Jake Disston begins the period by challenging his students to work at their tables to group the cards they have in front of them, creating as many different groupings as they can, and devising language to describe each group.

7th & 8th Grade Math – Graphs, Equations, & Tables*Jacob Disston and Jesse Ragent, Willard Middle School, Berkeley Unified School District, Berkeley, California*

Next Up: Lesson - Part 2

Previous: Pre Lesson

00:00 Ok, people have already started doing this which I'm really excited to see. What we'd like you to do is actually move them around and put them in groups.

00:07 And they can be groups that you see commonality between. So what we what you to do is to organize the graphs into groups,

00:14 create as many different groupings as you can and then be able to describe, come up with language to describe each group. Al right?

00:22 And I'll stop you in about five minutes to talk.

00:41 Do we have to find the equation?

00:45 …this one goes with that one.

01:05 This is like both.

01:10 This is too.

01:11 Not necessarily. Not that one. Positive.

01:23 These are just the X, or just the Y's.

01:40 I feel kind of left out.

01:49 There's one that starts at negative four maybe you compare those.

02:04 If they're the same but going separate ways it's a parallelogram.

02:21 Yeah. Yeah. Because this is like positive four and this is negative four.

02:27 So we are trying to make as many groups as we can, right?

02:41 Very good things.

02:44 Before he goes off, I've just got to say I'm so impressed both with the quality of the students but the hair. (laughter)

02:51 Thank you.

02:54 I heard it was crazy hair day too.

02:58 You caught us on our best day.

02:59 You should see us tomorrow.

03:00 What's tomorrow?

03:08 The whole shebang? No holds barred?

03:16 So here's what we want to do. What we'd like to do is hear some of what you guys noticed, what you thought was…what stood out about these graphs.

03:30 What are the things that you noticed? And I have these slightly larger copies and we will sort of get a sense so the whole class can see what different things you paid attention to.

03:37 So do you want to start over here?

03:39 Ah yeah. So for Jeff, we made a group of linear equation graphs.

03:45 O.K.

03:46 And for Sarah we had positive graphs.

03:50 So hold on one sec. So let me…and how did you determine linear? Let's call it linear. Give me some of the ones that are in linear.

04:00 So we have a Y equals minus X.

04:04 Just give me the graph numbers.

04:06 G9. G5. G12.

04:20 and G11

04:25 So G9, G7, G5, G12

04:30 No not seven.

04:31 Oh. O.K. and what did you give this name?

04:33 Linear equation graphs.

04:34 Linear equation graphs. O.K. Great. And why did you call them that?

04:38 We did you call them that?

04:39 Well, we call these…basically equation graphs…well linear equation graphs because you find the points by an equation

04:47 so say for G9 you have the equation Y=-X squared. So you could use X and find the points on a graph.

04:59 What do you guys think? Good? So we are talking about equations. Do you guys…So he used this word linear, how many people know what linear means?

05:10 Well, sort of.

05:11 So what does it mean? We call things linear equations or linear graphs. Yeah.

05:16 I think it has to do with order and being in order.

05:20 Tell me more.

05:22 Just so if you have Y=X squared or negative X squared then you would plug in one for X, two for X, three for X…

05:32 O.K. so I'll go to what you said Y=X squared. You'd plug in values. Do you have a way of organizing that?

05:38 Yes, so you would put Y=1 squared. One squared equals one.

05:44 And I heard somebody say T-chart, is T-chart familiar? Yes.

05:47 So a table or a T-chart you have something like this. You'd say X=1 so Y=1.

05:55 X equals 2, Y equals 4 excetera. And so on and so on.

06:02 I see the equation and I see the relationship between X and Y but I want to go back to this because you called it linear. Yeah.

06:08 So linear basically means lines, right?

06:10 So does everybody…would that make sense? That linear would mean lines? Why?

06:16 Because it has the word line in it?

06:20 Yeah. Because it has got the word line in there. Line. And what's the opposite of linear?

06:26 Non-linear?

06:27 Say it. Loud.

06:28 Non-linear?

06:29 Which would mean non…so what's something that's not a line?

06:35 A parabola?

06:36 A parabola. A curve. So what do we have up here?

06:39 Parabolas.

06:40 So these are actually…

06:43 Parabolas.

06:44 Parabolas, which are non-linear. O.K. So we made a mistake. Here let's put this.

06:54 Alright so…and they can be formed by using equations and tables etcetera. Great.

07:00 Do we have an eraser?

07:05 Yep.

07:08 Can I hear from a different group about a different grouping? Ah so let's go to Sam. What did you guys pull out?

