Three students continue their mathematical discussions.

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 Se salió x igual a -5….

X came out equal to -5.

00:05 y x se vale 7, tambien, y y es igual a -7. ¿Verdad? O sea, estos.

And x has a value of 7, also, and y is equal to -7. Right? That is, these.

00:15 Esto tambien tiene que hacer…

This also has to be...

00:16 Aja, mm hm.

A ha. Mmhm.

00:23 Aja, asi, te lo da.

A ha, like so, you get it.

00:37 x, x. -2, +6. x^2, -2x, 6x, -12. Y = (x-2)(x +6)

x, x. -2, +6. x^2, -2x, 6x, -12. Y = (x-2)(x +6)

01:01 Ya vamos.

Let's go…

01:08 y, -2?

y, -2?

01:10 Aja. Es este. Te da este aquí, está bien.

A ha. It's this one. You get this here, it's good.

01:15 Siempre te va a dar este numero de aquí.

You're always going to get this number, here.

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.