Erika Isomura’s 5th-grade students continue to work in pairs to classify, sort, and glue down decimal representations in numerical order. Students discuss the reasonableness of their results with their partner prior to gluing down their cards. Erika connects pairs with each other to have them share their thinking, discussion, and results (“Why did you put this card here?” “Are both of these answers possible?” “What do you think of their idea? Does it make sense?”)
STUDENT: So it either has to be this one, that one, or that one, or maybe one of these are wrong. Let's make sure...this one we think is right. Yeah.
STUDENT: This one...this one, yeah. Wait. This is...yeah, two point five. Yeah. Um, one point three, yeah. And the zero point nine, yes. Uh, eighty divided by a hundred, yes. Right? Yeah. Or is this one...or is this one that...that one? Because see, this is the hundredths place over here. So then it should be this, right?
STUDENT: Yeah, you're right. Just take away these two dots. Just put the fraction over there [inaudible] over there. Yeah, should be equivalent to this. Like eight times ten, eighty.
STUDENT: Uh-huh. So then this one, zero point twenty-five times zero point seventy-five, so that's for sure. Zero point five, yeah, that's for sure. Let's see, um, this one equals zero point thirty, so that's for sure. And then one fourth, yeah, that's for sure. And that one...
STUDENT: This goes over here, right?
STUDENT: Five point four goes all the way over here. And then where was...where is one thirty?
STUDENT: One thirty?
STUDENT: The one right there. Take it off.
STUDENT: All right. Here.
STUDENT: And this...wait. This is one...yeah. Wait, are you sure?
STUDENT: Yeah. I think, yeah. Probably, yeah. All right, thirteen tenths, I think goes over here because... Let me see the paper. All right. You have ten, five...yeah, it's this one.
STUDENT: What? Yeah, it's that one because it has...
STUDENT: I don't think it is.
STUDENT: Yeah, it is. You take away the three and that's ten over ten, that's one whole. And then you put the whole right there...
STUDENT: Oh, I thought you meant...oh, yeah.
STUDENT: Peel that one off.
STUDENT: The twenty-five?
STUDENT: Yeah. Two point twenty-five.
STUDENT: Like it's easier for us to solve compared to the yellow sheet. Like, it's harder because it's a picture.
STUDENT: We can get the answer like we did this.
STUDENT: Yeah. Like, we could look at them and see, but, like, it's hard to tell if we're actually right or wrong so we should do the blue one first, which is more helpful to be honest.
ERIKA ISOMURA: Ruchita and Federico, can you guys come here for a sec?
STUDENT: Here you go. Thank you!
ERIKA ISOMURA: So remember I asked you guys about the one-fourth of four?
ERIKA ISOMURA: And you said it was here because, like, four quarters make a dollar, and so one-fourth of those four quarters would be twenty-five cents?
ERIKA ISOMURA: But then you put it here. Why did you put yours here?
STUDENT: Because you said it was like a number of a number.
ERIKA ISOMURA: Uh-huh.
STUDENT: It's a multiplication question.
ERIKA ISOMURA: So one-fourth times...four.
STUDENT: Four. Which equals one whole. Because, um, you have one piece...um, it takes four pieces to make a whole. Then one piece times four, that's four pieces and that makes a whole.
ERIKA ISOMURA: So that one times four is the four pieces to make a whole.
ERIKA ISOMURA: Hm. So I'm a little puzzled because you totally made sense. I agree, four quarters make a dollar, so one of those four quarters would be twenty-five cents. But then we did study that for a long time, that of...in this context, it's a multiplication problem, and if I have four things and I take one-fourth of it, that's one whole left.
Who's correct, even? Or are you both correct? Is it possible for it to be both one whole and twenty-five hundredths? What do you think of their idea, Dylan? Does that make sense? Not do you agree with it, but does that make sense that there's four quarters in a dollar, and if I took one-fourth of those four quarters, I have one quarter?
ERIKA ISOMURA: And then one quarter is twenty-five cents?
ERIKA ISOMURA: Do you think Dylan's make sense that one-fourth times four wholes would give you one whole at the end?
ERIKA ISOMURA: So that's really weird because I haven't seen a problem really where it could be both one whole and twenty-five hundredths, when it's the same card. So have you two glued this one yet?
ERIKA ISOMURA: I'd like your two groups to hold off on gluing this part for now.
ERIKA ISOMURA: Kind of think about it for a little while and maybe we're going to bring it to the whole group and talk about that particular part before we all glue it, because it looks like you can see the other side but [inaudible]. All right? And if you decide to change your mind or you could come up with something new, let me know. Okay? I think that's an interesting question. I'm going to write it down on the board so we can talk about it later.
I felt like that was a place to step in and just say this is the way it works. It's not a choice here. Multiplication, yes we have choices, addition we have choices, subtracting, no, it goes in a particular order. Division, no, you have to go in a particular order.
I'm not really good at saying, "Here are the rules, you must follow them." They know that, so when I come down and say, "These are the rules that you must follow," it's a little bit tricky because that's not how I teach. When I say that they're kind of like, "Really, are you sure because ….”
Even though I said it, I didn't feel like they were convinced, so I wanted to talk them through it. Also because I knew that if they could get that and really make sure that their understanding of division worked, then the big idea was that important idea, [that] the small divided by large equals a fraction.