Pam  0:00  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam Harris, a former mimicker turned mather.

Kim  0:10  
And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  0:18  
We know that algorithms are amazing human achievements, but they are terrible teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:32  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:39  
And we invite you to join us to make math more figure-out-able.

Pam  0:44  
Hey, Kim! 

Kim  0:45  
Hey. We kind of just jumped on today and didn't really shoot the breeze at all.

Pam  0:49  
We didn't even say hi or anything. How's it going? 

Kim  0:52  
It's good. How are you? 

Pam  0:53  
You're very pink today.

Kim  0:55  
I am. I know. We should tell listeners

Kim  0:58  
that... 

Pam  0:59  
I feel like you don't wear pink very often.

Kim  1:01  
I don't. And listen, we've been telling people that we don't see each other, but we recently made a switch.

Pam  1:07  
Yeah, so I think people think we're in the same room when we record. And we're not. You're at your house. I'm at my house. But we have turned on the video, so we can actually (unclear). 

Kim  1:16  
Just recently, yeah. 

Pam  1:18  
Yeah, just recently. So, whether that's good or bad. We... Yeah, we didn't do it at the beginning. Very purposely, right? Because we weren't convinced that if we could see each other that we wouldn't describe well enough, so that audio... You know, we would assume. You would nod. I, you know, wink or whatever. I don't wink. But

Pam  1:38  
whatever. 

Kim  1:39  
We wouldn't say enough because we nod like, "I got it. You can move on."

Pam  1:42  
Exactly, yeah. So, we purposely didn't have video on, so that we would make sure that everybody could understand what we meant. We would describe enough, right? 

Kim  1:50  
Yeah.

Pam  1:51  
That's going to be part of our trick today, I think, in our episode is to make sure we describe enough.

Kim  1:56  
Now, we just make faces at each other. Alright, let's get on with it. Okay, so listen. Several months ago. I mean, it's been quite a while. MatherInCO. I'm hoping Colorado. Mather in Colorado left a review. And the title is, "A Must Listen Weekly." Five stars. Fantastic. And they said, "The best math education podcast out there right now. Learn real math yourself, and then have FUN teaching it with your students."

Pam  2:28  
And "fun" is in all caps. (unclear). 

Kim  2:29  
I know. Great. And, Mather,

Kim  2:32  
I sure hope that you still think so. It's been a while. Thank you for listening. And for giving us a review.

Pam  2:39  
Yeah. Yeah, you know the way the algorithms work, the more reviews you get and the more... 

Kim  2:44  
Yeah.

Pam  2:44  
If people will rate, then it shows it to more people, and then we can spread the word that Math is Figure-Out-Able to more people. So, yeah, thanks, MatherInCO. I like

Pam  2:52  
it. 

Kim  2:53  
I need to be in Colorado right now. 

Pam  2:55  
It's pretty, isn't it? Yeah.

Kim  2:56  
So nice. Okay, so this is a special episode because we often. Like, super, super often get asked by our Journey members and leaders about the best Problem String to do with teachers. Like, what's the one? And there are a few go tos. But a favorite by far is one that we affectionately call Sticks of Gum. In fact, when we call it that, people who have heard you mostly know exactly what you mean. But it really doesn't refer to the math that's happening. So, for you leaders, or for teachers who are wanting to share with your colleagues, but haven't ever had a chance to hear Pam walk through the string and describe her choices, this episode is for

Kim  3:40  
you. 

Pam  3:40  
Yeah, so Sticks of Gum. Yeah, when our coaching people say, "Hey, Sticks of Gum." When we refer to that, it's because for a long time, I used the context of a pack of gum having so many sticks of gum in it. Lately, I've done bags of M&Ms. But it has everything to do with one carton, or container, or box, or bag having so many items in it. And yeah, for a long time I did Sticks of Gum. So, hey, Kim, just for nostalgia sake, we'll do Sticks of Gum today. 

Kim  4:12  
Okay.

