This Problem String has the helper, clunker, clunker structure, where a problem like 10 × 13 can help with both 11 × 13 and 9 × 13. Then the next problem, 20 × 13 can also use the earlier 10 × 13, by doubling the 10 × 13 or adding the 11 × 13 and the 9 × 13. Then the 20 × 13 can help with both the 21 × 13 and the 18 × 13. You might note that the 21 × 13 could also be a combination of 10 × 13 and 11 × 13.

The model for this Problem String is the open array, using the factors as the dimensions of the rectangle and the product as the area. Write the problems as expressions, but record the strategies on the open arrays.

Notice that the 13 is constant in each problem, which means that the side length of the rectangle will also be consistent. As you draw the rectangles, keep the length of 13 as close to the same as you can. Also, keep the other dimensions in proportion, ie. the 20 should look twice as long as the 10.

The number 13 is purposefully chosen because it is a small double digit number that is not 10, 11, 12 (which students may already know multiples of).

As you prepare to lead your class through this Problem String, consider the following sample language, strategies to pull out from students, and ideas to highlight:

What is 10 times 13? What does a 10 by 13 rectangle look like?

Maybe you could think about using problems you know to help you with problems that are just a little over or just a little under.

Draw the rectangles in proportion. Label the dimensions and the area. When the class has agreed on an answer, write the equation, such as 10 × 13 = 130 and fill in the area.

As you circulate, look for students who are using the previous rectangles to help them.

Consider what you could ask at key times in the Problem String to help students build models, strategies, and big ideas:

How can you use the 10 by 13 to help with this problem?

Is there anything we've already done in this string that could help?

If you combine these two rectangles, does that combination represent this multiplication problem?

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. Multiplication problems can be represented by rectangular arrays where the factors are the dimensions and the product is the area. The distributive property can be represented by chunking that area into pieces and adding them together or subtracting a smaller rectangle from the larger.

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.

Questions like, "Casey, what are you thinking?" 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:

Oh, interesting.

And how did you think about nine thirteens?

Give me a thumbs up when you have an idea.

William, how did you think about 20 times 13?

Just give me a thumbs up when you think you have an idea.

Interesting! That's so interesting.

Casey, how did you- you didn't write anything down for this weird problem, what did you think about? 

Did that make sense?

What do you think about that idea?

How did you think about that one?

I'm wondering how you got that.

Adele, what did you think about?

And how did you think about this problem?

That's an interesting idea.

What do you think about that idea? Does that work for you?

I'm just wondering if that would be an idea that would be useful to you at some point?

Kim asked, "Did anyone else get 132? Okay, any other answers?." and later, "Anyone else get 143? Couple of people. Anyone else get a different answer?" 

Later in the string, Kim asks, "Anyone else get 117?"

This is a purposeful move to normalize that students may have gotten different answers. This accomplishes at least a couple of helpful things. Because she asks if anyone got a different answer, Kim does not signal to students if the first answer given is correct or incorrect. Students are given the space to reconsider their answer's validity because the teacher did not deem it correct. Also, students who got a different answer learn that this can happen, there is a low cost of failure, we will work it out together. The emphasis is not on the answer, the emphasis then falls on the students' thinking.

A couple of times, Kim just straight out asked for the strategy she was looking for, but in such a way that it does not give away what the strategy is.

I'm wondering if anybody used a problem that's up there already to help them. Give me a thumbs up if you used a problem that's already up there to help you.

Statements like, "Wes talk to us about 132," helps students know that they are putting their ideas out to the group, not just to the teacher.

Statements and questions like the following can help students realize that they need to listen to each other and make sense of each other's strategies:

I just heard somebody say oh, that was an interesting idea.

Does what William said make sense?

Did anybody hear and understand what Casey just said?

Casey just said 'I know 20 times 13, which is 260'. And then what did he say Anna?

Lucy did you, could you hear what she said?

Lucy, do you know what she, she, she saw that there were two thirteens, but why two thirteens? 

Adele. Does that match with what, what you did?

