# Lesson - Part 10

## lesson - part 10

Expand +

A group of students explain their thinking to each other as they look to find the vertex of the parabola. Cabana joins the group to challenge them to move between their representations.

## lesson - part 10

Carlos Cabana, San Leandro High School, San Leandro Unified School District, San Leandro, California

Next Up:   Lesson - Part 11
Previous:  Lesson - Part 9

00:00 4, 5. 5 tambien es 0.

4, 5, it's also 0.

00:07 No, estos van a ser 0.

No, these are going to be 0.

00:11 El 0 se va a ser negativo, si.

0 is going to be negative, yes.

00:15 Aqui estan los dos coordinados, mira. Estas dos son… aqui, y aqui. Estas son. Negativo uno, es a 0, y 5 es 0 tambien.

Here are the two coordinates, look. These two are... here, and here. These are. Minus 1, it's at 0, and 5 is 0 too.

00:25 Y para la 0, es este numero, de aquí. El 5, negativo 5.

And for the 0, it's this number, from here. 5, negative 5.

00:39 Negativo 8, tambien, en la…

Negative 8, also, in the...

00:41 En la…uno, que sale? Negativo 8?

On the one... what comes out? Negative 8?

00:47 Y numero 2, negativo 9.

And number 2, negative 9.

00:49 Negativo 9. Y luego negativo 8 va a ser, ¿otra vez? O negativo 9.

Negative 9. And then it's going to be negative 8, again? Or negative 9.

00:55 No, negativo 8,

No, negative 8.

00:56 Negativo 8,

Negative 8.

00:58 Negativo 8, y negativo 5.

Negative 8, and negative 5.

Negative 8, right?

01:03 Sí.

Right.

01:05 Y el 20, que es este?

And 20, what's this one?

01:11 Cabaña.

Cabaña.

01:12 Ahora tenemos que encontrar. El numero que … donde menos 2…

Now we have to find it... the number that... where minus 2...

01:20 Este es el …

This is the...

01:22 No, negativo 8.

No, negative 8.

01:25 Ya despues, va a ser -8. Todos empiecen a ser negativo.

And then, it's going to be negative 8. All of them start to be negative.

01:29 ¿Que es este?

What's this one?

01:31 Mm hm. Pero es negativo 9.

Mm hm. But it's negative 9.

01:33 Por eso no va a ser…

But it's not going to be that...

01:37 Tienes que marcar aquí los coordinados.

You have towrite down the coordinates here.

01:47 En la…que tenemos que hacer?

In the... what do we have to do?

01:51 El de…? Falta el 0, 1, y 2. Faltan, pues marco. 0, 1, y 2.

That one.... 0, 1, and 2 are left. Those ones, I'll write down. 0, 1, and 2.

02:01 Y el 0 es negativo 3?

And the 0 is negative 3?

02:11 -3. Sale -4.

-3, turns out -4.

02:15 -4, y luego.

-4, and then...

02:16 Y despues negativo 3.

And then -3.

02:19 El vertex, ese,

The vertex, that one.

02:27 Y la parabola.

And the parabola.

02:31 Cual? Nada mas vamos a encontrar la pura vertex. En la C, es este de aquí. La que esta.

Which one? We're only going to find the vertex itself. In C, it's this one here. The one that's here.

02:40 Ya que empiece a repetir los numeros, pero, como los dos son iguales, este es el centro, en medio de estos.

Once it starts repeating the numbers, but since the two are the same, this is the center, in the middle of these.

02:47 No, aquí que sea una linea.

No, here it should be a line.

02:53 Cual numero, entonces, es lo que empiece a repetir? Mm hm. De alli, asi. Como los dos son iguales, marques una linea entre los dos.

Which number, then -- is this what starts to repeat? Mm hm. From here, like that. Like the two are equal, write down a line in between the two.

03:04 Y sigue la D. En el papel.

And D comes next. On the paper.

03:11 Todo es en la B que hacemos …

Everything is in B that we did...

03:17 Pero, con esta, no sabemos exactamente cual, esta aquí, pero, como atar una coordinada.

But with this one, we don't know exactly which, it's here, but, we have to attach a coordinate.

03:24 Mm hm.

Mm hm.

03:26 Porque nos dijo que… este, entre medio de estos…

Why did you tell us that... this, in the middle of these...

03:29 ¿Entre medio de que?

In the middle of what?

03:30 De…dos numeros que estan iguales.

Of... the two numbers that are equal.

03:33 Y, pero…no saben a que nivel va a bajar, ¿verdad? La coordinada y no saben. ¿La coordinada del x?

And, but... you don't know what level is going to come down, right? You don't know the coordinate. The x-coordinate?

