Four students continue their discussion, checking their thinking using graphing calculators.

9th - 12th Grade - Newcomer ELL Algebra - Graphing Quadratics*Carlos Cabana, San Leandro High School, San Leandro Unified School District, San Leandro, California*

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00:00 Dos, dos menos…. Dos, igual a 0.

2, 2 minus... 2, equals 0.

00:06 Y aca, menos…

And here, minus...

00:08 Cuantas…

How many...

00:11 Y aca, estos. Que aca es…3, 3 mas…0?

And here, these. That here is ...3 , 3 plus... 0?

00:17 Aja. Mas 3 igual a 0.

A ha. +3 equals 0.

00:22 And then, what's next?

00:24 So X, equals, 2 o -3. So x es igual a 2, o -3.

So x equals 2 or -3. So x equals 2 or -3.

00:33 x es igual a 2, ¿o?

x is equal to 2, or?

00:35 -3.

-3.

00:38 Hay dos x, ¿verdad?

There are two x's, right?

00:41 Entonces…que… 2?

So then.. what? 2?

00:45 -3.

-3

00:46 Son dos x.

They're two x.

00:48 Aja.

A ha.

00:54 Te ayudo, y me ayudas.

I'll help you, and you help me.

00:56 Eh, dios mio. ¿Ya lo tienes? Ahora, debes de poner esto. Que va aquí?

Oh my God. Do you have it already? Now, you should put this. What goes here?

01:06 0 es igual…o, lo arreglo?

0 equals... or, should I fix it?

01:10 Si.

Yes.

01:16 x mas 3…

X plus 3.

01:18 Y luego tenemos que hacer el numero 4.

And then we have to do number 4.

01:21 No tienes que dejar el numero que aqui es dos.

You don't have to leave the number that is 2.

01:26 2 menos 2.

2 minus 2.

01:30 Y en la otra, pa' que te de 0, -3.

And in the other, so that it gives you 0, -3.

01:42 Ahora…pon 2x, igual a dos. O, o, -3.

Now, put 2x, equal to 0. 0 or, -3.

01:58 ¿Lo entendiste?

Did you understand it?

02:03 Consigue 2a. Si.

Find 2a. Yes.

02:30 Entonces, va a ser… y igual a 0, aquí, verdad? 0 mas… I mean, mas 4, menos 12.

So then, it's going to be, y equals 0, here, right? Or more, I mean, plus 4, minus 12.

02:45 Y despues…lo tienes? Despues que ponemos y menos 12, ya nos podemos poner todo esto.

And afterwards, you have it? After we put y minus 12, then we can put all of this.

02:56 12.

12.

03:01 Ay, pero lo puse!

Oh, but I put it?

03:02 You don't need that.

You don't need that.

03:06 Ya la agarraste? Y es igual a 12, o -12.

Do you get it? y equals 12 or -12.

03:10 Y aca es x, y x^2, y es +2,

And here is x, y x^2, y is +2.

03:20 Y es igual a negativo 12, porque todo es igual a 0.

Y equals -12, because it all equals 0.

03:26 Ahora..

Now...

03:28 Tenemos que convertirla en 0.

We have to convert it to 0.

03:30 Pon esto, para substituir.

Put this, to substitute.

03:38 Y ahora negativo 12.

And now -12.

03:50 Ok. Ahora vamos a hacer una x.

Ok. Now we're going to make an x.

03:53 ¿Vuelvo a poner la misma cosa con y?

Do I put the same thing for y?

03:56 El del cuadro…

It's the square...

03:57 Vamos a poner el 0.

We're going to put 0.

03:58 En el cuadro, ¿qué?

In the square, what?

03:59 Tenemos que volver a poner toda la ecuacion, no mas que

We have to put the whole equation, not just that..

04:02 Aquí.

Here.

04:06 Y despues ya nos conocemos igual a …

And then we already know that it equals...

04:19 El numero que te de…que te de…12?

The number that it gives you, that it gives you... 12?

04:28 Cual le va…

Which goes there...

04:31 -2 por 6? -12.

-2 times 6? -12.

04:38 Pero tiene que dar 4.

But it has to give you 4.

04:45 Ya te da -6 por 2.

It already gives you -6 by 2.

04:51 -6 por 2. -12. Pero le sale 4.

-6 times 2. -12. But it comes out 4.

04:56 El 4?

The 4?

05:02 Entonces era…

So then it was…

05:02 3…

3...

