One goal here is to access prior knowledge and formatively assess student understanding and confidence levels about division.
BECCA SHERMAN: Think we can do some of that for division?
STUDENT: Yes! I love division!
BECCA SHERMAN: Awesome. Okay! What do we know about division. Can we … talk to your partner again first? Talk to your partner: what do you know about division. So, partners, just someone next to you.
STUDENT: Um, well, like, I have 4 friends at my house, and I only have 5 cookies.
STUDENT: It’s like multiplication, but like you, divide it? ‘Cause, if 9 divided by 3 would be 3,
STUDENT: ‘Cause 3 times 3 equals 9.
VISITOR: That’s awesome!
STUDENT: No… I’d have 3 friends in my house, and I only have 4 cookies.
STUDENT: Mmmmm, no.
STUDENT: That would be 3, remainder 1.
VISITOR: Will you guys talk about division?
STUDENT: Yeah. Um. We should…uh, I forgot.
STUDENT: Kind of like multiplication, but… divided by..
STUDENT: It’s like, but you can switch it.
STUDENT: Like 5 times 3 equals 15, and.. 15 divided by 3 equals 5.
STUDENT: Yeah, like that. And, um, we can like get,
BECCA SHERMAN: You guys have some ideas to share?
STUDENT: … like… make a more multiple…
BECCA SHERMAN: Great. I’m really excited about how you guys are talking to each other and listening to each other. Cause we’re gonna keep doing that. And now, if you’d like to show me a peaceful raised hand.. Let’s get some ideas out here for division! Maybe someone we haven’t heard from yet. Yeah! What do you think about division?
STUDENT: Division is like subtraction.
BECCA SHERMAN: How is it like subtracting? Some people…. People want to know! They’re not sure what you mean. Can you tell us more about that? You know, I don’t like this green. Can you guys see that green?
OTHER STUDENTS: No.
BECCA SHERMAN: Some no, some yes. Okay. Let’s do blue. Do you think, do you have something else you can add to your idea, or do you want to see if someone else can?
STUDENT: Someone else.
BECCA SHERMAN: Okay. Is there someone else who wants to add to this idea of how is division like subtracting? Go ahead and call on someone. Thomas.
BECCA SHERMAN: Did you have an idea about this?
BECCA SHERMAN: Okay. So, call on someone … only raise your hand right now …let’s just see if we have some ideas about how division is like subtracting. ‘Cause we’re gonna try some other bigger division problems maybe with pictures, and we might need to use this idea. So we want to see if we have that idea. We might not.
STUDENT: I think if you do a division problem you can make it into a subtraction problem. If you do 2 divided by 4, you know, the answer’s gonna be 2, so, you get the 2 when you divided from 4. You get the 2 from subtracting, uh, 3, oh no.
STUDENT: 2…. 4 minus 2 equals 2, so you put the 2 there, and then divided it by 4, and that equals 2.
BECCA SHERMAN: So I heard a couple different things. ‘Cause I heard this… I heard this…Um, and…I’m wondering, what do you guys think about some of his ideas so far?
BECCA SHERMAN: What do you think?
STUDENT: Well, um, can I just share what I think?
BECCA SHERMAN: Let’s hold on, I do want to hear some different ideas. This, I think, does anyone else have ideas, and about Ron’s idea…This, taking 2 things and dividing them into 4 parts. And we’ve talked a lot about this. Like if we could draw a picture? Can you think of a picture? ‘Cause you and I have done this a bunch. Try and draw a picture for that in your head. And I also want you to see if you can draw a picture for that in your head. And then get back to us on this idea. Or do you want some paper to try it out? Think on? Okay. There’s this. Just write on the back. See what you can come up with. Okay.
Several gaps in understanding arise. In retrospect, this is a key difference between teaching this lesson for a fourth grade class versus the fifth grade class it was planned for. Also, it serves as valuable assessment step before beginning a division bar model problem with students. I find a tentative confidence about 2÷4 versus 4÷2 or how to explain why 4÷2 is the same as 4-2. This makes me question the effect of teaching fact families (related multiplication and division equations) before solid investigations of division. To refocus on models, I choose to leave these ideas momentarily unjustified.