03:40 Mm?

Mm?

03:43 Este seria 2.1? 2.5.

This would be 2.1? 2.5.

03:48 Casi.

Almost.

03:48 2.8? .6? .5.

2.8? .6? .5.

03:53 Leticia?

Leticia?

03:54 1.5?

1.5?

03:55 Hm mm. 1.5 estaria aquí.

Hm mm. 1.5 would be here.

03:58 O, es negativo 2.5!

Oh, it's negative 2.5!

04:01 ¿Por que 2.5 y no 2.8 o algo asi?

Why 2.5 and not 2.8 or something like that?

04:04 Porque…Aquí crece por 1, de aquí, y el otro, la mitad solamente. Para que se hagan 2.

Because... here it increases by 1, from here, and the other, by a half only. So that it makes 2.

04:13 Tenemos que poner justo en medio, ¿verdad? Asi que, si saben que la coordinada de x es -2.5, como dijo Jesus, ¿cómo van a encontrar la coordinada de y?

We have to put it right in the middle, right? So that, if you know that the coordinate of x is -2.5, like Jesus said, how are you going to find the y coordinate?

04:23 -2, o, digo, -2.5 porque es la misma. Es 0?

-2, or, I said, -2.5 because it's the same. It's 0?

04:30 Tiene que, no, tiene que encontrar la coordinada exacta, con uno a 0.

You have to... you have to find the exact coordinate, with 1 at 0.

04:33 Usando, reemplazando las x con…

Using... replacing x with...

04:36 Aja, usando la regla. Ok, pero vamos a ver las otras.

A ha, using the rule. Ok, but let's see the others.

04:39 Aquí no se falta hacer una tabla tan grande.

Here it's not necessary to make such a large table.

04:42 ¿Por qué?

Why?

04:43 Porque sin resulto, sin todo esta informacion, ok, abase de lo que esta aquí, ¿qué saben sobre el vertex?

Because without the answer, without all this information, ok, based on what is here, what do you all know about the vertex?

04:51 Que el vertex…

That the vertex...

04:53 Que tiene

That it has...

04:54 Si…

Yes...

04:55 Que tiene dos, dos coordinados, que tiene—que es lo que quise decir?

That it has two, two coordinates, that it has... what was I going to say?

05:02 El mismo valor de x?

The same x value?

05:04 ¿Cómo?

What?

05:05 El mismo valor de x.

The same x-value.

05:07 Tiene…

It has...

05:08 Esto si, pero yo…

Just so, but I...

05:09 Que tiene dos x.

That is has two x.

05:10 Pero yo te pregunte sobre el vertex.

05:14 Que es una coordinada… que es un punto…

What's a coordinate, what's a point...

05:19 Es un punto que … que da…

It's a point that... that gives

05:11 Aja, asi. Leticia sabe que el vertex es el punto menor, ¿verdad? Pero ¿qué mas saben sobre el vertex?

A ha, like this. Leticia knows that the vertex is the lowest point, right? But what more do you know about the vertex?

05:31 Que el vertex es el numero que… es el… es como, aquí crece en la.

That the vertex is the number that... it's the... it's like, here it increases in the...

05:37 En una tabla como lo pueden encontrar?

In a table how can you find it?

05:41 ¿Cómo?

What?

05:42 Lo que borro,

What he erased.

05:44 ¿Cómo lo encontraste aquí?

How did you find it here?

05:45 Pues, poniendo los dos, las dos coordinadas y los demas numeros.

Well, putting the two, the two coordinates and the rest of the numbers.

05:50 Aja. Pero, no queremos hacer todo este trabajo, ¿verdad? Abase, ¿cómo saben cual es el vertex, aparte de buscar el valor menor?

A ha. But, we don't want to do all that work, right? Based on that, how do you know which is the vertex, aside from looking for the lowest value?

06:00 Substituyendo valor de x!

Substituting the x value!

06:02 Aja. ¿Cuál valor de x? Eso tiene razon. Abase de … si sabes el valor de x que, de acuerdo que va a ser 2, como tenian.

A ha. Which x value? That's right. Based on... if you know the x value, it's going to be 2, like you had.

06:12 ¿Cómo supe yo que iba a ser 2? Sin esta tabla.

How did I know that it was going to be 2? Without this table.

06:16 Abase de lo que tienen aquí nada mas, ¿cómo se que la coordinada tiene que ser dos?

Based on what we all had here and nothing else, how do I know that the coordinate has to be two?

06:26 Porque aquí estan 5, y a lo demas…

Because here are 5, and for the rest....

06:30 ¿Y? ¿Y?

And? And?

06:33 No mas.

Nothing more.

