work time, the teacher circulates to see what students are doing or to ask students what they are thinking. If the teacher cannot see or hear the desired strategy(ies) quickly enough, the teacher can ask a question that draws out that strategy. Questions like:

Did anyone use the 10 workers to get to the 40 workers? Did anyone do something multiplicative?

The teacher models and requests a private response signal. Students using a private response signal allow their peers enough work time, are not distracting other students, and give the teacher valuable information about who is done and who needs more time. In the video, Pam verbally requested a private signal, "Give me a thumbs up when you think you know..." and she physically demonstrates the signal by holding her thumb up as she asked questions.

In a Problem String that is set in a context, introduce the context and then keep referring back to it. When students say things like, "Double 20, so halve 6," you can reply with, "Ah, so if you double the number of workers, that makes sense that you halve the time it takes to get the job done." The context can help students realize what they are doing. This is especially important if students are thinking additively.

For example, if a student is trying to find the time it takes 10 workers they cannot "double the 10 to 20, so double the 6 to 12", so double the workers takes double the time. If they halve the number of workers (from 20 to 10), then the time it takes to do the job is twice as long (from 6 hours to 12). Staying in context now helps establish a base you can lean back on, and come back to later when students are dealing with inverse variation and  tables that are not in context.

This Problem String is a concept/big idea building string—it's purpose is to build the sense of inverse variation in a table, that the number of hours varies inversely proportionally to the number of workers. The emphasis is on sense making. If you have more people, you can get more done in the same amount of time. If you have fewer people, you can get less done in the same amount of time. However, in this scenario, the amount to get done is fixed, the orchard needs to be cleared, the whole orchard, all completed. The time it takes to complete the job is what is varying.  This sets the stage for students to use already obtained entries to find others.

Through the rest of this Problem String, take multiple strategies because that will help students get used to the inverse relationships. The purpose is not to build a certain strategy, so don't focus on any certain one. Instead focus on the scenario and the inverse variation, by asking for and representing different strategies.

Since you are representing multiple strategies for each problem in a table, the board (display) can get very busy. Plan strategically at which points you will erase scaling marks so that students can focus on the relationships being used, but they can also focus on the values in the ratio table as needed.

The Problem String introduces a context that students are not familiar with. This is the first time that they have considered an inverse variation scenario. Pam lets them struggle. She pushes up against their ideas. She doesn't let them push her for a top down answer. 

At the beginning of the Problem String, Pam asks the first question, "Given 20 workers take 6 hours, how long will it take 10 workers?" It then takes several minutes of students talking to a partners and the class talking, before Pam says, "Go back to the 20 workers. When the 20 workers got it done in six hours, how much did 10 workers do?"  This helps focus the question and prompts a discussion where Vivanie says, "The less workers the less, I mean, the more hours and the more workers, the less hours." Pam confirms that most students are good with that generalization, but they are still not quite sure how many hours it takes 10 workers.

Pam decides to move on to the next problem: "I wonder if we could do another one. What if I said to you that we know that there's 24 hours until there's a hard freeze, and I don't know if you guys know, but when it freezes, then you lose all of it. The fruit rots. It's not good, can't sell it. So I know I have 24 hours, but I'm cheap. I got 24 hours to do it. I don't have just six hours. I have a whole day of 24 hours to do it. I want to only hire the number of workers I need."

Moving on to the next problem is a bold move and Pam is betting that getting more of the scenario on the table will help students confirm their beginning theories.

Pam hears students making conjectures about adding 6 and 3, but because the students are reasoning with naked numbers (not in context), their reasoning does not fit the scenario. To encourage them to use the context to help, Pam says, "So I hear you saying six and three but not six what and three what. So I want you to stay in workers and fields." Then she goes to the board and draws the "orchard".

"So it's like got the whole field and 20 guys go out there. Here's the field right? And 20 guys go out there and they do and they can clear the whole field in six hours. But today only 10 people showed up."

The detail of the drawing is less important then the fact that it represents the whole field. This problem is not about getting half of an orchard cleared. It's always the number of workers and time it takes to clear the WHOLE ORCHARD.

Pam says things like the following to help students stay in context: "So I hear you, but when you guys are talking, I don't hear workers" and "Six, six what, six what?"

Pam starts the string with reminders about writing or not as a reminder to not get into the mode of just copying stuff down. She also makes it clear that students' thinking and reasoning is valued:

[R]emember, if you want to keep track of your mental thinking, totally legal to keep track of your mental steps. But don't get caught in copy mode. Like, don't get your brain turned off and you're just copying what's going on. Keep your brain going. I like the way you guys think, and that's what I'm interested in hearing about.

Statements and questions like, "Don't let me interrupt, I was just wanted to listen" send the message that what students are thinking and discussing is worth listening to. Here are additional statements and questions that send that message:

What do you guys think?

I'm not convinced. These guys had a really interesting conversation over here. And then you guys did as well.

