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