06:34 No, quieres decir, well, tal vez, ¿quieres decir, Leticia?

No, you wanted to say, well, maybe... do you want to say, Leticia?

06:39 Ya se…por que

I know.... why

06:43 Leti! Come on.

Leti! Come on.

06:45 Bueno. Yo pienso..

Fine. I think...

06:46 Aja.

A ha...

06:48 Que usted supo substituir…

That you knew to substitute...

06:50 Si.. Esto lo que van a hacer. Pero, mi pregunta fue, ¿cómo supe que esta coordinada de x iba a ser 2, abase de esta informacion?

Yes. This is what you're going to do. But my question was, how did I know that this x coordinate was going to be 2, based on this information?

07:03 Por qué 2, y no 2.5, o 3, o 1. ¿Por qué tiene que ser dos?

Why 2, and not 2.5, or 3, or 1. Why does it have to be 2?

07:11 Porque …. Porque sigue esta… la regla?

Because... because it follows this... the rule?

07:15 Mmm. Arreglalo. Piensa en eso.

Mmm. Fix that. Think about it.

07:19 What's up… what's that?

What's up, what's that?

07:21 Juan?

Juan?

07:22 He's not here.

He's not here.

07:29 Ok. Disculpa.

Ok. Excuse me.

07:33 ¿Qué tiene de especial el vertex, aparte de ser el punto mas bajo?

What's special about the vertex, besides it being the lowest point?

07:37 Es…que, es donde se…

It's... that, it's where...

07:39 ¿Qué mas?

What else?

07:40 Que sea el punto mas bajo.

That it's the lowest point.

07:42 ¿Qué mas?

What else?

07:44 Que es un punto que tiene coordinados asi.

It's a point that has coordinates like this.

07:48 Siga… muy bien. Sigue, que esta todo correcto hasta esta hora. Solo que no les ayude en encontrar el 2. ¿Verdad?

Keep going. Very good. Keep going, all is correct up to now. It's just that it doesn't help you find the 2. Right?

07:55 ¿El vertex es donde empiecen a repetir los numeros?

The vertex is where the numbers start to repeat?

07:58 Si, ¿y que mas? Pero esto ya me acuerdo, es donde empieza a repetir, ¿verdad? No es que sigue con este 2.

Yes, and what else? But this I already remember, it's where it starts to repeat, right? It's not that it goes on with this 2.

08:07 El punto…. No sé que decir.

The point... I don't know what to say.

08:10 Aparte de eso. Tampoco…eso tampoco me ayuda saber la coordinada de x a ser 2. ¿Cómo supe que esto va a ser 2 exacto?

Apart from this. Neither... this doesn't help me to find the x-coordinate to be 2 either. How did I know that this was going to be 2 exactly?

08:17 Porque, para marcamos aquí con el 5…

Because... we wrote down the 5 here.

08:21 Pero…

But...

08:23 Y despues pones estas rayas… ¿que?

And then you put down these rays... what?

08:25 Pues que sabe que es… 2 viene aca.

So you know that it's .... 2 that goes here.

08:29 Pero no se exactamente donde estan las rayas, ¿verdad? No sé… no voy a saber con las rayas si va a ser exactamente en 2, o 1.5, 2.1…

But I don't know exactly where the rays are, right? I don't know... I'm not going to know with the rays if they're going to be exactly on 2, or 1.5, or 2.1....

08:39 Hm.

Hm.

08:41 Entonces… porque usted dice. No se que decir.

Then… why did you say so. I don't know what to say.

08:48 Porque hay dos puntos en x.

Because there are two points on x.

08:50 Mmm! ¿Y? ¿Qué tiene que ver el 2 con estos dos puntos?

Mmm! And? What does the 2 have to do with these two points?

08:54 Que asi la continua.

That that's how it continues.

08:56 ¿Cómo supe?

How did I know?

08:58 Porque…. Porque dividimos y…

Because... because we divided and...

09:03 ¿Me puedes dar otro handout?

Can you give me another handout?

09:16 Eduardo, you better not be talking outside your group again.

Eduardo, you better not be talking outside your group again.

09:23 ¿Qué saben sobre el vertex?

What do you know about the vertex?

09:25 Que es negativo.

That it's negative.

09:27 ¿Qué mas?

What else?

The y coordinate.

09:30 10 y el 20?

10 and the 20?

09:31 Aja…

A ha.

09:32 Que…

That...

09:33 ¿A qué valor?

At what value?

09:35 En el…

In the...

09:36 ¿Cómo sabes que esta el 15?

How do you know that it's 15.

09:39 No mas sale…porque esta… porque esta en la raya.

It just comes out... because... because it's on the line.

