Cabana meets with a group of students and gives them evaluative feedback about the strength and robustness of their approaches to solving the problems. He challenges them to write down the necessary steps, excluding those they can accomplish through mental math.
00:01 Good, ok. So, Maria, you organized your work so carefully that I don't need to ask you any questions, because I can see what you're thinking.
00:08 The only thing I would say is that this step right here, is not wrong, but it's not necessary. Because you did it in your… and same thing with this.
00:15 Right? It's not wrong, but did you do … all of this, can you do it in your head?
00:20 Mm hm.
00:21 Can you do it using the mental math strategies from before? Because if you can, then you should.
00:26 There might be harder ones, like …. This one, you could not do in your head. I assume, right? Right.
00:33 So that one, you can do algebraically. But if you can do it in your head, do it in your head.
00:40 Another A+.
00:45 ¿Donde esta la C?
Where is the c?
00:56 Good. Good. Yeah. Perfect. Good, Maira. So, same thing.
01:03 Me gusta lo que has hecho.
I like what you've done.
01:07 Cuando estas en este paso, si puedes encontrar las respuestas mentalmente, no hace falta…
When you're on this step, if you can find the answers mentally, you don't need to...
01:14 Pues si, ya no…no mas para…
But yes, not yet, just for...
01:15 Lo puedes hacer si quieres, pero no hace falta. ¿Ok?
You can do it if you want, but it's not required. Ok?
01:19 So… D.
01:23 9, 12, good. Lo factorizaron. Todo esto se llama el mismo. Ok. Ok. La E?
9, 12, good. You factored it. All of this is called the same. Ok, ok. E?
01:38 Give me one second to finish with them. All right? Yeah, you guys are doing too much work.
01:43 Your goal is to be able to find the vertex, da me un segundo, por favor. To find the vertex of this parabola, without a big long t-table.
01:52 All right? So if you made a big long t-table, that's too much work.
01:55 You told us to show work, though. To show work.
01:58 But I didn't say, show too much work. Like you could show..
02:03 Show ALL your work.
02:04 But, but that doesn't mean you have to… like, if I ask you to graph an equation that looks like y = mx + b, you're not going to use a t-table, right?
02:13 You use rise and run and the y-intercept. Right? Ok. Same thing here, you're shooting for shortcuts.
02:19 Tenemos que…
We have to...
02:20 So let's, so let's talk. So, what do you guys know about the vertex in this situation? ¿Qué saben sobre el vertex?
What do you know about the vertex?
02:27 Que es negativo..
That it's negative.
02:28 It's negative.
02:29 What's negative?
02:30 The point on the vertex.
02:31 Which part of the point?
02:33 The… x and y.
02:35 The y.
02:36 The y coordinate is negative, right? Ok. Maria, what else did you say you knew about the x?
02:43 Negativo, positivo, que? ¿verdad?
Negative, positive, what? Right?
02:45 Entonces es…
02:48 Ok. Maria, what do you know about the x. What did you say about the vertex?
02:53 That it's the point in the middle.
02:54 Do you guys agree wth that?
02:56 Melina, are you caught up? Yeah?
02:59 So, Maria dijo que el vertex es el punto en el medio, all right? En el medio de que?
So, Maria said that the vertex is the point in the middle, all right? In the middle of what?
03:03 De la parabola.
Of the parabola.
03:04 ¿Y de que mas?
And what else?
03:05 Donde se cruza el… de..
Where it crosses the.... of...
03:07 Para arriba..
03:08 Not… almost.
03:09 Donde es casi…
Where it's almost...
03:11 En el medio de que?
In the middle of what?
03:12 De los x-intercepts.
Of the x-intercepts.
03:13 ¡En el medio de los x-intercepts! Exacto. El punto esta justo aqui en el medio de los x-intercepts. Good, Eduardo.
In the middle of the x-intercepts! Exactly. The point is right there in the middle of the x-intercepts. Good. Eduardo.
03:18 So… lo que tienen que encontrar es el valor de x, que esta en medio de los x-intercepts. No hace falta usar una tabla completa.
So, what you have to find is the value of x, that is in the middle of the x-intercepts. You don't need to use a complete table.
03:26 ¿Por que 2?
03:28 Porque esta en el dos y…
Because it's on the 2 and...
03:30 Jesus penso que era uno, ¿por que es 2?
Jesus thought that it was 1, why is it 2?
03:35 Porque esta en medio del 1 y … como a traves de los…
Because it's in the middle of 1 and.... like between the…..