So, who understood what Michael and Barker are thinking about?

So, Damien help me, because I'm not sure what they were saying.

So Liliana and I were just talking and we were both having a hard time saying again what you said.

Cool, hey let's see if that holds.

You think it might be 18?

What're you thinking?

Okay, let's let's hear it out. So Barker, you guys ready to, are you guys ready to listen to someone else talking for a sec? Do you guys have it down? So, Omar and Mario have something to tell everybody. You said you have this whole argument, let's hear it.

So that's tricky. If 10 workers can get it done in 12 hours. More workers, it shouldn't take more time.

This is such an interesting conversation and you guys are doing such a good job figuring this out. 

You're saying it's got to be multiplicative. But if we double the number of workers, it's going to take half the time. If we half the number of workers, it's going to take them twice the amount of time, that seems really important.

That makes sense to you, yeah? I wonder if that 120 shows up anywhere else.

[Pam] You don't want to be wrong?

[Damien] Yeah.  [Pam] You know what? If we just keep talking and keep working it out, then it kind of doesn't matter, because we just keep talking and working it out. You could make a conjecture. You could say I'm wondering about, and then we can wonder about it together.

[Damien] Hold on, I need to think about this.

[Pam] And you know what, you can think about it. Because math is about thinking about what you know, and as you think and as you guys make sense of this stuff, you're literally building new connections in your head. I'm going to suggest that this is what we should've been doing with you guys all the way on. Ever since you were learning math, we should've been having you guys think like you are today, because your brains are going today. Like literally you're getting more dense in your brains, because you have more connections, because you're actually involved in the thinking. When you guys go onto your class next year, remember that! Remember that it's not about memorizing what your teacher gives you. It's about you guys thinking and reasoning. And you can, like, you could think just like you are with this one. And so I'm going to encourage you to keep thinking.

Pam asks students to weigh different answers against each other and discuss their reasoning. Here are some examples:

I'm not convinced. These guys had a really interesting conversation over here. And then you guys did as well.

But I am curious, I heard some people say three, right? And I heard some people say 12. So who's thinking three? So anybody justify the three? Damien just put his thumb down. I did hear three, but you guys changed your mind. Can you tell me about that? So, you're thinking it's not three anymore, why not?

So, we have two answers up here. We have the one from the last period and we have the one from you guys. What do you think, nine hours, eight hours?

At this point in the Problem String, Pam moved on to see if the next questions would help:

So we still have 12 and nine? Okay, cool, I wonder if we could do another one. What if I said to you that we know that there's 24 hours until there's a hard freeze?

Students had experienced the "Sticks of Gum" ratio table Problem String the day before. Because we were going to video the next day, I chose the "Sticks of Gum" Problem String because it is a really good string to get students talking and reasoning. This also had the effect of helping students move from additive thinking to multiplicative thinking. But, now students are also playing with the idea of splitting the difference, which works when the scenario has a constant rate of change, one pack has 17 sticks so 17 sticks per pack. If you know that 2 packs have 34 sticks and 4 packs have 68 sticks, you can reason that since 3 packs is exactly in the middle of 2 and 4 packs, then we can find the exact middle number of sticks between 34 and 68 to find that 3 packs have 51 sticks. This scenario can be represented by 𝑦 = 17𝑥, where 𝑦 is the number of sticks in 𝑥 packs.

The scenario in the orchard Problem String does not have a constant rate of change. It is not a directly proportional situation, it is an inversely proportional situation. This scenario can be represented by 𝑦 = 120/𝑥. 

As students were reasoning about the amount of time it would take for 10 workers to clear the orchard, Pam chose to introduce the next problem in the string, 40 workers. Now students can consider what happens when half the number of workers show up, 10, or double the number of workers show up, 40. By keeping context at the fore, students began to develop the sense that numbers were varying inversely. Double the number of workers take half the time. Half the number of workers take double the time. 

Indeed, moving on to the next problem did help students to make sense of both the multiplicative and also the inverse nature of the situation.

This is the transcript of the video, with bold comments of Pam reflecting on her teacher moves during the Problem String, inserted to highlight purposeful teacher moves.

- [Pam] Hey, we're going to do a quick problem string today. I don't think you're going to need to write anything down, but remember, if you want to keep track of your mental thinking, totally legal to keep track of your mental steps. But don't get caught in copy mode. Like, don't get your brain turned off and you're just copying what's going on. Keep your brain going. I like the way you guys think, and that's what I'm interested in hearing about. So what if we had... Actually, quick story, you guys ever been to an orchard? Setting the stage for students' responsibility during Problem String; the expectation of thinking over recording

- [Students]  (mixed responses)

- [Pam]  Like where trees are grown and there's fruit on the trees?

- [Students] Oh yeah.

- [Pam] So when I was growing up, we used to go pick cherries. Has anyone ever picked apples, or can you picture it? Can you picture a bunch of trees?