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

- [Kim] So today I want to do a Problem String with you guys, which is just a series of problems, right? You have paper and pencil. If you want to record something, you're welcome to. This is not the thing where you have to like sketch out everything I sketch out and record everything that I record. Um, but sometimes I might come around and I might like peek at your paper and see what you're kind of thinking about, okay. So if you want to write something down, it's totally fine. Generally, I really like it when people write the problem and the answer, right? But you don't have to like write a bunch of stuff just for the sake of writing, okay? When I do a Problem Strings with students I like to clarify this idea that they absolutely do not have to copy down what I am recording. All too often students feel like they need to take notes. Note taking is not the purpose of having paper available during a string. However, knowing the students and the problems in the string, I will at times ask them to write down the problem and answer. It often encourages additional recording of ideas when they know how to sketch their thinking. These students have participated in a few Problem Strings in the past, but the teacher had indicated that many of them always did place value partial products to solve 2-digit by 2-digit multiplication problems and I wanted to see who was thinking about my goal as it unfolded. So, let's get started. Let me ask you um, let's just start with this. What if I asked you 10 times 13? Just give me a thumbs up A quick gentle reminder of the private signal I'll be looking for. when you, when you've got it, 10 times 13. Some of you are writing the problem and the answer, that's fine. 10 times 13. Um, I don't think that one's necessarily challenging for you. What is 10 times 13 everybody? They have done prior work with times ten and therefore this is a fine time for a choral response.

- [Students] 130.

- [Kim] 130, okay 130. And I'm going to represent that on an array or an area model. And I'm going to record it like this - 10 times 13. My 13 is going to be a little bit longer. It's hard for me to see sometimes. 10 by 13, might look something like this. A 10 times 13 and the area is 130. I want to use this model to represent the area as we build throughout the string. I also want to attempt to model making rectangles proportional because the teacher indicated that students had been working with some area models but often split into four squares rather than making their place value splits proportional. Cool. What if I asked you, 11 times 13? Don't shout out. Remember! Mathematizing is not all about who is the fastest. Let's give everyone time to chew on the problem! 11 times 13, Hmm.

- [Kim] Casey, what are you thinking? Waiting, waiting... I didn't want to make him uncomfortable but I could tell he was thinking and not checked out. I walked on and looked back to see him smile and I gave him a thumbs up.

- [Kim] Wes, what did you get for 11 times 13? 55 seconds went by between me asking the problem and calling on Wes. While it feels a little long, I recognize that in that time I got to see how each student was solving the problem. Many of them were automatically splitting into 4 chunks. I wanted them to get part of the way through solving but intentionally didn't allow enough time for them all to finish before calling on a student who had used the target strategy.

- [Wes] 132.

- [Kim] 132. Did anyone else get 132? Okay, any other answers? I knew there were. Asking for other answers normalizes that this will happen and it's okay because we will work through it together. I don't always remember to do this, so I'm happy that i did today.

- [Student] I did

- [Kim] Yes, ma'am?

- [Student] 143.

- [Kim] 143.

- Anyone else get 143? Couple of people. Anyone else get a different answer? Good for me asking again.  Okay, Wes talk to us about 132. Talk to us...the group. But I really didn't do a good job of bringing other people into the conversation. 

- [Wes] Um, I got 132 because first I did 11 times 10,

- [Kim] Okay.

- [Wes] which is 110. And then I did um, 11 times 2, which is 22 and I added those together which was 132.

- [Kim] Okay. Can you say that again? And I'm going to kind of sketch what you just said. Okay, so you did what? 11? I'm going to hear you out and then I can decide how I want to record what you said. 

- [Wes] times 10 is 110

- [Kim] 11 times 10.

- [Wes] Is 110.

- [Kim] Which is 110. Okay, so the problem was 11 times 13. Let's not lose sight of what we are solving. So how much more do you have here?

- [Wes] Um 2.

- [Kim] This is 10, ten elevens but I need thirteen elevens. Some extra elevens. I said it this way on purpose rather than 10 times 11 and 13 times 11. This language of elevens can be helpful for students to unitize the elevens, treat them like one unit that we have 13 of.

- [Wes] So then uh, 3.

- [Kim] Three more elevens. We don't have a context but the number of elevens seemed helpful for him to hear. And what is three elevens?

- [Wes] Uh, um, 33.

- [Kim] 33, okay.

- [Wes] And then it will be 143.

- [Kim] Oh, so you did get 143. Like it never happened! Okay, cool. Is this an eraser? Yeah, 143, 143. Did anyone use this problem up here? Now that we have settled the answer, I want to hear from someone who used my target strategy. But to be honest, I could not remember who I was going to call on so I specifically asked for it. Give me thumbs up. Did anybody use this problem up here to help them with eleven thirteens? Yeah? You did?

- [Alden] Yeah I added this um 13 more.

- [Kim] You just added the 13 more?

- [Alden] Yeah because it would be 13 times 10 and then uh, 13 times 11, so you just add one more 13.

- [Kim] I just heard somebody say oh, that was an interesting idea. SO many kids did not see that connection. So if you know ten thirteens, which is 130, then eleven thirteens would just be 13 more?