Pam  4:13  
Even though I've been doing M&M's a lot lately. So, I would say to people, "Hey, have you ever... You know, anybody chew gum?" Whatever. Have a brief conversation about gum. And then I'll say, "Can you picture a pack of gum that has 27 sticks in it?" So, they're the long, skinny, kind of bendy. Not the fat, squat. You know, and I do enough conversation about it, so that people can kind of picture the pack. It's not about real world. It's really about just having people be able to sort of realize what's happening. I want them to be able to picture this pack of gum. And then I'll draw a table wherever I am on the whiteboard, or on my iPad, or whatever. And I'll write... Bags. See I'm in bags of M&M's. I'll write, "Pack". And then "Sticks". So, above the one column on the left, I'll write "Pack". And I'll write "Sticks" in the right hand column. And then I'll put 1 pack has 27 sticks. So, I have table now. Packs. Sticks. 1 to 27. But as I'm writing that, I'm talking about the story about gum. I might say something about who chews gum. "Raise your hand if you chew gum." I'll even sometimes say, "I think it's the one with the stripes on it." And then people will like, "Oh, yeah." And then they'll tell me brands. And so, not a long time, but enough. Enough time to get sort of some context, get kids kind of in, in the story enough that we're like, "Oh, yeah, we're talking about a pack of gum. And then I'll say, "Well, okay, so if 1 pack..." You can picture it. 1 pack. Remember long, skinny, kind of bendy sticks. Not those fat, squatty ones. If 1 pack has 27 sticks of gum in it, how many packs would 2? Sorry, how many sticks would 2 packs? And then I'll write the 2. So, as I said, "How many sticks would 2 packs have?" I'll actually write the 2 underneath the 1. So, so far I have a table. Pack. Sticks. 1 to 27. And then 2. And then I'll just sort of wait. Now, I'm not going to wait too long on this one because if I'm doing this string, kids are going to be able to double 27 without too much stress. It's not a speed game. But we've talked about before, we want to have Problem Strings be kind of snappy, right? So, I'm going to say, "How many sticks would be in 2 packs?" And I'm going to kind of watch. I'm going to look at students. And as soon as enough kids kind of light up, I might ask them, "Give me a thumbs up when you know the number of sticks in 2 packs." As soon as enough kids have chewed on that just enough that I might even say, "Just say it out loud, everybody." Or I might, "Turn and whisper shout to the person next to you how many sticks in 2 packs?" Just kind of get everybody going for this first one. People will often say 54. And then when they say 54, then I'll say, "Oh, so if you double the number of packs," and then I draw, kind of an arrow between the 1 to 2 packs. And next to it, I'll write times 2. Then you double the number of sticks. And I'll write an arrow from the 27 to the 54. And I'll write times 2. So, I'll say, "So, if you double the number of packs, then you double the number of sticks," as I draw those scaling arrows. And I'll say, "And that got you 54 sticks." Then, just recently, I've added a little bit of a line here where I want people to tell me how they were thinking about doubling 27. And the line that I've kind of added recently. So, if you haven't heard me do Sticks of Gum in a hot minute, here's a little bit of a new twist that I'll say. I'll say, "I don't... A lot of you just said 54. I don't think that 2 times 54 was one of those flash cards that you memorized in third grade." (unclear).

Kim  7:39  
You mean

Kim  7:39  
2 times 27? 

Pam  7:41  
Oh, gosh. 2 times 27. Thanks. Whoo. "I don't think that was a flash card that you memorized. It wasn't a fact that you like drilled. So, I think your brain had to do something. Most of us. Our brain had to do something to double 27.What does your brain do?" And I'll pause just a little bit, and then I'll say, "Like, did anybody think about doubling 27 as doubling 20?" And then I'll just trail off. "Did you double 20?" And I'll look. So, right now I'm trying to pull out one of the major strategies, one of the important strategies for doubling 27. So, Kim, if I were to ask, you know, somebody, "Hey, did you use 20?" What might that person say if they used 20?"

Kim  8:20  
They probably said, "I doubled 20 to get 40. And then I doubled the other 7 to get 14. And then I added 40 and 14 together."

Pam  8:29  
Nice. To get that 54. So, since we're talking about facilitation moves, you might find it interesting that often they'll say, "Well, I doubled 20, and I doubled 7." As if that's enough. 

Kim  8:43  
Mmhm.

Pam  8:43  
So, then I will say, "And double 20 is?"

Kim  8:46  
Mmhm. 

Pam  8:46  
And when they say 40, I'll write down. So, I have now written down, off to the side, 27 equals 20 plus 7. And as I then pull out of them "Double 20 is?" And they say 40, I'll write 40 down below 20. "And double 7 is?" And they say 14, I'll write 14 below the 7. And then I'll say, "And 40 and 14 is?" And then they'll say 54. "So, I'm going to represent, model what they just were doing. But what's interesting to me is how often I have to actually pull out of them more than they were going to say.

Kim  9:18  
Mmhm. 

Pam  9:18  
Like, they'll just say, "Well, you know, you double 20. You double 7." I'm like, "Well, what is double 20? What is double 7? Oh, and then you added those together." Similarly, then at that point, I'll say, "Did anybody not use 20?" And almost always people in the room will kind of smile and nod. And then I'll say, "Did anybody use 30?" And I'll kind of trail off. Now, at this point, many people who were just smiling like, "Oh, yeah, I did something different," actually then go, "Oh, no. I didn't use 30." Often, people will use 25. 

Kim  9:48  
Mmhm.

Pam  9:49  
Which is a fine strategy. But for my purposes in this string, I'm actually not going to pull out the 25. And so, Kim, don't let me forget. I'm going to talk about why in just a second. 

Kim  9:57  
Okay.

Pam  9:57  
I want to pull out the 30. And so, then I'll look. And if nobody is sparking on the 30, then I'll say, "Could you?"

