Uploaded by VideoTSBVI on 04.02.2010

Transcript:

A TSBVI Outreach Tutorial.

APH Graphic Aid for Mathematics

a.k.a. Graph Board.

Part 7: Graphing the quadratic (parabola) y = x squared and y = -x squared.

Presented by Susan Osterhaus,

a Texas School for the Blind and Visually Impaired Outreach Math Consultant.

Okay. I've cleared the graph board, except for the x and y axes that we have.

Again, the y axis and the x axis

are just rubber bands hooked down with thumbtacks.

And we have the first, second, third and fourth quadrants.

We're gonna move on from straight lines to graphing quadratics.

These are sometimes called parabolas,

and I'm gonna graph a very simple one: y equals x squared.

Now, normally, we would go ahead and have a t table

for this particular type of graphic...

And we'll just start simple though

'cause I'm gonna try to remember this in my head.

Maybe we'll put something up online for you.

Y equals x squared.

I always tell the students,

"Let's start with my favorite number,"

x value, which is 0.

And it looks like an O, kind of, for Miss O. So that's why I want it.

We're gonna start with x equals 0.

Well, if x is 0, and y is x squared, 0 squared is 0.

So, truthfully, our first point is: 0,0.

So I'm gonna take the thumbtack out

just so that we remember it's not the origin, just the origin anymore.

It's now actually a part of this graph.

So we've got one point at 0,0.

And then I... We've already talk...

We're gonna pretend we've already talked about parabolas and so forth.

So the students are going to already know what, probably, they should pick next.

They're gonna have... They're gonna go ahead and let x be 1

because it's the first integer, positive integer after 0.

So if we plug in 1 for x, we get y equals x squared

or y equals 1 squared, which is 1.

So, now, we have 1,1. Boy, this is simple. Alright.

Then, we're gonna go ahead and go with -1.

-1 squared is 1, so we can go on ahead and graph that point.

And then, of course, they're gonna try for 2.

If x is 2, 2 squared is 4.

So we're gonna go two to the right. One, two, three, four, up.

And by this time, hopefully, your students have figured out,

"You know, I don't even need to do that 'cause I've already figured out that these,

these points are symmetric with regard to the y axis.

So I know there's another point that's just right there.

And then they might pick 3.

3 squared is 9. One, two, three.

One, two, three, four, five, six, seven...nine.

Oops. Put that push pin in there.

And again, rather than going all the way down here and starting over again,

the clever ones just go like this.

And then they might say, "Uh-oh. Are we gonna have room for another one or not?"

Let's try x equals 4.

4 squared is 16. Well, let's find out.

One, two, three, four.

One, two... Oh, do we need to?

Actually, if this is 4, we know this is 9.

They start getting very clever and fast at this.

Nine, ten, eleven, twelve, thirteen, fourteen, fifteen.

Whoo! Did Miss O luck out?

Okay, we got it up there, and again, we're gonna come over here.

Now, if you're very, very talented at this...

Let's see if I'm talented or not today.

We're gonna ask your students to go ahead and connect the dots.

I'm gonna put one rubber band going that direction

And notice that these are the rubber bands

where I've made them longer by knotting two together.

And we now have the graph of the parabola y equals x squared.

Now, at this point, we could have used...

If you wanted to, you could have used the wires that come with this.

And I'm gonna reach over and get a wire.

So some people might prefer using the wire but I kind of like it like that.

I think we graphed enough points that it's relatively rounded or curved there.

So we've graphed y equals x squared.

But again, if you needed to, you could use the wires or you could use yarn

or anything else if you felt like you really wanted to emphasize the fact

that this is going around that vertex and coming back up a very nice curve.

Now, what I love to do, this is my favorite thing,

is I have the students graph this.

And you, probably, if you're timing me, you figured out how long it took me to do that.

And I have students, you know, working on this when I have other people observing.

And then I sound like I'm very mean because I say,

"Okay, that took you maybe a minute or two minutes.

And now, you have two seconds to graph y equals negative x squared."

And anybody who's observing just gasps and says, "How could she be so mean?"

And guess what my students do?

As I mentioned, this is y equals x squared,

and I asked them to graph y equals negative x squared.

There. Two seconds.

All they had to do was turn it upside down.

And they think they're pretty clever, and they like to show that off.

And what they're learning here

is that when you have a negative in front of the x squared portion,

that you know have a parabola opening downward.

When it's positive, y equals x squared, there's no positive there,

but it would open upward.

But they learn that really quickly, and they love showing that off.

Then if you wanna get into talking about parent functions,

you can have something like y equals x squared minus 10.

And that will drop this... You will have the same parabola

but it will be the same shape, but it will be 10 units down.

Or if you do y equals x squared plus 2, it would bring it up two units:

The vertex would come up two units.

Anyway, you can do all kinds of neat things, again, manually graphing.

And the students really get it when they're actually having to move points

through space and they're hands...

Very good for fine motor skills.

A lot of times, I have students that have that type of problem.

Again this really helps in that way, as well

Orientation and mobility, instructors always come talk to me if their student is having a problem.

'Cause they know my students will be having similar problems.

Again, graphing and graphing parabolas and all types of other things: wonderful.

Graph, graph, graph. That's my motto.

Again this is a quadratic.

And all of the quadratics are gonna have a shape like this,

and it will either move down or up or it will move this way or that way.

Basically, it's still going to have the same shape.

