Lesson - Part 4

lesson - part 4

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Both of these groups have literally never seen a problem like this worked out, and raise questions about mathematical tools and how to show and organize their work.

lesson - part 4

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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00:03 ¿Y con qué fin? ¿Qué están…buscando? Es todo parte del mismo problema, ¿o son dos?

And to what end? What are you... looking for? Is it all part of the same problem, or are there two?

00:09 Es… es el mismo.

It's... it's the same.

00:11 ¿Esto, para qué sirve and esto para qué sirve?

So this, what's this for, and this, what it's for?

00:14 O…


00:15 No, dejale a Leticia.

No, let Leticia go.

00:20 Esto no se. Esto tiene que encontra el y.

This, I don't know. This I have to find the y.

00:22 Ok, asi que es el resultado. ¿Puedes poner una pequeña t-table, por favor?

Ok, so that's the result. Can you put a small t-table, please?

00:25 Ah, muy bien. Y..

Oh, very good. And...

00:31 Así que… y esto aquí-- ¿para que sirve?

So that, and this here, what's it for?

00:33 Para encontrar la x-intercept.

To find the x-intercept.

00:35 Ah hah, pero…. Ok.

Uh huh, but... Ok.

00:39 Y esto… a ha. No. Tienes razon, los x-intercepts. Primero, les falta algo. A todos le falta algo.

And this, uh huh. No. You're right, the x-intercepts. But, you missed something. Everyone missed something.

00:47 Si. Esto está incompleto.

Yes. This is incomplete.

00:53 ¿Tenemos que mostrar la substitucion?

We have to show the substitution?

00:56 ¿Que substitución?

What substitution?

00:58 ¿Este?


00:58 No necesariamente… estos son problemas de mental math. ¿Verdad? Esto se hace en su cabeza. So esto está correcto.

Not necessarily. These are problems of mental math, right? This can be done in your head. So this is right.

01:05 Pero, cuando esta buscando un x-intercept, ¿cómo se empieza?

But when you're looking for an x-intercept, how do you start?

01:07 ¿A emplazar la y por cero?

To replace the y with 0?

01:10 Y ¿adonde hicieron eso?

And where did you all do that?

01:12 Si es cierto.

That's right.

01:15 Porque esto solo es, ¿verdad? Si el objetivo es tener 0.

Because that's all it is, right? If the objective is to have 0.

01:20 Ok. Asi que para mostrar el procedimiento bien, lo que sugiero es empiezen… a ver la cuestion aqui…ah.

Ok. So to show your procedure well, what I suggest to you is that you start... to look at the equation here. Ah.

01:28 Ok. Aquí lo hicieron muy bien. Aquí tienen la cuestion original, la misma ecuación con cero substituido, y la respuesta.

Ok. Here you did it really well. Here you have the original question, the same equation with zero substituted, and the answer.

01:34 Pero el resto del procedimiento, para los x-intercepts, esta todo deparramado y sin resolucion.

But the rest of the procedure, for the x-intercepts, is all scattered and unsolved.

01:39 Tenemos que hacer lo mismo que hicimos para la y para la ….

We have to do the same that we did for the y for the...

01:41 Aja, escriban la ecuacion original, y luego subtituyen 0 como hicieron, except con la y, y luego ¿qué?

Uh huh, write the original equation, and then substitute 0 like you did, except with the y, and then what?

01:52 Despues que substituye 0, ¿qué van a hacer?

After I substitute 0, what are you going to do?

01:55 Substituye…vamos a poner 0…

Substitute- we're going to put 0.

02:00 Aja, pero en esta ecuacion, la original, o sea, la primera cosa que hace… asi que despues de substituir 0, ¿qué van a hacer?

Uh huh, but in this equation, the original, that is, the first thing to do, so that after you substitute 0, what are you going to do?

02:10 Pon el..justo al…

Put the... right at...

02:15 Mm mm. ¿Por qué? ¿Cuál es la regla original?

Mm mm. Why? What is the original rule?

02:21 En vez de… ¿Parentésis?

Instead of … parentheses?

02:22 No.


02:23 Si, este es la original. Este.

Yes, this is the original. This.

02:26 A, ¡perdón, me equivoqué! Ok. All right.

Oh no, sorry, I made a mistake! Ok. All right.

02:31 So. X menos 1, x menos 4. Por eso. Gerardo, get to work. Ok. Es la ecuacion.

So, x-1, x-4. For this. Gerardo, get to work. Ok. This is the equation.

02:40 Lo primero que hicé fue hacer este rectangulo para encontrarla de esta forma. La ecuación.

The first thing I did was make this rectangle to find it this way. The equation.

02:46 A! Ahora entendí bien. Para que le diera el y…. ahora. Ok. Porque tu le dijiste que eran los x-intercepts.

Ah, now I understand it. So that it would give you the y. Now. Ok. So why did you say they were x-intercepts?

02:57 Pero no sirvieron. Well. Tienes que preguntar cuando estan trabajando este proprio problema.

But that didn't work. Well. You have to ask when you're working on this problem itself.