Pam  10:09  
And almost always. In fact, I don't think it's happened yet that someone hasn't sparked then. When I say, "Could you?" someone will raise their eyebrows and go, "Well..." And if it's just enough, I'll go, "Ooh, tell us about that." And then I'll call on that person. So, Kim, often when I call on a person, "How could you use 30 to double 27?" what might a person say at that point?

Kim  10:32  
Yeah, so they'd say, "I doubled 30." And if they were listening when you pulled out the double 20, double 7, then they'll say, "And that was 60. And then I doubled 3, and that was 6. So, then..."

Pam  10:47  
And I'm

Pam  10:47  
going to pause you there. I'm going to pause you there because I would pause the person as well. 

Kim  10:50  
Yeah. 

Pam  10:50  
I would say "6. 6? Where..." In fact if I can even back up just a little bit when they say 30 if... Oh, should I even... Sorry. If I didn't have to pull it out of them. If somebody actually did use 30, I'll say, "You used 30? 30 is not even in the problem." Like, I'll kind of push back a little bit, and sometimes people be like, "Uh, uh." They almost get flustered. Like, "Well, you asked if I use 30." I'm like, "Well, I know. But it's not even in the problem." They're like, "I know, but it's like almost 27." So, when you said then I was going to subtract 3. I would usually pause the person and go, "Whoa, whoa, whoa. Where's this 3 coming from?" Just so they can say, "Well, you know, 27 is 30 minus 3." And then, like you said, double the 3 to get 6. And then I cut you off. Sorry 60 minus 6 is?

Kim  11:34  
54. 

Pam  11:35  
Also that 54. Cool. So, now I have off to the side these two ways of thinking about doubling 27. And I'll often say, "So, nice. These are two of the major four..." Oh. Now, I should be clear. If I'm working with teachers, I will say this part. If I'm working with students... Let me do students first. If I'm working with students, I'll go, "Nice strategies, you guys. Nice way of using what you know to think about doubling 27. Cool." If I'm working with teachers, I'll say, "Hey, teachers. you might find it interesting. Here are two of the major four. There's only a few. Of the major four strategies that we want to build in kids to really think and reason about addition. These are two of the... There's only a few." Because I want to help teachers realize there's only a small set. But, Kim, that just made me think. Should I be focusing on facilitating this with kids? 

Kim  12:31  
Yeah.

Pam  12:32  
Or teachers? Or both?

Kim  12:33  
I've actually been sketching some notes. So, what we said was that this would be a great one for leaders to do with teachers. And for teachers to go with their colleagues. But I think we could go either way. But I did write down "student", and I crossed it out and wrote "teacher" because I wanted you to mention the idea of 17 versus 27 and for how long you would do this additive bit if you were working with adults versus working with students. Because I know it sounds like you were talking for a really long time. And part of that is just for the purpose of the podcast to really parse out all the moves and the questions.

Pam  13:09  
Yeah. 

Kim  13:10  
But with students, it really does not take that long.

Pam  13:13  
No, in fact, at this point, I may be two minutes into the string. 

Kim  13:17  
Yeah, yeah.

Pam  13:17  
Maybe not even that. Like, it's pretty quick. 

Kim  13:20  
Yeah.

Pam  13:20  
I'm keeping it snappy. Yeah.

Kim  13:21  
But with teachers, it is a bit longer because you are pulling more and you're solidifying the idea that this is just a few of the major strategies. In general, I think, you have talked a lot. Maybe even in the podcast about taking longer with adult strings because you don't get to see them very often, and you want to bring in the mathematics, but also the moves that are being made.

Pam  13:48  
Mmhm. Yeah, very true, very true. So, just so I'm clear moving forward. Can I talk about both students and teachers? Or do you want me to focus on if we're doing it, if a leader is doing it with teachers.

Kim  13:59  
Hmm. 

Pam  13:59  
You should see the look on my face right now. I'm like, "Um..."

Kim  14:01  
Let's do... Because we have fantastic Problem String books that teachers can grab. Let's go, leaders with teachers.

Pam  14:09  
Okay, okay. Because then you can go look...

Kim  14:10  
Or teachers with their colleagues. Yeah.

Pam  14:11  
Gotcha. Okay, so teachers with colleagues. So, we now have on the board Pack, Sticks, 1 to 27, 2 to 54 with these scaling arrows. I'm going to then say, "Okay, cool. We got the number of sticks in 1 pack, the number sticks in 2 packs. How about the number of sticks in 4 packs?"

Kim  14:28  
Mmhm. 

