Erika Isomura engages her 4th and 5th graders in a mental math talk, inviting multiple iterations of dividing by 10, e.g., 8,000/10, 800/10, 80/10, 8/10, (8/10)/10. She asks students to justify and revise their answers if needed. She reviews norms for whole-class participation, and asks students to evaluate the reasonableness of their results. Erika then presents multiple iterations of dividing 8 by multiples of 10 (e.g., 8/10, 8/100, 8/1,000, 8/10,000), noting similarities and patterns.
ERIKA ISOMURA: So we have another math talk and this is another number string. We've been doing our number strings for a while. You will probably recognize a lot of it, but let's still show good manners. Quiet thumbs if you have something to say instead of shouting out. Right? Eight thousand divided by ten. Ruchita?
STUDENT: Eight hundred.
ERIKA ISOMURA: Okay, we'll just leave it. We may or may not agree, but we'll leave it for now. Sofia?
ERIKA ISOMURA: Adam?
ERIKA ISOMURA: Eight divided by ten. Jesus?
ERIKA ISOMURA: Ten-eighths? Okay. You want to disagree with your own answer? Why?
ERIKA ISOMURA: Now you're saying eight-tenths. Which one and why?
STUDENT: Eight-tenths because you're cutting it (inaudible) pieces.
ERIKA ISOMURA: You're cutting it into the ten pieces?
ERIKA ISOMURA: So why do we want the ten in the denominator...denominator?
STUDENT: Because that's...
ERIKA ISOMURA: Rosa Linda would like to help you.
STUDENT: (Inaudible). Those are the pieces.
ERIKA ISOMURA: These are the pieces...where?
STUDENT: Um, in the whole.
ERIKA ISOMURA: In the whole. So using ten pieces from the whole thing. Okay. So Jesus revised his answer. That's fine, we can always revise. All right, eight-tenths divided by ten. Antonio?
ERIKA ISOMURA: Eight-hundredths. Again, we may or may not agree but we're going to leave it for now. And if I do eight-hundredths divided by ten, which I'm sure you all saw coming. Diego?
STUDENT: Eight one-thousandths.
ERIKA ISOMURA: Eight one-thousandths. Okay. And eight one-thousandths divided by ten. So, I'm expecting all of you to participate. Shyla, any thoughts? Lizzie?
STUDENT: Eight ten-thousandths.
ERIKA ISOMURA: Eight ten-thousandths. All right, take a moment to look at our string and decide if you are happy with all of our results, or if there's anything that you'd like to have us talk a little more about. Okay. We'll leave that for the moment. We're going to do a second number string. There may be some similarities, there may be some differences. We'll start with this one which we already know from Jesus and Rosa Linda. What about this? Alex?
STUDENT: Eight one-hundredths.
ERIKA ISOMURA: Eight one-hundredths. Up here! We've got eight one-hundredths there and eight one-hundredths there. Let's take a moment. All right, what about eight divided by one thousand? Rosa Linda?
STUDENT: Eight one-thousandths.
ERIKA ISOMURA: Eight one-thousandths. Also weird. This is eight one-thousandths and that one is too. Very odd.
ERIKA ISOMURA: Ooh, something happening in your brain, Adam?
ERIKA ISOMURA: Yay! And eight divided by ten thousand. Jerry?
STUDENT: Eight hundred-thousandths.
ERIKA ISOMURA: Eight hundred-thousandths.
STUDENT: Wait, no. No, eight ten-thousandths.
ERIKA ISOMURA: Which one do you think?
STUDENT: Eight ten-thousandths. Eight over ten thousand.
ERIKA ISOMURA: Why? Why did you change your mind?
STUDENT: Uh, it's just...I looked at the pattern and it showed, like, the same...the denominator is the same as the...
ERIKA ISOMURA: Oh, so based on the pattern we're seeing, you're feeling like it's probably ten thousand? Okay, we like patterns. Patterns are a great way to help us see something we're not sure about. We can often use patterns to help us confirm what we think we know. Now, for the hard part. Well, might not actually be harder. Let's see. Ten thousand, one thousand, one hundred, ten. I think I might do eight divided by one. Dylan?