09:42 Aja…¿y el vertex siempre esta en medio de la parabola? Si, ¿verdad? Tiene …sido, tiene que repetir, asi que hay la misma candidad de numeros a uno que del otro.

A ha. And the vertex is always in the middle of the parabola? Yes, right? It has.... it's been, it has to repeat, so that there's the same quantity of numbers on one side as the other.

09:52 Ok, asi que, abase de eso, como…

Ok, so that, based on this... how....

09:56 Porque, de aquí, aca son dos… no.

Because, from here, here are two... no.

09:58 Si!

Yes!

09:59 Son de aquí.

They're from here.

10:00 ¿Son cuatros?

They're 4?

10:02 En los tres, y de aquí tambien son tres.

On the 3, and from here it's 3 also.

10:04 Tambien son 3. Exacto. Asi que yo supe que de 5 a menos 1 son 6, asi que acaba de ver 3. Muy bien, Dulce.

They're also 3. Exactly. So I knew that from 5 to negative 1 are 6, so it just has to be 3. Very good, Dulce.

10:12 Es este, aquí?

It's this one, here?

10:18 Ok. Asi que eso quiere decir que no faltas en una tabla grande. En vez de hacer eso, pueden buscar el, solamente buscar en medio de los dos x-intercepts,

Ok. So this means that we're missing a large table. Instead of doing that, you can look for... only look in the middle of the two x-intercepts.

10:26 Y luego, abase de te acuerda de x, vas a encontrar la coordinada y. De parentesis no puede ser 9.

And then, based on what you remember of x, you're going to find the y-coordinate. From the parenthesis it can't be 9.

10:33 Porque no le sale aquí.

Because it doesn't come out here.

10:35 Negativo 9… aquí estaba

Negative 9... here it was.

10:37 Y eso… ah. Y eso, ¿cómo lo encontraron?

And that...ah. And that, how did you find it?

10:41 Porque en todo reemplazar las x por 2.

Because in the whole thing we replaced the x's by 2.

10:44 Por 2. Exacto. Ok? Ok, asi que aquí tampoco hace falta en encontrar todos estos valores, aunque no esta tan mal,

By 2. Exactly. Ok? Ok, so that here you don't need to find all these values either, although it's not bad.

10:52 Porque lo que tienen que hacer es buscar el valor de x, que esta justo en medio de los dos x-intercepts. Mm hm?

Because what you have to do is find the value of x, that's just in the middle of the two x-intercepts. Mm hm?

10:59 Ok. Aquí ya lo supe hacer, le falten encontrar eso, y esto no lo hicieron todavia, ¿verdad?

Ok. Here I know what to do, you just need to find that, and you haven't done that yet, right?

11:04 Es igual que eso.

It's the same as that one.

11:06 Ok, pero eso quiere… eso quiero que lo hagan sin esa tabla.

Ok, but you want... that I want you to do without that table.

11:11 Entonces…

So...

11:12 Porque, well, hagan lo que hicieron antes. Aquí, va abase de esto…

Because, well, do what you did before. Here, it goes based on this...

11:19 De los intercepts.

Of the intercepts.

11:20 Aja.

A ha.

11:21 Y luego encontrar el centro, para encontrar el valor de vertex.

And then finding the center, to find the value of the vertex.

11:23 Exacto. Ok?

Exactly. Ok?

This is a good example of kids being used to thinking that the algebra gets them to the graph or to the geometry, but not that that process also works backwards. I don't think that's something I made explicit to this class. They're looking at the vertex and they're not sure-- they can find it with a long T-table and they have some sense of, "I have to look in between these two numbers on the T-table" because they've done that before. They know what a vertex looks like, or at least Jesus knows what a vertex looks like in the table, but they're not translating that information to a graph. They're not using the visuals on the graph to help inform the conversation.

I think that speaks to the importance of not looking at multiple representations as a linear process, but in fact, as an interconnected web of similar ideas, where as a teacher I can choose which representation to start with and which one I want classes to generate.

That helps me frame the illuminator or big idea without being repetitive. But I also want students to realize that they can do that as well, which in this case they're not doing, and it's completely understandable-- they're in the middle of their thinking process and it goes in one direction.

They do understand what the vertex is, but they don't sort of see it because of multiple representations. And I think that happens often, that if students seem to understand one concept then they think they understand the whole thing, but there's more there and I need to push them to consider even more. Teachers need to identify what the big idea is, and then think about "What questions can I ask that will illuminate that big idea from multiple perspectives?" Multiple perspectives tend to be about representations in the traditional algebraic sense of tables, equations, and graphs. But it can also be thinking of algebra not only numerically but thinking about it geometrically.