Pam  14:29  
And at that point, I will let people think. I've written down the 4. I'm going to go erase those scaling marks times 2 on both sides. So, I've kind of cleaned up the ratio table a little bit. And now all you've got are sort of the 1 to 27, 2 to 54, 4 to blank. Let people think. Now, I'm going to go really fast here because nobody needs to think about doubling 54. Like, very rarely. So, when I say 4, I erase the scaling marks. I'm instantly going, "Ya'll, that's not... We don't have to wait very long on that one. Just say it out loud, everybody." And then they'll just say 108 because double 54 is... Now, at this point, I'll pause a little bit, and I'll go, "Okay, so did you do the same thing? Did you scale times 4?" And I'll draw the scaling arrow. And then the times 4. And then people are like, "No." And I go, "Or were you kind of looking to be lazy. I mean efficient. Because mathematicians often seek to be lazy. I mean efficient." So, I'm trying to joke a little bit here. But I'm also trying to help people realize that this instinct they just had to just double the two packs is a good instinct. That it's a mathematical behavior to want to be efficient. And I'll joke a little bit about how it's lazy. So, then I'll erase those scaling times 4. I'm just drew that really quick. And I'll say, "Oh, so you just doubled? You just doubled. Oh, okay. And then double 54. We're not even really going to talk about that. Next problem." Then I'll say, "Okay, how about 8? 8 packs of gum. What are you thinking about there?" And again, I'm not going to wait very long because doubling 108 is not too much. That's not... Nobody's going to... But I'll say, "Oh, so you're just going to double from the 4 to the 8. Double the 108. And so, is that 216? Okay, everybody, cool." And then I'll just sort of note, "Interesting that we just found 8 times 27 not by really multiplying by 8 at all but by multiplying by 2, doubling, doubling. Now, that's interesting. I wonder if you could think about multiplying 8 times anything by doubling it, doubling it, and doubling it again. Huh, interesting." And I'll just kind of just sit there for just a second. Just, "Huh. Okay, that's kind of interesting. Nice." Then, I might say, "Anybody want to guess the next problem?" Now, this is an attempt to kind of get people, what, sort of out of the string a little bit not just... Out of the string. This is an attempt to get people rising sort of above the idea of, "I'm just going to do what the teacher tells me. I'm going to do what the leader says," and instead, kind of like thinking ahead, guessing a little bit, sort of using... Not random guessing, but like, you know, well, we double, double, doubled. Often, people will smile, and they think that I'm going to go for 16. Why not, right? We just did 2. We did 4. We did 8. We've been doubling. So, then I'll just say, "Yeah, anybody want to guess?" I'll pause, and then I'll go, "10. 10 packs." And then people smile a little bit. They're like, "Oh, we thought you were going to go 16." Just a moment where you get to kind of just, you know, get people out of the mode of just doing and kind of, you know like, "What might be next? Oh." Then, they can chuckle just a little bit. Then I'll say, "I wonder. I wonder if there's anything up here that could help you find the number of sticks in 10 packs?"

Pam  17:40  
Then, I'll let people. Now, here's the first time in the string, Kim, where I'm actually going to pause longer than just a second. So, now I'm kind of looking. Oh, I probably should have said if I'm doing this Problem String with adults, I will probably before we get too far. And sometimes I forget, and so it gets a little... It might even be here in the string, but ideally it would be sooner. In fact, I think maybe ideally, it might be before I even start the string. I might invite everybody in the room. "Hey, would everybody just pick up a pencil. Just pick up a pen." Kim just smile because I said pencil. "Pick up something to write with." And then I'll do a little spiel on, "It's not if you write, it's what you write. So, I just want everybody to have something in your hand, because if you want to keep track of your mental thinking as we do this mathing experience, I want you to feel free as you look around the people next to you, 'Oh, they're writing. Okay. Socially it's okay for me to just jot something down.' So, that it's not this pressure on the cognitive load. I don't have too much of... I'm not trying to keep all this stuff in my head. I can write down what I need to to kind of keep track of the relationships I'm using. It's not if I write. It's what I write." And I'll also say to adults, "If you find yourself like mimicking steps that you memorized or that your teacher taught you in second or third grade, maybe in that case, put your pencil down. Maybe in that case, put it down and say to yourself, 'If I wasn't writing, what would I do? What could I use? Could I use relationships to figure? Ooh, ooh, I could do that? Okay.' Once you've got some relationships used, then pick your pencil back up. Keep track of those relationships. Don't just... If you find yourself in mimicking mode, put your pencil down. But if you're keeping track of your mental relationships, totally legal to do that." So, having everybody have something to write. So, at this point, when I've said 10 packs, some people are writing some things. Some people are just smiling. I'm going to say something like, "Is anybody interested to use the 8? Did anybody use the 8 packs?" Kim, almost always. It is very rare that somebody doesn't say, "Yeah, yeah. I could use the 8 packs," because they're kind of looking up at the table and they're saying to themselves, "Well, I got stuff I could use." So I'll say, "Well, how did you use the 8 packs?" And they'll say, "Well, I got 8 packs and 2 packs." Now, I'm going to draw a bracket between 8 and the 2 on that left side. And then I'll write 8 plus 2 next to it. And an arrow down to that 10. "So, if you had the sticks in 2 packs and the sticks in 8 packs, then you added those together?" "Yeah, so I added the 54 sticks plus the 216 sticks." So, off to the right now, I'm drawing that same bracket. 54 plus the 216. And then a little arrow down. And I'll say. "And what did you get?" And they'll say, "270." And I'll say, "Oh, did everybody else get 270? A lot of people got 270." Then I'll say did anybody not use the 8 packs? And someone will smile and say, "Well, I just added a 0." To which point then I get to, you know, go off on, "Did you really add a 0?" And often, what I'll say is, "What mathematically did you do?" And they'll say, "Well, you know, if you..." Now, if they don't have words for it, I'll say, "Are you scaling? Like, if you had 10 times the number of packs?" Now, I've just drawn a scaling arrow from the 1 pack to the 10 packs, and then written times 10.  "If you had 10 times the number of packs, do you have 10 times..." Now, I've drawn the scaling arrow on the right side, "...the number of sticks. So, 27 times 10. Oh, 270." Now, everybody in the room is like, "Oh, yeah. There's the 0 thing. Times 10. And I'll say, "Okay, cool. Because there's this times 10 thing in our number system. Your students might not know why that works. You might not know. Maybe we can talk about it later. But for now, because we have it, we're just going to use it. We got this times 10 thing in the 0. We're going to use it. Cool. Times 10." Before I go on. Right now on my ratio table, I've got a bunch of stuff. I've got those brackets with the 8 plus 2 and the 54 plus 216. I've got the scaling marks from the 1 to 27 down to the 10 to 270. I'm going to erase all of those. I'm going to clean up the ratio table. The only thing left in it are sort of the entries, the 1 to 27, 2 to 54, 4 to 108, 8 to 216, 10 to 270. That's all I have left with the Packs and Sticks on the top. Now, I'll say, "Cool. Alright, next one. Ready? How about 9 packs? I wonder. I wonder if there's anything up there you could help you? I don't know, maybe not." Then I let people work. People work. They're circulating. Kim, what do you think the 2 strategies are I'm going to go for here? And any thoughts on which one I'm going to try to get first?

