Erika Isomura’s 5th-grade students continue to work in pairs to classify, sort, and glue down decimal representations in numerical order. Erika invites her students to compare their work to their work for prior investigations and other problems. She reminds each partner to contribute equally to the pair work. She asks “How did you get that? Can you show it to me in a picture or with numbers?” She challenges partners to contribute to their pair’s mutual work: “Make sure he's proving it to you. Don't just let him talk it out.” As partners finish their work, Erika invites them to do a “gallery walk” of other partners’ work to check and compare their work.

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

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ERIKA ISOMURA: The one fourth of four and then you put that there?

STUDENT: So you have that "of" also means times. [Inaudible].

ERIKA ISOMURA: And then I see you use that idea again that we were working on when you were helping Alex develop his fraction idea. So where do you think the four...one-fourth of four goes?

STUDENT: Um, right here.

ERIKA ISOMURA: How come you're not writing down any of the problems and doing the calculating? How come you're counting on Rosa Linda to do them all?

ERIKA ISOMURA: How do you know one-fourth of four is the same as two-tenths or one-fifth?

STUDENT: Because "of" is times.

ERIKA ISOMURA: Uh-huh.

STUDENT: So four times one-fourth is two.

ERIKA ISOMURA: How did you get that? Can you show it to me in a picture or show it to me with numbers?

STUDENT: Minus one.

ERIKA ISOMURA: Are you sure?

STUDENT: Yeah. Positive.

ERIKA ISOMURA: So you know what your thing is, Diego?

STUDENT: What?

ERIKA ISOMURA: Too fast. Slow it down. Make sure you're checking.

STUDENT: Okay.

ERIKA ISOMURA: And you as his partner, make sure he's proving it to you. Don't just let him talk it out.

STUDENT: Because seven times five equals thirty-five, and we're doing this thing where you have to put the number here on top.

STUDENT: That's why seventy-five under tenth. Is that right?

STUDENT: Yeah!

STUDENT: Seventy-five and ten.

STUDENT: So now we're done. All we have to do is start gluing.

STUDENT: It's under ten, right?

STUDENT: Yes. I agree. Now it's time to start gluing.

STUDENT: Now we need for the...now we need some for these.

STUDENT: These! Those go to these.

STUDENT: No they don't.

STUDENT: Yes, look.

STUDENT: No they don't.

STUDENT: I think that...

STUDENT: These go together.

ERIKA ISOMURA: No, those go on that chart.

STUDENT: Yes, those go on this chart and I think these is to describe these, I think.

STUDENT: No, no. It's because, um, when...

ERIKA ISOMURA: So remember you guys said you lost some white cards?

STUDENTS: Yeah.

ERIKA ISOMURA: So that means that you're going to need to put your yellows on...your golds on and you need to write a white card to match them.

STUDENT: Oh!

ERIKA ISOMURA: So I want a decimal to go with the golds.

STUDENT: A decimal to match with these.

ERIKA ISOMURA: Yeah. So this doesn't go there. These were to make white cards. That's why they're white papers.

STUDENT: Oh, okay. Can I please go to the restroom?

ERIKA ISOMURA: Give me just a second, Najee. So these need to be matched up. Whichever ones don't have a white match, you need to make a white. Okay? So while he goes to the restroom, why don't you girls take a quick walk around. Do your little gallery walk. Check in and see what other people are doing.

STUDENT: Okay, you guys can do that.

ERIKA ISOMURA: Yeah. And then you can go, that way you won't miss out on what they're doing.

STUDENT: I'm still figuring out, um, if this is like, um, twenty-five...twenty-five hundreds or just one whole. And, um, I pretty much covered everything. I'm just looking around now to see, um, if it's correct or not. And I'm looking around for tweaks every now and then here. So yeah, that's it.

STUDENT: When did it...

STUDENT: It is. It has ten sections.

STUDENT: But it's supposed to be...

STUDENT: It is for this one.

STUDENT: It's supposed to be like the other ones too. It has to have eight hundred.

