< Back to Building the Ratio Table: 12 Sticks of Gum

## Annotated Transcript

This is the transcript of the video, with green comments of Kim reflecting on her teacher moves during the Problem String, inserted to highlight purposeful teacher moves.
- [Kim] So I wanna do another problem string with you guys today, okay. Remember you have some paper in front of you. If you wanna record your thinking, you're welcome to do that. You do not have to just copy down what I have on the board just for the sake of copying. But if you wanna record some things, you can do that as well, okay. You ready? Okay, so do you know that I have a couple of kids here at this school?

- [Kim] I think I've mentioned that. Two boys and one of 'ems name is Luke. And Luke's favorite thing of all time is to chew gum. Anyone like to chew gum here? Yeah, lots of gum chewing. And the kind of gum that he likes has 12 sticks in the pack. Setting context

- [Student] I like it.

- [Kim] You like gum too? 12 sticks in a pack. So I'm going to record some questions that I'm going to ask you. One pack has 12 sticks, okay, does that make sense? What if I asked you how many sticks of gum there were in two packs? You think you know? Give me a thumbs up Asking for a private signal allows for wait time when you think you know how many sticks of gum are in just two packs? Sam you think you know that one? What you think you know is enough. We're about thinking here, not just answer getting.

- [Sam] 24

- [Kim] 24. How do you know that's got to be 24? Because how you know is important. We are curious!

- [Sam] Because 12 plus 12 equals 24.

- [Kim] Okay. Two packs would just be 12 plus 12 equals 24. Does anyone agree with Sam? Okay, cool. Could I also think about that like if I double the number of packs then I double the number of sticks? Want to start using the word/thinking about doubling rather than additive thinking. Yeah I could think about that. So, I'm going record that like this: if I double the number packs, then I can double the number of sticks. Yeah? What if I said that Luke had four packs of gum? 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?

- [Bella] 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? Want them to start thinking about looking to the ratio table. Logan, what did you think about?

- [Logan] I did 4 times 12.

- [Kim] Okay. Is that what I'm asking you, 4 times 12? What's the problem anyway? 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"? Looking specifically for someone who doubled. Brooks?

- [Brooks] So I know that two--

- [Kim] Two what? Stay in context, my friend.

- [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? Clarifying

-[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?" Didn't get what I wanted so I decided to start with what Brooks told us and mention it to see if it pings for anyone in the room. Is that true? If I double the number of packs, can you picture that? Context. If I double the number of packs, do I need to double the number of sticks? And how many was that, how many sticks?

- [Student] 48.

- [Kim] 48, interesting. What if I asked you about eight packs of gum? Hmmm. Eight packs of gum. What if I asked you about eight packs of gum? Is there anything up here that might be helpful to you? Hint, hint. Eight packs of gum. Bless you.

-[Student] Thank you.

-[Kim] Colton, what did you get for eight packs of gum?

- [Colton] 96.

-[Kim] 96. Did anyone else get 96? Emily, you got 96. Colton, how did you think about 96 sticks of gum?

- [Colton] Because, we had... I counted by 12s.

- [Kim] Counted by 12's. Okay, did anyone think of these amounts that I have up here? Nudging to multiplicative thinking rather than additive nature of skip counting. Emily, you did? You thought about one of these amounts. Which amount did you think about?

- [Emily] 28.

- [Kim] twenty...48?

- [Emily] Yes.

- [Kim] And what did you think about that?

- [Emily] I knew that four plus four is eight. So I can just add 48 plus 48.

- [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? Made sense to me but acknowledging that Emily was out of context and using chunks that others had not thought of. Checking to see who followed her thinking.

- [Ava] I used the 12.

- [Kim] Can you tell us what Emily did? Did you hear what Emily did? Let's focus on this great strategy of doubling that already came up!

- [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? Help me! Make sense with me, mathematicians!

-[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? CONTEXT

-[Scott] You're adding two more packs, like, two times more.

- [Kim] Double the number of packs. Let me clarify your language

- [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? Making sense of what they are saying in regard to packs and sticks. Hmm. What if I asked you about 10 packs? Is there anything up here that could help you? I know one pack, I know two packs, I know four packs, I know eight packs. Can I use anything here to help me with 10 packs? Look kids! There are lots of helpers here! I'm going to let think about that for just a second.

-[Kim whispers] What are you thinking?

-[Ella] The 26, because it takes eight to get to 60, 96. So then you can just add two more. She means it takes 8 packs to get to 96, and then you can add two more packs (24) to get ten packs. Listening and understand is HARD sometimes.

- [Kim] Interesting.

-[Kim] Um, Ella, how did you think about the 10 packs?

- [Ella] What I did is, I saw 96 and then so I know it takes eight to get to ninety-

- [Kim] Eight what?

- [Ella] 12's to get to 96, so after that--

- [Kim] Eight what? We need context still

- [Ella] Eight 12's.

- [Kim] Eight 12's or eight packs?

- [Ella] Yes.

- [Kim] Okay, eight packs was 96 sticks.

- [Ella] Then could just add two more 12s.

- [Kim] Two more packs, which was 24 sticks. Can anybody picture what Ella just said? You can picture that Brooks?

- [Brooks] Yes.