Kim  22:14  
Yeah, I think people are going to use the 8 pack and the 1 pack.

Pam  22:18  
So, how could I ask for that instead of just hoping I get it? 

Kim  22:23  
Ah, that's

Kim  22:24  
good question. So, I think you're going to say, "Did anyone use the 8 pack again?"

Pam  22:28  
Again.

Pam  22:29  
Yeah. And often I'll say, "So, did anybody use the 8 pack this time? You know, we used 8 last time." Somebody's going to nod. I'm going to, "I mean, that's nice. The 8 packs plus the 1 pack." Again, I'm drawing the brackets over on the left. 8 plus 1. Arrow down to the 9. Then I'm on the right hand side the corresponding bracket from the 27 to the 216. And I'm like, "What is 27 and 216?" And they'll say... What did you get,

Pam  22:52  
Kim? 

Kim  22:53  
Oh, I don't know. Sorry, I wasn't doing it.

Kim  22:54  
What did you? 

Pam  22:56  
That's okay.

Kim  22:57  
216... 

Pam  22:58  
We're finding the number of sticks in 9 packs, so we're adding either 27 to 216. Is that... 

Kim  23:04  
243. 

Pam  23:06  
200 and... Yeah, 243. Cool. Thanks. Thanks, for... I didn't want to do the math. Thanks for that. Or what was the second strategy that people might use?

Kim  23:15  
For sure they're going to go back to the 10. Particularly because it's right above the 9. Somebody will notice that it's so close.

Pam  23:23  
Nice. And so, then 10 packs minus 1 pack. Now, I'm drawing a bracket between the 10. Oh, I just erased the 8 plus the 1. So, that's gone. Now, I've got the bracket between the 10 to the 1. So, 10 minus 1 packs. Arrow down to the 9. You know, sometimes, Kim, I will actually not do that. I will say, "Oh, so you did 10 minus 1 to get 9." And I'll just write next to the 10 a minus 1 to get 9. And then I'll say, "So, 270 minus 1. So, you got 269 sticks." And then I'll pause.

Kim  23:55  
Mmhm. 

Pam  23:55  
And I'll nod. And I'll smile.

Kim  23:56  
Mmhm. 

Pam  23:57  
And people will nod and smile. And then they'll start to shake their head.

Kim  24:00  
Mmhm.