STUDENT: No no. Exactly, right there.

STUDENT: No, it's supposed to have a hundred little squares.

STUDENT: No. Look at these, they're different. They're different.

STUDENT: Yeah, they're different but that doesn't...

STUDENT: Exactly. They're the same thing. [Inaudible]. Okay, one done.

STUDENT: Let me draw the next one. I'm going to draw nine-tenths. Nine-tenths?

STUDENT: There's nine-tenths here.

STUDENT: Oh.

STUDENT: Wait. Well, this one was for a different one. Different one. Where's nine-tenths? I think this one's for here.

STUDENT: Yeah. There's no five-tenths, is there?

STUDENT: Five-tenths? Uh, five-tenths.

ERIKA ISOMURA: How do you know that's a match?

STUDENT: Because it's zero...it's twenty-hundredths.

ERIKA ISOMURA: Uh-huh.

STUDENT: And there's no twenty-hundredths.

ERIKA ISOMURA: Hm. How do you know it's twenty-hundredths?

STUDENT: Because the...it's one and anything times one it's still twenty. So it's twenty right here.

ERIKA ISOMURA: Uh-huh.

STUDENT: And then it's in the hundredths so it's twenty-hundredths.

ERIKA ISOMURA: Okay. Can you write that down for me? Write the answer down. Okay. So you've convinced me that the multiplication works. Can that be simplified? Is there any other equivalent fraction that that could be?

STUDENT: Ten fifties.

ERIKA ISOMURA: Okay, write that down. Is there any other equivalent fraction that could be?

STUDENT: Five twenty-fifths.

ERIKA ISOMURA: You know what question I'm about to ask, right?

STUDENT: Yeah, another one?

ERIKA ISOMURA: Mm-hm.

STUDENT: I don't know.

ERIKA ISOMURA: So how did you go from twenty over a hundred to ten over five?

STUDENT: I halved it.

ERIKA ISOMURA: You halved it? How did you go from here to here?

STUDENT: [Inaudible].

ERIKA ISOMURA: Okay, so you divided by two. Is there anything else you could divide the numerator and denominator by?

STUDENT: One.

ERIKA ISOMURA: Anything else?

STUDENT: Oh! Okay, so it's...zero.

ERIKA ISOMURA: No no. Write the division sign or actually write it out "five divided by."

STUDENT: Divided by five.

ERIKA ISOMURA: And then what do you do with the denominator next to keep it the same? What would you do in the denominator? You're going to divide by five in the numerator, what will you do in the denominator?

STUDENT: Divide by five.

ERIKA ISOMURA: Okay. Because we have to divide by a form of one.

STUDENT: Twenty-five so...oh, it's one.

ERIKA ISOMURA: It's one whole?

STUDENT: Wait. Five and then zero. Oh, it's...

ERIKA ISOMURA: Antonio, what's five divided by five?

STUDENT: [Inaudible].

ERIKA ISOMURA: What's two divided by two?

STUDENT: Oh, one! Sorry, sorry.

ERIKA ISOMURA: What's eight divided by eight?

STUDENT: One.

ERIKA ISOMURA: What's ten divided by ten?

STUDENT: One.

ERIKA ISOMURA: What's fifty million divided by fifty million?

STUDENT: One.

ERIKA ISOMURA: What's five divided by five?

STUDENT: One.

ERIKA ISOMURA: And you don't see one-fifth anywhere in your chart, huh?

STUDENT: No. Oh! I didn't notice that. So it's right here.

ERIKA ISOMURA: So once again, what is the thing that you need to do?

STUDENT: Slow down.

ERIKA ISOMURA: Uh-huh.

STUDENT: Antonio, you're supposed to be my [inaudible] man.

STUDENT: I am but you don't listen to me.

STUDENT: You never said that, you're like...

One of the things was I really wanted to find out was whether they understood quantity in decimal form. Understood that hundredths are smaller than tenths, because they kind of sort of look the same, so maybe they're kind of sort of the same.

I really wanted to get a better understanding of their understanding of magnitude and size.