- [Students] Yeah.

- [Pam] And you go in the orchard and you pick all this stuff. So pretend that we have an orchard, okay? We've got an orchard, it's got a bunch of fruit on the trees, and we've got people who are interested to earn money. So, like maybe you guys have a summer job or something. And so if we have in this orchard, a bunch of workers that show up. So, we've got a bunch of workers show up. I don't know, we can call them fruit picker, whatever, workers. And if we had 20 workers show up, we know that it will take them six hours to clear the orchard, to pick all the fruit. So, we've done it often enough that we know that as long as we have 20 workers, they can get the job done in six hours. Can you picture that? Okay, cool. Setting context that is realizable. If that's true, if we know that that's true. What if I only had 10 workers show up one day? So I'm a little curious how long it would take us to clear the orchard. So, I'm hearing lots of people yell out. Remember to give me a thumbs up. Reminding them of the signal we used the day before when doing the first Problem String. Lots of people are yelling. Will you turn to the person next to you and will you convince them of your answer? Turn and talk, go. Don't let me interrupt, I was just wanted to listen. Keep talking, keep talking.

- [Liliana] Oh yeah! No, because 20 and then 10, but if there's more people then it's like six hours. You know what I'm saying?

- [Pam] It's hard to say isn't it? Keep talking, keep talking. What do you guys think? Encouraging students to keep at it even though we know that what happens in our heads is sometimes difficult to verbalize!

- [Vivianie] Well, I was figuring out, how like, since there's going to be, because you changed it to 10 workers, that's half of 20. So then it would be half of 6 which is 3.

- [Pam] Okay, so if I have half the number of workers, it's going to take half the amount of time to clear the orchard? Summarize, restate their thinking back to them so they can hear it, consider it.

- [Vivianie] Yeah. I think so, I don't know.

- [Amber] Wait no, no, no, I think it be more. Because there's less workers.

- [Vivianie] Oh, shoot!

- [Amber] Actually it'd be 12.

- [Vivianie] Oh!

- [Pam] Go ahead and finish the sentence you are on. I want to honor their thinking and discussing. I'm not convinced, I'm not convinced. These guys had a really interesting conversation over here. And then you guys did as well. I didn't, I didn't, I didn't get a chance to listen, hang on, hang on. I didn't get a chance to listen to anybody else. But I am curious, I heard some people say three, right? And I heard some people say 12. So who's thinking three? So anybody justify the three? Damien just put his thumb down. I did hear three, but you guys changed your mind. Can you tell me about that? So, you're thinking it's not three anymore, why not? Justify! Share your thinking! THIS is what mathematics is.

- [Vivianie] Because, well we thought it was three because 10 is half of 20. And so we figured like, half of six is three. But then

- [Pam] Oh, so that would make this three?

- [Vivianie] Yeah.

- [Pam] Okay.

- [Vivianie] But then, if there's less workers it would take more time, so it'd take longer.

- [Pam] So, it would take longer, how much longer?

- [Amber] Double the six.

- [Pam] So, you think you want to double that? And that's how you guys were getting the 12. Everybody's okay on that, that makes sense? Barker, did you want to say something?

- [Barker] Wait, hold on a minute.

- [Pam] No, let him think, let him think, don't interrupt, let him think. 

- [Barker] I was thinking, because you take away half of the workers, you would only add half the time.

- [Pam] Okay.

- [Barker] So, half of the time it took, it'd be three, if you add it to get 9.

- [Pam] So, you're back to three?

- [Barker] No, no, no.

- [Pam] No?

- [Student] He said three plus six.

- [Pam] Oh, three plus six, so nine. Oh wait, I think we better-

- [Micheal] Hey, that's what I was thinking too.

- [Pam] Is that what you were thinking too?

- [Micheal] Yeah.

- [Pam] So, who understood what Michael and Barker are thinking about? You did?

- [Damien] I was thinkin' that too.

- [Pam] You were thinking that too. So, Damien help me, because I'm not sure what they were saying. I knew what they were saying, but wanted to draw this idea of an additive relationship versus multiplicative relationship out into the open.

- [Damien] So like, since it's half the workers, you don't really double the time, because it is just half the workers. So, you just like, just like add three hours. Like, I don't know, the way I thought it, like the way I thought it was different than how I explained it. I don't know how to explain it.

- [Pam] Does that sound familiar to you?

- [Barker] Yeah, but I'm trying to find a better way to explain it. Yeah, because you only take half the workers. You don't take 75% of the workers. So if you took 75% of the workers, it would be three hours. I don't know how to explain it.

- [Damien] So wait, if you have like five workers, then you would double the time, because...

- [Pam] Why five workers?

- [Damien] No because I'm...

- [Pam] Gabriel, Gabriel, come on. 

- [Mia] We're struggling.

- [Damien] Can you erase the 10 and 12?