Pam  24:00  
They'll be like, "No." I'm like, "What do you mean no?" So, I want teachers to realize I recognize that kids might do that. And I want to put it up in front of them and go, "What are you going to do when a kid does that? What do you..." And then some teacher will say, "But if you subtract a pack, you have to subtract 27 sticks." And I'll say, "Oh, if you subtract a pack, you have to subtract all the sticks in that pack, so you can't just subtract 1 and subtract 1." Teachers, when you have a kid that gets in a ratio table and they do the same thing to both sides, but it's not... They haven't stayed proportional. Then, all you have to do is do experiences like this. When you get in a naked ratio table that doesn't have context, you can literally just say to them, "Okay, we just subtracted a pack. How many sticks are in that pack? Oh." You remind them of this experience. So, now I'm going to do 10 minus 1 with the bracket pointing down to the 9. So, now we're going to do 270 minus all the sticks in that pack, 27. Now, at that point, I may ask them how they did that subtraction. We may then go off down to the side. Think about 270 minus 27. That would be a good moment for me to sort of support additive reasoning with teachers. In that moment, I will ask somebody did you think about subtracting 20?" I'll draw a number line. 270 on the right. I'll pull out of somebody that they subtract 20. So, 270 minus 20 is 250. Then I'll say, "Okay, we're supposed to subtract 27. Oh, yeah. We got to subtract 7 more." I'll draw a shorter jump of 7. What's 250 minus 7? Is that 243? Yep, that's what we got before. Nice. Then I'll say, "I'm kind of curious. Did anybody not subtract 27 as 20 and 7? But did anybody subtract 27 as 30?" Then I pause. Look. If no one sparks, then I'll say, "Could you?" And I'll wait to see if anybody sparks. If no one sparks, we got more work to do with this group of teachers. Kim, I don't think I've ever had anybody not spark. Someone in the room goes, "Well, yeah. "You could do 270 minus 30." Now, I have below that first number line drawn a second number line. Started at the same horizontal location, so the 270s are lined up. Now, I've jumped further back than the 20 and the 7. But only a little bit further back. Bnd I've written minus 30. And I'll say, "Okay, what is 270 minus 30?" And they'll say, that's 240.

Pam  26:34  
Then we have a little fun where I'm like, "Okay, what are you going to do now?" If the person confidently says, "Well, I subtracted too much, so I have to add 3 back," I'm going to try to catch them saying "add", and I'm going to push on that just a little bit. I'm going to go, "Whoa, whoa, whoa. We're subtracting. Why are you adding?" And I'll look kind of confused. Sometimes people will back up. They're like, "Uh, I don't know. Uh." But sometimes people hold their ground. In that moment, it's very important that I just kind of just look at them like patiently like, "I don't know. What are you thinking? Like, this is we're thinking here." If they back... You know, if they sort of, "Ah, whatever!" Then, I'll be like, "Is anybody agree with them?" Like, we're subtracting. Can you add?" Somebody will say, "Well, yeah. You subtracted too much, so we need to adjust up." So, then I'll say, "Okay, adjust up how much?" I'll draw that little jump of 3. And I'll say, "Okay, here's the hard part. What is 240 and 3? Bam.  Hey, look at that. We're at a great place where the question is the answer." And when I land where that jump of adding up 3 should land lined up with the Problem String or with the number line above it, so that the 243s are vertically aligned. Horizontally aligned? How do you say that? They're in the same horizontal location? They're aligned. They're lined up. 

Kim  27:45  
They're lined. 

Pam  27:46  
So, that the jump of 27 in both number lines should be about the same. And I'm purposely trying to make that true.

Kim  27:54  
Before

Kim  27:54  
you go on to the next problem. 

Pam  27:56  
Yeah?

Kim  27:56  
You asked me what do you think people will do and what order am I going to be looking for them? And you didn't talk about that yet. Why a particular order?

Pam  28:05  
So, I'm going to try to pull out of teachers the additive, the add the 8 plus the 1 first because everybody can latch on to that. Even if they did the 10 minus 1, it will make sense to them to do the 8 plus 1." If I do the 10 minus 1 first, that has the potential for people to be like, "Wait, wait, wait what?" But if I do the 8 plus 1 first, everybody's grounded. We're all confident the answer is 243. Now, I can introduce this idea that maybe not everybody's thought about. You're like, "Pam, are you sure?" Yeah, I wouldn't have. So, I'm clear, I wouldn't have thought of that when I was a young in numeracy. So, then I'm going to pull out the strategy that I don't think everybody may have figured, and we get a chance to dive into that one.

Kim  28:45  
Yeah, also sometimes in these conversations, you talk about how they use the 8 and the one because they happen to be on the ratio table already.

Pam  28:55  
Mmm, mmhm. 

Kim  28:56  
But that many of us can find 10 of a pack. Or, you know, 10 packs of a thing, and then back up a pack because we readily hang on to 10 packs and 1 pack. But not so often do we just walk around with 8

Kim  29:08  
packs. 

Pam  29:09  
Yeah. In fact, a way that I'll try to pull out that conversation is to as soon as we've got it all on the board, I'll say, "So, two great strategies. Add the 8 and the 1. Take the 10 minus 1 pack. Which of those, if you were walking down the street, and bam, you ran into the problem. 9 packs. 27 sticks in a pack. How many sticks? Which of those do you carry with you?"

Kim  29:30  
Mmhm. 

Pam  29:30  
And then we can have a conversation about being lazy. I mean, efficient.

Kim  29:34  
Mmhm. 

