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. Adding equivalent ratios can be represented with brackets.

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.

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, Kim verbally requested a private signal, "Give me a thumbs up when you think you know..." and she physically demonstrated 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 2, so double 12," you can reply with, "Ah, so if you double 2 packs, that makes sense that you double the number of sticks, so double 12 sticks." The context can help students realize what they are doing. This is especially important when students are adding values.

For example, if a student is trying to find 9 packs by adding 8 packs and 1 pack, they cannot just "add 1", 96 sticks and one stick. They must add the corresponding number of sticks in one pack (12) to the corresponding number of sticks in eight packs (96). Staying in context now helps establish a base you can lean back on, and come back to later when students are dealing with ratio tables that are not in context.

This Problem String is a model building string—it's purpose is to build the model of a ratio table. In the beginning of the string, the emphasis is on doubling because the doubling can get students using what they know to find other amounts. This sets the stage for students to use already obtained entries to find others. Through the rest of this model-building string, take multiple strategies because that will help students get used to the model. You will be looking for certain strategies to pull out (doubling, over) but since the purpose is not to build a certain strategy, don't focus on any certain strategy. Instead focus on the model, by asking for and representing different strategies.

Since you are representing multiple strategies for each problem in a ratio table building string, the board (display) can get very busy. Plan strategically at which points you will erase scaling and adding 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.

Questions like, "Sam, you think you know that one?" instead of "What is the answer?" can help students relax into thinking instead of worrying only about answer getting.

The rest of these quotes each have elements of suggesting that math is more about reasoning and communicating justifications and less about answer getting or mimicking procedures:

"Did anyone think about something that was up here, to help them figure out how many?"

"Logan, what did you think about?"

"Did anyone think about what we know about two packs and just say, 'Well, if I know two packs then I can use that to help me figure out four packs?'"

"Interesting."

"Is there anything up here up here that might be helpful to you?"

"Did anyone else get 96?"

" I see a few confused looks on people's faces. She just said, 'I know 4 plus 4 is 8 so I can just add 48 and 48'. Does anyone think they know what Emily is talking about? Ava, you think you have an idea of what Emily is talking about? Can you say that for us in your own words?"

"Can you picture that?"

"Interesting, did anyone else add two packs plus eight packs? Bella did. Did anyone think about the 10 packs in a different way?"

"Sounds like you did something like Emily did."

"Let me ask you one more, you ready? We know how much one pack would be, we know how much 10 packs would be, two packs would be, four packs, eight packs, 10 packs. I wonder if you could use any of those amounts to help you to figure out nine packs of gum. Nine packs of gum, is there anything up there that might help you think of nine packs of gum?"

"Who understands what Sam did?"

Hannah Mae responded with, "I'm thinking that 12 times 9 equals 108."

Kim said, "So, you just know 12 times 9?"

Hannah Mae answered, "Yes," and then Kim moved on. She acknowledged that Hannah Mae just knew it and then asked the next student, "Great. Sam, what are you thinking about?" By doing so, she sends the message that it's fine that Hannah Mae just knew 12 times 9, but that she continues to be interested in students' thoughts.

Near the beginning of the Problem String, Kim asks students to find the number of sticks in four packs. She is looking to find someone who doubled the two packs. Notice how she interacts with students as she continues to ask questions to find someone who doubled the two packs:

Kim: What if I said that Luke had four packs of gum? How many sticks would that be? How many sticks would that be? Bella, you think you know?

Bella:  48?

Kim: 48. Anyone agree with Bella? Bella, how did you do that so quickly?

Bella: Because I know how to do skip count by 12's.

Kim: Oh, so you thought about 12 and 12 and 12, that was pretty fast. Did anyone think about something that was up here, to help them figure out how many? Logan, what did you think about?

Logan: I did 4 times 12.

Kim: Okay. Is that what I'm asking you, 4 times 12? Yeah. Did anyone think about what we know about two packs and just say, "Well, if I know two packs then I can use that to help me figure out four packs"? Brooks?

Brooks: So I know that two--

Kim: Two what?

Brooks: Two times 24 equals 48. And so I got, so I used two times 24 and then for the fact 'cause I know that two times 24, you just skip count, it equals 48.

Kim: And why did you do two 24's?

Brooks: Because if you, well if you do two 24's and four 12's, you get the same answer as 48.

Kim: Interesting. So if you have two packs of gum, it's 24 sticks. Then you can just double that and say, "Well if I double the number of packs then I can double the number of sticks?" Is that true? If I double the number of packs, can you picture that? If I double the number of packs, do I need to double the number of sticks? And how many was that, how many sticks?

Brooks: 48.

Kim: 48, interesting.

Kim asks gradually more and more specific questions while still honoring what students are saying. Then she spends some time questioning about and restating the doubling relationship because she knows not many of the students were using it. And ending with "interesting," gives students the idea that she is interested in their thinking, especially when they are using relationships from the ratio table.

Kim referenced confused faces and asked students to repeat what other students said or restate in their own words. Notice how she uses the natural complexity of the math to bring the disequilibrium to the forefront to talk about.

Kim: I see a few confused looks on people's faces. She just said I know 4 plus 4 is 8 so I can just add 48 and 48. Does anyone think they know what Emily is talking about? Ava, you think you have an idea of what Emily is talking about? Can you say that for us in your own words?

Ava: I used the 12.

Kim: Can you tell us what Emily did? Did you hear what Emily did?

Ava: Yeah.

Kim: Okay.

Ava: So she added 48 plus 48.

Kim: Why? Why did she add 48 and 48? That's what I'm confused about. Is anybody else confused about that?

Ava: Because 24 plus 24 equals 48. So you add them both together and you get 96.

Kim: Why did she add 48 and 48 though? Scott, you think you have an idea?

Scott: Yeah, because half of eight is four, that means you're adding two. And like you said you're adding two to the--

Kim: Two what?

Scott: Two more packs, like, two times more.

Kim: Double the number of packs.

Scott: Yeah.

Kim: Is it true that if I have four packs, I can use that amount of four packs to help me figure out eight packs? Is it true that if four packs of gum is 48 sticks, then if I double the number of packs, can you picture that? If I double the number of packs, then is that true that I'll have double the number of sticks?