- [Pam] Absolutely, yep! Oh, except I think we knew the 10. That's what we know. We know that if 20 workers show up, it takes six hours, but today, only half the number of workers showed up. So then the workers are like, dang, how long is it going to take us to do that? Going back to context. Only HALF of the workers show up...

- [Damien] It would take them...

- [Mia] It would be 12, wouldn't it?

- [Pam] 12.

- [Amber] I think it would.

- [Pam] So, she's saying 12, let's have it out. She's saying 12, you're saying nine I think, nine? Which one is it? At this point I wanted to put it back on the class to argue their thinking. I wanted all students to consider both ideas and make sense out of both. This sense making takes effort. Having both of these answers is a clear indicator that we have students at different levels of mathematical reasoning: some are thinking additively and others multiplicatively. 

- [Omar] Wouldn't it be 12 because if you have...

- [Students] I think nine.

- [Omar] If you have 20 workers.

- [Pam] Listen, listen, listen.

- [Omar] If you have 20 workers and it only takes six hours, and you split the amount of workers into half. So you would add another, another... Like, you would add the other half like... How do I explain this? If it's the other half of the hours and add them up with the six, so it would be 12.

- [Pam] Did you follow that?

- [Barker] Yes, I get that, but at the same time, there was something else I was thinking, I just dunno how to explain it.

- [Amber] How'd you get nine? Like just...

- [Mia] I can explain it.

- [Amber] Yeah, explain it to him.

- [Mia] Okay, since you have six and half of six is three. So you're going to add six and three, which is nine and then it's nine.

- [Pam] So I hear you, but when you guys are talking, I don't hear workers-

- [Mia] Because half of six is three.

- [Amber] Oh, oh!

- [Pam] But what is the six and what is the three? I hear you guys-

- [Barker] If there's 20 workers, it takes them six hours and then you take half the workers away.

- [Vivianie] Yeah, six plus three, like you said, but like, why would you add them?

- [Pam] Six, six what, six what? Stay in context! 

- [Barker] It would be like half the time. Six what?

- [Vivianie] Hours. But like why would you add them?

- [Pam] So I have 10 workers and they go out in the field. Hang on, so I hear you saying six and three, but not six what and three what? So I want you to like stay in workers and field. So if I got the whole field and 20 guys go out there, here's the field, right? And 20 guys go out there and they do, and they can clear the whole field in six hours. But today only 10 people showed up. Trying to summarize what we already know; giving a visualization that may help students clarify that it is always the whole orchard getting cleared - that is the job. Not half an orchard or part of the orchard. That's not what's varying. The time it takes and the numbers of workers are what is varying.

- [Students] So it'd take more time.

- [Pam] So, I think everybody knew that it has to be more.

- [Student] So, I think it's 12.

- [Pam] You think it should double as 12? Dylan's nodding his head, yes, it's doubles. Let's vote, let's vote. How many of you think it's double?

- [Vivianie] Oh, wait!

- [Pam] How many of you think it's nine? Oh, Mario just said something helpful, go ahead.

- [Mario] I dunno how to explain it but what I know is like, say half of six, adding three more hours, since it's half of the workers. I can't explain it.

- [Vivianie] Yeah, that's what I was thinking, because Barker was saying that too and now...

- [Pam] I'm a little curious, how much of the work, go back to the 20 workers, when the 20 workers got it done in six hours, how much did 10 workers do?

- [Students] Three.

- [Pam] How much of the work did 10 workers do?

-[Student] They did three, three hours.

- [Vivianie] Half of it.

- [Pam] No, they worked for six hours.

- [Vivianie] They did half of the work.

- [Pam] They all worked, for all six hours. So, when they were all working for all six hours, if I only looked at the part of the field that the 10 workers were in, how much did they get done? Trying to express the work in total man hours, 120 hours.

- [Vivianie] Half of it.

- [Pam] Did they get half of it done?

- [Vivianie] Yeah.

- [Pam] Do you not understand my question?

- [Damien] Oh wait, oh wait, I think I know how to explain it.

- [Pam] Okay.

- [Damien] So like, okay, if you have 40 workers, it's only going to take them three hours, right?

- [Pam] Why? Yes! But let's bring the rest of the students here too. Explain your thinking!

- [Damien] Because there's, because Yeah, there's more workers, so- Since 20 workers take six hours, 40 workers will take three hours. But if what's it called? Since 20 workers, take six hours... Wait, hold on. From 40 to 20, you add three hours. And then from 20 to 10, you add another three hours.

- [Vivianie] Oh yeah, that makes sense. The less workers, the less, I mean the more hours and the more workers, the less hours.

- [Amber] Yeah.