Pam  29:34  
You know, like you just said, we could recreate the 8 to add it to the one. But we carry that times 10 around with us. We want to have that efficiency conversation.

Kim  29:43  
Mmhm. 

Pam  29:44  
Yeah, with teachers and students. Cool. So, we're almost done with the Problem String. At that point, then I will erase everything except just the numbers, the 1 to 27, 2 to 54, down to 9 to 243. But all the brackets in the scaling marks. We cleaned everything up. And then I'll say, "Alright, gum. Yum, gum. Guys, let's have 100 What about 100 packs of gum?" Again, I don't wait very long on this one. I might have people say that number out loud just so people can hear that that number has 2 names 2700 and 2,700. I might take a minute to just talk about how it has two names. But why doesn't 10 times 27, 270? We call that 270. Could we also call that twenty-seven 10s? I might mention that. But for today's purposes, then I'm going to say, "Alright, you guys. Ready? Last problem of the string. How about 99? 99 packs?

Pam  30:37  
Can you find the number of... Can you use what you know to reason and find the number of sticks in 99 packs? Go." Then I'll let people do that. I'll usually circulate. I'm looking to see how people are doing the subtraction. Kim, I'll also pop in. Leaders if you have played I Have, You Need with your teachers before you do this Problem String. You could, right before I did the 99, you could have said, "Hey, I meant to play a round of I Have, You Need with me. Oops, totally forgot. Completely unrelated. Sorry. Hey, just really quick. If I had... Random, if I had 27. Just totally random. If I had 27, how many would you need to make 100?" And I'll let people just sort of think about that a little. They'll tell me. In fact, what is that, Kim? If I have 27, what do you need to make 100?

Kim  31:24  
73. 

Pam  31:25  
Okay, so I'll just, "You know, totally unrelated. Sorry, I meant to play I Have, You Need. I know we've been playing that. Thanks." Then I'll go, "Alright, 99 packs. 99 packs. Go." Now, at this point, there's a couple people usually in the room that smile because they know I played I Have, You Need for just a reason there. But then people will say, "Alright, if I have 100 packs." And I'll pull. "You know, how did you do it?" They'll say, "100 packs minus 1 pack." I'll draw the bracket again. 100 minus 1. Arrow down to the 99. And then you have to take the sticks. So, 2,700 minus the 27. And I'll say, "Mmm, 2,700 sticks minus 27 sticks. I mean. We could do subtraction. We could go over there and borrow, regroup, whatever. But can we think about?" And then someone will smile. And I might go represent that on a number line. Subtract 20. Subtract 7. Leaders, if you've got teachers who've never thought about 2,700 minus 20, this is a good moment for them to think about that. So, don't go too fast here. Like, let them. I would have, you know, draw the number line on the board. What is 2,700 minus 20? That's good place value work. That's good like magnitude work. What are these numbers mean? What is 2,700 minus 20? And then subtract another 7. You're feeling that partner of 10 when you subtract that 7. All good stuff to do. If you've had teachers that have a little bit more experience, in that moment, you might say, "Ooh, why did we just play I Have, You Need. Can we use that partner of 100 to help us think about 2,700 minus 27. Ooh, there's 100 in there. And if I can think about that 100 minus 27 as being 73, can I think about 2,700

Pam  33:05  
minus 27 as being 2,673? Bam, and we have the number of sticks in 99 packs." And at that point, I might stop the string and just go, "Yall, what problem did you just solve?" And smile. And they will go, "Uh..." And sometimes they'll say, "Well, it's 2,700 minus 27." And I'll say, "Yeah, yeah. But why? Why were you doing that? We were doing..." And then I'll write down 99 times 27. And I'll say, "Check it out. You guys just did 99 times 27 thinking and reasoning using what you know, using relationships,. Well done."

Kim  33:39  
Yeah, nice. I know we're going a little bit long here. 

Pam  33:43  
This is a long episode, yeah.

Kim  33:44  
It is. But I think this is really, really useful. And I'm going to ask you two questions, and then I'm going to recap. So, you started off by sketching your ratio table, and I want you to talk for just a second about when you do horizontal versus vertical ratio tables because I know people have seen you do it both ways. And is there a rhyme or reason?

Pam  34:02  
Yeah, it's interesting. When you and I started presenting together, I noticed you always did it horizontally and I always did it vertically. 

Kim  34:07  
Yeah.

Pam  34:08  
And I had to ask myself why?

Kim  34:10  
Mmhm. 

Pam  34:10  
The reason... So, I was doing a vertical ratio table as I was drawing it here on my iPad next to me. And I will do that when I present almost all the time. In fact, I'd be hard pressed to think of a horizontal time. Only, because everything I write, almost all the time, has a horizontal ratio table simply for space.

Kim  34:30  
Yeah. 

Pam  34:30  
So, in all the books, you'll see a ton of horizontal ratio tables because I need to get more on the page.

Kim  34:35  
Yeah.