ERIKA ISOMURA: Okay. Ten thousand, one thousand, one hundred, ten, one, one-tenth. How about eight divided by one-tenth? Quickly, whisper to a partner what you think it is. All right, any thoughts? Lizzie?
ERIKA ISOMURA: Eighty. Hm, interesting! Did anybody say eighty to their partners? Okay. Eight divided by...let's see. Ten thousand, one thousand, one hundred, ten, one, one-tenth, one-hundredth. Hm. Quickly, whisper to your partner what you think. Roberto, what do you think it was?
STUDENT: Eight hundred.
ERIKA ISOMURA: Is that what you whisper to your partner, or do you think it is something else? Something else? What did you guys think it was?
STUDENT: Um, we thought it was eight hundredths.
ERIKA ISOMURA: Eight hundredths. So this?
STUDENT: No, eight eight-hundredths. Eight and then eight hundred.
ERIKA ISOMURA: Oh! We'll put that a question. We'll have to investigate that further. Okay. So I do see some patterns. Now, here's my next question. Yesterday the fifth graders and the fourth graders began to do some work with calculators using a new type of number called decimals. And some of you have been very irritated with our number talks because you felt like there might be another way to say eight-tenths, and there might be another way to say eight-hundredths, or eight-thousandths, or so on. So does anybody want to talk about what they think maybe we could also write for eight-tenths? Another way we might want to write that down. Dylan?
STUDENT: Zero point eight.
ERIKA ISOMURA: Zero point eight. So not eight wholes but eight and that's the tenths place. Hm. How about this? What would that look like? How do we show that we want to volunteer our answer? Alex?
STUDENT: Zero point zero eight.
ERIKA ISOMURA: Zero point zero eight. So I still have eight but I'm not in the tenths place, I'm in the hundredths place. Oh, interesting! Is there another way to write this? Antonio?
ERIKA ISOMURA: Zero...
ERIKA ISOMURA: point...
STUDENT: zero zero eight.
ERIKA ISOMURA: Hm, is there a pattern happening here? We saw a pattern with our fractions, is there any kind of pattern we're seeing with our decimal representation? Turn to your partner, what do you see? All right, coming back together. Eight ten-thousandths. Now that we're starting to write decimal representation, what would that look like? You can use patterns or you can use the decimal work we started yesterday. [Inaudible], what do you think?
STUDENT: Zero point zero zero zero eight.
ERIKA ISOMURA: Hm, interesting! So let's see. Not tenths, not hundredths, not thousandths, but eight in the ten thousandths place. Nice! Very cool! All right. So, obviously we can write these since we noticed that they're the matches, but for now we are going to be moving on to our work. So, fourth graders quietly slip out, fifth graders stay.
The number talks were to give 4th graders a touch on moving towards decimals but not necessarily expect them to produce anything, just to get it in their heads. The 5th graders then turn around and use it. At first, I thought, “Decimals are a nonissue if they know fractions really well and if they can understand the patterns in place value." My idea was to really open up this series of lessons using a bunch of number talks. The number strings are a series of equations that are all related to each other. Typically, when I do them they tend to be related to place value. The ones I'm most comfortable with are times 10, 100, 1,000. I've done them with fractions where you do something like 1/2, 1/4, 1/8, but I'm personally most comfortable when I'm using the base 10 number system in the talk because I think that gives them a lot of understanding of why the numbers are turning out the way they are and how the number system that we work with really is helpful. This particular number string strategy helped them see how the numbers are growing in those relationships. The day before this they had started doing some decimal work on their calculators. We hadn't really named them, so I still heard them calling them 0 point whatever.
How do we think about the ones place, and how is it different from the tens place, or the hundreds, and so on? They talked about each time you move over, it's ten times bigger. Then, we talked about here's this decimal. What's this place called? They were able to tell me, because we've played with it. They named it as the tenths and the hundredths. Then, I asked what would be next. Thousandths. They were able to name it, and they wanted to go keep going. Then, we wrote it out. They were more familiar still with the fraction notation, so we went with that. The tenth, the hundredth. How do we go this direction? They said, "If you're dividing by ten each time you go, you're going ten parts smaller."