- [Pam] Do you guys agree with that? You guys agree with that? I think everybody's agreeing with that, which is cool. Because that's kind of, that's totally different than some scenarios where if some, like yesterday, when we talked about, if this one was times two, it doubled the number of bags, we had double the number of M&M's. But that's not happening today. If you have double the number of workers, you're saying it takes less time and it took exactly half the time? Yesterday, we had worked with a ratio table, where students were scaling in tandem. This is the first time these students have encountered inverse relationships on a ratio table and we are taking some serious time to work it out. Giving them time to work out the relationships with sense making is an important goal!

- [Students] Yeah.

- [Pam] Yeah? So, we're still a little unclear on the 10. Are we still, some people still say 12, some people still say nine, yes or no?

- [Dylan]  I still think it's nine.

- [Pam] So we still have 12 and nine? Okay, cool, I wonder if we could do another one. Even though we aren't still quite there with an answer we have raised some important ideas and I know we can revisit this particular problem in the string. I made a choice to move on at this point. What if I said to you that we know that there's 24 hours until there's a hard freeze? And I don't know if you guys know, but when it freezes, then you lose all the fruit, all the fruit rots. It's not good, you can't sell it. So if I know I have 24 hours, but I'm cheap. I got 24 hours to do it, I don't have just six hours. I have a whole day, I have 24 hours to do it. I want to only hire the number of workers I need. So think, think. You look like you want to talk to your partner, do you? Yeah, turn and talk to your partner, go. Partner talk gives all students more time and space to work out this new twist to the scenario. Okay. What are you thinking about?

- [Mario] That's like adding more workers, I can't explain it.

- [Pam] What's adding more?

- [Mario] You're adding more workers.

- [Pam] You want to add more workers? So if you have 24 hours you have more time than you've ever had. 

- [Mario] I know, so that we don't have to subtract the workers

- [Pam] Do you subtract the workers or do you divide the workers? Subtracting workers is additive thinking. I "drop" the word "divide" to see if it is within the students' zone of proximal development - within their grasp. 

- [Omar] I mean the hours, and if you. Then just hire two, because if it takes them that long to do the field then it takes that time then we'd just hire two workers instead of having

- [Pam] You think two workers could do the whole field in 24 hours?

- [Omar] Yeah, I'm pretty sure. I think it's 100 or 80, 100 or 80.

- [Pam] Why 100 and why 80?

- [Mario] No, because we like adding in some more workers instead of the-

- [Pam] Oh, you're adding more workers. But do we need more workers if you have more time? Reminding students of what was agreed upon; more time means less workers needed.

- [Omar] No.

- [Pam] You need fewer workers. But now you're not sure how many fewer? Yeah, keep working on that. This sends the message, "I believe in you. Now that we've clarified, you can keep making sense."  What are you guys thinking about? Where do you come up with that?

- [Liliana]
Okay, so less workers, the more um...

- [Pam] That you need, to clear the whole field, yeah.

- [Liliana] But I would still be under the 24 hour mark.

- [Pam] Can you say what you just said? That everybody, that about if you have more time than you need... I think you said if you have fewer workers you can, how did you say that? You don't remember? Let's see if we can do it. What are you guys thinking about if you have a ton more time? So Liliana and I were just talking and we were both having a hard time saying again what you said. If I have a lot more time, do I need fewer workers to get the whole job done or more workers to get the whole job done?

- [Students] Fewer.

- [Pam] Fewer, everybody's good on fewer? What do you think? How many workers do you think we need if we have a whole day? How many do you think Omar?

- [Omar] Five.

- [Pam] Five! Liliana, she also said five, where's five coming from? Notice that I didn't say, "Five. Finally! Correct! Let's move on." Instead, I tried to keep students thinking, justifying, making sense of the 5 workers.

- [Omar] Because if you divide 20 by four, you'll get five.

- [Pam] Okay, so you just said, if you divide 20 by four, you get five. Where's the divided by four coming from?

- [Omar] From six times four.

- [Pam] So if you quadruple the amount of time you have, if you have four times as much time, you're saying you need a fourth the number of workers? Restating the multiplicative inverse relationship, using some words that might help students clarify what they mean. This is an example of just in time vocabulary: giving students words to describe what they are thinking about. How does that feel? Does that help anybody think about if we have half the number of workers? Dylan, why are you saying 12 times two? Oh sorry.

- [Dylan] I don't know.

- [Pam] You don't know? Okay, you were asking. What do you guys think about the 10 hours now? Or sorry, yeah, the 10, sorry the 10 workers? If half the number of workers show up? Repeating what they have been saying in multiplicative language, "half".

- [Cindy] Four!

- [Damien] I think like the way this works is like, the more you go down in workers, like the more the time's going to like go up.

- [Pam] With fewer workers, it's going to take us-

- [Damien] So like, yeah, one worker is going to take more than 24 hours.

- [Pam] I think, I think we agree. Because if it's five workers, that takes 24 hours. If I only have one worker, that's going to take a lot more. How about, what if I have... Lets go from the five-

- [Cindy] Oh my gosh! Five is the half of 10, so it's just add six more and that's 12. I'm sorry for that, it's like freakin'- LOVE these aha moments. Worth the wait time.

