Erika Isomura’s 5th-grade students work in pairs to classify, sort, and glue down decimal representations in numerical order. She challenges her students to explain and defend their thinking.
STUDENT: We need to simplify to get this.
STUDENT: [Inaudible] which one is which. So this one is right here.
STUDENT: This is also obvious because it's two wholes, and it's five.
ERIKA ISOMURA: So you can pretend like they're cakes and [inaudible]. Okay? So in this cake, how many pieces would you be giving out?
STUDENT: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
ERIKA ISOMURA: Okay, so then what...
STUDENT: I gave out one.
ERIKA ISOMURA: And you gave out one, so what fraction would that be?
ERIKA ISOMURA: One-tenth. All right, nice! I didn't even think about that. I need to make a different shape next time that doesn't look so cake like.
STUDENT: Then I'll do a pie.
ERIKA ISOMURA: Do a pie?
ERIKA ISOMURA: Not a cake? Okay.
STUDENT: I think one-tenth goes here for...because on the poster it said that when the decimal is here, that's a tenth and that's a hundredth. So I think this goes here.
STUDENT: Which one? The one-tenth?
STUDENT: Yeah, with the one point.
STUDENT: I think this goes here. The one-tenth goes here. So it's the tenth [inaudible]. So when you put it...like, if you switch it...
STUDENT: Oh yeah, you switch it around.
STUDENT: Here you go.
STUDENT: Thank you. So it goes here and then we just need to find zero point two.
STUDENT: That's why we have to go across the right.
STUDENT: No but those says we're missing a card, not a make one up.
STUDENT: We're missing...we're not missing. It's just that...
STUDENT: So these are the ones. 1, 2, 3, 4, 5, 6, 7...
STUDENT: The zero point two stands for two-tenths and...
STUDENT: We only have six, Lizzie.
STUDENT: There's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.
STUDENT: No. 1, 2...
STUDENT: These, I'm talking about these. I'm talking about the white ones.
STUDENT: 1, 2, 3, 4, 5, 6, 7. We need seven more.
STUDENT: I'm talking about the white ones.
STUDENT: Oh no, she's talking about the ones that need a...
STUDENT: I'm trying to count the white ones to see [inaudible].
STUDENT: Seven and then 6, 7, 8, 9. There is nine...
MIA BULJAN: How is that related to Diego's idea where he moves the decimal points?
ERIKA ISOMURA: How come Diego is talking about moving decimal points and adding in zeroes and things?
STUDENT: Well because he's doing, I think, division and multiplication.
ERIKA ISOMURA: And since we've already studied whole numbers, we know that they sometimes function a little differently.
ERIKA ISOMURA: Nice! All right. And this one I thought would be a little bit harder for people. How do you know that that's five-tenths or one half? STUDENT: You're just multiplying one-tenth by five. Because that's pretty easy, it's five-tenths. Because, like, you just have one-tenth, so just separate the tenth...like the tenth, and then just leave the one, and then, like, just take off these two.
ERIKA ISOMURA: Mm-hm.
STUDENT: And just leave the one right over there, and then one times five. So then, like, technically you're doing five times ten and five times one, so we know it's five-tenths.
ERIKA ISOMURA: Okay. Do you agree?
ERIKA ISOMURA: All right. And can you tell me about this one?
STUDENT: Um, like, yesterday we told you about the quarters, and like, four quarters equal one dollar. And one-fourth of four is twenty-five cents, so it'll be zero point twenty-five.
STUDENT: So we're trying to find the ones that work with this because...because Lizzie had came up with this, like this method that, like, when you take away the decimal, just leave the number there.
ERIKA ISOMURA: Mm-hm. STUDENT: [Inaudible]. Or something similar to it.
ERIKA ISOMURA: So can you show me an example on this one? What do you mean take the decimal away?
STUDENT: Like, um, if you move the decimal, that number, where you push the numbers together is twenty-five.
ERIKA ISOMURA: Twenty-five. So you're looking for something with a twenty-five?
STUDENT: Yes or close, or similar.
ERIKA ISOMURA: So how did that push the numbers together without a decimal work to get sixteen over sixteen?
STUDENT: Oh yeah, because we're trying to figure out because we're missing one more, that's why. I just did that as a mistake, that's why.
ERIKA ISOMURA: Okay. Does anybody think that sixteen over sixteenth goes with one and three-tenths?
ERIKA ISOMURA: So I can see you take the decimal away. I see the five, four. I see the thirty. Okay, I can see some of that, but that's the one that I don't see how that number works. What do you think of that idea with this one, Najee?
STUDENT: Yes, because...because, um, because you guys had made a poster...I mean, like, because our class had made a poster with the decimal and the left is the ones, and the right is the hundredths or tenths.
ERIKA ISOMURA: Okay, so the wholes are on the left and our parts or decimals...
STUDENT: So that means we're going to make a twenty-five tenths.
ERIKA ISOMURA: Twenty-five tenths? Where would that go?
STUDENT: That will go here.
ERIKA ISOMURA: Okay.
STUDENT: And then five times two equals ten. That's like the multiplication symbol but at the dot, and five times two equals ten and you put that over two-tenths. Now we need to do the card.
Almost all of the lessons that we do, like formal lessons, are embedded in a story. That means that because they [my students] get attached to the story there's a character, there's a situation. They remember stories. They don't remember, "Do you remember the day we talked about these numbers?” When their thinking is attached to a story we’ve thought about before, it makes a lot more sense to them because it's nothing new. It really isn't. It's the same thing with this little twist.