05:06 Que es..2, no. 4 por 1, 4. 4 por 2, 8. 4 por 3, 12.

What is … 2, no. 4 times 1, 4. 4 times 2, 8. 4 times 3, 12.

05:16 Por que dije? 2, por -2, es -4.

Why did I say… 2, times -2, is -4.

05:25 Pero no va a salir -2.

But it's not going to turn out -2.

05:28 Si.

Yes.

05:33 4 por 3, 12.

4 times 3, 12.

05:36 2 por -6,

2 times -6.

05:39 Ponlo?

Put it?

05:41 -6.

-6.

05:42 Dos, por…-12. Entonces vamos a hacer… 6, mas -2, 4. Si son. Asi se sale.

2 times.... -12. So then we have to make... 6, plus -2, 4. They are. That's how it turns out.

05:54 No, a ver a ver.

No, let's see, let's see.

05:55 A ver, cual me dijiste?

Let's see. Which did you tell me?

05:56 Dos… y.

Two... and

05:58 Alli esta. Lo sumamos? Sumalos y ves. ¿Que sale? Sumalos.

There it is. Should we add them? Add those and see. What comes out? Add them.

06:10 2, me equivoqué. 2 y -6.

2, I made a mistake. 2 and -6.

06:16 Si, te digo que… aca es. +2, -6….. -6x, 2x,

Yes, I'm telling you that... here it is. +2, -6.... -6x, 2x.

06:28 Sale -12.

It comes out -12.

06:30 Alli esta. Sumalos. A ver. Sumalo.

There is it. Add them. To see. Add it.

06:33 Si.

Yes.

06:34 Sale como? Sale 4, no?

How does it come out? It comes out 4, right?

06:37 No, sale -4. Que 2, mas 6, negativo 4.

No, it comes out -4. That 2, plus 6, is negative 4.

06:44 No, ya lo saque, asi.

No, I already got it, like this.

06:46 A ver. Yo voy a hacerlo al reves.

Let's see. I'm going to do it the other way.

06:48 Cuando?

When?

06:49 Al reves.

Backwards.

07:00 Y este al reves. Como es…igual a 2, va a ser – 6. Entonces, -6.

And this backwards. How is it... equal to 2, it's going to be -6. So, -6.

07:10 Aqui esta -6, arriba con…

Here is -6, above with...

07:12 Aqui.viene, como si tiene abajo, -6. Y arriba es +2.

Here comes, like we have below, -6. And above is +2.

07:20 O, si, si si. Mas dos?

Oh, yes, yes yes. Plus 2?

07:23 Dos.

Two.

07:23 Y si sale este? So entonces, era al reves.

And that comes out? So then, it was backwards.

07:33 Como va?

How's it going?

07:35 Vuelve a estar -6.

It's back to -6.

07:37 x, 2. 2x,

x, 2. 2x.

07:42 Es +2 arriba,

It's +2 above.

07:44 2…si, sale -2 arriba.

2... yes, it turns out -2 above.

07:50 O, es menos 2. Y abajo es +6?

Oh, it's -2. And above is +6?

07:57 Abajo es menos 6.

Below it's -6.

08:00 -2, abajo menos 6, no?

=2, below is -6, no?

08:04 -2?

-2?

08:08 Aqui va a ver…Que teniamos, +2? Mira, vuelvelo a hacer.

Here there's going to be, what do we have, +2? Look, do it again.

08:12 Mira. Es…2 mas, menos…

Look. It's... 2 plus... minus....

08:16 Arriba tiene que ser negativo 2. Y abajo positivo 6.

Above it has to be -2. And below +6.

08:21 Abajo es negativo! Arriba es negativo y abajo ….

Below it's negative! Above it's negative and below...

08:26 Positivo. So, x

Positive. So, x

08:33 Es 6x todo

Is 6x all..

08:36 6x arriba, cuantos? Menos 12? Igual a …

6x above, how many? -12? Equals...

08:41 Esto si va a hacer. 6x – 12, -12…

This one is going to be... 6x-12, -12

08:55 So, esto va a ser así. Igual a 0. Igual a 0.

So, this is going to be like this. Equals 0. Equals 0.

09:00 Tenemos que poner todo esto, ¿verdad?

We have to put all this, right?

09:02 Igual a x-2 …

Equal to x-2.

09:04 Falto ponerle esto…. Y entonces que va a ser.. x -2, parentesis…

I missed putting this. And then it's going to be, x-2, parenthesis.