- [Pam] So you think that might be 12?

- [Cindy] Yes, it is 12, oh my god! 

- [Pam] Cool, hey let's see if that holds. Math is about thinking and reasoning. Let's see if our conjecture holds up with new information. Here's the next question I have. What if 15 workers show up? And I'm the boss and I'm like, "Oh, I wonder how long it's going to take them." Keeping it in context. Helping students think about workers and time, not just a naked 15.  What're you guys talking about?

- [Damien] Yeah, we think this one is nine.

- [Pam] How?

- [Damien] Because 20 and 10 workers is six to 12 hours. If you have 20 workers that's six hours. If you have 10 workers that's 12 hours and between that it would be-

- [Pam] Split the difference, you're right between, all right. Hold on to that thought. Splitting the difference is a strategy that showed up the day before in a directly proportional situation, where there is a constant rate of change. Today is inversely proportional, with a non constant rate of change, so splitting the difference does not work. Let's get some more thinking on the table.  What're you guys thinking?

- [Barker] 18.

- [Pam] You think it might be 18?  Justify your thinking! 

- [Barker] I dunno.

- [Student] No! Well, because then it'll be less. Because if it's more workers, it's less, like 10 and 12 hours. And If it's more workers, then it's got to be less than 12 too. It's not 2.

- [Pam] What're you thinking?

- [Mia] I dunno. Oh, because 40 is at the highest.

- [Student] 12 and six, so it's-

- [Pam] We already have so many workers, so many workers, it only took them three hours. What're you guys thinking?

- [Vivianie] 15 is I think is in between the 20. And so the number between six would be nine.

- [Pam] Okay, between those two. What're you guys thinking about?

- [Liliana] I really have no idea.

- [Pam] Some ideas, okay Gabriel what do you think?

- [Liliana] Gabe thinks it's nine.

- [Gabriel] So, 10 is 12 right? If you're going to add the five and then you're going to add three to it.

- [Pam] So, lots of you guys are splitting the difference. A couple other groups are doing that. Okay, so let's come together a little bit. I'm a little curious. A lot of you guys were pretty confident that if we halve the number of workers, then we would double the amount of time it would take, right? Because if you have half as many workers, it takes twice as much time. So I'm a little curious, a lot of you guys were talking about... So I'm going to go ahead and erase these, because I don't think we need that anymore. Then a lot of you guys were talking about using the 10 and the 20.  Let's see if we can get students to weigh the additive "splitting the difference" strategy against the multiplicative double/halve relationship.

- [Student] Eight or nine.

- [Cindy] Why, because they were listening?

- [Pam] Oh, shoot, we shouldn't have interrupted you because it was good. What you guys were doing was good. Can you tell everybody what you're talking about Barker?

- [Barker] I don't remember.

- [Cindy] Yes you do.

- [Pam] Can you, do you remember what you started saying?

- [Cindy] No, I didn't hear what he was were saying.

- [Pam] So, a lot of you guys were talking about between the 10 and the 20. You said the 15 was between the 10 and the 20. Who wants to go with it? You guys want to tell us about that? Damian and Joel were saying the same thing. So, the 15 is between the 10 and 20.

- [Vivianie] Yeah, and so between the six and 12 is nine.

- [Pam] And so you're thinking nine. Did anybody get a different answer than nine? We're going to pretend to keep talking so that they'll keep going. So, la, la, la, la, la. La, la, la, la, la

- [Vivianie] They said 18.

- [Pam] You guys are thinking 18. Did anybody get a different answer?

- [Mario] Well, we went from 17 to 18. We had this little argument when talking about it.

- [Pam] Okay, let's let's hear it out. So Barker, you guys ready to, are you guys ready to listen to someone else talking for a sec? Do you guys have it down? So, Omar and Mario have something to tell everybody. You said you have this whole argument, let's hear it. Let's hear it guys. What was it, what was your whole argument? Listen up, listen up.

- [Mario] But I was wrong.

- [Omar] Just say it, say it. I love that this is becoming a safe place to share thinking already.

- [Mario] Okay, I was thinking also five plus ten -

- [Pam] This five, five workers?

- [Mario] Wait, five minus 20 workers. And now I'm getting six minus 24, so that'd be 18 hours.

- [Pam] Oh, is that how you're getting the 18?

- [Mario] Yeah.

- [Vivianie] Oh.

- [Pam] So, if you take 20 workers and you say subtract five workers, that's only 15 workers. So you're saying if 20 workers can get the job done in six hours and five workers can get the job done in 24 hours, then you think 15 workers can get the job done in 18 hours?  Back to context. Let the context influence how you're thinking.

- [Damien] No. That doesn't make sense.

- [Pam] What doesn't make sense?

- [Damien] Because if less workers can get it done in faster time. Like 10 Workers can get it done in 12 hours, why wouldn't more workers get it done in less?