07:16 We had undefined in a pile.

07:21 And tell me what you mean by…or tell me which ones first of all so we can get them up…

07:24 G1.

07:28 G1. Only one, O.K. G1 and G1 looked like that and you called it what?

07:32 Undefined.

07:33 Undefined. So tell me about undefined.

07:36 Because it has no rise or runs so it doesn't move.

07:43 So no rise and runs. What do people think of that idea of undefined? No rise and run. Somehow different than… we actually don't have the linear ones up

07:53 But I assume that, Dylan, that if you pulled out non-linear than the other were linear.

07:57 And does this fit? Where does this fit? This G1.

08:00 It fits with the undefined…well it's a line so it would be in the linear group.

08:04 O.K. O.K. So…

08:09 So what's undefined? The whole line is undefined? Or… Yeah.

08:17 The slope is undefined.

08:18 Yeah. There's no slope.

08:19 Because you talked about rise and run or something.

08:20 Yeah.

08:24 So can I write slope undefined?

08:27 Yeah.

08:31 Great. And did anybody else find another one that fit that same category? Slope undefined? So the G1 was the only one with slope undefined?

08:38 G4

08:39 G4

08:43 That's zero.

08:43 I think everybody agrees with you.

08:46 Opposite day!

08:48 Opposite day! So is G4 undefined?

08:52 No.

08:53 Why?

08:58 So, it does have a run, so to speak, it goes across but I don't have to go up at all. And this has a rise but I don't go over,

09:07 So yeah that idea of why this is defined, and this isn't defined, is kind of an interesting thing to think about. Maybe we will come back to that.

09:16 So let's hold that out because that doesn't…we are going to say that doesn't fit there.

09:19 Any other categories that people came up with? Yes.

09:22 Linear equations.

09:23 Linear equations. And give me a couple that fit with…

09:29 G8, G3, G10, G6, G2, and G7. Yeah just the only ones that were left.

09:50 So, we've got some linear equations and again, linear means…line.

09:56 So, have we put them all up? The twelve? Now that doesn't mean that is the only way you could have sorted them, right?

10:03 Did people come up with other categories besides the three that we have? Slope undefined, non-linear, linear. Neil?

10:10 We came up with a positive linear, like, positive and negative graphs.

10:17 O.K. Do you have a linear? Cool. We are running out of magnets but that's O.K.

10:24 And then within here you had positive and negative as ways to, sort of, separate. O.K.? Any other sub-categories? Yeah.

10:39 The same Y intercept?

10:41 Same Y intercept. Tell me about that.

10:44 I have three groups here of Y-intercepts. In this one I have two on negative four and one on positive four.

10:53 O.K. Give me…

10:56 G2, G8 and G7.

10:57 O.K. so G2 has a Y intercept of positive four.

11:04 Positive four. And you saw other ones that had a Y intercept of positive four?

11:09 Negative four. Both the other ones, negative four.

11:11 Ah, got it. So you had positive four and negative four and…

11:14 Negative four.

11:16 O.K. Alright. Got it.

11:21 So we can divide those into different Y-intercepts. Alright?

11:25 And then this one, the Y-intercepts are all on zero.

11:29 Got it. So we have ones like this one and this one and this one, G6, which have Y-intercepts of zero.

11:35 Yeah. G6, G3 and G10.

11:36 O.K. Great. Any other things people noticed to help sort or to group these? We've got, sort of, Y-intercept, positive slope, negative slope, linear, non-linear, slope undefined.

11:49 Yeah.

11:50 Zero. A zero slope.

11:54 And a zero slope so among the linear there is positive slope, negative slope and zero slope.

11:59 Excellent. Which doesn't fit either positive or negative.

12:05 Yeah.

12:06 O.K. Alright. Good.

12:09 You guys are so impressive. I wish I had you for my class.

JESSE RAGENT: It is clear early on that there is lots of background knowledge that the students bring to the task. In the first few minutes I hear lots of relevant vocabulary "... just x, just y, slope, positive, negative, 0, undefined, intercept..." And yet, as the first student reports, it becomes clear that while there is some understanding of the information, there still remains plenty of misunderstanding and confusion.

his clip reminds me of the need to honor all responses in class discussion. Jake is able to elegantly bring the incorrect notion of Dylan's "linear" grouping of the quadratics into proper focus and clarity in a way that did not denigrate the student at all. Additionally, I am reminded of how I should provide opportunities like this that enable me as a teacher to see how students will bring prior knowledge to a novel task. Only then will I know how deep (or shaky) their learning is.