Erika Isomura’s students continue pair work on the problems, defending their thinking to each other. They make use of the sentence stems Erika gave them (e.g., “I think ____ because,” “This way is easier on our brains because______,” “What do you think?” “How did you feel about this problem?” “I knew it was this answer because I thought about the 0s”).
Erika engages pairs in explaining their thinking (e.g., “Show me in your answer,” “Do you think that will work every time?”). She reminds her students to note moments in their work that suggest that they might need to have a discussion about it.
STUDENT: That one is zero point zero two because, um, like, if you...if you do... Sometimes the zeroes can be helpful because zero point zero two...so you could take off these two zeroes and then put these two back on, but with the decimal.
STUDENT: Because look...oh yeah!
STUDENT: Yeah because then that will make it easier...easier for your brain to do it.
STUDENT: Okay, so let's keep on checking. I got ninety. Oh yeah, so it's [inaudible]. I only got one wrong.
STUDENT: So what do you think, uh... Okay, how did you feel about, like, um, seven thousand divided by ten?
STUDENT: That was easy because I just put...I knew I just, um, I knew it was seven hundred automatically because I only thought about the zeroes. Like, uh, I knew I had to find...uh, seven thousand...I knew I had to find, uh, number that's exactly like seven thousand but has one less zero. Like...it's hard to explain. It's hard to explain. But I do this all the time. So. Okay, I knew automatically seven hundred...wait. Yeah, I knew automatically seven hundred would get the answer because, uh, in my head I knew ten would go into seven thousand seven hundred times, because I did seven hundred times ten [inaudible].
STUDENT: Is a fraction the same as a decimal?
STUDENT: Oh yeah because...
STUDENT: Four-tenths equals zero point four.
STUDENT: We have different answers because you got...
STUDENT: I got them in fractions, you got them in decimals. All right.
STUDENT: So what did you write for that part?
STUDENT: What did you do to help yourself solve these problems mentally?
STUDENT: I talked and discussed all of the problems with my partner.
My students are starting to think that a pattern is happening. You really see linking it to the fractions and then the notation and then the patterns. It's no joke. This was the first time they really thought about how do decimals get written and how does that relate to the names. I wasn't sure when they would bring up the language. But they got really agitated about naming it and how it's written and how the names relate, because this looks different from that.
They couldn't see a connection between them, so I just told them: "Here's one way people sometimes think about the connection. So in the 10ths, some people think that that one zero is the ones place here. That's one way." Just a little mnemonical cheat sheet to remember how many places you go over. That afternoon was our first formal naming [of] the places, or decimals.
I was really impressed, because quite a few of them said, "If you think about the denominator, the tenths have a 10. If the ones had a denominator of one, it would just be that number, which is a whole. Therefore, it would never be on the parts side." I thought, "Cool! I think they've got some pretty good understanding about some fractions with this value.”