After her students work individually on various problems, Mia Buljan invites them to “find the other people who have your problem and see if they agree with you.” In their conversations, the students explain their thinking clearly to each other, striving to reach consensus.

3rd grade math - interpreting multiplication & division*Mia Buljan, Glassbrook Elementary School, Hayward Unified School District, Hayward, California*

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**Lesson Part 2c**

MIA BULJAN: Raise your hand if you have a green paper. Thank you. Raise your hand if you have a, uh, this orange paper. Esbin, if I don't see your hand raised, I'm going to come over and ask why you don't have a paper, so please follow my directions. You have one? Oh no, you just got here. You have one? You have one? Okay. Now boys and girls, you have one that you think describes your problem. Now, you can find the other people who have your problem and see if they agree with you. If they agree, I want you to explain, Amari, why it works, to each other. And if you disagree, I want you to come to a consensus. I want you to talk to each other, like Esbin and Enmy did, until they agreed on what the problem said. So, they didn't start out the same, but they worked it out together. Okay? So when you get together, you're looking at your green problems or your orange ones?

STUDENT: Orange.

MIA BULJAN: Orange. Everybody, which one are you looking at?

STUDENTS: Orange.

MIA BULJAN: And you're deciding if you picked the same one, and then you have to prove that it's right. You have to prove that that one makes sense for your problem. Does everybody understand?

STUDENTS: Yes.

MIA BULJAN: Okay, find the people that are working on the same problem as you.

STUDENT: All the sixes together. And then, I have six and I put the sixes together to make a twelve, and the only leftovers I had to do is just three, so then twelve plus three equals fifteen.

STUDENT: I want to show you my one. I have fifteen cookies so I break them in threes. And there are five.

ERIKA ISOMURA: Can you read the problem again?

STUDENT: Sally baked fifteen cookies. She put them in bags of three. How many bags of cookies did she make?

STUDENT: Okay, so this one. So, so, so. So, this is counted by threes. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12...45 (Counting)

STUDENT: We all have the same.

STUDENT: No, that's forty-five.

STUDENT: What's your answer?

STUDENT: But the real, but Adil and Celine, the real answer is fifteen because look at, I have all my threes and I put them into six. And then I have my other threes in the bag to make it twelve. And then I have three more, and how much does that equal all together?

STUDENT: But you don't know, you don't, you don't know if you have the right answer. See, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.

STUDENT: Fifteen. The answer is fifteen.

STUDENT: But why do you put these on top of this?

STUDENT: Because that's the bags. About to...

STUDENT: But all of this is the bags. This is the bag of threes. So this is the bag of threes, threes, and threes, and threes. So I have, so they said three bags. So these are the three bags, but this also equals fifteen. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. But it has three bags, so these are the three bags. One, two, three. And I'm counting by threes. Three, four, five, six, seven, eight, nine. So you count one of the blocks three times.

STUDENT: Oh, so it's three, six, nine. Twelve, fifteen, eighteen.

STUDENT: No. One, so that's 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18.

STUDENT: And then you keep on going to forty-five, so...

STUDENT: Until all the blocks are done, and then you can do it. So this is counting by fives to fifteen. And then that's done, I'm done with this.

STUDENT: Oh.

STUDENT: So that's how you get it.

STUDENT: All right.

STUDENT: So, three...

STUDENT: Like, it's telling you like, three, six, nine, twelve, fifteen, eighteen. And then you keep on going.

STUDENT: Keep going?

STUDENT: Yeah.

STUDENT: So, this is how it's...

MIA BULJAN: I am going to ask your group to present to me your findings. So, first of all, is there any group that does not agree right now. Every group agrees?

STUDENT: Yep.

MIA BULJAN: It's okay, he was, he was in a different class. Okay, so if every group agrees, right now I want you to get together and decide who's going to represent you. Someone is going to represent you explaining it. Go.

MIA BULJAN: The third thing I’m noticing on the footage is how much they [the students] struggled with picking the words that describe what they did. Almost a third of them chose to go with blank cards for the word form, and attempted to come up with an explanation on their own.

This is a formative assessment lesson, and I love that I knew, after doing this exercise with them, that they needed more work and experiences with the language of groupings, with the words that go with making and counting equal groups. Almost two-thirds of [the students in] this class were designated as English language learners, and I’m thinking a lot about how we use language to talk, but we also use it to think. In this case, I think their limited vocabulary in this area may also be limiting their thinking around equal groups.