And so your students can graph it similar to this.

**Captions by Project readOn**

APH Graphic Aid for Mathematics

a.k.a. Graph Board.

Part 7: Graphing the quadratic (parabola) y = x squared and y = -x squared.

Presented by Susan Osterhaus,

a Texas School for the Blind and Visually Impaired Outreach Math Consultant.

Okay. I've cleared the graph board, except for the x and y axes that we have.

Again, the y axis and the x axis

are just rubber bands hooked down with thumbtacks.

And we have the first, second, third and fourth quadrants.

We're gonna move on from straight lines to graphing quadratics.

These are sometimes called parabolas,

and I'm gonna graph a very simple one: y equals x squared.

Now, normally, we would go ahead and have a t table

for this particular type of graphic...

And we'll just start simple though

'cause I'm gonna try to remember this in my head.

Maybe we'll put something up online for you.

Y equals x squared.

I always tell the students,

"Let's start with my favorite number,"

x value, which is 0.

And it looks like an O, kind of, for Miss O. So that's why I want it.

We're gonna start with x equals 0.

Well, if x is 0, and y is x squared, 0 squared is 0.

So, truthfully, our first point is: 0,0.

So I'm gonna take the thumbtack out

just so that we remember it's not the origin, just the origin anymore.

It's now actually a part of this graph.

So we've got one point at 0,0.

And then I... We've already talk...

We're gonna pretend we've already talked about parabolas and so forth.

So the students are going to already know what, probably, they should pick next.

They're gonna have... They're gonna go ahead and let x be 1

because it's the first integer, positive integer after 0.

So if we plug in 1 for x, we get y equals x squared

or y equals 1 squared, which is 1.

So, now, we have 1,1. Boy, this is simple. Alright.

Then, we're gonna go ahead and go with -1.

-1 squared is 1, so we can go on ahead and graph that point.

And then, of course, they're gonna try for 2.

If x is 2, 2 squared is 4.

So we're gonna go two to the right. One, two, three, four, up.

And by this time, hopefully, your students have figured out,

"You know, I don't even need to do that 'cause I've already figured out that these,

these points are symmetric with regard to the y axis.

So I know there's another point that's just right there.

And then they might pick 3.

3 squared is 9. One, two, three.

One, two, three, four, five, six, seven...nine.

Oops. Put that push pin in there.

And again, rather than going all the way down here and starting over again,

the clever ones just go like this.

And then they might say, "Uh-oh. Are we gonna have room for another one or not?"

Let's try x equals 4.

4 squared is 16. Well, let's find out.

One, two, three, four.

One, two... Oh, do we need to?

Actually, if this is 4, we know this is 9.

They start getting very clever and fast at this.

Nine, ten, eleven, twelve, thirteen, fourteen, fifteen.

Whoo! Did Miss O luck out?

Okay, we got it up there, and again, we're gonna come over here.

Now, if you're very, very talented at this...

Let's see if I'm talented or not today.

We're gonna ask your students to go ahead and connect the dots.

I'm gonna put one rubber band going that direction

And notice that these are the rubber bands

where I've made them longer by knotting two together.

And we now have the graph of the parabola y equals x squared.

Now, at this point, we could have used...

If you wanted to, you could have used the wires that come with this.

And I'm gonna reach over and get a wire.

So some people might prefer using the wire but I kind of like it like that.

I think we graphed enough points that it's relatively rounded or curved there.

So we've graphed y equals x squared.

But again, if you needed to, you could use the wires or you could use yarn

or anything else if you felt like you really wanted to emphasize the fact

that this is going around that vertex and coming back up a very nice curve.

Now, what I love to do, this is my favorite thing,

is I have the students graph this.

And you, probably, if you're timing me, you figured out how long it took me to do that.

And I have students, you know, working on this when I have other people observing.

And then I sound like I'm very mean because I say,

"Okay, that took you maybe a minute or two minutes.

And now, you have two seconds to graph y equals negative x squared."

And anybody who's observing just gasps and says, "How could she be so mean?"

And guess what my students do?

As I mentioned, this is y equals x squared,

and I asked them to graph y equals negative x squared.

There. Two seconds.

All they had to do was turn it upside down.

And they think they're pretty clever, and they like to show that off.

And what they're learning here

is that when you have a negative in front of the x squared portion,

that you know have a parabola opening downward.

When it's positive, y equals x squared, there's no positive there,

but it would open upward.

But they learn that really quickly, and they love showing that off.

Then if you wanna get into talking about parent functions,

you can have something like y equals x squared minus 10.

And that will drop this... You will have the same parabola

but it will be the same shape, but it will be 10 units down.

Or if you do y equals x squared plus 2, it would bring it up two units:

The vertex would come up two units.

Anyway, you can do all kinds of neat things, again, manually graphing.

And the students really get it when they're actually having to move points

through space and they're hands...

Very good for fine motor skills.

A lot of times, I have students that have that type of problem.

Again this really helps in that way, as well

Orientation and mobility, instructors always come talk to me if their student is having a problem.

'Cause they know my students will be having similar problems.

Again, graphing and graphing parabolas and all types of other things: wonderful.

Graph, graph, graph. That's my motto.

Again this is a quadratic.

And all of the quadratics are gonna have a shape like this,

and it will either move down or up or it will move this way or that way.

Basically, it's still going to have the same shape.

And so your students can graph it similar to this.

**Captions by Project readOn**