As her 5th-grade students finish their card sorts and defend their thinking, Erika Isomura increases the rigor, inviting pairs to go beyond just solving the problem: “Make something challenging, make something that the other kids are going to be like ... then after they think about it like ‘oh yes, I got it.’ Give them a little bit of a push, okay?” She connects their thinking back to the previous lesson with fractional lengths of strings.

5th Grade Math – Decimal Place Value*Erika Isomura, Glassbrook Elementary School, Hayward Unified School District, Hayward, California*

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ERIKA ISOMURA: There's no white to go with those?

STUDENT: Nope.

ERIKA ISOMURA: Okay, great. So you figured out these are the ones that you must've lost. That's cool. So now we need to think of what would a decimal look like?

STUDENT: [Inaudible]

ERIKA ISOMURA: What?

STUDENT: That one.

ERIKA ISOMURA: [Inaudible]

STUDENT: And for one-fifth...I don't really know, but zero point five...no, that's ten. It's fifty. Oh my gosh.

ERIKA ISOMURA: Ignore that side, use the back.

STUDENT: I'm trying to find out what, um, one-fifth.

STUDENT: That I know that because...what would that be? Two? You can divide... Oh, you can divide but then what's going to be the fraction with one-tenth? Let's do drawing.

STUDENT: It's that we have to, um, do the, like, the math problem. Like [inaudible], or times, or division. And we...so we have to write it down on a blank piece of paper, and we have to place it on the ones that don't have any card numbers yet. So right now, doing eight-tenths, zero point eight, and I don't know [inaudible].

STUDENT: Um, you said you're going to do eight-tenths?

STUDENT: Yeah.

STUDENT: [Inaudible]

STUDENT: We have it right here. Wait, that's three. Oh, never mind. Remind me to take that off if we can't find it. Uh, you have five-tenths or something like that? Yeah. We have to make five-tenths, I believe. I'll make it.

STUDENT: No, I already made it.

STUDENT: All right, place it there. All right, we need twenty-five. Twenty-five. We need to make twenty-five too.

STUDENT: We already made twenty-five.

STUDENT: Where? That's twenty-five...well, there's...we need one for right here. All right. Do we have to draw, like, four sections? I'll use these here. I'll use these.

STUDENT: Um, we're doing the ones that we know for sure. Like, there are some maybes we don't know if they actually go there, but we're just doing the ones we're for sure go there. And we have some that we don't even know, so we just need to find out where these go. And then if they don't go in this one, like, in the ones we don't know about, then we can glue these on and find out where these actually go. So we glue the ones we actually for sure know.

STUDENT: So we solved the mystery of one-fourth [inaudible]. They finally agreed with me since, um, four pieces and it's four wholes, you need one piece which is one whole. So we finally solved the mystery.

ERIKA ISOMURA: You guys came to an agreement?

STUDENT: Yes.

ERIKA ISOMURA: Good. I don't want you to be bored.

STUDENT: Oh.

ERIKA ISOMURA: So for this one I definitely want you, and Ruchita, and Federico to talk because I think you are the only teams that's going to get to the pink cards. Okay?

STUDENT: Can I get an extra piece of paper?

ERIKA ISOMURA: What?

STUDENT: I might need an extra piece of paper.

ERIKA ISOMURA: Don't worry about it for now. Federico, Ruchita, I'd like the three of you guys to talk about your pink cards with Dylan, because you're the only team that's going to get the pink cards today. Don't cut them up yet. Just come back here and have a chat about what you notice about them, because there's some stuff there that I put just to kind of check in and see what things we still need to better understand.

STUDENT: When we write these do we write the problem?

ERIKA ISOMURA: Well, I don't know. What's on the blue ones?

STUDENT: Problem.

ERIKA ISOMURA: So what do you think you're going to write on the blue ones?

STUDENT: A fraction.

ERIKA ISOMURA: A fraction? If you think so.

STUDENT: It's a problem.

ERIKA ISOMURA: You do realize, you know, that your classmates are going to take a look at it at some point.

STUDENT: It's a problem.

ERIKA ISOMURA: So which ones are you working on?

STUDENT: This one, this one. That's it.

ERIKA ISOMURA: This one if you want to.

STUDENT: Oh yeah, that one.

ERIKA ISOMURA: It has a blue but since they seem to always be doubled, you might want to think about it. And if you need to, we'll just attach another paper so that you can make your problem a little longer.

STUDENT: All right.

ERIKA ISOMURA: All right, so who's doing this one? You? And who's doing this one?

STUDENT: Me.

ERIKA ISOMURA: You? So what are you thinking?

STUDENT: Um, two-tenths minus one-tenth.

ERIKA ISOMURA: Do you think that will give you one-tenth? Would it help for him to write it out for you to see it? Write it out. He'll check yours. And then Antonio, why don't you do the same thing? You write a problem for this then give it to Diego and let him check it. Okay?

STUDENT: Seventy-five cents equals seventy-five hundredths.

ERIKA ISOMURA: Okay. And this one, you guys said...that was the one I thought was going to be the hardest for everybody. So what do you think that one is?

STUDENT: Because these are five, um, five wholes because five smileys is one whole.

ERIKA ISOMURA: Uh-huh.

STUDENT: And then...and so for half, like, two will be four because...

STUDENT: Um, the five equals ten because there's ten pieces in [inaudible] whole or...

ERIKA ISOMURA: Okay.

STUDENT: And then, um, this was substitute as four double.

ERIKA ISOMURA: So if you double that to get your tens, you double that to get four?

STUDENT: Yeah.

ERIKA ISOMURA: Man, I thought this would be much more of a challenge. What about this one?

STUDENT: One whole and three-tenths.

ERIKA ISOMURA: Okay, how about that one?

STUDENT: That's, um, [inaudible].

ERIKA ISOMURA: Do you agree, Federico?

STUDENT: Yeah.

ERIKA ISOMURA: Prove it.

STUDENT: Okay, right now you just have, like, this is five cents and these three are pennies. So 1, 2, 3. Eight.

ERIKA ISOMURA: But, like, why isn't it this?

STUDENT: Because it's not eight-tenths.

ERIKA ISOMURA: Huh?

STUDENT: It's not eight-tenths.

ERIKA ISOMURA: But it's not an eight-hundredths either. It's eight cents.

STUDENT: Oh!

ERIKA ISOMURA: Oh!

STUDENT: One dollar is like, um, one hundred of hundredths, so then eight cents is like eight hundredths.

STUDENT: Oh yeah!

STUDENT: Oh!

ERIKA ISOMURA: Okay. Well, that makes me so sad because I really thought that would be a bigger challenge.

STUDENT: Please don't be more.

ERIKA ISOMURA: So when you guys feel like you know them, feel free to start cutting and gluing.

STUDENT: Okay.