09:14 x + 6,

x+6

09:20 Cero igual a x, menos 2 , x + 6…

Zero equals x, minus 2, x + 6...

09:25 Pero aca ponle, ponle el f.

But over here, put the f.

09:29 Arriba?

Above?

09:32 O, y despues…

Oh, and after…..

09:36 Pon uno que te de cero… Es lo mismo que aca.

Put one that gives you 0, it's the same as here.

09:41 Es igual que el otro problema.

It's the same as the other problem.

09:47 x-

X-

09:48 Y aca, menos 6, a 6, igual a 0.

And here, minus 6, to 6, equals 0.

09:53 Tenemos que hacer este, igual a 2, -2.

We have to do this, equals 2, -2.

09:56 Si, y a 6

Yes, and to 6

09:58 Igual a -6.

Equals -6.

10:02 Ya lo puse al reves!

I already put it backwards!

10:06 In 10 minutes, you should finish number 2, and start -- call me over to check it, and call the back. And start the back.

10:11 6 mas…

6 plus...

10:12 You guys are doing great! En diez minutos.

You guys are doing great! In 10 minutes.

10:16 Al reves! Es… menos, verdad?

Backwards, it's ... minus, right?

10:22 X, x…igual a 2 o -6.

X, x equals 2 or -6.

10:30 X igual a 2 o negativo 6.

X equals 2 or negative 6.

10:34 Luis, explica, no me digas.

Luis, explain, don't tell me.

10:43 A ver.

Let's see.

10:53 Ahora debes de poner la misma ecuacion aqui. Las 4x-2,

Now you need to put the same equation here. 4x-2.

11:07 No. Pon la x.

No, put the x.

11:12 Y ahora debes de ponerlo…. Como aca. Si.

And now you should put it... like here. Yes.

11:32 No. Ponle 0 igual.

No. Put 0 equal.

11:37 Y ahora debes de poner 0,

And now you should put 0.

11:40 Igual a 2.

Equals 2.

11:42 No, pero le parto con el 2 menos 2, porque …

No, but I divide it with the 2 minus 2, because...

11:46 Luego ponle x igual a 2,

Then I put x equals 2.

11:50 A dos..

2.

11:53 OR, -6.

Or -6.

11:57 Luego ponlo alli en la tabla. Ponla en la t-table.

Then put it there in the table. Put it on the t-table.

12:03 Pon 2x.

Put 2x.

12:06 Nos falta poner las tablas, verdad?

We still need to put the tables, right?

12:12 Es dos..

It's 2...

12:13 Que? Menos 6.

What? Minus 6.

12:16 No, puesto los 2x también.

No, put the 2x also.

12:19 Que lo pones a 2x!

Put the 2x!

12:20 Mas que son 2x. Ponla 2.

More that are 2x. Put 2 there.

12:24 Adonde?

Where?

12:27 No, la tabla esta abajo.

No, the table is below.

12:29 Pero lo de arriba tampoco le ha puesto de aqui.

But I haven't put the one on top either.

12:31 Y entonces el resultado, verdad?

And then the result, right?

12:33 Si.

Yes.

12:34 El otro lado es 2 y negativo 6.

The other side is 2 and negative 6.

12:38 Este -7.. Aqui es…no. Aqui es…2? Y negativo 3.

This -7. Here is... no. Here is 2? and negative 3.

12:46 Oye. Ponle 2.

Listen. Put 2 there.

This is a fairly representative discussion, in terms of how focused the group is on the mathematics. I think the only thing that's exceptional is how long they sustained. You can see toward the end some of that breaking down, with the students in the background doing something else. At that point in the school year, March, most of us had been together for quite a while, so they know what strong group work looks and sounds like and how it benefits them. They know that I don't like it when the class is breaking down.

In some ways it's too bad that the cameras weren't there at the beginning of the year when we were working on all those norms because there would've been many more teacher moves to see. Regarding this group's use of the calculators to test their thinking around positive and negative numbers, I think that's a common trap that teachers are told to fall into, especially beginning teachers, that the kids need to be able to do "______" so that they can be successful at algebra. But for the most part, I think students can get by without any of those, as long as they have something and as long as they continue to attend to those things. So I'm used to kids coming in not knowing very much, although something, about working with positive or negative numbers.

With their use of the calculator, they had to realize that, "Oh okay, we don't know this offhand, so we're going to figure it out this way." I think that's, in general, the appropriate use for a calculator, unless it's an essential part of the problem.