Jacob Disston introduces the day's task to his students.
JACOB DISSTON: Oliver, you had another whole category. Who are you standing with? All right, why don't you guys come up and pull your group out. (fade out). This is Vanessa, Derek, and Oliver. So tell us what you think. Why are you guys saying you guys are similar? You want to tell us?
STUDENT: Yeah, it's because all the problems have variables.
JACOB DISSTON: So they all have variables.
JACOB DISSTON: OK.
JACOB DISSTON: Those are all the distributive property?
STUDENT: They go like that (gesturing). There's one and there's two and they go like that (gesturing).
STUDENTS: Commutative? Associative?
STUDENT: Commutative, yeah.
JACOB DISSTON: They're all the commutative property?
JACOB DISSTON: Sarah?
STUDENT: Might be associative property?
JACOB DISSTON: They're all the associative property?
STUDENT: They all have a property. The first one has commutative, the second one has distributive and the third has associative.
JACOB DISSTON: Can you tell me what you mean by "this one has the commutative property in it"? What do you mean?
STUDENT: Because x plus y equals y plus x, that would be commutative because the variables are switched.
JACOB DISSTON: OK. So you're saying this represents something you know, which you're calling the commutative property. You had a comment about this.
STUDENT: Yeah. I'm the one who had the f times one equals f, and I don't think it's here. I think it should go back here.
JACOB DISSTON: OK, and why don't you think it's here?
STUDENT: Because it don't look nothing like none of them.
JACOB DISSTON: And did you find anybody that you think is similar?
JACOB DISSTON: OK, so you stand alone.
STUDENT: Yeah, I want that to go back over there.
JACOB DISSTON: All right, well, if you've got no one to stand with, I'm going to put it over here. OK, you have another group. The two of you? Come pull them out. Four of you? Come pull them out and put them right over here.... Is there one more?
STUDENT: There's one more, but my (inaudible) put out.
JACOB DISSTON: OK. All right. So tell us why you guys are a group.
STUDENT: I think they're all the same because all the numbers seem to have a variable after it.
JACOB DISSTON: OK. In every one, the number has a variable after it. We got a number, a variable, a number, a variable...numbers and variables paired together. OK. Yeah?
JACOB DISSTON: They all have addition. OK. Is that all they all have?
STUDENT: No. And subtraction. And an equals sign.
JACOB DISSTON: Do they all have an equals sign?
JACOB DISSTON: So if we were to say, among these, they are similar because they have pairs - that is, number and variable - Ron? We could say these are all similar because they're all paired. Number, variable, number, variable. They're all similar because they all involve addition and subtraction. Then you said they're all similar because they have equal signs. So if we wanted to pay attention to the equal sign, would these all be grouped?
STUDENTS: No. Yes. No. You don't...(inaudible)
JACOB DISSTON: I don't?
STUDENT: Never mind.
JACOB DISSTON: So if we wanted to pay attention to equals sign, this one is not like the others, right? Why not? Why is this different from these? I want you to think about this before you answer. Hold on. I know people are going to say this one has an equals sign and those don't. But why is that important for us to know? Why is that an important thing for seventh graders to know about, that sometimes - we're calling these symbols strings - sometimes symbol strings have equals signs and sometimes they don't. Do you have an idea, Jasmine?
JACOB DISSTON: OK, so ease up. Do you have an idea about why?
STUDENT: If it has an equals sign, there's an equation or something. And if there's not an equals sign, it's not a finished problem.
JACOB DISSTON: It's not a finished problem.
STUDENT: What is that called?
JACOB DISSTON: So it's an expression.
JACOB DISSTON: Say it again?
STUDENT: That's an equation.
JACOB DISSTON: This one? Are these ones equations?
JACOB DISSTON: Why not, Laura?
STUDENT: Because they don't have equals signs?
JACOB DISSTON: Because they don't have equals signs. These don't have equals signs? Are there any others that don't have equals signs that we haven't grouped? Maya?
STUDENT: I think that you should take the one with the equals sign out and put the one that says 3b plus 4b plus - yeah. You should put that one in because all of them -
STUDENTS: Uh-uh, uh-uh. No.
STUDENT: -- because they all have numbers. But I guess some people are going to say no because the seventeen doesn't have a variable.
JACOB DISSTON: Yeah, so if they want to group themselves as things that have pairs - number and variable - and this one doesn't have a pair, it's got that seventeen that's standing alone -anybody remember what we call a number that doesn't have a variable next to it?
STUDENTS: Outlier? (inaudible)
JACOB DISSTON: It's called a constant. So if it has a constant and these don't, then maybe that's not alike. But there's a reason you put these together. So what's the important reason that you put them together?
STUDENT: Because they all have numbers, addition signs, and variables - wait, I take that back. I think it should be them three because you can put them together, like, let me show you. (gesturing) You could put 3x plus 2x which equals 5x, that would be 5x plus 2y, and with this...
STUDENT: Yeah. (turns away)
JACOB DISSTON: Now, wait, what is happening right now, what I want to point to, is that you guys are starting to see deeper things than we started seeing, and I really like that. So we're going to keep going with this discussion, but I want to stop at this point of, you said this belongs and now you're saying it doesn't. Noah, you said this does belong. How come?
STUDENT: Because it's an expression.
JACOB DISSTON: Does everybody know the word "expression"?
JACOB DISSTON: So we're good with that word?
JACOB DISSTON: You can look up here and you can see which ones are expressions?
STUDENTS: Yes. No.
JACOB DISSTON: Somebody else called it - you called this one that has the equals sign something. What did you call it?
STUDENT: An equation?
JACOB DISSTON: Yeah, you called it an equation. And then we started with these guys. What do we call these guys? They don't have equals signs per se either.
STUDENTS: They're called, um, um. Inequalities.
JACOB DISSTON: Inequalities. OK. So I want to stop our discussion for just a sec.
COMMENTARY BY COACH LINDA FISHER: I like thinking about all the structures and moves designed into the lesson to promote student engagement and interaction. I like the movement and change of pace. One of the routines the teachers have been working on is the “huddle”, gathering students at the front of the class to have a discussion. How did the huddle contribute to the lesson?
My favorite part is watching the change in what students are noticing and the level of detail being discussed as the lesson progresses. In clip 2 Jake talks to students about it being okay to change their minds. So often students think of math as right or wrong. I like that they are encouraged to keep their minds open and fluid. I think it gives students more reason to be active listeners. In all the discussion there is this idea that the symbol strings could be grouped this way, but if you think some other attribute is important then you can group it another way. It forces students to really reflect on what is more important. Students need to evaluate attributes against each other: which similarities are more important and which differences can be ignored or not ignored? Any answer is acceptable if you have a reason.