Erika Isomura works with her 5th graders to develop their understanding of how decimals work. She begins her lesson having her students work in pairs to do a “calculate and check” mental math decimal task, using calculators to check, then evaluating their work with symbols and writing (e.g. “We disagree because,” “I have a question because”).
She reminds her students to do the mental math, then discuss what they got “in their brains,” and then check with their calculators. As the pairs work, Erika circulates around the classroom and probes their thinking (e.g. “Did you agree? Did that make sense?”).
ERIKA ISOMURA: All right, so today you're going to be doing some more pattern stuff. We started yesterday, and we're going to do this similar to what we did yesterday.
So we have some problems like two thousand divided by ten. In your head and then calculator check. Fifty divided by ten. In your head, calculator check. But instead of writing down your own thoughts just by yourself, I'd like you to talk to your partners at your tables -- and I'll be assigning your tables -- about what you noticed.
So star is going to be: So easy, there's nothing to say. "We both agree. What is there to say? We're both superstars with this."
Exclamation: What? What? "Oh my gosh, this is crazy! How did that happen?" If you've got an answer that shocked and surprised you.
Question mark: "I don't get it. I don't know what happened. What is this on my calculator? Where did that come from?" Okay?
And then heart. That's Ms. B's invention. Heart means we disagree with each other but we're really excited because that means we're going to come back together and talk about it later, and we're all going to learn something new, and we love it when we learn something new. So when we disagree with our partners at our tables, we're going to say, "That's a good thing because that gives us a chance to grow and learn more."
So you'll be putting your symbols and then a little bit of writing. "I think this is easy because... I think exclamation because... I think question because... I think heart because..." Okay? All right. Okay, you should have your calculators from earlier but if you didn't keep them at your desks, I'll come around with some calculators.
STUDENT: Two thousand and twenty.
STUDENT: Twenty divided by ten. Two.
STUDENT: It's two? Are you sure?
STUDENT: How? So if you do two times ten, it will make twenty. But twenty divided by ten is...
STUDENT: Which means twenty thousand, probably.
STUDENT: No, I'm going to do it. I think you did it wrong. Two thousand divided by ten...
STUDENT: It's two hundred.
STUDENT: It says two hundred. Never mind, I was wrong.
STUDENT: So...you can't erase it.
STUDENT: Oh yeah, we can't.
STUDENT: You've got to write it down...like, you've got to write down the right answer.
STUDENT: Two hundred. All right, fifty divided by ten.
STUDENT: I was surprised [inaudible].
STUDENT: Let's write exclamation.
STUDENT: You're going to do an exclamation point. Fifty divided by ten.
STUDENT: I'm pretty sure about this one.
STUDENT: Easy. Seven thousand divided by ten. Seven hundred.
ERIKA ISOMURA: All right, girls and boys, we're going to take a pause because we've noticed that some of you are calculator-ing immediately. So calculators aside.
You need to do all the mental work first on side one. Once the whole page is done, discuss what you've got in your brains first and then check together. Okay? So in your brains first. Share with your brains and then check.
STUDENT: Zero point zero four. No, zero point four.
STUDENT: I think it's four-tenths.
STUDENT: If we do that, like, into a decimal...
STUDENT: Oh yeah! So we're going to put it [inaudible]. What was the number again? Zero point...
STUDENT: I thought it was zero point four.
STUDENT: Okay. So then let's just put a question mark. We're still not sure.
STUDENT: Three divided by one hundred. I think it's zero point zero three.
STUDENT: Oh yeah, that's equal to. Okay, one divided by a thousand.
STUDENT: Well, I think it's zero point zero zero... Ones, tenths, hundredths, thousandths. Nine divided by zero point one...
ERIKA ISOMURA: Okay. Adam, do you agree with that?
ERIKA ISOMURA: Did you even hear what he said? So my question was, you said zero point four and Dylan said yeah, but I want you to make sure that you explain to each other why you think that is.
STUDENT: Okay, why do you think I'm correct?
STUDENT: Because you transfer the four wholes into [inaudible], that'll be forty pieces and forty divided by ten equals four.
STUDENT: Or you could just write four-tenths, right?
STUDENT: Could you just write four-tenths?
STUDENT: Yeah but we're doing decimals, so.
ERIKA ISOMURA: Did you agree?
ERIKA ISOMURA: Did that make sense?
ERIKA ISOMURA: We're not doing that until both of your partners have done everything mentally and you've talked about the mental answers, then you guys can check together. Okay?
STUDENT: Because one, you counted to a thousand so it'll be like, one, two, three...
STUDENT: But why is it one whole though?
STUDENT: I put one thousand.
STUDENT: Yeah but how is one divided by one thousand equals one thousand?
STUDENT: Let me check.
STUDENT: This one whole does not have one thousand.
STUDENT: One divided by one thousand. We could do it the old way. It equals the same thing.
STUDENT: One divided by one thousand... So one is supposed to be in here, one thousand is supposed to be...
STUDENT: My bad, my bad. You got me. You got me right there.
STUDENT: Wait, I think I did it, like, zero point...zero point eight divided by ten. All right, I think we're going to go forward, so I think it's going to be eighty. Yeah, I think it's going to be eighty. And zero point zero two divided by a hundred, I think it's going to be two hundred.
As my students initially shared their thinking, some of these answers were 100% in their heads. Some of these were just call-out answers.
We write it. If there's disagreement, we'll talk about it later. Some of these were, “we did it.” There comes a point later on where we calculator-check because we want to verify the pattern we think we're using is correct.
At the beginning, if they see the pattern or not, it's more me giving information on what's going on so that I can set up the next ones of “oh, they didn't get this. Let me poke a little harder and see where the ‘fall down‘ is” or “Oh yeah, they're getting it.” I really wanted to see, what if anything, they were internalizing.