- [Pam] So that's tricky. If 10 workers can get it done in 12 hours. More workers, it shouldn't take more time.

- [Cindy] Oh yeah it is 9 and 10-

- [Pam] So Mario, I think we agree with you. I want him to know that finding own errors is a thing to do! Let's agree with him then on what he now knows to be true. Okay, so we're still thinking nine. Is that the only answer we got out there?

- [Students]
Yes.

- [Pam] So last period, I heard a different answer. Last period, I heard people say something about the five workers and it took triple the time. This is something I do when I want to raise an idea that I have not heard in the class I am working with. It could be a correct or incorrect answer, a different strategy... something I want to explore. If five workers-

- [Cindy] Five times three-

- [Pam] Do you want to finish? It looks important, do you want to finish?

- [Cindy] I dunno what I'm saying.

- [Pam] Last period somebody said something about five workers taking triple the time. What would that mean... Yeah, five, sorry. If I had five workers and I tripled the number of workers, then what would that do to the time?  I purposefully use the multiplicative language of "triple".

- [Damien] Divide it by three.

- [Pam] If I triple the workers, then I would have to divide that by three? What's 24 divided by three? So, we have two answers up here. We have the one from the last period and we have the one from you guys. What do you think, nine hours, eight hours?  It's not about which is right or wrong or who is right or wrong, but what you think about each answer. What are you thinking?

- [Students]
Nine.

- [Pam] Can you give me a reason using workers and hours for nine? Because how does this like...

- [Barker] That makes sense, but-

- [Pam] What're you thinking about? Do you want to come up here? Oh sure. You guys really think hard. You think it's eight or nine and how are you going to justify that? You're the boss, you got to make sure you clear the field. - Ah, in order, okay.

- [Barker] So you divide this 9 right here. What would that be?

- [Pam] That's a good question. So, you're wondering if these are in order, than do these go in, and we call that arithmetically, do they go by the same amount additively? Do I add the same amount?

- [Barker] But if not, that would be because 12, 24 is

- [Pam] Oh, so you just answered the question. If we're confident in this, it can't be additively. Because we know it's multiplicative. Yes, let's do that together. This is such an interesting conversation and you guys are doing such a good job figuring this out. Barker, you decided, so Barker decided to fix me a little bit and he's like, let's put those in order. So Barker, can you tell me what your table looks like? I'm thinking that seeing the relationships in another way would help some of the students move from additive thinking to multiplicative thinking.

- [Barker] Oh, mine's 5, 10, 15, 20.

- [Student] That's what I was just doing right now.

- [Pam] Is that it? 40, okay.

- [Barker] 24, 12, blank

- [Student] We don't know that one.

- [Pam] Six, three, okay and then.

- [Vivianie] Three, six, nine, 12.

- [Amber] Oh, what about the 24?

- [Barker] 12 and 24 is not the same as three to six. Three to six, you're adding three.

- [Pam] So it looks like you're adding three, but in reality, are you multiplying by?

- [Barker] Two.

- [Cindy] Two? Whoa!

- [Pam] What's this, what's that?

- [Barker] Well, that can't be right either, because the 12 would shift down one.

- [Pam] Why?

- [Barker] Because a three times two is six, six times two would be 12, which would be at the 15 workers.

- [Pam] But wait a minute, if to get from three to six, you divided by two. So this has to be, or sorry. Three to six is times two. So this is divided by two? I didn't write that very neatly. But to get from 20 to 15, how do you get from 20 to 15?

- [Damien] You're subtracting, but that doesn't-

- [Pam] So, we need a multiplicative relationship.

- [Damien] I need Desmos

- [Pam] And maybe we'll pull that out tomorrow and keep exploring. A fine idea, but not right now. Can you get to 15 from any of these, multiplicatively? You can do this, students! 

- [Damien] No. Oh, five.

- [Vivianie] Wait!

- [Pam] So, what does it take to get from five, times three? So, what do you have to do here?

- [Damien] Times three.

- [Pam] Divided by three. So is that a reason for it to be eight?

- [Vivianie] Wait, what?

- [Omar] That's so crazy.

- [Pam] Is that crazy, why, what's crazy about it? Verbalize what you are noticing!

- [Omar] I dunno, but we like, started off with just 20 and six.

- [Pam] It's true.

- [Omar] Then we went from 10 to 12 and 40 and three and five and 24 and now we're ended up over here. We're multiplying and dividing.

- [Pam] It's pretty cool, pretty cool. We now have great multiplicative language happening. Let's see if we can continue to solidify it with the next question. I have one more question for you. What if we only had one worker show up?

- [Barker] Wait, so is that right?

- [Pam] You tell me. So Barker I'm gonna ask you a question. Is it consistent?  Let me drop the idea that it is important that the table is consistent, that the relationships are acting consistently in the scenario.

- [Barker] No.

