As the class wraps up the number talk, the learners mention many different ways of representing the rule: x3 – 3, times 3 minus 3, 3x – 3.
FRAN DICKINSON: All right. Ryan, it looks like you have a question.
STUDENT: Um, well I have, I have a question for Teo. If it was a x val, y value, wouldn’t you do the opposite of the x? So wouldn’t you divide, maybe?
STUDENT: I don’t know. Do you know, Griffin?
FRAN DICKINSON: Griffin, you want to respond to that?
STUDENT: yeah. Well, I did it backwards, and I said, plus 3 instead of minus 3, and then divided by 3. So 0 plus 3 is 3, divided by 3 is 1.
FRAN DICKINSON: So we’re doing a lot of talking about this rule. What is the rule? Can we write a rule here? That might help us to understand what Griffin was saying by working backwards. Maggie?
STUDENT: x, 3, minus 3. No. x 3 minus 3.
FRAN DICKINSON: x3-3. Can you just kind of walk us through, what this is here?
STUDENT: like, x3 is x times 3.
FRAN DICKINSON: Okay. So x3 means x times 3? And then take away 3. So, I see some silent disagreement around the room. Is there anyone who cares to make a comment about that? Sam. Adams.
STUDENT: Well, maybe you could change that, I agree with x3-3 equals y, but maybe you can add something to it? x3-3=y?
FRAN DICKINSON: Oh, okay. So you would just like to add “equals y.”
STUDENT: if some people could be unclear.
FRAN DICKINSON: My apologies for running out of space on my card here, but I’ve written “=y.” I think that’s a nice suggestion. I see some more silent disagreement in the room. Yes. Jonah?
STUDENT: I think it would make more sense if it said times 3 minus 3. Not x3 minus 3 because….
FRAN DICKINSON: So what do you propose, though?
STUDENT: Instead of, like saying x3, that could sound like, it’s like a number instead of 3. It should just be times 3 minus 3.
FRAN DICKINSON: On our chart. So you’re saying times 3, uh, minus 3.
FRAN DICKINSON: That’s how you want to define that rule?
STUDENT: Or put the dot there, or something…
FRAN DICKINSON: Morgan. So x • 3 – 3. All right.
STUDENT: I think that number’s mixed around. I think it’s 3x minus 3.
FRAN DICKINSON: Okay. So I can write it 3x-3. Why is this more right than this? Or is it more right than that? Do Maddie and Teo disagree? Turn to your partner and have that conversation.
STUDENT: I think this could be x groups of 3, or 3 groups of x.
STUDENT: it’s like 4 groups of 2, or 2 groups of 4. It’s the same thing, but …
STUDENT: It really matters how you look at it.
STUDENT: It matters in places, like if you have to get like, groups. Like if you have to get, like if you get 4 groups of 2, it’s different than getting 2 groups of 4.
STUDENT: Oh! I get it.
FRAN DICKINSON: Great. I heard some great conversations happening around the room. Would anyone care to repeat out what they heard at their tables? Caitlin.
STUDENT: well, we thought that like the x should go after the 3, because that was in CPM.
FRAN DICKINSON: Oh. So the book told you to do it.
FRAN DICKINSON: So it must be right.
STUDENT: The book made me do it.
Silent Signals Explained: In our class we silently agree by waving our arms parallel to each other – like a referee declaring a field goal good. To show silent disagreement, the kids wave their arms back and forth to show negation – like when a referee declares a field goal attempt no good!
We also use thumbs to show how we are thinking about concepts: thumbs up - I get it, thumbs down – I don’t get it, thumbs parallel to the ground- I’m getting there.