Pam  34:35  
So, when I present, I'll do it vertically, so that I hope people see them both, and that you will do both in your

Pam  34:42  
classroom.

Kim  34:42  
Right.

Pam  34:43  
Yep. 

Kim  34:43  
And it worked out because when you and I presented, then we each had both going on.

Pam  34:48  
We had both. Yeah, it was nice.

Kim  34:49  
Okay.

Pam  34:49  
Yeah.

Kim  34:50  
Another. I think one last question.

Pam  34:51  
Okay.

Kim  34:52  
Sometimes you do 17 sticks, sometimes 27. 

Pam  34:56  
Yeah.

Pam  34:57  
So, again, it was a little of you and me. I always did 27. You did 17. And one day you and I compared. I also sometimes did 37 and 47. But I stopped doing those. There's... 27 is brilliant because there's times doubling that are just go really quickly, and so I don't have to spend more time than I want to in this string.

Kim  35:19  
Right

Pam  35:20  
Doing the doubling. Not so true with 37 and 47. The doubling gets to whatever. 17 is great for younger students or even sometimes teachers of younger students.

Kim  35:32  
Mmhm. 

Pam  35:32  
But it's not as nice to have the doubling conversation because double 17, there's really... I don't know. It doesn't...

Kim  35:40  
Yeah.

Pam  35:41  
There's not as much to talk about. More people have to think about doubling 27. It's easier to double 17. You double 10. You double 7. And you're kind of done. You can double 20 and back up. But almost nobody thinks that's a good idea. Double 27 is a bit, quite a bit more like whoa. So, people are more willing to entertain the doubling. The last thing I'll say about 17 versus 27 is the subtraction. So, we end up doing subtraction, if I'm working with people who have had less numeracy experience, it could be... Is "easier" the word I want to use here? It could be an easier inroad, a better inroad. Because when I subtract 17, I don't have to think about the subtraction as much. But that's a tricky wicket because I also want people to have to think about the subtraction a little bit.

Kim  36:30  
Right.

Pam  36:30  
So, I tend to do 27 more often unless I'm dealing with kids, and then I might do 17.

Kim  36:37  
Yeah. 

Pam  36:38  
Yeah.

Kim  36:38  
So, one of the other things I'll note is that when we did start presenting together, you would set aside this ridiculous amount of time to do this string, and we would argue over like, "Wait, it's a string." And after several experiences of watching you deliver this string, I was able to step back and say, okay, with teachers, with leaders doing this with teachers, it's not just about experiencing the math that's happening. It is. But I just jotted down some things, some conversations, some things that you're able to lob in. The conversation about place value with add a 0, and twenty-seven 10s being 270. You get to have some conversations about am I subtracting 1 and 1 on both sides, so that sticks and packs. You're talking about purposeful order of strategies. You lob in the idea of an I Have, You Need. You talk about doubling strategies. You talk about strategies for subtraction, Remove a Friendly Number and Remove a Friendly Number, Over. And something else. Oh, and talk about why you choose a particular order. So, with leaders doing this with teachers, there's a lot of opportunity to stretch it out and to get a lot more meat out of this one particular string. Which is why we think it's such a great one to do when you are starting out.

Kim  37:58  
Yeah.

Pam  37:58  
Yeah. And I may not land on any of the things you just said very long at all.

Kim  38:04  
Right. 

Pam  38:04  
But if they come up a little bit, that's a differentiation thing. That's a way that all adults in the room are being stretched a little bit. I'm going to have teachers who don't notice or are able to hang on to most of what you just said, those extra little things that are coming out, but they're doing the math. They're having a good mathing experience. But I'm doing all those little things enough that I'm also stretching anybody in the room who's noticing the teacher moves, or the cadence, or... Yeah. I'll pop in one other thing. So, depending on how much time I've done and who the teachers are, I may throw out one more problem at the end.

Kim  38:38  
Mmhm. 

Pam  38:38  
Which would be, now I'm going to put the number on the other side of the ratio table. I've been giving you packs, and you had to figure out sticks. Now, I'm going to say, "What if there were 2,646 sticks? How many packs did I have?" 

Kim  38:51  
Yeah.

Pam  38:51  
And then that could be one more place that we could go with this Problem String that is a brilliant extension.

Kim  38:57  
Yeah. Alright, so long enough. I think on Sticks of

Kim  39:01  
Gum. It's one of our favorites.

Pam  39:02  
Thanks for hanging with us on this long one, yeah.

Kim  39:03  
For

Kim  39:04  
sure. If you would love to see what this string might look like written down. Hopefully you took some great notes. But if you'd like to see it, we are going to hand you a lovely download. It is at www.mathisfigureoutable.com/ratio-tables. mathisfigureoutable.com/ratio-tables. And we'll put that in the show notes. 

Pam  39:27  
Bam.

Pam  39:28  
Ya'll, thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. Thanks for spreading the word that Math is Figure-Out-Able!