03:41 Mm hm…
03:42 It's the number in between.
03:43 So right here, there's three numbers in between, and right here, there's also three numbers in between.
03:49 Ok, so. Si ahora, abase de eso, saben que … la coordinada de x es 2. ¿Está comiendo algo, Melina, o que?
Ok, so. If now, based on this, you know that... the coordinate of x is 2.... Are you eating something, Melina, or what?
03:57 Ok. Ahora, ¿qué van a hacer?
Ok, Now, what are you going to do?
03:58 Buscar el y-intercept.
Look for the y-intercept.
03:59 Mm hm! And how are you going to find that.
04:00 Por valor en la ecuacion.
By the value in the equation.
04:01 Using the rule, that's right. Ok? So. So, on all of these, if you can find the x-intercepts, you don't have to make a long t-table.
04:15 You can just follow this idea. It can't be 9.
04:19 2 times 2 is 4..
04:21 Negative 9.
04:22 Puede ser 9?
Can it be 9?
04:23 ¿Por que no?
04:25 Negativo 9.
04:26 Ok. Entendido lo que están haciendo?
Ok. Understood, what you're doing?
04:30 Si, no hace falta hacer una tabla grande. Ok. Go back to your teams.
Yes, you don't have to make a big table. Ok. Go back to your teams.
04:36 No hace falta en hacer una tabla grande, ¿por qué no? …
It's not required to make a big table, why not?
04:39 Porque es mucho trabajo.
Because it's a lot of work.
04:40 Aja, ¿y? En vez de hacer una tabla grande, ¿que van a hacer?
A ha, and? Instead of making a big table, what are you going to do?
04:43 Ok. ¿En cual están?
Ok, which one are we on?
04:44 En esta.
04:45 ¡No se!
I don't know!
04:47 No se? El … no tiene la respuesta, ¿verdad? Quiero el vertex. ¿Qué son las coordinadas?
I don't know? He... doesn't have the answer, right? I want the vertex. What are the coordinates?
04:53 Y no es esa.
And it's not that.
04:55 ¿No es del vertex?
It's not the vertex?
04:57 No, es este, al medio del -2 y -2.
No, this, in the middle of -2 and -2.
04:59 ¿Cómo sabes?
How do you know?
05:00 Porque, como aquí es este numero repetido, los dos.
Because, here is the number repeated, the two.
05:05 Did you hear that, Lana?
05:07 Ok, asi que tiene que encontrar las coordinadas exactas.
Ok, so you need to find the exact coordinates.
05:10 Oh, aquí, aquí aquí aquí.
Oh, here, here here here.
05:13 Aquí son los 3.5… no.
Here they're 3.5... no.
05:17 Si, 3.5
05:19 No, -3.5.
05:21 -3.5? Es entre -2 y -3?
-3.5? Is between -2 and -3?
05:28 Debe ser 2.5. No se.
It should be 2.5. I don't know.
05:31 ¿Qué numero esta entre -2 y -3?
What number is between -2 and -3?
05:38 ¿Negativo, verdad? Porque no?
Negative, right? Why not?
05:41 ¿Por qué no?
05:43 Nos equivoquemos con el 3.5.
We made a mistake with the 3.5.
05:46 Ok. Vamos a ayudarnos con el 2 de negativo. Entre 2 y 3…
Ok, we're going to help ourselves out with the negative 2. Between 2 and 3...
05:54 Es entre 2 y 3?
Is between 2 and 3?
05:58 Es -2.5.
06:00 Ponganse de acuerdo y…
Okay, come to an agreement and...
06:03 Positivo, negativo, o…
Positive, negative, or...
06:05 Y ¿por qué dijeron positivo, entonces?
So why did you say positive, then?
06:08 No, negativo.
06:09 Ok. Asi que es… apúntenla en la tabla, muy bien, Maira, háganlo asi. Ya tiene que encontrar la coordinada y.
Ok, so then you need to mark it on the table, very good, Maira, do it like so. Then you have to find the y coordinate.
06:22 No. La tienen que encontrar, esta coordinada.
No. You have to find that coordinate.
06:25 A, 1.5.
06:25 Ahora si.
Now that's right.
06:27 Where are you guys?
06:28 No sabemos que vamos a hacer.
We don't know what we're going to do.
06:30 ¿No saben que?
You don't know what?
06:31 No encontramos.
We didn't find it.
06:33 Revisamos, que nos ayude.
We looked it over, help us.
06:35 ¿Qué pasa en la otra? No. No quiero ver una tabla completa. ¿Qué dije?