03:01 Ok. So. Esto está totalmente correcto, aunque no es necesario. Porque, ¿cómo se encuentre un y-intercept?

Ok, so. This is totally correct, though it's not necessary. Because, how do you find a y-intercept?

03:10 Substituyendo pa' los de x.

Substituting for those of x.

03:11 Aja, y aquí está valor de x. ¿Verdad? Asi que antes de hacer todo este trabajo de multiplicar, vamos a ver si es necesario.

Uh huh, and here is the value of x, right? So that before we do all this work of multiplying, we're going to see if it's necessary.

03:17 Si substituimos cero por x, ¿qué nos da?

If we substitute 0 for x, what does it give us?

03:21 0 por x?

0 for x?

03:22 No, no cero por x.

No, not 0 for x.

03:26 Sustituimos x por 0. Ponemos 0 en vez x.

We substitute x for 0. We put 0 in place of x.

03:30 Mmm hmm. ¿Qué nos va a dar?

Mm hm. What's it going to give us?

03:33 ¿Quién es el resource manager de este equipo? Ok. Gracias, Alejandra, empieza.

Who is the resource manager for this team? Ok. Thank you, Alejandra. Begin.

03:42 ¿Cuál problema es este?

Which problem is this?

03:44 La 2a.


03:45 ¿Esto aquí? Ok. ¿Cómo empezaron?

This here? Ok. How did you start?

03:48 Así.

Like this.

03:49 Ya sacamos la y…

We took out the y…

03:51 El y-intercept? Ok.

The y-intercept? Ok.

03:53 Y empezamos con el del x-intercept.

And we started with the x-intercept.

03:57 Uh huh. Ok, porque les mostraron su procedimiento muy bien, pero les falta una cosa.

Uh huh. Ok, because you showed your procedure very well, but you're missing one thing.

04:03 Aquí, la x.

Here, the x.

04:05 No.


04:07 Ok.


04:10 Todo está bien organizado. Aquí veo la ecuacion, veo que empezaron substituyendo 0 por x, en ambos lados.

Everything is well organized. Here I see the equation, I see that you started substituting 0 for x, on both sides.

04:18 Esto les falta hacerlo pero no está mal, despues de la substitucion, la respuesta. Todo esto esta perfecto.

You still have this to do, but it's not bad, after the substitution, the answer. All this is perfect.

04:26 Ok. Aquí … no escribieron la ecuacion original, pero ni importa, puesta aquí, substituyeron 0 por y, lo factorizaron.

Ok. Here, you didn't write the original equation, but it's not important, well here, you substituted 0 for y, you factored.

04:34 No sé porque esto está aquí, porque es equivalente al…

I don't know why this is here, because it's equivalent to..

04:36 No, vamos todo esto…

No, we went all this..

04:37 Si, pero lo que nos importa son los factores, ¿verdad?

Yes, but what's important are the factors, right?

04:40 Estabamos chequeando si estaba bien.

We were checking to see if it was good.

04:42 Ok. No hace falta en mostrarlo, pero… no está mal.

Okay. It's not required to show it, but... it's not bad.

04:48 Esta expresion aquí, ¿qué tiene que ver con este problema? Porque les falta algo aquí.

This expression here, what does it have to do with this problem? Because there is something missing here.

04:55 Que es…esto aquí.

What is... this here.

04:56 Muy bien, Alejandra. Es exactamente igual a esta area. Aja. Les falta esto.

Very good, Alejandra. It's exactly equal to this area. A ha. You're missing this.

05:00 O, la y!

Oh, the y!

05:02 0, 0.


05:03 Y no.

Y no.

05:04 0 igual.

0 equal.

05:05 ¿Y ahora lo pueden resolver?

And now are you able to solve it?

05:07 Aja. Tenemos que hacerlo pa' que sea…

A ha. We have to do it so that it will be...

05:10 Nice job, you guys.

I think that questions about tools and questions about process go together, and I wanted to attend to how to show the work so that I could get to the point of asking the questions about why, making sure they just weren't doing things just by rote. What I was trying to provide for the first group is a model based on their bits and pieces of how they could show that work. For the second group who'd shown the work in a more organized way, it was a narrative about what that work might mean, mathematically, what the reasons might be underlying it. Some students get mixed up because they're forgetting about how to go about substituting in zero for x or y intercepts. They start from this very concrete process of finding the zero in a graph, and substituting it from there, to this quick "I'll just plug in zero," and they forget that there are reasons behind those zero choices that they figured out themselves.

I have to bring them back to that level of concreteness. It also comes up in this clip and others, where I have to tell kids to summarize their results in a little t-table, because they lose that representation and then they don't see the connection between all these numbers that they're figuring out and overall objective and what it might look like in a graphical representation.

So as the teacher, I'm thinking about am I going to have to attend to the big picture first and then the details and reasons, or the other way around? As a practitioner, I think about what's the big idea, and then what mathematics do kids need to figure out that big idea? In practice, different topics could work both ways and the teacher has to figure out which way helps students understand the whole picture.