Erika Isomura’s 5th grade students continue to work in pairs to classify, sort, and glue down decimal representations in numerical order. Erika circulates around the classroom, engaging students in sharing their thinking. She says to one pair, “This could be our prototype. This is our test run. We're kind of working on it, thinking. After we have some new ideas and maybe we have some better understanding of what we're doing, we can always come back to it. Okay?”
STUDENT: Five five-tenths.
STUDENT: Wait, remember...where did you leave the white ones?
STUDENT: I didn't have the white ones.
STUDENT: Well, I think five times...five times seven, five times seven equals thirty-five. I think that's seven thirty... Lizzie, I think that this is seven...
STUDENT: Where did you take them?
STUDENT: The white little blank papers. Remember you said...
STUDENT: I think that's seven thirty [inaudible].
STUDENT: Because the reason we have extras now that we're going to make those is because, um...
ERIKA ISOMURA: What's that?
STUDENT: I think they lost something and they think it's me.
ERIKA ISOMURA: So let's not fight about who lost something. Let's just do what we can. So look at this as a work in progress. We can learn from what we've got.
STUDENT: Oh, this is like our kites.
ERIKA ISOMURA: Yeah.
STUDENT: This is the prototype.
ERIKA ISOMURA: And then you can do...
STUDENT: And then next week we're going to do the real one.
ERIKA ISOMURA: Exactly. Okay?
STUDENT: The reason, um, why we were fighting is because, like, the white ones are just blank like these. Like, um, I left them on the table and then...
ERIKA ISOMURA: Yeah, so we can look at it the way Najee said. These could be our prototypes. These are our test runs. We're kind of working on our thinking. After we have some new ideas and maybe we have some better understanding of what we're doing, we can always come back into it. Okay?
STUDENT: Oh, can I please [inaudible].
ERIKA ISOMURA: And here's the other thing you can do, Lizzie. For these ones that don't seem to have partners, I'll go get you a piece of white paper and I'll cut up some, and you can always write your own.
STUDENT: Can you just turn them around?
ERIKA ISOMURA: We can put in some white parts here, where you write the decimals to match the fractions that need partners. Okay? So I'll go get a paper and I'll cut you up some squares. Guys, we're not changing the white ones now. It is what it is.
STUDENT: Two-tenths right there.
ERIKA ISOMURA: So Najee...
STUDENT: Because...because if you look up there, this right here is the tenths. Yeah, that's...no, that's the ones. That's the tenths and...yeah. That's the tenths.
STUDENT: This one goes here, right?
STUDENT: This is one whole because if you look right here... Camila, if you look right here, the decimal and the tenths and this is the wholes, which means we've got one-tenth...I mean one whole and the decimal. We have one whole, so that's why we need another paper to write one whole for this one.
STUDENT: Oh yeah!
STUDENT: Oh, so that's one whole. Like, one is just by itself.
STUDENT: These are the wholes and these are the parts, and that's one whole. And that's the tenths, hundredths, thousandths, and keep going.
STUDENT: One whole.
STUDENT: Yeah, that's whole.
STUDENT: So six and six stands for one whole too.
STUDENT: Which means that goes there because...
STUDENT: Six and six stands for one whole and that is one point zero, and it means one whole. Then that's one whole.
ERIKA ISOMURA: Nice job! I like how you guys worked it out. How you both had ideas and you talked about it [inaudible]. That's awesome, that conversation.
STUDENT: Now we need zero point seven. That's seventy-five tenths.
STUDENT: Zero point seventy-five tenths...
STUDENT: We can make new ones.
STUDENT: This one because we have these two left and they must be either one of this. So if we do this one and there's no...it's not possible that it could be one whole. So it should be this.
ERIKA ISOMURA: So that one could possibly be one whole?
ERIKA ISOMURA: On the other hand, I'm not unknown for sticking more than one thing together. Just saying. So your rationale and your logic totally make sense, but don't forget that I also sometimes just like messing with you all. So put it where you think you want to put it. Come to an agreement together.
ERIKA ISOMURA: How are you doing so far?
ERIKA ISOMURA: Are you feeling pretty happy with what you've done so far? Yeah?
STUDENT: Well, we did make mistakes on the first one but, like, we solved it but, like, we...
ERIKA ISOMURA: So Najee had a really cool idea. He said think of this as your kite prototype.
ERIKA ISOMURA: And then we can always come back into it later with a redesign. Okay? So don't get too caught up on that part. So which ones are you particularly proud of?
ERIKA ISOMURA: Which ones have you solved that you feel like, "Yeah, we got it."
STUDENT: This one because it's thirty, and thirty, and plus thirty equals ninety.
ERIKA ISOMURA: Uh-huh.
STUDENT: It says this is zero point ninety, so I think [inaudible].
ERIKA ISOMURA: So how are you seeing this as thirty, and thirty, and thirty?
STUDENT: Because the threes are in tenths place so that equals thirty. So I add it all to thirty, so then I got ninety.
ERIKA ISOMURA: Interesting. What do you think of that, Asia?
STUDENT: I think it's a great idea.
ERIKA ISOMURA: You think it's a great idea? Can you tell me about any of these? Choose one that you think you guys really think you got. And it could be in the blues or the golds.
STUDENT: This one.
ERIKA ISOMURA: That one? Okay, so the...you're pretty sure that this is two-tenths. How do you know that that one-fifth is the same as two-tenths?
STUDENT: Um, because...the two is in the tenths place and the five, I think, is in the tenths place too.
ERIKA ISOMURA: Hm, so the two is in the tenths place and the five is also. So if they're in the same place, is it okay for them to be different numbers?
ERIKA ISOMURA: Yeah? Okay.
There were some students who immediately saw "Well, we've done this. It may not look exactly the same, but we've done this." A couple of them had come from the last big unit we did on multiplying and dividing fractions. One reason a pair gave me was place value: you line up the ones together, you line up the tenths together, you make sure the decimal points are together, and then you just do it.