What happened with the other one? No. I don't want to see a complete table. What did I say?
06:40 No hace falta hacer todo esto.
That we don't need to do all this.
06:42 Well, no fue la intención.
Well, it wasn't the intention.
06:44 … como es de aquí?
How is it over here?
06:47 this one is 12.5.
06:48 ¿Por qué no nos avisaste?
Why didn't you warn us?
06:52 Bueno. Is this 12? Did you get that with a rule?
06:58 You guys stopped working together, I think.
07:01 No, we didn't! We're working together.
07:03 I know, but they didn't understand what to do here.
07:06 So, quien puede explicar como empezar este problema?
So, who can explain how to start this problem?
07:11 Buscando la x,
Looking for x.
07:12 Muy bien..
07:13 Ya cuando la llevo, bajarlo aquí, como, por la x, ¿no? Para encontrarlo en la y.
And when I've got it, I put it down here, like, for x, right? To find the y.
07:19 Ok. Lo que dijiste es, lo dijeste mal, pero parece que tu idea esta bien. Dijiste que tiene que encontrar el x-intercept,
Ok. What you said is... you said it incorrectly, but it seems that your idea is good. You said that you need to find the x-intercept.
07:27 Can I interrupt you for a second?
07:30 Ok. No van a encontrar el x-intercept. Los x-intercepts ya están aquí. Pero, a ¿qué te referias? ¿Qué tiene que encontrar?
Ok, you're not going to find the x-intercept. The x-intercepts are already here. But, what are you referring to? What do you have to find?
07:38 Quiero las coordinadas del vertex. ¿Cómo lo van a hacer? Here, move your hand. Para que alguien puede ver.
I want the coordinates of the vertex. How are you going to do it? Here. Move your hand, so that others can see.
07:44 Buscar al x. Find the x, right?
Look for x. Find the x, right?
07:45 What do you mean? X?
07:47 The number, here?
07:48 Oh, so the x coordinate of the vertex. Ok. Asi que este valor de x, ¿cómo lo van a encontrar?
Oh, so the x coordinate of the vertex. Ok. So that this value of x, how are you going to find it?
07:55 Contando lo de aca. X menos 1,
Counting the one from here. X minus 1.
07:59 No lo hice en escala.
I didn't do it in a scale.
08:00 ¿Por qué? 1, 2, 3.
08:06 El numero en medio de estos dos. Right? You said the number in the middle of these.
The number in the middle of these two. Right? You said the number is in the middle of these.
08:10 De aquí,
08:12 No lo hice en una escala.
I didn't do it in a scale.
08:13 No mas en la medida, no mas a la medida.
Just in the measurement, just in the measurement.
08:18 El numero en medio de estos dos números.
The number in the middle of these two numbers.
08:20 Which is…
08:24 Es el numero en medio de 1 a… 5.
It's the number in the middle of 1 to... 5.
08:29 Es como estamos decidiendo antes de encontrar el median. ¿Verdad? Encuentren el median.
It's like we're deciding before we find the median. Right? Find the median.
08:35 3 mas… 3 o 2. 1, 2. ¿Ya ves? Te dije.
3 plus, 3 or 2. 1, 2. You see? I told you.
08:40 Ok. ¿Estan de acuerdo? Alfonso? Ok. Y ahora que, ¿Eduardo?
Ok. Are you in agreement? Alfonso? Ok. And now what, Eduardo?
08:45 Mete al 2 al..equation.
Put the 2 in the ... equation.
08:48 Mueve la mano para Melina pueda..
Move your hand so that Melina can...
08:51 Ya! Ya!
08:52 ¿Y ahora?
08:54 Le metes el 2 al… the x right here. Substitute the 2 to the x.
You put the 2 on... the x right here. Substitute the 2 for the x.
09:01 Ya te digo.
I'm telling you.
09:03 And it gets -9.
09:04 2 por 2…
2 times 2...
09:07 22. Ok. Pero Eduardo ya les dijo, son -9.
Ok. But Eduardo already told you, they're -9.
09:13 Please finish what you're doing.
09:15 Estamos bien. Negativo 9.
We're fine. Negative 9.
09:17 Ok. Ok, eso… can I interrupt you, Eduardo?
09:19 Este, por negativo, sale… a positivo.
This, times a negative, comes out... a positive.
09:24 No puede ser numero positivo. What are you talking about? Ok. Pay attention.
It can't be a positive number. What are you talking about? Ok. Pay attention.