- [Kim] She said, well we know eight packs is 96. STICKS, Kim, STICKS. I realize now I missed an opportunity to stay with packs and sticks.

- [Brooks] Yeah.

- [Kim] And then two packs more packs is 24. STICKS. So what can I do to figure out the number of 10 packs? How can I use what Ella just said? Brooks what can I do?

- [Brooks] You can, you can add 24 and 96 or you can do 10 times 12.

- [Kim] How can I do what Ella did?

- [Brooks] You can, so if 24... Well two plus eight equals 10

- [Kim] Okay.

- [Brooks] So, you can do 24 plus 96.

- [Kim] Okay, let me record that and then you can tell me another way. So, I could think about two packs plus eight packs would give me 10 packs and then I would have to add the 24 sticks plus the 96 sticks, and does anyone know what that would give me? What would that give me, TW?

- [TW] 120.

- [Kim] 120 sticks. Interesting, did anyone else add two packs plus eight packs? Bella did. Did anyone think about the 10 packs in a different way? In a different way? You were like, hm, I don't know but I think I know 10 packs. Kyle what did you think about?

- [Kyle] So I did, the 12 song to 96.

- [Kim] Okay.

- [Kyle] Then I did two more which equaled 120.

- [Kim] How many 12's did you say you knew?

- [Kyle] I know that if there's eight packs it equals 96.

- [Kim] Okay. Pointing to how it's already on the board.

- [Kyle] And then you need two more to get to 10.

- [Kim] Okay. Also, on the board.

- [Kyle] So, I did 96 plus 24.

- [Kim] Okay.

- [Kyle] And I know that, that equals 120.

- [Brooks] You could do 120 minus 12.

- [Kim] Interesting. [Whispering] What are you thinking about?

- [Kyle] I did, so, I know that 10 is 120,  so minused 12 which equals 108.

- [Kim] Okay, interesting. What are you thinking about Hannah Mae? Did you see anything that would helpful?

-[Emily] I'm thinking that 12 times 9 equals 108.

- [Kim] So, you just know 12 times 9?

- [Hannah Mae] Yes.

- [Kim] Great.

-[Kim] Sam, what are you thinking about?

- [Sam] So I did this. I did nine plus one and then I--

- [Kim] Eight plus one?

- [Sam] Or eight plus one.

- [Kim] Its okay.

- [Sam] Yes and then...

- [Sam] So, I had 108. And then, I just... I just... So since like 10, I mean 12, is something like a yucky number, something like that,  I added 10 to get 8 to get 118, and then two more to get 120.

- [Kim] So you thought about "I gotta find 9 packs. So one pack that I know plus eight more packs that I know gives me the nine packs". Nicely done. Restate Sam's strategy back to him, because I am about to have him tell his strategy to the class and want to help him be more clear to his classmates.

-[Kim] Okay, I heard a couple of different strategies for the nine packs and I just found out that Sam just said that 12's kind of a yucky number to multiply by. So, for some people that's true. Some of us like 12's, some of us are like mmh, 12's. 12's are not as nice. But I heard a couple of different strategies for the nine, nine times 12. Sam start us off. What did you use, up here, to help you figure out nine 12's.

- [Sam] So I know that nine plus... or like, nice plus, or eight plus one gets me to nine.

- [Kim] So eight packs plus one pack would get me to the nine packs. Okay, so if I add one pack plus eight packs then what do I have to do with the sticks?

- [Sam] So, you would have 108. And then...

- [Kim] Wait, how did you get 108?

- [Sam] So, I have eight packs.

- [Kim] Which is how many sticks?

- [Sam] 96.

- [Kim] 96 sticks.

- [Sam] And then I multiplied by 12.

- [Kim] Added 12 sticks, okay. Fixing his language without making a big deal of it. I knew what he meant because we had just spoken. Important to confer with students sharing as often as possible.

- [Sam] And I got 108.

- [Kim] Okay, thank you, that makes sense to me. Who understands what Sam did? Who understands what Sam did? He said I got to figure out nine packs. So, I'm going to use one pack plus nine packs which equals, I mean, one pack plus eight packs which equals nine packs. Scott, is that what you did?

- [Scott] Uh, yeah.

- [Kim] You did the eight packs plus the one pack? Who am I thinking about? Kyle, is that what you did? You did eight packs plus one pack? Don't forget who you picked to share, Kim!

- [Kyle] No, so.

- [Kim] What did you do?

- [Kyle] I did 10 packs, we know that's 120 sticks.

- [Kim] Okay.

- [Kyle] So then I subtracted 12 and then I know that that equals 108 sticks.

- [Kim] So Kyle said, I'm going to start with 10 packs which is 120 sticks and I'm going to get rid of a pack, which is 12 sticks and does that give you nine packs? If I have 10 packs minus a pack?

-[Students] Yes.

-[Kim] 10 packs minus a pack, and I can subtract 120 sticks minus 12 sticks and, indeed, that gave us 108 sticks.

Reflections: I wish I had known these students and the instruction given a little better before I had chosen this particular string. Some of the students had already memorized their 12's facts or were singing a song to get to a particular fact (additively). This impacted how willing they were to make use of the known information on the ratio table and consider the problems multiplicatively. Going back, I would have chosen a number like 17 sticks in a pack so they would explore the relationships more readily because the problems would have been more challenging.