- [Pam] How's it not consistent?

- [Barker] Well, it's not consistent by the same number.

- [Pam] So is it consistent multiplicatively?

- [Barker] Yes.

- [Pam] Can I get in between all the numbers? If I double the number of workers, do I half the number of times?

- [Barker] Yes.

- [Pam] If I triple the workers, do I third time?

- [Barker] Yes.

- [Pam] If I quadruple the workers, in fact, can we find a quadruple? Can you go from five to 20?

- [Barker] Yes.

- [Pam] So let's do that. If I want to go from five to, where's 20? Five to 20, that's times four, is this divided by four over here? Yeah, sure enough.

- [Omar] So there's really no right or wrong.

- [Pam] It's looking to me like pretty consistent, that you were seeing this relationship isn't additive. A lot of you guys were trying to add back that three, but I think what you're saying is it's not additive. It can't be, you're saying it's got to be multiplicative. Giving students ownership of the ideas. But if we double the number of workers, it's going to take half the time. If we half the number of workers, it's going to take them twice the amount of time, that seems really important. In hindsight, I wonder if it would have been a good time to have students repeat that back, to have them solidify their understanding by verbalizing it. But the string was already a little longer than I had planned so I didn't. It probably would have helped more students if I had.  So how does that play into, what if I only have one worker?

- [Student] 24 hours.

- [Pam] We already had 24 hours, right? I can't pull up what you guys were saying. Can you, I think we decided this was eight, right? You guys decided that to be multiplicative it had to be eight? Let me ask you a question, maybe before I should've said the one. Is there a number that's sort of happening in here that we haven't talked about yet? I'm wondering if anyone can make use of the 120 worker-hours to help them determine how long it will take one worker. 

- [Damien] I want to say I know, but I don't want to be wrong. This is so raw. So honest. Good for you for voicing this Damien! He is not alone!

- [Pam] You don't want to be wrong?

- [Damien] Yeah.

- [Pam] You know what? If we just keep talking and keep working it out, then it kind of doesn't matter, because we just keep talking and working it out. You could make a conjecture. You could say I'm wondering about, and then we can wonder about it together. Such an important thing to share. I'm glad I got the opportunity to say this to this group of students who are working so hard at new ways to think.

- [Damien] I think if, okay, so since five workers is 24 hours, you divide five and you get one and then you multiply 24 by five and you get 120. So I think it's 120 hours.

- [Students] That makes sense.

- [Pam] That makes sense to you, yeah? I wonder if that 120 shows up anywhere else? Like what is 40 workers times three hours. Is that, is that 120? Does it show up anywhere else?  Using phrases like "that makes sense to you?" and "I wonder" is language that can help students realize that mathematics is all about sense making and wondering. But this has to happen when students are sense making for it to really sink in. "Oh this? This is what we should be doing?"

- [Vivianie] Oh, shoot.

- [Cindy] Oh my God!

- [Pam] Hey Barker, I wonder if we're right? Like, I wonder if we multiply all these together, five times 24, is that also 120? What's five times 20? It's 100, five times four, sure enough it's 120. Aha, this is feeling pretty good. So what, you ready, what if I only had an hour? I got to clear the field, I need an hour. How many workers?

- [Student]
You need a lot of workers.

- [Students]
120. 16.

- [Pam] You think, I heard a bunch of answers, somebody said 120, what do you think about 120 workers?

- [Barker] Because 40 takes three. 40 times three would give you what?

- [Pam] 40 times three, you mean divided by three?

- [Damien] Yeah. 60 workers, because if 40 takes three hours, twice takes six hours, if you add another 20 to 40.

- [Pam] Oh, you're back to adding. I think we decided that adding, yeah, I think

- [Damien] Hold on, I need to think about this.

- [Pam] And you know what, you can think about it. Because math is about thinking about what you know, and as you think and as you guys make sense of this stuff, you're literally building new connections in your head. I'm going to suggest that this is what we should've been doing with you guys all the way on. Ever since you were learning math, we should've been having you guys think like you are today, because your brains are going today. Like literally you're getting more dense in your brains, because you have more connections, because you're actually involved in the thinking. When you guys go onto your class next year, remember that! Remember that it's not about memorizing what your teacher gives you. It's about you guys thinking and reasoning. And you can, like, you could think just like you are with this one. And so I'm going to encourage you to keep thinking.

Reflections:  For a first shot at an inversely proportional situation, these 9th grade students were doing great! For only their second Problem String ever, they were doing great! I was a little surprised how long we spent getting the scenario down so that we could settle on answers to the first few questions. But once the thinking got going, students were really trying to make sense of what was going on. There were a few times I think I could have asked students what they meant in terms of workers and hours that might have been helpful earlier. There are also a few times that I used naked numbers instead of staying in context that I wish I could rewind. I also recognize that we did not hear from every student. In future lessons, I would definitely want to check in with more students and invite their voices.