09:30 Ok. Eso quiere decir que no hace falta encontrar hacer una tabla grande.
Ok. This means that you don't need to find... to make a large table.
09:33 Entonces, esta bien, ya?
So, that's good, right?
09:36 No, de aquí, del papel.
No, from here, from the paper.
09:37 Que dice. ¿Y? Can you pay attention? No hace falta hacer una tabla grande. Ok? Asi que para la B, no hace falta encontrar todos los valores intermediados.
What does it say. And? Can you pay attention? You don't need a large table. Ok? So for B, you don't need to find all the intermediate values.
09:49 Pues si, para sacarlos. Para ver.
But yes, to find those. To see.
09:51 ¡No hace falta!
It's not required!
09:53 Porque aquí no encontraron el valor de 0, de 1, de 3, de 4. Solo encontraron el valor de x al vertex, y luego la coordinada y.
Because here you didn't find the value of 0, of 1, of 3, of 4. You only found the value of x at the vertex, and then the y coordinate.
10:04 Y el dos, ¿ese? La coordinada de x, lo encontró Eduardo…abase de los x-intercepts.
And the two, that one? The x coordinate, that Eduardo found, based on the x-intercepts.
10:13 Es cierto? Ok, asi que aquí, ¿adonde están los x-intercepts?
Is that right? Ok, so from here, where are the x-intercepts?
10:18 No. Estos no son las coordinadas de x, los x-intercepts. Los circulan, por favor.
No. These aren't the x-coordinates, the x-intercepts. Circle them, please.
10:24 Aja. Asi que, ¿cuál es el numero que esta en medio de 3 y -1?
A ha. So that, what is the number in the middle of 3 and negative 1?
10:33 No, es cierto… 2.
No, it's right, 2.
10:35 Hm mm.
10:36 De 2.5?
10:42 Es 1!
10:45 1, le dije, Cabana!
1, I told you, Cabana!
10:47 Asi que no hace falta escribir todos estos valores, el 1 es suficiente.
So you don't need to write all these values, 1 is sufficient.
10:51 Lo meto en la ecuación, 1 menos 2?
Should I put it in the equation, 1 minus 2?
10:56 Mm hm.
Maria, the first student with whom I interact, is definitely much more bilingual than most kids. I think she was a ninth grader. Maira, who's in the red, with the white hair band, is older, I think she was a junior, and Alejandro was also older, so therefore more recent immigrants.
I remember that Maria was a quietly confident learner who doesn't impose herself on the group, and doesn't often ask questions of the group, but is willing to share her thinking (complete or incomplete) with everybody, so she creates a calm atmosphere within her team.
It felt like a really difficult team to manage, in retrospect. I felt like I needed to make sure that Maria understands, because she's got to be the one facilitating the conversation, no matter what, which I hesitate to do. I don't really want to think about the designated "teacher" in the group. But in this particular group, given the dynamics of the other three kids, I don't think they're going to have an easy conversation as a foursome. So Maria can facilitate it, but I'm not sure that as much learning would happen as needs to happen.
I think that's part of my mindset in circulating to teams, trying to attend to which student is doing the facilitating and what does that sound like. Are they just saying like "first you do this and then you do that?" Or are they giving reasons and are they putting questions in the middle or just making statements?
Then as I talk to both groups simultaneously, it's getting close to the end of the hour and they're both stuck in the same kind of place. I wanted to try to get them "unstuck" together, so that they get far enough that we can have a productive conversation the next day.
As teachers move toward the end of an instructional period, most of the time we're used to doing that by pulling the whole class together and doing a quick lecture or something that might feel like Socratic dialogue, but I try to not do that. Instead I'm trying to keep things focused within the group conversation and in this case my compromise is to try to pull two groups together.
I realized from the conversations with the other two groups, that kids were stuck on finding the coordinates of the vertex in a way that I maybe couldn't necessarily anticipate. I wanted to push them towards that, because otherwise they won't have a complete graph of the parabola. What I want the next day is the whole picture, everything you need to do to have an "A+" sketch of a parabola.
That includes not just procedural fluency but being able to give reasons along the way, and insight into strategic choices about which form of the equation, and how do you find the middle number. I had in my head the big picture of what we'll need to discuss the next day.
There's always this balance between pushing kids faster than they are going themselves, but I feel like in this clip I'm short-cutting paths a little bit of what I'm thinking their questions might be based on the other two groups. That way, they get far enough today that the next day's work won't feel like someone is presenting information that they haven't thought about themselves already.