- [Student] Yeah.

- [Kim] And that's 143? You liked that idea?

- [Student] Yeah I liked it.

- [Kim] Oh, interesting. Okay, let me ask you this one. What if I asked you 9 times 13? Hm. I wonder if there's anything up there that can help you with 9 times 13. Let me just put this comment out there are see if it settles with anyone.

- [Kim] Sorry bud.

- [Kim] How do you know this one?

- [Kim] Okay, it's, 127 is 3 less, so what would 13 less be? This is only 3 less, 127 is three less than 130. So what would 13 less be? Joseph had used the 10 by 13, but then was off when subtracting the 13 from 130. I decided to point it out but not spend a lot of time with him on it. Mental note and move on. 

- [Kim] Taylor. Talk to me about what you're thinking for nine times... What did you get for the answer?

- [Taylor] I got 117.

- [Kim] Anyone else get 117?

- [Student] Yeah.

- [Kim] Okay. Seeing lots of people that got 117, okay. And how did you think about nine thirteens?

- [Taylor] Um so what I thought about is the first problem that we did, the tens 13. So I just took one group away. So it'd be nine times 13.

- [Kim] One group of what? Important that we don't just say take one away. 

- [Taylor] 13.

- [Kim] Okay.

- [Taylor] And so all I did was 130 - 13.

- [Kim] Okay, pause for just a second. You said this first problem was 10 thirteens. Let me see if I can sketch that close. Close, but also floating up to the right :) 10 thirteens and you knew that, you kind of just know 10 thirteens, right? That's kind of like a nice friendly amount. This is a subtle suggestion that students might realize, "Ah, that is a nice friendly amount!" 10 thirteens is 130, but you said I don't need 10 thirteens. I only need?

- [Students] Nine

- [Kim] Nine thirteens. So if this is 10 thirteens Intentional hand motion to try to indicate rows stack on top of each other, could I just cut off a row of 13?

- [Students] Yes.

- [Kim] And that would be, and what is 130 minus 13? Well I know one 30 minus 10 is 120. Minus another three and I could even sketch that. Quick sketch of ONE way to subtract those. Several students had stacked them. Even though we are focused on multiplication, I want to encourage them to keep thinking abut the addition and subtraction within the problem. 130 minus 10 is 130, minus 3 is.

- [Wes] 127

- [Kim] Oh, sorry 120 minus three is?

- [Students] 117. 

- [Kim] 117, nicely done. What if I asked you um, 20 times 13? What if I asked you 20... Give me a thumbs up when you have an idea.

- [Kim] Did you just do that?

- [Student] Yeah I already had it.

- [Kim] How did you do that one?

- [Nick] This is double the first one.

- [Kim] Hmm, William, what did you get for 20 times 13?

- [William] Uh, 260.

- [Kim] Anyone else get 260? Okay just give me a thumbs up if you think, just give me a thumbs up if you agree. The hands! I'm seeing some use a thumbs up, but it is not yet their routine. Lots of 260s. And so William, how did you think about 20 times 13?

- [William] So what I did was I just wrote a simple problem that we did first. 10 times 13, and that equals 130. Then below it I did up and down problem, which I did 130 times two. And that got me to 260.

- [Kim] So it sounds to me like you kind of said to yourself, I know 10 thirteens. And so if I know 10 thirteens, I can just double that?

- [William] Yeah.

- [Kim] Does what William said make sense? If I know 10 thirteens I can double that, because 10 thirteens and another 10 thirteens makes 20 thirteens. So that's 260? I wish I hadn't restated here and asked another student to. I am not thrilled with how much the conversation ball goes back and forth between me and a single student. Something to work on.

- [Students] Yes

- [Kim] Yeah?

- So it sounds to me like you guys are doing a lot of things with this 10 times 13. In summary...

- [William] Yeah, it's helping a lot. :)

- [Kim] Is it helping a lot? Okay, let me ask you this one. I'm going to come over here because I ran out of room. What if I asked you this amount right here? Just give me a thumbs up when you think you have an idea. NOT "when you get an answer".

- [Kim] You didn't even have to write anything down. How do you know?

- [Casey] Because all I have to do is add 13.

- [Kim] Interesting. Lots of thumbs up so quickly. Casey, Casey just looked at the board for a second and then he went "bink". That's so interesting. Casey, how did you- you didn't write anything down for this weird problem, what did you think about? What did you think about? So much more interesting than "What is your answer?"

- [Casey] Well I thought about 20 times 13, which is 260. And 21 times 13, all you have to do is to add another 13.

- [Kim] Did anybody hear and understand what Casey just said? This is your job!

- [Student] Yeah.

- [Kim] Yeah. Did that make sense?

- [Student] Yeah

- [Kim] Okay, so what Casey just, I'm going to, I'm going to represent it. My job. Casey just said I know 20 times 13, which is 260. And then what did he say Anna?  I brought another student voice in!

- [Anna] Um plus another group of 13.

- [Kim] Plus one more 13 which is 273. What do you think about that idea? Let me give you one last problem, ready?

- [Students] Yes.

- [Kim] So, I gave you 10 times 13 and you used that. And then what happened here? You went a little bit over?

- [Students] Yeah.

- [Kim] What happened here, Joseph? So we went a little bit over for this problem. An extra 13, what happened here? Joseph? He used this strategy.

- [Joseph] You subtracted 13.

- [Kim] We want a little bit under this amount. And then what happened here? Alison, what happened here?

- [Alison] Um, we added 130 to it.

- [Kim] We, we doubled it, right? We did it another 10 groups, yeah. And then what happened here?

- [Kim] So this 10 by 13 was incredibly helpful. Hint hint. This is a purposeful recap and nudge. What if I asked you this? Last problem. One second to think about that. Hmm. Just give me a thumbs up when you think you have an idea.

- [Kim] How did you think about that one?

- [Alden] I did uh, 260 minus 26.

- [Kim] I'm wondering how you got that. His strategy was what I was looking for but he had subtracted incorrectly and was off a bit. How did you subtract 26?

- [Kim] Maybe write the 260 down, because that's where you started. Okay and then how would 26? Do you know 20 less than 260? I'm suggesting a strategy other than the algorithm I saw on his paper.

- [Alden] Yeah, 240.

- [Kim] And then 6 less than that.

- [Alden] Oh yeah. That would be 234.

- [Kim] I'm wondering if anybody used a problem that's up there already to help them. Give me a thumbs up if you used a problem that's already up there to help you. I'm being very specific at this point that it's all I want to hear about. There were a variety of strategies, but right now we are going to focus on the over/under strategy. Adele, what did you think about, what did you get for 18 times 13?

- [Adele] I got 234.

- [Kim] Give me a thumbs up if you got 234, so many. So many people got 234. And how did you think about this problem?

- [Adele] I used 13 plus 13 is 26, so I just removed it from 260, which is the answer to 20 times thirteen.

- [Kim] Adele was kind of quiet there did anyone hear? Lucy did you, could you hear what she said?

- [Lucy] Yes. Calling on Lucy because she did not use the targeted strategy.

- [Kim] What did she say?

- [Lucy] She, doubled 13 and then um, then she minused wait.

- [Kim] She did, she minused it I'm clearly not getting caught up in vocabulary at this point from something. Ava do you, are you with me babe? I probably should not call her babe. Terms of endearment are an old habit of mine that I need to work on. To solve this, Adele used something up here. Thumbs up. Lucy, do you know what she, she, she saw that there were two thirteens, but why two thirteens? I'm clear that not everyone knows why.

- [Lucy] Because if 20 times 13 is 260. Then if you minus 20. Minus, minus 3. Then you'd get 18 so you would just subtract two groups of 13.

- [Kim] Adele. Does that match with what, what you did? You used the 20 thirteens to help you? Is it true that 18 thirteens is the same as 20 thirteens, minus two thirteens?

- [Students] Yes.

- [Kim] Is that 18 thirteens?

- [Students] Yes.

- [Kim] Yeah, so she said, I'm going to go to a problem. I like. This is kind of a yuck problem. Can I go to a problem I like near it, which is 20 thirteens, and then just take a little bit off? Take two thirteens off. And what is two thirteens?

- [Students] 26

- [Kim] So 260 minus 26, which is?

- [Students] 234

- [Kim] That's an interesting idea. How many of you think that thinking about a fact that you know, a problem that you know, and then going maybe a little bit over it or a little bit under it would be helpful at some point? This is a by-product of seeing these students once. I was aware we were already over 15 minutes in and I wanted to wrap up. If given another opportunity to do another similar (sister/echo) string I would have ended with having them turn and talk about things they noticed but not summarize. Like given a problem, like 18 times 13, that's pretty yucky. But could you think about, I know 20 of them and then go a little bit under? Because you know 20 thirteens. What do you think about that idea? Does that work for you?

- [Nick] You could also do something else though.

- [Kim] You, you could do lots of things. I'm just wondering if that would be an idea that would be useful to you at some point. I think so, maybe go a little bit over or a little bit under, surrounding a problem that you know. Nicely done.